{"id":2914,"date":"2023-11-19T22:41:58","date_gmt":"2023-11-19T21:41:58","guid":{"rendered":"http:\/\/ff.mhrooz.xyz\/?p=2914"},"modified":"2023-11-29T11:34:45","modified_gmt":"2023-11-29T10:34:45","slug":"guan_yu_code-based_cryptography_de_lun_wen_bi_ji","status":"publish","type":"post","link":"https:\/\/blog.mhrooz.xyz\/index.php\/2023\/11\/19\/guan_yu_code-based_cryptography_de_lun_wen_bi_ji\/","title":{"rendered":"\u5173\u4e8ecode-based cryptography\u7684\u8bba\u6587\u7b14\u8bb0"},"content":{"rendered":"\n<!doctype html>\n<html>\n<head>\n<meta charset='UTF-8'><meta name='viewport' content='width=device-width initial-scale=1'>\n\n<style type='text\/css'>html {overflow-x: initial !important;}:root { --bg-color: #ffffff; --text-color: #333333; --select-text-bg-color: #B5D6FC; --select-text-font-color: auto; --monospace: \"Lucida Console\",Consolas,\"Courier\",monospace; --title-bar-height: 20px; }\n.mac-os-11 { --title-bar-height: 28px; }\nhtml { font-size: 14px; background-color: 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inherit;\n}\n\n.MathJax g[data-mml-node=\"xypic\"] path {\n  stroke-width: inherit;\n}\nmjx-container[jax=\"SVG\"] path[data-c], mjx-container[jax=\"SVG\"] use[data-c] {\n\t\t\t\t\t\t\tstroke-width: 0;\n\t\t\t\t\t\t} @media print { @page {margin: 0 0 0 0;} body.typora-export {padding-left: 0; padding-right: 0;} #write {padding:0;}}\n<\/style><title>paper reading<\/title>\n<\/head>\n<body class='typora-export'><div class='typora-export-content'>\n<div id='write'  class=''><h1 id='1-code-based-cryptography-lecture-notes'><a href='https:\/\/arxiv.org\/pdf\/2304.03541.pdf'><span>1. Code-based Cryptography: Lecture Notes<\/span><\/a><\/h1><h2 id='11-introduction'><span>1.1 Introduction<\/span><\/h2><p><span>A <\/span><strong><span>linear code<\/span><\/strong><span> <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.192ex\" height=\"1.652ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -705 527 730\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.057ex;\"><defs><path id=\"MJX-32-TEX-C-43\" d=\"M201 -25Q167 -25 136 -14T75 23T29 94T12 202Q12 290 50 394T161 574Q227 642 303 673T433 704Q435 705 457 705Q533 701 533 640Q533 606 507 548T464 474Q431 444 396 444Q381 444 381 453Q381 459 388 473T407 513T428 563Q433 580 433 594Q433 636 381 636Q314 636 260 594T175 489T128 363T112 247Q112 157 153 101T273 44Q347 44 398 121Q413 144 437 157T481 171Q496 171 496 160Q496 150 476 123Q426 56 350 16T201 -25Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"43\" xlink:href=\"#MJX-32-TEX-C-43\"><\/use><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow data-mjx-texclass=\"ORD\"><mi data-mjx-variant=\"-tex-calligraphic\" mathvariant=\"script\">C<\/mi><\/mrow><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">\\mathcal{C}<\/script><span> of length <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.357ex\" height=\"1.025ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -442 600 453\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.025ex;\"><defs><path id=\"MJX-1143-TEX-I-1D45B\" d=\"M21 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type=\"math\/tex\">n<\/script><span> and dimension <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.179ex\" height=\"1.595ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -694 521 705\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.025ex;\"><defs><path id=\"MJX-1790-TEX-I-1D458\" d=\"M121 647Q121 657 125 670T137 683Q138 683 209 688T282 694Q294 694 294 686Q294 679 244 477Q194 279 194 272Q213 282 223 291Q247 309 292 354T362 415Q402 442 438 442Q468 442 485 423T503 369Q503 344 496 327T477 302T456 291T438 288Q418 288 406 299T394 328Q394 353 410 369T442 390L458 393Q446 405 434 405H430Q398 402 367 380T294 316T228 255Q230 254 243 252T267 246T293 238T320 224T342 206T359 180T365 147Q365 130 360 106T354 66Q354 26 381 26Q429 26 459 145Q461 153 479 153H483Q499 153 499 144Q499 139 496 130Q455 -11 378 -11Q333 -11 305 15T277 90Q277 108 280 121T283 145Q283 167 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transform=\"translate(311,-150) scale(0.707)\"><use data-c=\"1D45E\" xlink:href=\"#MJX-36-TEX-I-1D45E\"><\/use><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mo stretchy=\"false\">[<\/mo><mi>n<\/mi><mo>,<\/mo><mi>k<\/mi><msub><mo stretchy=\"false\">]<\/mo><mi>q<\/mi><\/msub><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">[n,k]_q<\/script><span>-code, is a subspace of <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"2.53ex\" height=\"2.414ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -683 1118.3 1067.2\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.869ex;\"><defs><path id=\"MJX-7-TEX-D-1D53D\" d=\"M584 499Q569 490 566 490Q558 490 552 497T546 515Q546 535 533 559Q526 574 506 593T469 621Q415 648 326 648Q293 648 287 647T275 641Q264 630 263 617Q262 609 260 492V370L275 372Q323 376 350 392T393 441Q409 473 409 506Q409 529 427 529Q437 529 442 519Q444 511 444 362Q444 212 442 206Q436 197 426 197Q409 197 409 217Q409 265 375 299Q346 328 280 335H260V206Q260 70 262 63Q265 46 276 41T326 35Q362 35 366 28Q377 17 366 3L360 -1H24Q12 5 12 16Q12 35 51 35Q92 38 97 52Q102 60 102 341T97 632Q91 645 51 648Q12 648 12 666Q12 675 24 683H573Q576 678 584 670V499ZM137 341Q137 131 136 89T130 37Q129 36 129 35H182Q233 35 233 39Q226 54 225 92T224 346L226 623L231 635L235 648H129Q132 641 133 638T135 603T137 517T137 341ZM549 603V648H495L506 641Q531 621 533 619L549 603ZM409 317V395L400 386Q390 376 375 366L357 355L373 346Q394 331 397 328L409 317Z\"><\/path><path id=\"MJX-7-TEX-I-1D45B\" d=\"M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z\"><\/path><path id=\"MJX-7-TEX-I-1D45E\" d=\"M33 157Q33 258 109 349T280 441Q340 441 372 389Q373 390 377 395T388 406T404 418Q438 442 450 442Q454 442 457 439T460 434Q460 425 391 149Q320 -135 320 -139Q320 -147 365 -148H390Q396 -156 396 -157T393 -175Q389 -188 383 -194H370Q339 -192 262 -192Q234 -192 211 -192T174 -192T157 -193Q143 -193 143 -185Q143 -182 145 -170Q149 -154 152 -151T172 -148Q220 -148 230 -141Q238 -136 258 -53T279 32Q279 33 272 29Q224 -10 172 -10Q117 -10 75 30T33 157ZM352 326Q329 405 277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q233 26 290 98L298 109L352 326Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"msubsup\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D53D\" xlink:href=\"#MJX-7-TEX-D-1D53D\"><\/use><\/g><\/g><g data-mml-node=\"mi\" transform=\"translate(644,363) scale(0.707)\"><use data-c=\"1D45B\" xlink:href=\"#MJX-7-TEX-I-1D45B\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(644,-247) scale(0.707)\"><use data-c=\"1D45E\" xlink:href=\"#MJX-7-TEX-I-1D45E\"><\/use><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><msubsup><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"double-struck\">F<\/mi><\/mrow><mi>q<\/mi><mi>n<\/mi><\/msubsup><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">\\mathbb{F}_q^n<\/script><span> of dimension <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.179ex\" height=\"1.595ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -694 521 705\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.025ex;\"><defs><path id=\"MJX-1790-TEX-I-1D458\" d=\"M121 647Q121 657 125 670T137 683Q138 683 209 688T282 694Q294 694 294 686Q294 679 244 477Q194 279 194 272Q213 282 223 291Q247 309 292 354T362 415Q402 442 438 442Q468 442 485 423T503 369Q503 344 496 327T477 302T456 291T438 288Q418 288 406 299T394 328Q394 353 410 369T442 390L458 393Q446 405 434 405H430Q398 402 367 380T294 316T228 255Q230 254 243 252T267 246T293 238T320 224T342 206T359 180T365 147Q365 130 360 106T354 66Q354 26 381 26Q429 26 459 145Q461 153 479 153H483Q499 153 499 144Q499 139 496 130Q455 -11 378 -11Q333 -11 305 15T277 90Q277 108 280 121T283 145Q283 167 269 183T234 206T200 217T182 220H180Q168 178 159 139T145 81T136 44T129 20T122 7T111 -2Q98 -11 83 -11Q66 -11 57 -1T48 16Q48 26 85 176T158 471L195 616Q196 629 188 632T149 637H144Q134 637 131 637T124 640T121 647Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D458\" xlink:href=\"#MJX-1790-TEX-I-1D458\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>k<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">k<\/script><span>. <\/span><\/p><p><span>To represent an <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"5.724ex\" height=\"2.347ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 2529.9 1037.2\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.65ex;\"><defs><path id=\"MJX-36-TEX-N-5B\" d=\"M118 -250V750H255V710H158V-210H255V-250H118Z\"><\/path><path id=\"MJX-36-TEX-I-1D45B\" d=\"M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z\"><\/path><path id=\"MJX-36-TEX-N-2C\" d=\"M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z\"><\/path><path id=\"MJX-36-TEX-I-1D458\" d=\"M121 647Q121 657 125 670T137 683Q138 683 209 688T282 694Q294 694 294 686Q294 679 244 477Q194 279 194 272Q213 282 223 291Q247 309 292 354T362 415Q402 442 438 442Q468 442 485 423T503 369Q503 344 496 327T477 302T456 291T438 288Q418 288 406 299T394 328Q394 353 410 369T442 390L458 393Q446 405 434 405H430Q398 402 367 380T294 316T228 255Q230 254 243 252T267 246T293 238T320 224T342 206T359 180T365 147Q365 130 360 106T354 66Q354 26 381 26Q429 26 459 145Q461 153 479 153H483Q499 153 499 144Q499 139 496 130Q455 -11 378 -11Q333 -11 305 15T277 90Q277 108 280 121T283 145Q283 167 269 183T234 206T200 217T182 220H180Q168 178 159 139T145 81T136 44T129 20T122 7T111 -2Q98 -11 83 -11Q66 -11 57 -1T48 16Q48 26 85 176T158 471L195 616Q196 629 188 632T149 637H144Q134 637 131 637T124 640T121 647Z\"><\/path><path id=\"MJX-36-TEX-N-5D\" d=\"M22 710V750H159V-250H22V-210H119V710H22Z\"><\/path><path id=\"MJX-36-TEX-I-1D45E\" d=\"M33 157Q33 258 109 349T280 441Q340 441 372 389Q373 390 377 395T388 406T404 418Q438 442 450 442Q454 442 457 439T460 434Q460 425 391 149Q320 -135 320 -139Q320 -147 365 -148H390Q396 -156 396 -157T393 -175Q389 -188 383 -194H370Q339 -192 262 -192Q234 -192 211 -192T174 -192T157 -193Q143 -193 143 -185Q143 -182 145 -170Q149 -154 152 -151T172 -148Q220 -148 230 -141Q238 -136 258 -53T279 32Q279 33 272 29Q224 -10 172 -10Q117 -10 75 30T33 157ZM352 326Q329 405 277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q233 26 290 98L298 109L352 326Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mo\"><use data-c=\"5B\" xlink:href=\"#MJX-36-TEX-N-5B\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(278,0)\"><use data-c=\"1D45B\" xlink:href=\"#MJX-36-TEX-I-1D45B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(878,0)\"><use data-c=\"2C\" xlink:href=\"#MJX-36-TEX-N-2C\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1322.7,0)\"><use data-c=\"1D458\" xlink:href=\"#MJX-36-TEX-I-1D458\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(1843.7,0)\"><g data-mml-node=\"mo\"><use data-c=\"5D\" xlink:href=\"#MJX-36-TEX-N-5D\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(311,-150) scale(0.707)\"><use data-c=\"1D45E\" xlink:href=\"#MJX-36-TEX-I-1D45E\"><\/use><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mo stretchy=\"false\">[<\/mo><mi>n<\/mi><mo>,<\/mo><mi>k<\/mi><msub><mo stretchy=\"false\">]<\/mo><mi>q<\/mi><\/msub><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">[n,k]_q<\/script><span>-code, we may take any basis of it, namely a set of <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.179ex\" height=\"1.595ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -694 521 705\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.025ex;\"><defs><path id=\"MJX-1790-TEX-I-1D458\" d=\"M121 647Q121 657 125 670T137 683Q138 683 209 688T282 694Q294 694 294 686Q294 679 244 477Q194 279 194 272Q213 282 223 291Q247 309 292 354T362 415Q402 442 438 442Q468 442 485 423T503 369Q503 344 496 327T477 302T456 291T438 288Q418 288 406 299T394 328Q394 353 410 369T442 390L458 393Q446 405 434 405H430Q398 402 367 380T294 316T228 255Q230 254 243 252T267 246T293 238T320 224T342 206T359 180T365 147Q365 130 360 106T354 66Q354 26 381 26Q429 26 459 145Q461 153 479 153H483Q499 153 499 144Q499 139 496 130Q455 -11 378 -11Q333 -11 305 15T277 90Q277 108 280 121T283 145Q283 167 269 183T234 206T200 217T182 220H180Q168 178 159 139T145 81T136 44T129 20T122 7T111 -2Q98 -11 83 -11Q66 -11 57 -1T48 16Q48 26 85 176T158 471L195 616Q196 629 188 632T149 637H144Q134 637 131 637T124 640T121 647Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D458\" xlink:href=\"#MJX-1790-TEX-I-1D458\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>k<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">k<\/script><span> linearly independent vectors. We use them as rows to form the <\/span><strong><span>generator matrix<\/span><\/strong><span> <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"9.419ex\" height=\"2.801ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -853.7 4163.4 1237.9\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.869ex;\"><defs><path id=\"MJX-70-TEX-B-1D406\" d=\"M465 -10Q281 -10 173 88T64 343Q64 413 85 471T143 568T217 631T298 670Q371 697 449 697Q452 697 459 697T470 696Q502 696 531 690T582 675T618 658T644 641T656 632L732 695Q734 697 745 697Q758 697 761 692T765 668V627V489V449Q765 428 761 424T741 419H731H724Q705 419 702 422T695 444Q683 520 631 577T495 635Q364 635 295 563Q261 528 247 477T232 343Q232 296 236 260T256 185T296 120T366 76T472 52Q481 51 498 51Q544 51 573 67T607 108Q608 111 608 164V214H464V276H479Q506 273 680 273Q816 273 834 276H845V214H765V113V51Q765 16 763 8T750 0Q742 2 709 16T658 40L648 46Q592 -10 465 -10Z\"><\/path><path id=\"MJX-70-TEX-N-2208\" d=\"M84 250Q84 372 166 450T360 539Q361 539 377 539T419 540T469 540H568Q583 532 583 520Q583 511 570 501L466 500Q355 499 329 494Q280 482 242 458T183 409T147 354T129 306T124 272V270H568Q583 262 583 250T568 230H124V228Q124 207 134 177T167 112T231 48T328 7Q355 1 466 0H570Q583 -10 583 -20Q583 -32 568 -40H471Q464 -40 446 -40T417 -41Q262 -41 172 45Q84 127 84 250Z\"><\/path><path id=\"MJX-70-TEX-D-1D53D\" d=\"M584 499Q569 490 566 490Q558 490 552 497T546 515Q546 535 533 559Q526 574 506 593T469 621Q415 648 326 648Q293 648 287 647T275 641Q264 630 263 617Q262 609 260 492V370L275 372Q323 376 350 392T393 441Q409 473 409 506Q409 529 427 529Q437 529 442 519Q444 511 444 362Q444 212 442 206Q436 197 426 197Q409 197 409 217Q409 265 375 299Q346 328 280 335H260V206Q260 70 262 63Q265 46 276 41T326 35Q362 35 366 28Q377 17 366 3L360 -1H24Q12 5 12 16Q12 35 51 35Q92 38 97 52Q102 60 102 341T97 632Q91 645 51 648Q12 648 12 666Q12 675 24 683H573Q576 678 584 670V499ZM137 341Q137 131 136 89T130 37Q129 36 129 35H182Q233 35 233 39Q226 54 225 92T224 346L226 623L231 635L235 648H129Q132 641 133 638T135 603T137 517T137 341ZM549 603V648H495L506 641Q531 621 533 619L549 603ZM409 317V395L400 386Q390 376 375 366L357 355L373 346Q394 331 397 328L409 317Z\"><\/path><path id=\"MJX-70-TEX-I-1D458\" d=\"M121 647Q121 657 125 670T137 683Q138 683 209 688T282 694Q294 694 294 686Q294 679 244 477Q194 279 194 272Q213 282 223 291Q247 309 292 354T362 415Q402 442 438 442Q468 442 485 423T503 369Q503 344 496 327T477 302T456 291T438 288Q418 288 406 299T394 328Q394 353 410 369T442 390L458 393Q446 405 434 405H430Q398 402 367 380T294 316T228 255Q230 254 243 252T267 246T293 238T320 224T342 206T359 180T365 147Q365 130 360 106T354 66Q354 26 381 26Q429 26 459 145Q461 153 479 153H483Q499 153 499 144Q499 139 496 130Q455 -11 378 -11Q333 -11 305 15T277 90Q277 108 280 121T283 145Q283 167 269 183T234 206T200 217T182 220H180Q168 178 159 139T145 81T136 44T129 20T122 7T111 -2Q98 -11 83 -11Q66 -11 57 -1T48 16Q48 26 85 176T158 471L195 616Q196 629 188 632T149 637H144Q134 637 131 637T124 640T121 647Z\"><\/path><path id=\"MJX-70-TEX-N-D7\" d=\"M630 29Q630 9 609 9Q604 9 587 25T493 118L389 222L284 117Q178 13 175 11Q171 9 168 9Q160 9 154 15T147 29Q147 36 161 51T255 146L359 250L255 354Q174 435 161 449T147 471Q147 480 153 485T168 490Q173 490 175 489Q178 487 284 383L389 278L493 382Q570 459 587 475T609 491Q630 491 630 471Q630 464 620 453T522 355L418 250L522 145Q606 61 618 48T630 29Z\"><\/path><path id=\"MJX-70-TEX-I-1D45B\" d=\"M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z\"><\/path><path id=\"MJX-70-TEX-I-1D45E\" d=\"M33 157Q33 258 109 349T280 441Q340 441 372 389Q373 390 377 395T388 406T404 418Q438 442 450 442Q454 442 457 439T460 434Q460 425 391 149Q320 -135 320 -139Q320 -147 365 -148H390Q396 -156 396 -157T393 -175Q389 -188 383 -194H370Q339 -192 262 -192Q234 -192 211 -192T174 -192T157 -193Q143 -193 143 -185Q143 -182 145 -170Q149 -154 152 -151T172 -148Q220 -148 230 -141Q238 -136 258 -53T279 32Q279 33 272 29Q224 -10 172 -10Q117 -10 75 30T33 157ZM352 326Q329 405 277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q233 26 290 98L298 109L352 326Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D406\" xlink:href=\"#MJX-70-TEX-B-1D406\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(1181.8,0)\"><use data-c=\"2208\" xlink:href=\"#MJX-70-TEX-N-2208\"><\/use><\/g><g data-mml-node=\"msubsup\" transform=\"translate(2126.6,0)\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D53D\" xlink:href=\"#MJX-70-TEX-D-1D53D\"><\/use><\/g><\/g><g data-mml-node=\"TeXAtom\" transform=\"translate(644,363) scale(0.707)\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D458\" xlink:href=\"#MJX-70-TEX-I-1D458\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(521,0)\"><use data-c=\"D7\" xlink:href=\"#MJX-70-TEX-N-D7\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1299,0)\"><use data-c=\"1D45B\" xlink:href=\"#MJX-70-TEX-I-1D45B\"><\/use><\/g><\/g><g data-mml-node=\"mi\" transform=\"translate(644,-247) scale(0.707)\"><use data-c=\"1D45E\" xlink:href=\"#MJX-70-TEX-I-1D45E\"><\/use><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">G<\/mi><\/mrow><mo>\u2208<\/mo><msubsup><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"double-struck\">F<\/mi><\/mrow><mi>q<\/mi><mrow data-mjx-texclass=\"ORD\"><mi>k<\/mi><mo>\u00d7<\/mo><mi>n<\/mi><\/mrow><\/msubsup><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">\\mathbf{G}\\in\\mathbb{F}_q^{k\\times n}<\/script><span>, then <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.192ex\" height=\"1.652ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -705 527 730\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.057ex;\"><defs><path id=\"MJX-32-TEX-C-43\" d=\"M201 -25Q167 -25 136 -14T75 23T29 94T12 202Q12 290 50 394T161 574Q227 642 303 673T433 704Q435 705 457 705Q533 701 533 640Q533 606 507 548T464 474Q431 444 396 444Q381 444 381 453Q381 459 388 473T407 513T428 563Q433 580 433 594Q433 636 381 636Q314 636 260 594T175 489T128 363T112 247Q112 157 153 101T273 44Q347 44 398 121Q413 144 437 157T481 171Q496 171 496 160Q496 150 476 123Q426 56 350 16T201 -25Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"43\" xlink:href=\"#MJX-32-TEX-C-43\"><\/use><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow data-mjx-texclass=\"ORD\"><mi data-mjx-variant=\"-tex-calligraphic\" mathvariant=\"script\">C<\/mi><\/mrow><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">\\mathcal{C}<\/script><span> can be written as <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"24.081ex\" height=\"2.801ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -853.7 10643.6 1237.9\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.869ex;\"><defs><path id=\"MJX-13-TEX-C-43\" d=\"M201 -25Q167 -25 136 -14T75 23T29 94T12 202Q12 290 50 394T161 574Q227 642 303 673T433 704Q435 705 457 705Q533 701 533 640Q533 606 507 548T464 474Q431 444 396 444Q381 444 381 453Q381 459 388 473T407 513T428 563Q433 580 433 594Q433 636 381 636Q314 636 260 594T175 489T128 363T112 247Q112 157 153 101T273 44Q347 44 398 121Q413 144 437 157T481 171Q496 171 496 160Q496 150 476 123Q426 56 350 16T201 -25Z\"><\/path><path id=\"MJX-13-TEX-N-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path id=\"MJX-13-TEX-N-7B\" d=\"M434 -231Q434 -244 428 -250H410Q281 -250 230 -184Q225 -177 222 -172T217 -161T213 -148T211 -133T210 -111T209 -84T209 -47T209 0Q209 21 209 53Q208 142 204 153Q203 154 203 155Q189 191 153 211T82 231Q71 231 68 234T65 250T68 266T82 269Q116 269 152 289T203 345Q208 356 208 377T209 529V579Q209 634 215 656T244 698Q270 724 324 740Q361 748 377 749Q379 749 390 749T408 750H428Q434 744 434 732Q434 719 431 716Q429 713 415 713Q362 710 332 689T296 647Q291 634 291 499V417Q291 370 288 353T271 314Q240 271 184 255L170 250L184 245Q202 239 220 230T262 196T290 137Q291 131 291 1Q291 -134 296 -147Q306 -174 339 -192T415 -213Q429 -213 431 -216Q434 -219 434 -231Z\"><\/path><path id=\"MJX-13-TEX-B-1D41C\" d=\"M447 131H458Q478 131 478 117Q478 112 471 95T439 51T377 9Q330 -6 286 -6Q196 -6 135 35Q39 96 39 222Q39 324 101 384Q169 453 286 453Q359 453 411 431T464 353Q464 319 445 302T395 284Q360 284 343 305T325 353Q325 380 338 396H333Q317 398 295 398H292Q280 398 271 397T245 390T218 373T197 338T183 283Q182 275 182 231Q182 199 184 180T193 132T220 85T270 57Q289 50 317 50H326Q385 50 414 115Q419 127 423 129T447 131Z\"><\/path><path id=\"MJX-13-TEX-B-1D426\" d=\"M40 442Q217 450 218 450H224V365Q226 367 235 378T254 397T278 416T314 435T362 448Q376 450 400 450H406Q503 450 534 393Q545 376 545 370Q545 368 555 379Q611 450 716 450Q774 450 809 434Q850 414 861 379T873 276V213V198V62H942V0H933Q915 3 809 3Q702 3 684 0H675V62H744V194V275Q744 348 735 373T690 399Q645 399 607 370T557 290Q555 281 554 171V62H623V0H614Q596 3 489 3Q374 3 365 0H356V62H425V194V275Q425 348 416 373T371 399Q326 399 288 370T238 290Q236 281 235 171V62H304V0H295Q277 3 171 3Q64 3 46 0H37V62H106V210V303Q106 353 104 363T91 376Q77 380 50 380H37V442H40Z\"><\/path><path id=\"MJX-13-TEX-B-1D406\" d=\"M465 -10Q281 -10 173 88T64 343Q64 413 85 471T143 568T217 631T298 670Q371 697 449 697Q452 697 459 697T470 696Q502 696 531 690T582 675T618 658T644 641T656 632L732 695Q734 697 745 697Q758 697 761 692T765 668V627V489V449Q765 428 761 424T741 419H731H724Q705 419 702 422T695 444Q683 520 631 577T495 635Q364 635 295 563Q261 528 247 477T232 343Q232 296 236 260T256 185T296 120T366 76T472 52Q481 51 498 51Q544 51 573 67T607 108Q608 111 608 164V214H464V276H479Q506 273 680 273Q816 273 834 276H845V214H765V113V51Q765 16 763 8T750 0Q742 2 709 16T658 40L648 46Q592 -10 465 -10Z\"><\/path><path id=\"MJX-13-TEX-N-3A\" d=\"M78 370Q78 394 95 412T138 430Q162 430 180 414T199 371Q199 346 182 328T139 310T96 327T78 370ZM78 60Q78 84 95 102T138 120Q162 120 180 104T199 61Q199 36 182 18T139 0T96 17T78 60Z\"><\/path><path id=\"MJX-13-TEX-N-2208\" d=\"M84 250Q84 372 166 450T360 539Q361 539 377 539T419 540T469 540H568Q583 532 583 520Q583 511 570 501L466 500Q355 499 329 494Q280 482 242 458T183 409T147 354T129 306T124 272V270H568Q583 262 583 250T568 230H124V228Q124 207 134 177T167 112T231 48T328 7Q355 1 466 0H570Q583 -10 583 -20Q583 -32 568 -40H471Q464 -40 446 -40T417 -41Q262 -41 172 45Q84 127 84 250Z\"><\/path><path id=\"MJX-13-TEX-D-1D53D\" d=\"M584 499Q569 490 566 490Q558 490 552 497T546 515Q546 535 533 559Q526 574 506 593T469 621Q415 648 326 648Q293 648 287 647T275 641Q264 630 263 617Q262 609 260 492V370L275 372Q323 376 350 392T393 441Q409 473 409 506Q409 529 427 529Q437 529 442 519Q444 511 444 362Q444 212 442 206Q436 197 426 197Q409 197 409 217Q409 265 375 299Q346 328 280 335H260V206Q260 70 262 63Q265 46 276 41T326 35Q362 35 366 28Q377 17 366 3L360 -1H24Q12 5 12 16Q12 35 51 35Q92 38 97 52Q102 60 102 341T97 632Q91 645 51 648Q12 648 12 666Q12 675 24 683H573Q576 678 584 670V499ZM137 341Q137 131 136 89T130 37Q129 36 129 35H182Q233 35 233 39Q226 54 225 92T224 346L226 623L231 635L235 648H129Q132 641 133 638T135 603T137 517T137 341ZM549 603V648H495L506 641Q531 621 533 619L549 603ZM409 317V395L400 386Q390 376 375 366L357 355L373 346Q394 331 397 328L409 317Z\"><\/path><path id=\"MJX-13-TEX-I-1D458\" d=\"M121 647Q121 657 125 670T137 683Q138 683 209 688T282 694Q294 694 294 686Q294 679 244 477Q194 279 194 272Q213 282 223 291Q247 309 292 354T362 415Q402 442 438 442Q468 442 485 423T503 369Q503 344 496 327T477 302T456 291T438 288Q418 288 406 299T394 328Q394 353 410 369T442 390L458 393Q446 405 434 405H430Q398 402 367 380T294 316T228 255Q230 254 243 252T267 246T293 238T320 224T342 206T359 180T365 147Q365 130 360 106T354 66Q354 26 381 26Q429 26 459 145Q461 153 479 153H483Q499 153 499 144Q499 139 496 130Q455 -11 378 -11Q333 -11 305 15T277 90Q277 108 280 121T283 145Q283 167 269 183T234 206T200 217T182 220H180Q168 178 159 139T145 81T136 44T129 20T122 7T111 -2Q98 -11 83 -11Q66 -11 57 -1T48 16Q48 26 85 176T158 471L195 616Q196 629 188 632T149 637H144Q134 637 131 637T124 640T121 647Z\"><\/path><path id=\"MJX-13-TEX-I-1D45E\" d=\"M33 157Q33 258 109 349T280 441Q340 441 372 389Q373 390 377 395T388 406T404 418Q438 442 450 442Q454 442 457 439T460 434Q460 425 391 149Q320 -135 320 -139Q320 -147 365 -148H390Q396 -156 396 -157T393 -175Q389 -188 383 -194H370Q339 -192 262 -192Q234 -192 211 -192T174 -192T157 -193Q143 -193 143 -185Q143 -182 145 -170Q149 -154 152 -151T172 -148Q220 -148 230 -141Q238 -136 258 -53T279 32Q279 33 272 29Q224 -10 172 -10Q117 -10 75 30T33 157ZM352 326Q329 405 277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q233 26 290 98L298 109L352 326Z\"><\/path><path id=\"MJX-13-TEX-N-7D\" d=\"M65 731Q65 745 68 747T88 750Q171 750 216 725T279 670Q288 649 289 635T291 501Q292 362 293 357Q306 312 345 291T417 269Q428 269 431 266T434 250T431 234T417 231Q380 231 345 210T298 157Q293 143 292 121T291 -28V-79Q291 -134 285 -156T256 -198Q202 -250 89 -250Q71 -250 68 -247T65 -230Q65 -224 65 -223T66 -218T69 -214T77 -213Q91 -213 108 -210T146 -200T183 -177T207 -139Q208 -134 209 3L210 139Q223 196 280 230Q315 247 330 250Q305 257 280 270Q225 304 212 352L210 362L209 498Q208 635 207 640Q195 680 154 696T77 713Q68 713 67 716T65 731Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"43\" xlink:href=\"#MJX-13-TEX-C-43\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(804.8,0)\"><use data-c=\"3D\" xlink:href=\"#MJX-13-TEX-N-3D\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(1860.6,0)\"><use data-c=\"7B\" xlink:href=\"#MJX-13-TEX-N-7B\"><\/use><\/g><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\" transform=\"translate(2360.6,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D41C\" xlink:href=\"#MJX-13-TEX-B-1D41C\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(3149.3,0)\"><use data-c=\"3D\" xlink:href=\"#MJX-13-TEX-N-3D\"><\/use><\/g><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\" transform=\"translate(4205.1,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D426\" xlink:href=\"#MJX-13-TEX-B-1D426\"><\/use><use data-c=\"1D406\" xlink:href=\"#MJX-13-TEX-B-1D406\" transform=\"translate(958,0)\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(6344.9,0)\"><use data-c=\"3A\" xlink:href=\"#MJX-13-TEX-N-3A\"><\/use><\/g><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\" transform=\"translate(6900.7,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D426\" xlink:href=\"#MJX-13-TEX-B-1D426\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(8136.4,0)\"><use data-c=\"2208\" xlink:href=\"#MJX-13-TEX-N-2208\"><\/use><\/g><g data-mml-node=\"msubsup\" transform=\"translate(9081.2,0)\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D53D\" xlink:href=\"#MJX-13-TEX-D-1D53D\"><\/use><\/g><\/g><g data-mml-node=\"mi\" transform=\"translate(644,363) scale(0.707)\"><use data-c=\"1D458\" xlink:href=\"#MJX-13-TEX-I-1D458\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(644,-247) scale(0.707)\"><use data-c=\"1D45E\" xlink:href=\"#MJX-13-TEX-I-1D45E\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(10143.6,0)\"><use data-c=\"7D\" xlink:href=\"#MJX-13-TEX-N-7D\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow data-mjx-texclass=\"ORD\"><mi data-mjx-variant=\"-tex-calligraphic\" mathvariant=\"script\">C<\/mi><\/mrow><mo>=<\/mo><mo fence=\"false\" stretchy=\"false\">{<\/mo><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">c<\/mi><\/mrow><mo>=<\/mo><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">mG<\/mi><\/mrow><mo>:<\/mo><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">m<\/mi><\/mrow><mo>\u2208<\/mo><msubsup><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"double-struck\">F<\/mi><\/mrow><mi>q<\/mi><mi>k<\/mi><\/msubsup><mo fence=\"false\" stretchy=\"false\">}<\/mo><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">\\mathcal{C}=\\{\\mathbf{c}=\\mathbf{mG}:\\mathbf{m}\\in\\mathbb{F}_q^k\\}<\/script><span>. <\/span><\/p><p><span>Conversely, any matrix <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"9.419ex\" height=\"2.801ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -853.7 4163.4 1237.9\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.869ex;\"><defs><path id=\"MJX-70-TEX-B-1D406\" d=\"M465 -10Q281 -10 173 88T64 343Q64 413 85 471T143 568T217 631T298 670Q371 697 449 697Q452 697 459 697T470 696Q502 696 531 690T582 675T618 658T644 641T656 632L732 695Q734 697 745 697Q758 697 761 692T765 668V627V489V449Q765 428 761 424T741 419H731H724Q705 419 702 422T695 444Q683 520 631 577T495 635Q364 635 295 563Q261 528 247 477T232 343Q232 296 236 260T256 185T296 120T366 76T472 52Q481 51 498 51Q544 51 573 67T607 108Q608 111 608 164V214H464V276H479Q506 273 680 273Q816 273 834 276H845V214H765V113V51Q765 16 763 8T750 0Q742 2 709 16T658 40L648 46Q592 -10 465 -10Z\"><\/path><path id=\"MJX-70-TEX-N-2208\" d=\"M84 250Q84 372 166 450T360 539Q361 539 377 539T419 540T469 540H568Q583 532 583 520Q583 511 570 501L466 500Q355 499 329 494Q280 482 242 458T183 409T147 354T129 306T124 272V270H568Q583 262 583 250T568 230H124V228Q124 207 134 177T167 112T231 48T328 7Q355 1 466 0H570Q583 -10 583 -20Q583 -32 568 -40H471Q464 -40 446 -40T417 -41Q262 -41 172 45Q84 127 84 250Z\"><\/path><path id=\"MJX-70-TEX-D-1D53D\" d=\"M584 499Q569 490 566 490Q558 490 552 497T546 515Q546 535 533 559Q526 574 506 593T469 621Q415 648 326 648Q293 648 287 647T275 641Q264 630 263 617Q262 609 260 492V370L275 372Q323 376 350 392T393 441Q409 473 409 506Q409 529 427 529Q437 529 442 519Q444 511 444 362Q444 212 442 206Q436 197 426 197Q409 197 409 217Q409 265 375 299Q346 328 280 335H260V206Q260 70 262 63Q265 46 276 41T326 35Q362 35 366 28Q377 17 366 3L360 -1H24Q12 5 12 16Q12 35 51 35Q92 38 97 52Q102 60 102 341T97 632Q91 645 51 648Q12 648 12 666Q12 675 24 683H573Q576 678 584 670V499ZM137 341Q137 131 136 89T130 37Q129 36 129 35H182Q233 35 233 39Q226 54 225 92T224 346L226 623L231 635L235 648H129Q132 641 133 638T135 603T137 517T137 341ZM549 603V648H495L506 641Q531 621 533 619L549 603ZM409 317V395L400 386Q390 376 375 366L357 355L373 346Q394 331 397 328L409 317Z\"><\/path><path id=\"MJX-70-TEX-I-1D458\" d=\"M121 647Q121 657 125 670T137 683Q138 683 209 688T282 694Q294 694 294 686Q294 679 244 477Q194 279 194 272Q213 282 223 291Q247 309 292 354T362 415Q402 442 438 442Q468 442 485 423T503 369Q503 344 496 327T477 302T456 291T438 288Q418 288 406 299T394 328Q394 353 410 369T442 390L458 393Q446 405 434 405H430Q398 402 367 380T294 316T228 255Q230 254 243 252T267 246T293 238T320 224T342 206T359 180T365 147Q365 130 360 106T354 66Q354 26 381 26Q429 26 459 145Q461 153 479 153H483Q499 153 499 144Q499 139 496 130Q455 -11 378 -11Q333 -11 305 15T277 90Q277 108 280 121T283 145Q283 167 269 183T234 206T200 217T182 220H180Q168 178 159 139T145 81T136 44T129 20T122 7T111 -2Q98 -11 83 -11Q66 -11 57 -1T48 16Q48 26 85 176T158 471L195 616Q196 629 188 632T149 637H144Q134 637 131 637T124 640T121 647Z\"><\/path><path id=\"MJX-70-TEX-N-D7\" d=\"M630 29Q630 9 609 9Q604 9 587 25T493 118L389 222L284 117Q178 13 175 11Q171 9 168 9Q160 9 154 15T147 29Q147 36 161 51T255 146L359 250L255 354Q174 435 161 449T147 471Q147 480 153 485T168 490Q173 490 175 489Q178 487 284 383L389 278L493 382Q570 459 587 475T609 491Q630 491 630 471Q630 464 620 453T522 355L418 250L522 145Q606 61 618 48T630 29Z\"><\/path><path id=\"MJX-70-TEX-I-1D45B\" d=\"M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z\"><\/path><path id=\"MJX-70-TEX-I-1D45E\" d=\"M33 157Q33 258 109 349T280 441Q340 441 372 389Q373 390 377 395T388 406T404 418Q438 442 450 442Q454 442 457 439T460 434Q460 425 391 149Q320 -135 320 -139Q320 -147 365 -148H390Q396 -156 396 -157T393 -175Q389 -188 383 -194H370Q339 -192 262 -192Q234 -192 211 -192T174 -192T157 -193Q143 -193 143 -185Q143 -182 145 -170Q149 -154 152 -151T172 -148Q220 -148 230 -141Q238 -136 258 -53T279 32Q279 33 272 29Q224 -10 172 -10Q117 -10 75 30T33 157ZM352 326Q329 405 277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q233 26 290 98L298 109L352 326Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D406\" xlink:href=\"#MJX-70-TEX-B-1D406\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(1181.8,0)\"><use data-c=\"2208\" xlink:href=\"#MJX-70-TEX-N-2208\"><\/use><\/g><g data-mml-node=\"msubsup\" transform=\"translate(2126.6,0)\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D53D\" xlink:href=\"#MJX-70-TEX-D-1D53D\"><\/use><\/g><\/g><g data-mml-node=\"TeXAtom\" transform=\"translate(644,363) scale(0.707)\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D458\" xlink:href=\"#MJX-70-TEX-I-1D458\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(521,0)\"><use data-c=\"D7\" xlink:href=\"#MJX-70-TEX-N-D7\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1299,0)\"><use data-c=\"1D45B\" xlink:href=\"#MJX-70-TEX-I-1D45B\"><\/use><\/g><\/g><g data-mml-node=\"mi\" transform=\"translate(644,-247) scale(0.707)\"><use data-c=\"1D45E\" xlink:href=\"#MJX-70-TEX-I-1D45E\"><\/use><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">G<\/mi><\/mrow><mo>\u2208<\/mo><msubsup><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"double-struck\">F<\/mi><\/mrow><mi>q<\/mi><mrow data-mjx-texclass=\"ORD\"><mi>k<\/mi><mo>\u00d7<\/mo><mi>n<\/mi><\/mrow><\/msubsup><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">\\mathbf{G}\\in\\mathbb{F}_q^{k\\times n}<\/script><span> with rank <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.179ex\" height=\"1.595ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -694 521 705\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.025ex;\"><defs><path id=\"MJX-1790-TEX-I-1D458\" d=\"M121 647Q121 657 125 670T137 683Q138 683 209 688T282 694Q294 694 294 686Q294 679 244 477Q194 279 194 272Q213 282 223 291Q247 309 292 354T362 415Q402 442 438 442Q468 442 485 423T503 369Q503 344 496 327T477 302T456 291T438 288Q418 288 406 299T394 328Q394 353 410 369T442 390L458 393Q446 405 434 405H430Q398 402 367 380T294 316T228 255Q230 254 243 252T267 246T293 238T320 224T342 206T359 180T365 147Q365 130 360 106T354 66Q354 26 381 26Q429 26 459 145Q461 153 479 153H483Q499 153 499 144Q499 139 496 130Q455 -11 378 -11Q333 -11 305 15T277 90Q277 108 280 121T283 145Q283 167 269 183T234 206T200 217T182 220H180Q168 178 159 139T145 81T136 44T129 20T122 7T111 -2Q98 -11 83 -11Q66 -11 57 -1T48 16Q48 26 85 176T158 471L195 616Q196 629 188 632T149 637H144Q134 637 131 637T124 640T121 647Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D458\" xlink:href=\"#MJX-1790-TEX-I-1D458\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>k<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">k<\/script><span> defines an <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"5.724ex\" height=\"2.347ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 2529.9 1037.2\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.65ex;\"><defs><path id=\"MJX-36-TEX-N-5B\" d=\"M118 -250V750H255V710H158V-210H255V-250H118Z\"><\/path><path id=\"MJX-36-TEX-I-1D45B\" d=\"M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z\"><\/path><path id=\"MJX-36-TEX-N-2C\" d=\"M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z\"><\/path><path id=\"MJX-36-TEX-I-1D458\" d=\"M121 647Q121 657 125 670T137 683Q138 683 209 688T282 694Q294 694 294 686Q294 679 244 477Q194 279 194 272Q213 282 223 291Q247 309 292 354T362 415Q402 442 438 442Q468 442 485 423T503 369Q503 344 496 327T477 302T456 291T438 288Q418 288 406 299T394 328Q394 353 410 369T442 390L458 393Q446 405 434 405H430Q398 402 367 380T294 316T228 255Q230 254 243 252T267 246T293 238T320 224T342 206T359 180T365 147Q365 130 360 106T354 66Q354 26 381 26Q429 26 459 145Q461 153 479 153H483Q499 153 499 144Q499 139 496 130Q455 -11 378 -11Q333 -11 305 15T277 90Q277 108 280 121T283 145Q283 167 269 183T234 206T200 217T182 220H180Q168 178 159 139T145 81T136 44T129 20T122 7T111 -2Q98 -11 83 -11Q66 -11 57 -1T48 16Q48 26 85 176T158 471L195 616Q196 629 188 632T149 637H144Q134 637 131 637T124 640T121 647Z\"><\/path><path id=\"MJX-36-TEX-N-5D\" d=\"M22 710V750H159V-250H22V-210H119V710H22Z\"><\/path><path id=\"MJX-36-TEX-I-1D45E\" d=\"M33 157Q33 258 109 349T280 441Q340 441 372 389Q373 390 377 395T388 406T404 418Q438 442 450 442Q454 442 457 439T460 434Q460 425 391 149Q320 -135 320 -139Q320 -147 365 -148H390Q396 -156 396 -157T393 -175Q389 -188 383 -194H370Q339 -192 262 -192Q234 -192 211 -192T174 -192T157 -193Q143 -193 143 -185Q143 -182 145 -170Q149 -154 152 -151T172 -148Q220 -148 230 -141Q238 -136 258 -53T279 32Q279 33 272 29Q224 -10 172 -10Q117 -10 75 30T33 157ZM352 326Q329 405 277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q233 26 290 98L298 109L352 326Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mo\"><use data-c=\"5B\" xlink:href=\"#MJX-36-TEX-N-5B\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(278,0)\"><use data-c=\"1D45B\" xlink:href=\"#MJX-36-TEX-I-1D45B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(878,0)\"><use data-c=\"2C\" xlink:href=\"#MJX-36-TEX-N-2C\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1322.7,0)\"><use data-c=\"1D458\" xlink:href=\"#MJX-36-TEX-I-1D458\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(1843.7,0)\"><g data-mml-node=\"mo\"><use data-c=\"5D\" xlink:href=\"#MJX-36-TEX-N-5D\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(311,-150) scale(0.707)\"><use data-c=\"1D45E\" xlink:href=\"#MJX-36-TEX-I-1D45E\"><\/use><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mo stretchy=\"false\">[<\/mo><mi>n<\/mi><mo>,<\/mo><mi>k<\/mi><msub><mo stretchy=\"false\">]<\/mo><mi>q<\/mi><\/msub><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">[n,k]_q<\/script><span>-code.<\/span><\/p><blockquote><p><span>A <\/span><strong><span>subspace<\/span><\/strong><span> of a vector space is a nonempty subset that satisfies the requirements for a <\/span><strong><span>vector space<\/span><\/strong><span>: Linear combinations stay in the subspace.<\/span>\n<span>(i) If we add any vector <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.294ex\" height=\"1.025ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -442 572 453\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.025ex;\"><defs><path id=\"MJX-212-TEX-I-1D465\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D465\" xlink:href=\"#MJX-212-TEX-I-1D465\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>x<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">x<\/script><span> and <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.109ex\" height=\"1.464ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -442 490 647\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.464ex;\"><defs><path id=\"MJX-18-TEX-I-1D466\" d=\"M21 287Q21 301 36 335T84 406T158 442Q199 442 224 419T250 355Q248 336 247 334Q247 331 231 288T198 191T182 105Q182 62 196 45T238 27Q261 27 281 38T312 61T339 94Q339 95 344 114T358 173T377 247Q415 397 419 404Q432 431 462 431Q475 431 483 424T494 412T496 403Q496 390 447 193T391 -23Q363 -106 294 -155T156 -205Q111 -205 77 -183T43 -117Q43 -95 50 -80T69 -58T89 -48T106 -45Q150 -45 150 -87Q150 -107 138 -122T115 -142T102 -147L99 -148Q101 -153 118 -160T152 -167H160Q177 -167 186 -165Q219 -156 247 -127T290 -65T313 -9T321 21L315 17Q309 13 296 6T270 -6Q250 -11 231 -11Q185 -11 150 11T104 82Q103 89 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D466\" xlink:href=\"#MJX-18-TEX-I-1D466\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>y<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">y<\/script><span> in the subspace, then <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"5.168ex\" height=\"1.783ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -583 2284.4 788\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.464ex;\"><defs><path id=\"MJX-19-TEX-I-1D465\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><path id=\"MJX-19-TEX-N-2B\" d=\"M56 237T56 250T70 270H369V420L370 570Q380 583 389 583Q402 583 409 568V270H707Q722 262 722 250T707 230H409V-68Q401 -82 391 -82H389H387Q375 -82 369 -68V230H70Q56 237 56 250Z\"><\/path><path id=\"MJX-19-TEX-I-1D466\" d=\"M21 287Q21 301 36 335T84 406T158 442Q199 442 224 419T250 355Q248 336 247 334Q247 331 231 288T198 191T182 105Q182 62 196 45T238 27Q261 27 281 38T312 61T339 94Q339 95 344 114T358 173T377 247Q415 397 419 404Q432 431 462 431Q475 431 483 424T494 412T496 403Q496 390 447 193T391 -23Q363 -106 294 -155T156 -205Q111 -205 77 -183T43 -117Q43 -95 50 -80T69 -58T89 -48T106 -45Q150 -45 150 -87Q150 -107 138 -122T115 -142T102 -147L99 -148Q101 -153 118 -160T152 -167H160Q177 -167 186 -165Q219 -156 247 -127T290 -65T313 -9T321 21L315 17Q309 13 296 6T270 -6Q250 -11 231 -11Q185 -11 150 11T104 82Q103 89 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D465\" xlink:href=\"#MJX-19-TEX-I-1D465\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(794.2,0)\"><use data-c=\"2B\" xlink:href=\"#MJX-19-TEX-N-2B\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1794.4,0)\"><use data-c=\"1D466\" xlink:href=\"#MJX-19-TEX-I-1D466\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>x<\/mi><mo>+<\/mo><mi>y<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">x+y<\/script><span> is in the subspace.<\/span><\/p><p><span>(ii) If we multiply any vector <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.294ex\" height=\"1.025ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -442 572 453\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.025ex;\"><defs><path id=\"MJX-212-TEX-I-1D465\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D465\" xlink:href=\"#MJX-212-TEX-I-1D465\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>x<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">x<\/script><span> in the subspace by any scalar <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"0.98ex\" height=\"1.025ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -442 433 453\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.025ex;\"><defs><path id=\"MJX-21-TEX-I-1D450\" d=\"M34 159Q34 268 120 355T306 442Q362 442 394 418T427 355Q427 326 408 306T360 285Q341 285 330 295T319 325T330 359T352 380T366 386H367Q367 388 361 392T340 400T306 404Q276 404 249 390Q228 381 206 359Q162 315 142 235T121 119Q121 73 147 50Q169 26 205 26H209Q321 26 394 111Q403 121 406 121Q410 121 419 112T429 98T420 83T391 55T346 25T282 0T202 -11Q127 -11 81 37T34 159Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D450\" xlink:href=\"#MJX-21-TEX-I-1D450\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>c<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">c<\/script><span>, then <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"2.274ex\" height=\"1.025ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -442 1005 453\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.025ex;\"><defs><path id=\"MJX-22-TEX-I-1D450\" d=\"M34 159Q34 268 120 355T306 442Q362 442 394 418T427 355Q427 326 408 306T360 285Q341 285 330 295T319 325T330 359T352 380T366 386H367Q367 388 361 392T340 400T306 404Q276 404 249 390Q228 381 206 359Q162 315 142 235T121 119Q121 73 147 50Q169 26 205 26H209Q321 26 394 111Q403 121 406 121Q410 121 419 112T429 98T420 83T391 55T346 25T282 0T202 -11Q127 -11 81 37T34 159Z\"><\/path><path id=\"MJX-22-TEX-I-1D465\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D450\" xlink:href=\"#MJX-22-TEX-I-1D450\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(433,0)\"><use data-c=\"1D465\" xlink:href=\"#MJX-22-TEX-I-1D465\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>c<\/mi><mi>x<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">cx<\/script><span> is in the subspace.<\/span><\/p><\/blockquote><p><span>The <\/span><strong><span>dual code<\/span><\/strong><span> of an <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"5.724ex\" height=\"2.347ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 2529.9 1037.2\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.65ex;\"><defs><path id=\"MJX-36-TEX-N-5B\" d=\"M118 -250V750H255V710H158V-210H255V-250H118Z\"><\/path><path id=\"MJX-36-TEX-I-1D45B\" d=\"M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z\"><\/path><path id=\"MJX-36-TEX-N-2C\" d=\"M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z\"><\/path><path id=\"MJX-36-TEX-I-1D458\" d=\"M121 647Q121 657 125 670T137 683Q138 683 209 688T282 694Q294 694 294 686Q294 679 244 477Q194 279 194 272Q213 282 223 291Q247 309 292 354T362 415Q402 442 438 442Q468 442 485 423T503 369Q503 344 496 327T477 302T456 291T438 288Q418 288 406 299T394 328Q394 353 410 369T442 390L458 393Q446 405 434 405H430Q398 402 367 380T294 316T228 255Q230 254 243 252T267 246T293 238T320 224T342 206T359 180T365 147Q365 130 360 106T354 66Q354 26 381 26Q429 26 459 145Q461 153 479 153H483Q499 153 499 144Q499 139 496 130Q455 -11 378 -11Q333 -11 305 15T277 90Q277 108 280 121T283 145Q283 167 269 183T234 206T200 217T182 220H180Q168 178 159 139T145 81T136 44T129 20T122 7T111 -2Q98 -11 83 -11Q66 -11 57 -1T48 16Q48 26 85 176T158 471L195 616Q196 629 188 632T149 637H144Q134 637 131 637T124 640T121 647Z\"><\/path><path id=\"MJX-36-TEX-N-5D\" d=\"M22 710V750H159V-250H22V-210H119V710H22Z\"><\/path><path id=\"MJX-36-TEX-I-1D45E\" d=\"M33 157Q33 258 109 349T280 441Q340 441 372 389Q373 390 377 395T388 406T404 418Q438 442 450 442Q454 442 457 439T460 434Q460 425 391 149Q320 -135 320 -139Q320 -147 365 -148H390Q396 -156 396 -157T393 -175Q389 -188 383 -194H370Q339 -192 262 -192Q234 -192 211 -192T174 -192T157 -193Q143 -193 143 -185Q143 -182 145 -170Q149 -154 152 -151T172 -148Q220 -148 230 -141Q238 -136 258 -53T279 32Q279 33 272 29Q224 -10 172 -10Q117 -10 75 30T33 157ZM352 326Q329 405 277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q233 26 290 98L298 109L352 326Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mo\"><use data-c=\"5B\" xlink:href=\"#MJX-36-TEX-N-5B\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(278,0)\"><use data-c=\"1D45B\" xlink:href=\"#MJX-36-TEX-I-1D45B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(878,0)\"><use data-c=\"2C\" xlink:href=\"#MJX-36-TEX-N-2C\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1322.7,0)\"><use data-c=\"1D458\" xlink:href=\"#MJX-36-TEX-I-1D458\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(1843.7,0)\"><g data-mml-node=\"mo\"><use data-c=\"5D\" xlink:href=\"#MJX-36-TEX-N-5D\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(311,-150) scale(0.707)\"><use data-c=\"1D45E\" xlink:href=\"#MJX-36-TEX-I-1D45E\"><\/use><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mo stretchy=\"false\">[<\/mo><mi>n<\/mi><mo>,<\/mo><mi>k<\/mi><msub><mo stretchy=\"false\">]<\/mo><mi>q<\/mi><\/msub><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">[n,k]_q<\/script><span>-code <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.192ex\" height=\"1.652ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -705 527 730\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.057ex;\"><defs><path id=\"MJX-32-TEX-C-43\" d=\"M201 -25Q167 -25 136 -14T75 23T29 94T12 202Q12 290 50 394T161 574Q227 642 303 673T433 704Q435 705 457 705Q533 701 533 640Q533 606 507 548T464 474Q431 444 396 444Q381 444 381 453Q381 459 388 473T407 513T428 563Q433 580 433 594Q433 636 381 636Q314 636 260 594T175 489T128 363T112 247Q112 157 153 101T273 44Q347 44 398 121Q413 144 437 157T481 171Q496 171 496 160Q496 150 476 123Q426 56 350 16T201 -25Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"43\" xlink:href=\"#MJX-32-TEX-C-43\"><\/use><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow data-mjx-texclass=\"ORD\"><mi data-mjx-variant=\"-tex-calligraphic\" mathvariant=\"script\">C<\/mi><\/mrow><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">\\mathcal{C}<\/script><span> is defined as <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"34.378ex\" height=\"2.566ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 15195 1134.2\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.869ex;\"><defs><path id=\"MJX-25-TEX-C-43\" d=\"M201 -25Q167 -25 136 -14T75 23T29 94T12 202Q12 290 50 394T161 574Q227 642 303 673T433 704Q435 705 457 705Q533 701 533 640Q533 606 507 548T464 474Q431 444 396 444Q381 444 381 453Q381 459 388 473T407 513T428 563Q433 580 433 594Q433 636 381 636Q314 636 260 594T175 489T128 363T112 247Q112 157 153 101T273 44Q347 44 398 121Q413 144 437 157T481 171Q496 171 496 160Q496 150 476 123Q426 56 350 16T201 -25Z\"><\/path><path id=\"MJX-25-TEX-N-2217\" d=\"M229 286Q216 420 216 436Q216 454 240 464Q241 464 245 464T251 465Q263 464 273 456T283 436Q283 419 277 356T270 286L328 328Q384 369 389 372T399 375Q412 375 423 365T435 338Q435 325 425 315Q420 312 357 282T289 250L355 219L425 184Q434 175 434 161Q434 146 425 136T401 125Q393 125 383 131T328 171L270 213Q283 79 283 63Q283 53 276 44T250 35Q231 35 224 44T216 63Q216 80 222 143T229 213L171 171Q115 130 110 127Q106 124 100 124Q87 124 76 134T64 161Q64 166 64 169T67 175T72 181T81 188T94 195T113 204T138 215T170 230T210 250L74 315Q65 324 65 338Q65 353 74 363T98 374Q106 374 116 368T171 328L229 286Z\"><\/path><path id=\"MJX-25-TEX-N-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path id=\"MJX-25-TEX-N-7B\" d=\"M434 -231Q434 -244 428 -250H410Q281 -250 230 -184Q225 -177 222 -172T217 -161T213 -148T211 -133T210 -111T209 -84T209 -47T209 0Q209 21 209 53Q208 142 204 153Q203 154 203 155Q189 191 153 211T82 231Q71 231 68 234T65 250T68 266T82 269Q116 269 152 289T203 345Q208 356 208 377T209 529V579Q209 634 215 656T244 698Q270 724 324 740Q361 748 377 749Q379 749 390 749T408 750H428Q434 744 434 732Q434 719 431 716Q429 713 415 713Q362 710 332 689T296 647Q291 634 291 499V417Q291 370 288 353T271 314Q240 271 184 255L170 250L184 245Q202 239 220 230T262 196T290 137Q291 131 291 1Q291 -134 296 -147Q306 -174 339 -192T415 -213Q429 -213 431 -216Q434 -219 434 -231Z\"><\/path><path id=\"MJX-25-TEX-B-1D41C\" d=\"M447 131H458Q478 131 478 117Q478 112 471 95T439 51T377 9Q330 -6 286 -6Q196 -6 135 35Q39 96 39 222Q39 324 101 384Q169 453 286 453Q359 453 411 431T464 353Q464 319 445 302T395 284Q360 284 343 305T325 353Q325 380 338 396H333Q317 398 295 398H292Q280 398 271 397T245 390T218 373T197 338T183 283Q182 275 182 231Q182 199 184 180T193 132T220 85T270 57Q289 50 317 50H326Q385 50 414 115Q419 127 423 129T447 131Z\"><\/path><path id=\"MJX-25-TEX-N-2208\" d=\"M84 250Q84 372 166 450T360 539Q361 539 377 539T419 540T469 540H568Q583 532 583 520Q583 511 570 501L466 500Q355 499 329 494Q280 482 242 458T183 409T147 354T129 306T124 272V270H568Q583 262 583 250T568 230H124V228Q124 207 134 177T167 112T231 48T328 7Q355 1 466 0H570Q583 -10 583 -20Q583 -32 568 -40H471Q464 -40 446 -40T417 -41Q262 -41 172 45Q84 127 84 250Z\"><\/path><path id=\"MJX-25-TEX-D-1D53D\" d=\"M584 499Q569 490 566 490Q558 490 552 497T546 515Q546 535 533 559Q526 574 506 593T469 621Q415 648 326 648Q293 648 287 647T275 641Q264 630 263 617Q262 609 260 492V370L275 372Q323 376 350 392T393 441Q409 473 409 506Q409 529 427 529Q437 529 442 519Q444 511 444 362Q444 212 442 206Q436 197 426 197Q409 197 409 217Q409 265 375 299Q346 328 280 335H260V206Q260 70 262 63Q265 46 276 41T326 35Q362 35 366 28Q377 17 366 3L360 -1H24Q12 5 12 16Q12 35 51 35Q92 38 97 52Q102 60 102 341T97 632Q91 645 51 648Q12 648 12 666Q12 675 24 683H573Q576 678 584 670V499ZM137 341Q137 131 136 89T130 37Q129 36 129 35H182Q233 35 233 39Q226 54 225 92T224 346L226 623L231 635L235 648H129Q132 641 133 638T135 603T137 517T137 341ZM549 603V648H495L506 641Q531 621 533 619L549 603ZM409 317V395L400 386Q390 376 375 366L357 355L373 346Q394 331 397 328L409 317Z\"><\/path><path id=\"MJX-25-TEX-I-1D45B\" d=\"M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z\"><\/path><path id=\"MJX-25-TEX-I-1D45E\" d=\"M33 157Q33 258 109 349T280 441Q340 441 372 389Q373 390 377 395T388 406T404 418Q438 442 450 442Q454 442 457 439T460 434Q460 425 391 149Q320 -135 320 -139Q320 -147 365 -148H390Q396 -156 396 -157T393 -175Q389 -188 383 -194H370Q339 -192 262 -192Q234 -192 211 -192T174 -192T157 -193Q143 -193 143 -185Q143 -182 145 -170Q149 -154 152 -151T172 -148Q220 -148 230 -141Q238 -136 258 -53T279 32Q279 33 272 29Q224 -10 172 -10Q117 -10 75 30T33 157ZM352 326Q329 405 277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q233 26 290 98L298 109L352 326Z\"><\/path><path id=\"MJX-25-TEX-N-3A\" d=\"M78 370Q78 394 95 412T138 430Q162 430 180 414T199 371Q199 346 182 328T139 310T96 327T78 370ZM78 60Q78 84 95 102T138 120Q162 120 180 104T199 61Q199 36 182 18T139 0T96 17T78 60Z\"><\/path><path id=\"MJX-25-TEX-N-2200\" d=\"M0 673Q0 684 7 689T20 694Q32 694 38 680T82 567L126 451H430L473 566Q483 593 494 622T512 668T519 685Q524 694 538 694Q556 692 556 674Q556 670 426 329T293 -15Q288 -22 278 -22T263 -15Q260 -11 131 328T0 673ZM414 410Q414 411 278 411T142 410L278 55L414 410Z\"><\/path><path id=\"MJX-25-TEX-N-2C\" d=\"M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z\"><\/path><path id=\"MJX-25-TEX-N-27E8\" d=\"M333 -232Q332 -239 327 -244T313 -250Q303 -250 296 -240Q293 -233 202 6T110 250T201 494T296 740Q299 745 306 749L309 750Q312 750 313 750Q331 750 333 732Q333 727 243 489Q152 252 152 250T243 11Q333 -227 333 -232Z\"><\/path><path id=\"MJX-25-TEX-N-27E9\" d=\"M55 732Q56 739 61 744T75 750Q85 750 92 740Q95 733 186 494T278 250T187 6T92 -240Q85 -250 75 -250Q67 -250 62 -245T55 -232Q55 -227 145 11Q236 248 236 250T145 489Q55 727 55 732Z\"><\/path><path id=\"MJX-25-TEX-N-30\" d=\"M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 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transform=\"translate(560,363) scale(0.707)\"><use data-c=\"2217\" xlink:href=\"#MJX-25-TEX-N-2217\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(1241.3,0)\"><use data-c=\"3D\" xlink:href=\"#MJX-25-TEX-N-3D\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2297.1,0)\"><use data-c=\"7B\" xlink:href=\"#MJX-25-TEX-N-7B\"><\/use><\/g><g data-mml-node=\"msup\" transform=\"translate(2797.1,0)\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D41C\" xlink:href=\"#MJX-25-TEX-B-1D41C\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(544,363) scale(0.707)\"><use data-c=\"2217\" xlink:href=\"#MJX-25-TEX-N-2217\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(4022.4,0)\"><use data-c=\"2208\" xlink:href=\"#MJX-25-TEX-N-2208\"><\/use><\/g><g data-mml-node=\"msubsup\" transform=\"translate(4967.2,0)\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use 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data-mml-node=\"mi\"><use data-c=\"43\" xlink:href=\"#MJX-25-TEX-C-43\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(9735.6,0)\"><use data-c=\"2C\" xlink:href=\"#MJX-25-TEX-N-2C\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(10180.3,0)\"><use data-c=\"27E8\" xlink:href=\"#MJX-25-TEX-N-27E8\"><\/use><\/g><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\" transform=\"translate(10569.3,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D41C\" xlink:href=\"#MJX-25-TEX-B-1D41C\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(11080.3,0)\"><use data-c=\"2C\" xlink:href=\"#MJX-25-TEX-N-2C\"><\/use><\/g><g data-mml-node=\"msup\" transform=\"translate(11524.9,0)\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D41C\" xlink:href=\"#MJX-25-TEX-B-1D41C\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(544,363) scale(0.707)\"><use data-c=\"2217\" xlink:href=\"#MJX-25-TEX-N-2217\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(12472.5,0)\"><use data-c=\"27E9\" xlink:href=\"#MJX-25-TEX-N-27E9\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(13139.3,0)\"><use data-c=\"3D\" xlink:href=\"#MJX-25-TEX-N-3D\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(14195,0)\"><use data-c=\"30\" xlink:href=\"#MJX-25-TEX-N-30\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(14695,0)\"><use data-c=\"7D\" xlink:href=\"#MJX-25-TEX-N-7D\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><msup><mrow data-mjx-texclass=\"ORD\"><mi data-mjx-variant=\"-tex-calligraphic\" mathvariant=\"script\">C<\/mi><\/mrow><mo>\u2217<\/mo><\/msup><mo>=<\/mo><mo fence=\"false\" stretchy=\"false\">{<\/mo><msup><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">c<\/mi><\/mrow><mo>\u2217<\/mo><\/msup><mo>\u2208<\/mo><msubsup><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"double-struck\">F<\/mi><\/mrow><mi>q<\/mi><mi>n<\/mi><\/msubsup><mo>:<\/mo><mi mathvariant=\"normal\">\u2200<\/mi><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">c<\/mi><\/mrow><mo>\u2208<\/mo><mrow data-mjx-texclass=\"ORD\"><mi data-mjx-variant=\"-tex-calligraphic\" mathvariant=\"script\">C<\/mi><\/mrow><mo>,<\/mo><mo fence=\"false\" stretchy=\"false\">\u27e8<\/mo><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">c<\/mi><\/mrow><mo>,<\/mo><msup><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">c<\/mi><\/mrow><mo>\u2217<\/mo><\/msup><mo fence=\"false\" stretchy=\"false\">\u27e9<\/mo><mo>=<\/mo><mn>0<\/mn><mo fence=\"false\" stretchy=\"false\">}<\/mo><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">\\mathcal{C}^*=\\{\\mathbf{c}^*\\in\\mathbb{F}_q^n:\\forall\\mathbf{c}\\in\\mathcal{C},\\langle\\mathbf{c},\\mathbf{c}^*\\rangle=0\\}<\/script><span>. Since <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"8.386ex\" height=\"1.762ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -697 3706.6 779\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.186ex;\"><defs><path id=\"MJX-26-TEX-B-1D41C\" d=\"M447 131H458Q478 131 478 117Q478 112 471 95T439 51T377 9Q330 -6 286 -6Q196 -6 135 35Q39 96 39 222Q39 324 101 384Q169 453 286 453Q359 453 411 431T464 353Q464 319 445 302T395 284Q360 284 343 305T325 353Q325 380 338 396H333Q317 398 295 398H292Q280 398 271 397T245 390T218 373T197 338T183 283Q182 275 182 231Q182 199 184 180T193 132T220 85T270 57Q289 50 317 50H326Q385 50 414 115Q419 127 423 129T447 131Z\"><\/path><path id=\"MJX-26-TEX-N-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path id=\"MJX-26-TEX-B-1D426\" d=\"M40 442Q217 450 218 450H224V365Q226 367 235 378T254 397T278 416T314 435T362 448Q376 450 400 450H406Q503 450 534 393Q545 376 545 370Q545 368 555 379Q611 450 716 450Q774 450 809 434Q850 414 861 379T873 276V213V198V62H942V0H933Q915 3 809 3Q702 3 684 0H675V62H744V194V275Q744 348 735 373T690 399Q645 399 607 370T557 290Q555 281 554 171V62H623V0H614Q596 3 489 3Q374 3 365 0H356V62H425V194V275Q425 348 416 373T371 399Q326 399 288 370T238 290Q236 281 235 171V62H304V0H295Q277 3 171 3Q64 3 46 0H37V62H106V210V303Q106 353 104 363T91 376Q77 380 50 380H37V442H40Z\"><\/path><path id=\"MJX-26-TEX-B-1D406\" d=\"M465 -10Q281 -10 173 88T64 343Q64 413 85 471T143 568T217 631T298 670Q371 697 449 697Q452 697 459 697T470 696Q502 696 531 690T582 675T618 658T644 641T656 632L732 695Q734 697 745 697Q758 697 761 692T765 668V627V489V449Q765 428 761 424T741 419H731H724Q705 419 702 422T695 444Q683 520 631 577T495 635Q364 635 295 563Q261 528 247 477T232 343Q232 296 236 260T256 185T296 120T366 76T472 52Q481 51 498 51Q544 51 573 67T607 108Q608 111 608 164V214H464V276H479Q506 273 680 273Q816 273 834 276H845V214H765V113V51Q765 16 763 8T750 0Q742 2 709 16T658 40L648 46Q592 -10 465 -10Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D41C\" xlink:href=\"#MJX-26-TEX-B-1D41C\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(788.8,0)\"><use data-c=\"3D\" xlink:href=\"#MJX-26-TEX-N-3D\"><\/use><\/g><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\" transform=\"translate(1844.6,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D426\" xlink:href=\"#MJX-26-TEX-B-1D426\"><\/use><use data-c=\"1D406\" xlink:href=\"#MJX-26-TEX-B-1D406\" transform=\"translate(958,0)\"><\/use><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">c<\/mi><\/mrow><mo>=<\/mo><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">mG<\/mi><\/mrow><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">\\mathbf{c}=\\mathbf{mG}<\/script><span>, <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"2.18ex\" height=\"1.652ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -705 963.6 730\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.057ex;\"><defs><path id=\"MJX-27-TEX-C-43\" d=\"M201 -25Q167 -25 136 -14T75 23T29 94T12 202Q12 290 50 394T161 574Q227 642 303 673T433 704Q435 705 457 705Q533 701 533 640Q533 606 507 548T464 474Q431 444 396 444Q381 444 381 453Q381 459 388 473T407 513T428 563Q433 580 433 594Q433 636 381 636Q314 636 260 594T175 489T128 363T112 247Q112 157 153 101T273 44Q347 44 398 121Q413 144 437 157T481 171Q496 171 496 160Q496 150 476 123Q426 56 350 16T201 -25Z\"><\/path><path id=\"MJX-27-TEX-N-2217\" d=\"M229 286Q216 420 216 436Q216 454 240 464Q241 464 245 464T251 465Q263 464 273 456T283 436Q283 419 277 356T270 286L328 328Q384 369 389 372T399 375Q412 375 423 365T435 338Q435 325 425 315Q420 312 357 282T289 250L355 219L425 184Q434 175 434 161Q434 146 425 136T401 125Q393 125 383 131T328 171L270 213Q283 79 283 63Q283 53 276 44T250 35Q231 35 224 44T216 63Q216 80 222 143T229 213L171 171Q115 130 110 127Q106 124 100 124Q87 124 76 134T64 161Q64 166 64 169T67 175T72 181T81 188T94 195T113 204T138 215T170 230T210 250L74 315Q65 324 65 338Q65 353 74 363T98 374Q106 374 116 368T171 328L229 286Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"msup\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"43\" xlink:href=\"#MJX-27-TEX-C-43\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(560,363) scale(0.707)\"><use data-c=\"2217\" xlink:href=\"#MJX-27-TEX-N-2217\"><\/use><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><msup><mrow data-mjx-texclass=\"ORD\"><mi data-mjx-variant=\"-tex-calligraphic\" mathvariant=\"script\">C<\/mi><\/mrow><mo>\u2217<\/mo><\/msup><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">\\mathcal{C}^*<\/script><span> is the nullspace of <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"2.045ex\" height=\"1.6ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -697 904 707\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.023ex;\"><defs><path id=\"MJX-44-TEX-B-1D406\" d=\"M465 -10Q281 -10 173 88T64 343Q64 413 85 471T143 568T217 631T298 670Q371 697 449 697Q452 697 459 697T470 696Q502 696 531 690T582 675T618 658T644 641T656 632L732 695Q734 697 745 697Q758 697 761 692T765 668V627V489V449Q765 428 761 424T741 419H731H724Q705 419 702 422T695 444Q683 520 631 577T495 635Q364 635 295 563Q261 528 247 477T232 343Q232 296 236 260T256 185T296 120T366 76T472 52Q481 51 498 51Q544 51 573 67T607 108Q608 111 608 164V214H464V276H479Q506 273 680 273Q816 273 834 276H845V214H765V113V51Q765 16 763 8T750 0Q742 2 709 16T658 40L648 46Q592 -10 465 -10Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D406\" xlink:href=\"#MJX-44-TEX-B-1D406\"><\/use><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">G<\/mi><\/mrow><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">\\mathbf{G}<\/script><span>, and is an <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"9.847ex\" height=\"2.347ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 4352.4 1037.2\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.65ex;\"><defs><path id=\"MJX-29-TEX-N-5B\" d=\"M118 -250V750H255V710H158V-210H255V-250H118Z\"><\/path><path id=\"MJX-29-TEX-I-1D45B\" d=\"M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z\"><\/path><path id=\"MJX-29-TEX-N-2C\" d=\"M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z\"><\/path><path id=\"MJX-29-TEX-N-2212\" d=\"M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z\"><\/path><path id=\"MJX-29-TEX-I-1D458\" d=\"M121 647Q121 657 125 670T137 683Q138 683 209 688T282 694Q294 694 294 686Q294 679 244 477Q194 279 194 272Q213 282 223 291Q247 309 292 354T362 415Q402 442 438 442Q468 442 485 423T503 369Q503 344 496 327T477 302T456 291T438 288Q418 288 406 299T394 328Q394 353 410 369T442 390L458 393Q446 405 434 405H430Q398 402 367 380T294 316T228 255Q230 254 243 252T267 246T293 238T320 224T342 206T359 180T365 147Q365 130 360 106T354 66Q354 26 381 26Q429 26 459 145Q461 153 479 153H483Q499 153 499 144Q499 139 496 130Q455 -11 378 -11Q333 -11 305 15T277 90Q277 108 280 121T283 145Q283 167 269 183T234 206T200 217T182 220H180Q168 178 159 139T145 81T136 44T129 20T122 7T111 -2Q98 -11 83 -11Q66 -11 57 -1T48 16Q48 26 85 176T158 471L195 616Q196 629 188 632T149 637H144Q134 637 131 637T124 640T121 647Z\"><\/path><path id=\"MJX-29-TEX-N-5D\" d=\"M22 710V750H159V-250H22V-210H119V710H22Z\"><\/path><path id=\"MJX-29-TEX-I-1D45E\" d=\"M33 157Q33 258 109 349T280 441Q340 441 372 389Q373 390 377 395T388 406T404 418Q438 442 450 442Q454 442 457 439T460 434Q460 425 391 149Q320 -135 320 -139Q320 -147 365 -148H390Q396 -156 396 -157T393 -175Q389 -188 383 -194H370Q339 -192 262 -192Q234 -192 211 -192T174 -192T157 -193Q143 -193 143 -185Q143 -182 145 -170Q149 -154 152 -151T172 -148Q220 -148 230 -141Q238 -136 258 -53T279 32Q279 33 272 29Q224 -10 172 -10Q117 -10 75 30T33 157ZM352 326Q329 405 277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q233 26 290 98L298 109L352 326Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mo\"><use data-c=\"5B\" xlink:href=\"#MJX-29-TEX-N-5B\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(278,0)\"><use data-c=\"1D45B\" xlink:href=\"#MJX-29-TEX-I-1D45B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(878,0)\"><use data-c=\"2C\" xlink:href=\"#MJX-29-TEX-N-2C\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1322.7,0)\"><use data-c=\"1D45B\" xlink:href=\"#MJX-29-TEX-I-1D45B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2144.9,0)\"><use data-c=\"2212\" xlink:href=\"#MJX-29-TEX-N-2212\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(3145.1,0)\"><use data-c=\"1D458\" xlink:href=\"#MJX-29-TEX-I-1D458\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(3666.1,0)\"><g data-mml-node=\"mo\"><use data-c=\"5D\" xlink:href=\"#MJX-29-TEX-N-5D\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(311,-150) scale(0.707)\"><use data-c=\"1D45E\" xlink:href=\"#MJX-29-TEX-I-1D45E\"><\/use><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mo stretchy=\"false\">[<\/mo><mi>n<\/mi><mo>,<\/mo><mi>n<\/mi><mo>\u2212<\/mo><mi>k<\/mi><msub><mo stretchy=\"false\">]<\/mo><mi>q<\/mi><\/msub><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">[n,n-k]_q<\/script><span>-code. <\/span><\/p><p><span>Its generator matrix is the <\/span><strong><span>parity-check matrix<\/span><\/strong><span> of <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.192ex\" height=\"1.652ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -705 527 730\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.057ex;\"><defs><path id=\"MJX-32-TEX-C-43\" d=\"M201 -25Q167 -25 136 -14T75 23T29 94T12 202Q12 290 50 394T161 574Q227 642 303 673T433 704Q435 705 457 705Q533 701 533 640Q533 606 507 548T464 474Q431 444 396 444Q381 444 381 453Q381 459 388 473T407 513T428 563Q433 580 433 594Q433 636 381 636Q314 636 260 594T175 489T128 363T112 247Q112 157 153 101T273 44Q347 44 398 121Q413 144 437 157T481 171Q496 171 496 160Q496 150 476 123Q426 56 350 16T201 -25Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"43\" xlink:href=\"#MJX-32-TEX-C-43\"><\/use><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow data-mjx-texclass=\"ORD\"><mi data-mjx-variant=\"-tex-calligraphic\" mathvariant=\"script\">C<\/mi><\/mrow><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">\\mathcal{C}<\/script><span>, i.e., <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"12.859ex\" height=\"3.024ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -1060.7 5683.9 1336.8\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.625ex;\"><defs><path id=\"MJX-551-TEX-B-1D407\" d=\"M400 0Q376 3 226 3Q75 3 51 0H39V62H147V624H39V686H51Q75 683 226 683Q376 683 400 686H412V624H304V388H595V624H487V686H499Q523 683 673 683Q824 683 848 686H860V624H752V62H860V0H848Q824 3 674 3Q523 3 499 0H487V62H595V326H304V62H412V0H400Z\"><\/path><path id=\"MJX-551-TEX-N-2208\" d=\"M84 250Q84 372 166 450T360 539Q361 539 377 539T419 540T469 540H568Q583 532 583 520Q583 511 570 501L466 500Q355 499 329 494Q280 482 242 458T183 409T147 354T129 306T124 272V270H568Q583 262 583 250T568 230H124V228Q124 207 134 177T167 112T231 48T328 7Q355 1 466 0H570Q583 -10 583 -20Q583 -32 568 -40H471Q464 -40 446 -40T417 -41Q262 -41 172 45Q84 127 84 250Z\"><\/path><path id=\"MJX-551-TEX-D-1D53D\" d=\"M584 499Q569 490 566 490Q558 490 552 497T546 515Q546 535 533 559Q526 574 506 593T469 621Q415 648 326 648Q293 648 287 647T275 641Q264 630 263 617Q262 609 260 492V370L275 372Q323 376 350 392T393 441Q409 473 409 506Q409 529 427 529Q437 529 442 519Q444 511 444 362Q444 212 442 206Q436 197 426 197Q409 197 409 217Q409 265 375 299Q346 328 280 335H260V206Q260 70 262 63Q265 46 276 41T326 35Q362 35 366 28Q377 17 366 3L360 -1H24Q12 5 12 16Q12 35 51 35Q92 38 97 52Q102 60 102 341T97 632Q91 645 51 648Q12 648 12 666Q12 675 24 683H573Q576 678 584 670V499ZM137 341Q137 131 136 89T130 37Q129 36 129 35H182Q233 35 233 39Q226 54 225 92T224 346L226 623L231 635L235 648H129Q132 641 133 638T135 603T137 517T137 341ZM549 603V648H495L506 641Q531 621 533 619L549 603ZM409 317V395L400 386Q390 376 375 366L357 355L373 346Q394 331 397 328L409 317Z\"><\/path><path id=\"MJX-551-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-551-TEX-I-1D45B\" d=\"M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z\"><\/path><path id=\"MJX-551-TEX-N-2212\" d=\"M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z\"><\/path><path id=\"MJX-551-TEX-I-1D458\" d=\"M121 647Q121 657 125 670T137 683Q138 683 209 688T282 694Q294 694 294 686Q294 679 244 477Q194 279 194 272Q213 282 223 291Q247 309 292 354T362 415Q402 442 438 442Q468 442 485 423T503 369Q503 344 496 327T477 302T456 291T438 288Q418 288 406 299T394 328Q394 353 410 369T442 390L458 393Q446 405 434 405H430Q398 402 367 380T294 316T228 255Q230 254 243 252T267 246T293 238T320 224T342 206T359 180T365 147Q365 130 360 106T354 66Q354 26 381 26Q429 26 459 145Q461 153 479 153H483Q499 153 499 144Q499 139 496 130Q455 -11 378 -11Q333 -11 305 15T277 90Q277 108 280 121T283 145Q283 167 269 183T234 206T200 217T182 220H180Q168 178 159 139T145 81T136 44T129 20T122 7T111 -2Q98 -11 83 -11Q66 -11 57 -1T48 16Q48 26 85 176T158 471L195 616Q196 629 188 632T149 637H144Q134 637 131 637T124 640T121 647Z\"><\/path><path id=\"MJX-551-TEX-N-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><path id=\"MJX-551-TEX-N-D7\" d=\"M630 29Q630 9 609 9Q604 9 587 25T493 118L389 222L284 117Q178 13 175 11Q171 9 168 9Q160 9 154 15T147 29Q147 36 161 51T255 146L359 250L255 354Q174 435 161 449T147 471Q147 480 153 485T168 490Q173 490 175 489Q178 487 284 383L389 278L493 382Q570 459 587 475T609 491Q630 491 630 471Q630 464 620 453T522 355L418 250L522 145Q606 61 618 48T630 29Z\"><\/path><path id=\"MJX-551-TEX-I-1D45E\" d=\"M33 157Q33 258 109 349T280 441Q340 441 372 389Q373 390 377 395T388 406T404 418Q438 442 450 442Q454 442 457 439T460 434Q460 425 391 149Q320 -135 320 -139Q320 -147 365 -148H390Q396 -156 396 -157T393 -175Q389 -188 383 -194H370Q339 -192 262 -192Q234 -192 211 -192T174 -192T157 -193Q143 -193 143 -185Q143 -182 145 -170Q149 -154 152 -151T172 -148Q220 -148 230 -141Q238 -136 258 -53T279 32Q279 33 272 29Q224 -10 172 -10Q117 -10 75 30T33 157ZM352 326Q329 405 277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q233 26 290 98L298 109L352 326Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D407\" xlink:href=\"#MJX-551-TEX-B-1D407\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(1177.8,0)\"><use data-c=\"2208\" xlink:href=\"#MJX-551-TEX-N-2208\"><\/use><\/g><g data-mml-node=\"msubsup\" transform=\"translate(2122.6,0)\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D53D\" xlink:href=\"#MJX-551-TEX-D-1D53D\"><\/use><\/g><\/g><g data-mml-node=\"TeXAtom\" transform=\"translate(644,530.4) scale(0.707)\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mo\"><use data-c=\"28\" xlink:href=\"#MJX-551-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(389,0)\"><use data-c=\"1D45B\" xlink:href=\"#MJX-551-TEX-I-1D45B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(989,0)\"><use data-c=\"2212\" xlink:href=\"#MJX-551-TEX-N-2212\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1767,0)\"><use data-c=\"1D458\" xlink:href=\"#MJX-551-TEX-I-1D458\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2288,0)\"><use data-c=\"29\" xlink:href=\"#MJX-551-TEX-N-29\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2677,0)\"><use data-c=\"D7\" xlink:href=\"#MJX-551-TEX-N-D7\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(3455,0)\"><use data-c=\"1D45B\" xlink:href=\"#MJX-551-TEX-I-1D45B\"><\/use><\/g><\/g><g data-mml-node=\"mi\" transform=\"translate(644,-138.9) scale(0.707)\"><use data-c=\"1D45E\" xlink:href=\"#MJX-551-TEX-I-1D45E\"><\/use><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">H<\/mi><\/mrow><mo>\u2208<\/mo><msubsup><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"double-struck\">F<\/mi><\/mrow><mi>q<\/mi><mrow data-mjx-texclass=\"ORD\"><mo stretchy=\"false\">(<\/mo><mi>n<\/mi><mo>\u2212<\/mo><mi>k<\/mi><mo stretchy=\"false\">)<\/mo><mo>\u00d7<\/mo><mi>n<\/mi><\/mrow><\/msubsup><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">\\mathbf{H}\\in\\mathbb{F}_q^{(n-k)\\times n}<\/script><span> , whose rows are basis of the nullspace.<\/span><\/p><p><span>The linear code <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.192ex\" height=\"1.652ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -705 527 730\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.057ex;\"><defs><path id=\"MJX-32-TEX-C-43\" d=\"M201 -25Q167 -25 136 -14T75 23T29 94T12 202Q12 290 50 394T161 574Q227 642 303 673T433 704Q435 705 457 705Q533 701 533 640Q533 606 507 548T464 474Q431 444 396 444Q381 444 381 453Q381 459 388 473T407 513T428 563Q433 580 433 594Q433 636 381 636Q314 636 260 594T175 489T128 363T112 247Q112 157 153 101T273 44Q347 44 398 121Q413 144 437 157T481 171Q496 171 496 160Q496 150 476 123Q426 56 350 16T201 -25Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"43\" xlink:href=\"#MJX-32-TEX-C-43\"><\/use><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow data-mjx-texclass=\"ORD\"><mi data-mjx-variant=\"-tex-calligraphic\" mathvariant=\"script\">C<\/mi><\/mrow><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">\\mathcal{C}<\/script><span> can also be written as <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"23.653ex\" height=\"2.566ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 10454.6 1134.2\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.869ex;\"><defs><path id=\"MJX-33-TEX-C-43\" d=\"M201 -25Q167 -25 136 -14T75 23T29 94T12 202Q12 290 50 394T161 574Q227 642 303 673T433 704Q435 705 457 705Q533 701 533 640Q533 606 507 548T464 474Q431 444 396 444Q381 444 381 453Q381 459 388 473T407 513T428 563Q433 580 433 594Q433 636 381 636Q314 636 260 594T175 489T128 363T112 247Q112 157 153 101T273 44Q347 44 398 121Q413 144 437 157T481 171Q496 171 496 160Q496 150 476 123Q426 56 350 16T201 -25Z\"><\/path><path id=\"MJX-33-TEX-N-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path id=\"MJX-33-TEX-N-7B\" d=\"M434 -231Q434 -244 428 -250H410Q281 -250 230 -184Q225 -177 222 -172T217 -161T213 -148T211 -133T210 -111T209 -84T209 -47T209 0Q209 21 209 53Q208 142 204 153Q203 154 203 155Q189 191 153 211T82 231Q71 231 68 234T65 250T68 266T82 269Q116 269 152 289T203 345Q208 356 208 377T209 529V579Q209 634 215 656T244 698Q270 724 324 740Q361 748 377 749Q379 749 390 749T408 750H428Q434 744 434 732Q434 719 431 716Q429 713 415 713Q362 710 332 689T296 647Q291 634 291 499V417Q291 370 288 353T271 314Q240 271 184 255L170 250L184 245Q202 239 220 230T262 196T290 137Q291 131 291 1Q291 -134 296 -147Q306 -174 339 -192T415 -213Q429 -213 431 -216Q434 -219 434 -231Z\"><\/path><path id=\"MJX-33-TEX-B-1D41C\" d=\"M447 131H458Q478 131 478 117Q478 112 471 95T439 51T377 9Q330 -6 286 -6Q196 -6 135 35Q39 96 39 222Q39 324 101 384Q169 453 286 453Q359 453 411 431T464 353Q464 319 445 302T395 284Q360 284 343 305T325 353Q325 380 338 396H333Q317 398 295 398H292Q280 398 271 397T245 390T218 373T197 338T183 283Q182 275 182 231Q182 199 184 180T193 132T220 85T270 57Q289 50 317 50H326Q385 50 414 115Q419 127 423 129T447 131Z\"><\/path><path id=\"MJX-33-TEX-N-2208\" d=\"M84 250Q84 372 166 450T360 539Q361 539 377 539T419 540T469 540H568Q583 532 583 520Q583 511 570 501L466 500Q355 499 329 494Q280 482 242 458T183 409T147 354T129 306T124 272V270H568Q583 262 583 250T568 230H124V228Q124 207 134 177T167 112T231 48T328 7Q355 1 466 0H570Q583 -10 583 -20Q583 -32 568 -40H471Q464 -40 446 -40T417 -41Q262 -41 172 45Q84 127 84 250Z\"><\/path><path id=\"MJX-33-TEX-B-1D405\" d=\"M425 0L228 3Q63 3 51 0H39V62H147V618H39V680H644V676Q647 670 659 552T675 428V424H613Q613 433 605 477Q599 511 589 535T562 574T530 599T488 612T441 617T387 618H368H304V371H333Q389 373 411 390T437 468V488H499V192H437V212Q436 244 430 263T408 292T378 305T333 309H304V62H439V0H425Z\"><\/path><path id=\"MJX-33-TEX-I-1D45B\" d=\"M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z\"><\/path><path id=\"MJX-33-TEX-I-1D45E\" d=\"M33 157Q33 258 109 349T280 441Q340 441 372 389Q373 390 377 395T388 406T404 418Q438 442 450 442Q454 442 457 439T460 434Q460 425 391 149Q320 -135 320 -139Q320 -147 365 -148H390Q396 -156 396 -157T393 -175Q389 -188 383 -194H370Q339 -192 262 -192Q234 -192 211 -192T174 -192T157 -193Q143 -193 143 -185Q143 -182 145 -170Q149 -154 152 -151T172 -148Q220 -148 230 -141Q238 -136 258 -53T279 32Q279 33 272 29Q224 -10 172 -10Q117 -10 75 30T33 157ZM352 326Q329 405 277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q233 26 290 98L298 109L352 326Z\"><\/path><path id=\"MJX-33-TEX-N-3A\" d=\"M78 370Q78 394 95 412T138 430Q162 430 180 414T199 371Q199 346 182 328T139 310T96 327T78 370ZM78 60Q78 84 95 102T138 120Q162 120 180 104T199 61Q199 36 182 18T139 0T96 17T78 60Z\"><\/path><path id=\"MJX-33-TEX-B-1D407\" d=\"M400 0Q376 3 226 3Q75 3 51 0H39V62H147V624H39V686H51Q75 683 226 683Q376 683 400 686H412V624H304V388H595V624H487V686H499Q523 683 673 683Q824 683 848 686H860V624H752V62H860V0H848Q824 3 674 3Q523 3 499 0H487V62H595V326H304V62H412V0H400Z\"><\/path><path id=\"MJX-33-TEX-N-22BA\" d=\"M318 -182Q302 -212 280 -212H278H275Q249 -212 239 -182L238 84V351H162L87 352Q57 362 57 391T84 429Q89 431 280 431H470L474 429Q477 427 479 426T484 423T490 417T495 410T499 402T500 391Q500 365 470 352L394 351H318V-182Z\"><\/path><path id=\"MJX-33-TEX-B-1D7CE\" d=\"M266 654H280H282Q500 654 524 418Q529 370 529 320Q529 125 456 52Q397 -10 287 -10Q110 -10 63 154Q45 212 45 316Q45 504 113 585Q140 618 185 636T266 654ZM374 548Q347 604 286 604Q247 604 218 575Q197 552 193 511T188 311Q188 159 196 116Q202 87 225 64T287 41Q339 41 367 87Q379 107 382 152T386 329Q386 518 374 548Z\"><\/path><path id=\"MJX-33-TEX-N-7D\" d=\"M65 731Q65 745 68 747T88 750Q171 750 216 725T279 670Q288 649 289 635T291 501Q292 362 293 357Q306 312 345 291T417 269Q428 269 431 266T434 250T431 234T417 231Q380 231 345 210T298 157Q293 143 292 121T291 -28V-79Q291 -134 285 -156T256 -198Q202 -250 89 -250Q71 -250 68 -247T65 -230Q65 -224 65 -223T66 -218T69 -214T77 -213Q91 -213 108 -210T146 -200T183 -177T207 -139Q208 -134 209 3L210 139Q223 196 280 230Q315 247 330 250Q305 257 280 270Q225 304 212 352L210 362L209 498Q208 635 207 640Q195 680 154 696T77 713Q68 713 67 716T65 731Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"43\" xlink:href=\"#MJX-33-TEX-C-43\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(804.8,0)\"><use data-c=\"3D\" xlink:href=\"#MJX-33-TEX-N-3D\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(1860.6,0)\"><use data-c=\"7B\" xlink:href=\"#MJX-33-TEX-N-7B\"><\/use><\/g><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\" transform=\"translate(2360.6,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D41C\" xlink:href=\"#MJX-33-TEX-B-1D41C\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(3149.3,0)\"><use data-c=\"2208\" xlink:href=\"#MJX-33-TEX-N-2208\"><\/use><\/g><g data-mml-node=\"msubsup\" transform=\"translate(4094.1,0)\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D405\" xlink:href=\"#MJX-33-TEX-B-1D405\"><\/use><\/g><\/g><g data-mml-node=\"mi\" transform=\"translate(757,363) scale(0.707)\"><use data-c=\"1D45B\" xlink:href=\"#MJX-33-TEX-I-1D45B\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(757,-247) scale(0.707)\"><use data-c=\"1D45E\" xlink:href=\"#MJX-33-TEX-I-1D45E\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(5603.2,0)\"><use data-c=\"3A\" xlink:href=\"#MJX-33-TEX-N-3A\"><\/use><\/g><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\" transform=\"translate(6158.9,0)\"><g data-mml-node=\"msup\"><g data-mml-node=\"mi\"><use data-c=\"1D407\" xlink:href=\"#MJX-33-TEX-B-1D407\"><\/use><use data-c=\"1D41C\" xlink:href=\"#MJX-33-TEX-B-1D41C\" transform=\"translate(900,0)\"><\/use><\/g><g data-mml-node=\"TeXAtom\" transform=\"translate(1444,413.1) scale(0.707)\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mo\"><use data-c=\"22BA\" xlink:href=\"#MJX-33-TEX-N-22BA\"><\/use><\/g><\/g><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(8323.9,0)\"><use data-c=\"3D\" xlink:href=\"#MJX-33-TEX-N-3D\"><\/use><\/g><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\" transform=\"translate(9379.6,0)\"><g data-mml-node=\"mn\"><use data-c=\"1D7CE\" xlink:href=\"#MJX-33-TEX-B-1D7CE\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(9954.6,0)\"><use data-c=\"7D\" xlink:href=\"#MJX-33-TEX-N-7D\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow data-mjx-texclass=\"ORD\"><mi data-mjx-variant=\"-tex-calligraphic\" mathvariant=\"script\">C<\/mi><\/mrow><mo>=<\/mo><mo fence=\"false\" stretchy=\"false\">{<\/mo><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">c<\/mi><\/mrow><mo>\u2208<\/mo><msubsup><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">F<\/mi><\/mrow><mi>q<\/mi><mi>n<\/mi><\/msubsup><mo>:<\/mo><mrow data-mjx-texclass=\"ORD\"><msup><mi mathvariant=\"bold\">Hc<\/mi><mrow data-mjx-texclass=\"ORD\"><mo>\u22ba<\/mo><\/mrow><\/msup><\/mrow><mo>=<\/mo><mrow data-mjx-texclass=\"ORD\"><mn mathvariant=\"bold\">0<\/mn><\/mrow><mo fence=\"false\" stretchy=\"false\">}<\/mo><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">\\mathcal{C}=\\{\\mathbf{c}\\in\\mathbf{F}_q^n:\\mathbf{Hc^{\\intercal}}=\\mathbf{0}\\}<\/script><span>.<\/span><\/p><p><span>Conversely, any matrix <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"12.859ex\" height=\"3.024ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -1060.7 5683.9 1336.8\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.625ex;\"><defs><path id=\"MJX-551-TEX-B-1D407\" d=\"M400 0Q376 3 226 3Q75 3 51 0H39V62H147V624H39V686H51Q75 683 226 683Q376 683 400 686H412V624H304V388H595V624H487V686H499Q523 683 673 683Q824 683 848 686H860V624H752V62H860V0H848Q824 3 674 3Q523 3 499 0H487V62H595V326H304V62H412V0H400Z\"><\/path><path id=\"MJX-551-TEX-N-2208\" d=\"M84 250Q84 372 166 450T360 539Q361 539 377 539T419 540T469 540H568Q583 532 583 520Q583 511 570 501L466 500Q355 499 329 494Q280 482 242 458T183 409T147 354T129 306T124 272V270H568Q583 262 583 250T568 230H124V228Q124 207 134 177T167 112T231 48T328 7Q355 1 466 0H570Q583 -10 583 -20Q583 -32 568 -40H471Q464 -40 446 -40T417 -41Q262 -41 172 45Q84 127 84 250Z\"><\/path><path id=\"MJX-551-TEX-D-1D53D\" d=\"M584 499Q569 490 566 490Q558 490 552 497T546 515Q546 535 533 559Q526 574 506 593T469 621Q415 648 326 648Q293 648 287 647T275 641Q264 630 263 617Q262 609 260 492V370L275 372Q323 376 350 392T393 441Q409 473 409 506Q409 529 427 529Q437 529 442 519Q444 511 444 362Q444 212 442 206Q436 197 426 197Q409 197 409 217Q409 265 375 299Q346 328 280 335H260V206Q260 70 262 63Q265 46 276 41T326 35Q362 35 366 28Q377 17 366 3L360 -1H24Q12 5 12 16Q12 35 51 35Q92 38 97 52Q102 60 102 341T97 632Q91 645 51 648Q12 648 12 666Q12 675 24 683H573Q576 678 584 670V499ZM137 341Q137 131 136 89T130 37Q129 36 129 35H182Q233 35 233 39Q226 54 225 92T224 346L226 623L231 635L235 648H129Q132 641 133 638T135 603T137 517T137 341ZM549 603V648H495L506 641Q531 621 533 619L549 603ZM409 317V395L400 386Q390 376 375 366L357 355L373 346Q394 331 397 328L409 317Z\"><\/path><path id=\"MJX-551-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-551-TEX-I-1D45B\" d=\"M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z\"><\/path><path id=\"MJX-551-TEX-N-2212\" d=\"M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z\"><\/path><path id=\"MJX-551-TEX-I-1D458\" d=\"M121 647Q121 657 125 670T137 683Q138 683 209 688T282 694Q294 694 294 686Q294 679 244 477Q194 279 194 272Q213 282 223 291Q247 309 292 354T362 415Q402 442 438 442Q468 442 485 423T503 369Q503 344 496 327T477 302T456 291T438 288Q418 288 406 299T394 328Q394 353 410 369T442 390L458 393Q446 405 434 405H430Q398 402 367 380T294 316T228 255Q230 254 243 252T267 246T293 238T320 224T342 206T359 180T365 147Q365 130 360 106T354 66Q354 26 381 26Q429 26 459 145Q461 153 479 153H483Q499 153 499 144Q499 139 496 130Q455 -11 378 -11Q333 -11 305 15T277 90Q277 108 280 121T283 145Q283 167 269 183T234 206T200 217T182 220H180Q168 178 159 139T145 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data-mml-node=\"mi\"><use data-c=\"1D53D\" xlink:href=\"#MJX-551-TEX-D-1D53D\"><\/use><\/g><\/g><g data-mml-node=\"TeXAtom\" transform=\"translate(644,530.4) scale(0.707)\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mo\"><use data-c=\"28\" xlink:href=\"#MJX-551-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(389,0)\"><use data-c=\"1D45B\" xlink:href=\"#MJX-551-TEX-I-1D45B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(989,0)\"><use data-c=\"2212\" xlink:href=\"#MJX-551-TEX-N-2212\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1767,0)\"><use data-c=\"1D458\" xlink:href=\"#MJX-551-TEX-I-1D458\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2288,0)\"><use data-c=\"29\" xlink:href=\"#MJX-551-TEX-N-29\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2677,0)\"><use data-c=\"D7\" xlink:href=\"#MJX-551-TEX-N-D7\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(3455,0)\"><use data-c=\"1D45B\" xlink:href=\"#MJX-551-TEX-I-1D45B\"><\/use><\/g><\/g><g data-mml-node=\"mi\" transform=\"translate(644,-138.9) scale(0.707)\"><use data-c=\"1D45E\" xlink:href=\"#MJX-551-TEX-I-1D45E\"><\/use><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">H<\/mi><\/mrow><mo>\u2208<\/mo><msubsup><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"double-struck\">F<\/mi><\/mrow><mi>q<\/mi><mrow data-mjx-texclass=\"ORD\"><mo stretchy=\"false\">(<\/mo><mi>n<\/mi><mo>\u2212<\/mo><mi>k<\/mi><mo stretchy=\"false\">)<\/mo><mo>\u00d7<\/mo><mi>n<\/mi><\/mrow><\/msubsup><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">\\mathbf{H}\\in\\mathbb{F}_q^{(n-k)\\times n}<\/script><span> of rank <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"5.302ex\" height=\"1.756ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -694 2343.4 776\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.186ex;\"><defs><path id=\"MJX-1297-TEX-I-1D45B\" d=\"M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z\"><\/path><path id=\"MJX-1297-TEX-N-2212\" d=\"M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z\"><\/path><path id=\"MJX-1297-TEX-I-1D458\" d=\"M121 647Q121 657 125 670T137 683Q138 683 209 688T282 694Q294 694 294 686Q294 679 244 477Q194 279 194 272Q213 282 223 291Q247 309 292 354T362 415Q402 442 438 442Q468 442 485 423T503 369Q503 344 496 327T477 302T456 291T438 288Q418 288 406 299T394 328Q394 353 410 369T442 390L458 393Q446 405 434 405H430Q398 402 367 380T294 316T228 255Q230 254 243 252T267 246T293 238T320 224T342 206T359 180T365 147Q365 130 360 106T354 66Q354 26 381 26Q429 26 459 145Q461 153 479 153H483Q499 153 499 144Q499 139 496 130Q455 -11 378 -11Q333 -11 305 15T277 90Q277 108 280 121T283 145Q283 167 269 183T234 206T200 217T182 220H180Q168 178 159 139T145 81T136 44T129 20T122 7T111 -2Q98 -11 83 -11Q66 -11 57 -1T48 16Q48 26 85 176T158 471L195 616Q196 629 188 632T149 637H144Q134 637 131 637T124 640T121 647Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D45B\" xlink:href=\"#MJX-1297-TEX-I-1D45B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(822.2,0)\"><use data-c=\"2212\" xlink:href=\"#MJX-1297-TEX-N-2212\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1822.4,0)\"><use data-c=\"1D458\" xlink:href=\"#MJX-1297-TEX-I-1D458\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>n<\/mi><mo>\u2212<\/mo><mi>k<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">n-k<\/script><span> defines an <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"5.724ex\" height=\"2.347ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 2529.9 1037.2\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.65ex;\"><defs><path id=\"MJX-36-TEX-N-5B\" d=\"M118 -250V750H255V710H158V-210H255V-250H118Z\"><\/path><path id=\"MJX-36-TEX-I-1D45B\" d=\"M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z\"><\/path><path id=\"MJX-36-TEX-N-2C\" d=\"M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z\"><\/path><path id=\"MJX-36-TEX-I-1D458\" d=\"M121 647Q121 657 125 670T137 683Q138 683 209 688T282 694Q294 694 294 686Q294 679 244 477Q194 279 194 272Q213 282 223 291Q247 309 292 354T362 415Q402 442 438 442Q468 442 485 423T503 369Q503 344 496 327T477 302T456 291T438 288Q418 288 406 299T394 328Q394 353 410 369T442 390L458 393Q446 405 434 405H430Q398 402 367 380T294 316T228 255Q230 254 243 252T267 246T293 238T320 224T342 206T359 180T365 147Q365 130 360 106T354 66Q354 26 381 26Q429 26 459 145Q461 153 479 153H483Q499 153 499 144Q499 139 496 130Q455 -11 378 -11Q333 -11 305 15T277 90Q277 108 280 121T283 145Q283 167 269 183T234 206T200 217T182 220H180Q168 178 159 139T145 81T136 44T129 20T122 7T111 -2Q98 -11 83 -11Q66 -11 57 -1T48 16Q48 26 85 176T158 471L195 616Q196 629 188 632T149 637H144Q134 637 131 637T124 640T121 647Z\"><\/path><path id=\"MJX-36-TEX-N-5D\" d=\"M22 710V750H159V-250H22V-210H119V710H22Z\"><\/path><path id=\"MJX-36-TEX-I-1D45E\" d=\"M33 157Q33 258 109 349T280 441Q340 441 372 389Q373 390 377 395T388 406T404 418Q438 442 450 442Q454 442 457 439T460 434Q460 425 391 149Q320 -135 320 -139Q320 -147 365 -148H390Q396 -156 396 -157T393 -175Q389 -188 383 -194H370Q339 -192 262 -192Q234 -192 211 -192T174 -192T157 -193Q143 -193 143 -185Q143 -182 145 -170Q149 -154 152 -151T172 -148Q220 -148 230 -141Q238 -136 258 -53T279 32Q279 33 272 29Q224 -10 172 -10Q117 -10 75 30T33 157ZM352 326Q329 405 277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q233 26 290 98L298 109L352 326Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mo\"><use data-c=\"5B\" xlink:href=\"#MJX-36-TEX-N-5B\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(278,0)\"><use data-c=\"1D45B\" xlink:href=\"#MJX-36-TEX-I-1D45B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(878,0)\"><use data-c=\"2C\" xlink:href=\"#MJX-36-TEX-N-2C\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1322.7,0)\"><use data-c=\"1D458\" xlink:href=\"#MJX-36-TEX-I-1D458\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(1843.7,0)\"><g data-mml-node=\"mo\"><use data-c=\"5D\" xlink:href=\"#MJX-36-TEX-N-5D\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(311,-150) scale(0.707)\"><use data-c=\"1D45E\" xlink:href=\"#MJX-36-TEX-I-1D45E\"><\/use><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mo stretchy=\"false\">[<\/mo><mi>n<\/mi><mo>,<\/mo><mi>k<\/mi><msub><mo stretchy=\"false\">]<\/mo><mi>q<\/mi><\/msub><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">[n,k]_q<\/script><span>-code. <\/span><\/p><blockquote><p><span>Solutions to <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"7.737ex\" height=\"1.762ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -697 3419.6 779\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.186ex;\"><defs><path id=\"MJX-52-TEX-B-1D406\" d=\"M465 -10Q281 -10 173 88T64 343Q64 413 85 471T143 568T217 631T298 670Q371 697 449 697Q452 697 459 697T470 696Q502 696 531 690T582 675T618 658T644 641T656 632L732 695Q734 697 745 697Q758 697 761 692T765 668V627V489V449Q765 428 761 424T741 419H731H724Q705 419 702 422T695 444Q683 520 631 577T495 635Q364 635 295 563Q261 528 247 477T232 343Q232 296 236 260T256 185T296 120T366 76T472 52Q481 51 498 51Q544 51 573 67T607 108Q608 111 608 164V214H464V276H479Q506 273 680 273Q816 273 834 276H845V214H765V113V51Q765 16 763 8T750 0Q742 2 709 16T658 40L648 46Q592 -10 465 -10Z\"><\/path><path id=\"MJX-52-TEX-B-1D431\" d=\"M227 0Q212 3 121 3Q40 3 28 0H21V62H117L245 213L109 382H26V444H34Q49 441 143 441Q247 441 265 444H274V382H246L281 339Q315 297 316 297Q320 297 354 341L389 382H352V444H360Q375 441 466 441Q547 441 559 444H566V382H471L355 246L504 63L545 62H586V0H578Q563 3 469 3Q365 3 347 0H338V62H366Q366 63 326 112T285 163L198 63L217 62H235V0H227Z\"><\/path><path id=\"MJX-52-TEX-N-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path id=\"MJX-52-TEX-B-1D7CE\" d=\"M266 654H280H282Q500 654 524 418Q529 370 529 320Q529 125 456 52Q397 -10 287 -10Q110 -10 63 154Q45 212 45 316Q45 504 113 585Q140 618 185 636T266 654ZM374 548Q347 604 286 604Q247 604 218 575Q197 552 193 511T188 311Q188 159 196 116Q202 87 225 64T287 41Q339 41 367 87Q379 107 382 152T386 329Q386 518 374 548Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D406\" xlink:href=\"#MJX-52-TEX-B-1D406\"><\/use><use data-c=\"1D431\" xlink:href=\"#MJX-52-TEX-B-1D431\" transform=\"translate(904,0)\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(1788.8,0)\"><use data-c=\"3D\" xlink:href=\"#MJX-52-TEX-N-3D\"><\/use><\/g><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\" transform=\"translate(2844.6,0)\"><g data-mml-node=\"mn\"><use data-c=\"1D7CE\" xlink:href=\"#MJX-52-TEX-B-1D7CE\"><\/use><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">Gx<\/mi><\/mrow><mo>=<\/mo><mrow data-mjx-texclass=\"ORD\"><mn mathvariant=\"bold\">0<\/mn><\/mrow><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">\\mathbf{Gx}=\\mathbf{0}<\/script><span> form the <\/span><strong><span>nullspace<\/span><\/strong><span> of <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"2.045ex\" height=\"1.6ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -697 904 707\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.023ex;\"><defs><path id=\"MJX-44-TEX-B-1D406\" d=\"M465 -10Q281 -10 173 88T64 343Q64 413 85 471T143 568T217 631T298 670Q371 697 449 697Q452 697 459 697T470 696Q502 696 531 690T582 675T618 658T644 641T656 632L732 695Q734 697 745 697Q758 697 761 692T765 668V627V489V449Q765 428 761 424T741 419H731H724Q705 419 702 422T695 444Q683 520 631 577T495 635Q364 635 295 563Q261 528 247 477T232 343Q232 296 236 260T256 185T296 120T366 76T472 52Q481 51 498 51Q544 51 573 67T607 108Q608 111 608 164V214H464V276H479Q506 273 680 273Q816 273 834 276H845V214H765V113V51Q765 16 763 8T750 0Q742 2 709 16T658 40L648 46Q592 -10 465 -10Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D406\" xlink:href=\"#MJX-44-TEX-B-1D406\"><\/use><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">G<\/mi><\/mrow><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">\\mathbf{G}<\/script><span>.<\/span><\/p><p><span>Elimination simplifies a system of linear equations without changing the solutions, i.e., if <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"2.045ex\" height=\"1.6ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -697 904 707\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.023ex;\"><defs><path id=\"MJX-44-TEX-B-1D406\" d=\"M465 -10Q281 -10 173 88T64 343Q64 413 85 471T143 568T217 631T298 670Q371 697 449 697Q452 697 459 697T470 696Q502 696 531 690T582 675T618 658T644 641T656 632L732 695Q734 697 745 697Q758 697 761 692T765 668V627V489V449Q765 428 761 424T741 419H731H724Q705 419 702 422T695 444Q683 520 631 577T495 635Q364 635 295 563Q261 528 247 477T232 343Q232 296 236 260T256 185T296 120T366 76T472 52Q481 51 498 51Q544 51 573 67T607 108Q608 111 608 164V214H464V276H479Q506 273 680 273Q816 273 834 276H845V214H765V113V51Q765 16 763 8T750 0Q742 2 709 16T658 40L648 46Q592 -10 465 -10Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D406\" xlink:href=\"#MJX-44-TEX-B-1D406\"><\/use><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">G<\/mi><\/mrow><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">\\mathbf{G}<\/script><span> is row reduced to <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"11.688ex\" height=\"2.283ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -759 5166.2 1009\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-40-TEX-B-1D406\" d=\"M465 -10Q281 -10 173 88T64 343Q64 413 85 471T143 568T217 631T298 670Q371 697 449 697Q452 697 459 697T470 696Q502 696 531 690T582 675T618 658T644 641T656 632L732 695Q734 697 745 697Q758 697 761 692T765 668V627V489V449Q765 428 761 424T741 419H731H724Q705 419 702 422T695 444Q683 520 631 577T495 635Q364 635 295 563Q261 528 247 477T232 343Q232 296 236 260T256 185T296 120T366 76T472 52Q481 51 498 51Q544 51 573 67T607 108Q608 111 608 164V214H464V276H479Q506 273 680 273Q816 273 834 276H845V214H765V113V51Q765 16 763 8T750 0Q742 2 709 16T658 40L648 46Q592 -10 465 -10Z\"><\/path><path id=\"MJX-40-TEX-V-2032\" d=\"M79 43Q73 43 52 49T30 61Q30 68 85 293T146 528Q161 560 198 560Q218 560 240 545T262 501Q262 496 260 486Q259 479 173 263T84 45T79 43Z\"><\/path><path id=\"MJX-40-TEX-N-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path id=\"MJX-40-TEX-N-5B\" d=\"M118 -250V750H255V710H158V-210H255V-250H118Z\"><\/path><path id=\"MJX-40-TEX-B-1D408\" d=\"M397 0Q370 3 218 3Q65 3 38 0H25V62H139V624H25V686H38Q65 683 218 683Q370 683 397 686H410V624H296V62H410V0H397Z\"><\/path><path id=\"MJX-40-TEX-B-1D424\" d=\"M32 686L123 690Q214 694 215 694H221V255L377 382H346V444H355Q370 441 476 441Q544 441 556 444H562V382H476L347 277L515 62H587V0H579Q564 3 476 3Q370 3 352 0H343V62H358L373 63L260 206L237 189L216 172V62H285V0H277Q259 3 157 3Q46 3 37 0H29V62H98V332Q98 387 98 453T99 534Q99 593 97 605T83 620Q69 624 42 624H29V686H32Z\"><\/path><path id=\"MJX-40-TEX-N-7C\" d=\"M139 -249H137Q125 -249 119 -235V251L120 737Q130 750 139 750Q152 750 159 735V-235Q151 -249 141 -249H139Z\"><\/path><path id=\"MJX-40-TEX-B-1D400\" d=\"M296 0Q278 3 164 3Q58 3 49 0H40V62H92Q144 62 144 64Q388 682 397 689Q403 698 434 698Q463 698 471 689Q475 686 538 530T663 218L724 64Q724 62 776 62H828V0H817Q796 3 658 3Q509 3 485 0H472V62H517Q561 62 561 63L517 175H262L240 120Q218 65 217 64Q217 62 261 62H306V0H296ZM390 237L492 238L440 365Q390 491 388 491Q287 239 287 237H390Z\"><\/path><path id=\"MJX-40-TEX-N-5D\" d=\"M22 710V750H159V-250H22V-210H119V710H22Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"msup\"><g data-mml-node=\"mi\"><use data-c=\"1D406\" xlink:href=\"#MJX-40-TEX-B-1D406\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(937,363) scale(0.707)\"><use data-c=\"2032\" xlink:href=\"#MJX-40-TEX-V-2032\"><\/use><\/g><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(1459.2,0)\"><use data-c=\"3D\" xlink:href=\"#MJX-40-TEX-N-3D\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2515,0)\"><use data-c=\"5B\" xlink:href=\"#MJX-40-TEX-N-5B\"><\/use><\/g><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\" transform=\"translate(2793,0)\"><g data-mml-node=\"msub\"><g data-mml-node=\"mi\"><use data-c=\"1D408\" xlink:href=\"#MJX-40-TEX-B-1D408\"><\/use><\/g><g data-mml-node=\"TeXAtom\" transform=\"translate(469,-150) scale(0.707)\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D424\" xlink:href=\"#MJX-40-TEX-B-1D424\"><\/use><\/g><\/g><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(3741.2,0) translate(0 -0.5)\"><use data-c=\"7C\" xlink:href=\"#MJX-40-TEX-N-7C\"><\/use><\/g><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\" transform=\"translate(4019.2,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D400\" xlink:href=\"#MJX-40-TEX-B-1D400\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(4888.2,0)\"><use data-c=\"5D\" xlink:href=\"#MJX-40-TEX-N-5D\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow data-mjx-texclass=\"ORD\"><msup><mi mathvariant=\"bold\">G<\/mi><mo data-mjx-alternate=\"1\">\u2032<\/mo><\/msup><\/mrow><mo>=<\/mo><mo stretchy=\"false\">[<\/mo><mrow data-mjx-texclass=\"ORD\"><msub><mi mathvariant=\"bold\">I<\/mi><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">k<\/mi><\/mrow><\/msub><\/mrow><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">A<\/mi><\/mrow><mo stretchy=\"false\">]<\/mo><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">\\mathbf{G'}=[\\mathbf{I_{k}}|\\mathbf{A}]<\/script><span>, then <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"7.737ex\" height=\"1.762ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -697 3419.6 779\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.186ex;\"><defs><path id=\"MJX-52-TEX-B-1D406\" d=\"M465 -10Q281 -10 173 88T64 343Q64 413 85 471T143 568T217 631T298 670Q371 697 449 697Q452 697 459 697T470 696Q502 696 531 690T582 675T618 658T644 641T656 632L732 695Q734 697 745 697Q758 697 761 692T765 668V627V489V449Q765 428 761 424T741 419H731H724Q705 419 702 422T695 444Q683 520 631 577T495 635Q364 635 295 563Q261 528 247 477T232 343Q232 296 236 260T256 185T296 120T366 76T472 52Q481 51 498 51Q544 51 573 67T607 108Q608 111 608 164V214H464V276H479Q506 273 680 273Q816 273 834 276H845V214H765V113V51Q765 16 763 8T750 0Q742 2 709 16T658 40L648 46Q592 -10 465 -10Z\"><\/path><path id=\"MJX-52-TEX-B-1D431\" d=\"M227 0Q212 3 121 3Q40 3 28 0H21V62H117L245 213L109 382H26V444H34Q49 441 143 441Q247 441 265 444H274V382H246L281 339Q315 297 316 297Q320 297 354 341L389 382H352V444H360Q375 441 466 441Q547 441 559 444H566V382H471L355 246L504 63L545 62H586V0H578Q563 3 469 3Q365 3 347 0H338V62H366Q366 63 326 112T285 163L198 63L217 62H235V0H227Z\"><\/path><path id=\"MJX-52-TEX-N-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path id=\"MJX-52-TEX-B-1D7CE\" d=\"M266 654H280H282Q500 654 524 418Q529 370 529 320Q529 125 456 52Q397 -10 287 -10Q110 -10 63 154Q45 212 45 316Q45 504 113 585Q140 618 185 636T266 654ZM374 548Q347 604 286 604Q247 604 218 575Q197 552 193 511T188 311Q188 159 196 116Q202 87 225 64T287 41Q339 41 367 87Q379 107 382 152T386 329Q386 518 374 548Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D406\" xlink:href=\"#MJX-52-TEX-B-1D406\"><\/use><use data-c=\"1D431\" xlink:href=\"#MJX-52-TEX-B-1D431\" transform=\"translate(904,0)\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(1788.8,0)\"><use data-c=\"3D\" xlink:href=\"#MJX-52-TEX-N-3D\"><\/use><\/g><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\" transform=\"translate(2844.6,0)\"><g data-mml-node=\"mn\"><use data-c=\"1D7CE\" xlink:href=\"#MJX-52-TEX-B-1D7CE\"><\/use><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">Gx<\/mi><\/mrow><mo>=<\/mo><mrow data-mjx-texclass=\"ORD\"><mn mathvariant=\"bold\">0<\/mn><\/mrow><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">\\mathbf{Gx}=\\mathbf{0}<\/script><span> and <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"8.364ex\" height=\"1.903ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -759 3697 841\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.186ex;\"><defs><path id=\"MJX-42-TEX-B-1D406\" d=\"M465 -10Q281 -10 173 88T64 343Q64 413 85 471T143 568T217 631T298 670Q371 697 449 697Q452 697 459 697T470 696Q502 696 531 690T582 675T618 658T644 641T656 632L732 695Q734 697 745 697Q758 697 761 692T765 668V627V489V449Q765 428 761 424T741 419H731H724Q705 419 702 422T695 444Q683 520 631 577T495 635Q364 635 295 563Q261 528 247 477T232 343Q232 296 236 260T256 185T296 120T366 76T472 52Q481 51 498 51Q544 51 573 67T607 108Q608 111 608 164V214H464V276H479Q506 273 680 273Q816 273 834 276H845V214H765V113V51Q765 16 763 8T750 0Q742 2 709 16T658 40L648 46Q592 -10 465 -10Z\"><\/path><path id=\"MJX-42-TEX-V-2032\" d=\"M79 43Q73 43 52 49T30 61Q30 68 85 293T146 528Q161 560 198 560Q218 560 240 545T262 501Q262 496 260 486Q259 479 173 263T84 45T79 43Z\"><\/path><path id=\"MJX-42-TEX-B-1D431\" d=\"M227 0Q212 3 121 3Q40 3 28 0H21V62H117L245 213L109 382H26V444H34Q49 441 143 441Q247 441 265 444H274V382H246L281 339Q315 297 316 297Q320 297 354 341L389 382H352V444H360Q375 441 466 441Q547 441 559 444H566V382H471L355 246L504 63L545 62H586V0H578Q563 3 469 3Q365 3 347 0H338V62H366Q366 63 326 112T285 163L198 63L217 62H235V0H227Z\"><\/path><path id=\"MJX-42-TEX-N-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path id=\"MJX-42-TEX-B-1D7CE\" d=\"M266 654H280H282Q500 654 524 418Q529 370 529 320Q529 125 456 52Q397 -10 287 -10Q110 -10 63 154Q45 212 45 316Q45 504 113 585Q140 618 185 636T266 654ZM374 548Q347 604 286 604Q247 604 218 575Q197 552 193 511T188 311Q188 159 196 116Q202 87 225 64T287 41Q339 41 367 87Q379 107 382 152T386 329Q386 518 374 548Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"msup\"><g data-mml-node=\"mi\"><use data-c=\"1D406\" xlink:href=\"#MJX-42-TEX-B-1D406\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(937,363) scale(0.707)\"><use data-c=\"2032\" xlink:href=\"#MJX-42-TEX-V-2032\"><\/use><\/g><\/g><g data-mml-node=\"mi\" transform=\"translate(1181.5,0)\"><use data-c=\"1D431\" xlink:href=\"#MJX-42-TEX-B-1D431\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(2066.2,0)\"><use data-c=\"3D\" xlink:href=\"#MJX-42-TEX-N-3D\"><\/use><\/g><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\" transform=\"translate(3122,0)\"><g data-mml-node=\"mn\"><use data-c=\"1D7CE\" xlink:href=\"#MJX-42-TEX-B-1D7CE\"><\/use><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow data-mjx-texclass=\"ORD\"><msup><mi mathvariant=\"bold\">G<\/mi><mo data-mjx-alternate=\"1\">\u2032<\/mo><\/msup><mi mathvariant=\"bold\">x<\/mi><\/mrow><mo>=<\/mo><mrow data-mjx-texclass=\"ORD\"><mn mathvariant=\"bold\">0<\/mn><\/mrow><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">\\mathbf{G'x}=\\mathbf{0}<\/script><span> give the same solutions <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.373ex\" height=\"1.005ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -444 607 444\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: 0px;\"><defs><path id=\"MJX-43-TEX-B-1D431\" d=\"M227 0Q212 3 121 3Q40 3 28 0H21V62H117L245 213L109 382H26V444H34Q49 441 143 441Q247 441 265 444H274V382H246L281 339Q315 297 316 297Q320 297 354 341L389 382H352V444H360Q375 441 466 441Q547 441 559 444H566V382H471L355 246L504 63L545 62H586V0H578Q563 3 469 3Q365 3 347 0H338V62H366Q366 63 326 112T285 163L198 63L217 62H235V0H227Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D431\" xlink:href=\"#MJX-43-TEX-B-1D431\"><\/use><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">x<\/mi><\/mrow><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">\\mathbf{x}<\/script><span>. Therefore, the nullspace of <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"2.045ex\" height=\"1.6ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -697 904 707\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.023ex;\"><defs><path id=\"MJX-44-TEX-B-1D406\" d=\"M465 -10Q281 -10 173 88T64 343Q64 413 85 471T143 568T217 631T298 670Q371 697 449 697Q452 697 459 697T470 696Q502 696 531 690T582 675T618 658T644 641T656 632L732 695Q734 697 745 697Q758 697 761 692T765 668V627V489V449Q765 428 761 424T741 419H731H724Q705 419 702 422T695 444Q683 520 631 577T495 635Q364 635 295 563Q261 528 247 477T232 343Q232 296 236 260T256 185T296 120T366 76T472 52Q481 51 498 51Q544 51 573 67T607 108Q608 111 608 164V214H464V276H479Q506 273 680 273Q816 273 834 276H845V214H765V113V51Q765 16 763 8T750 0Q742 2 709 16T658 40L648 46Q592 -10 465 -10Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D406\" xlink:href=\"#MJX-44-TEX-B-1D406\"><\/use><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">G<\/mi><\/mrow><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">\\mathbf{G}<\/script><span> is the same as the nullspace of <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"2.673ex\" height=\"1.74ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -759 1181.5 769\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.023ex;\"><defs><path id=\"MJX-46-TEX-B-1D406\" d=\"M465 -10Q281 -10 173 88T64 343Q64 413 85 471T143 568T217 631T298 670Q371 697 449 697Q452 697 459 697T470 696Q502 696 531 690T582 675T618 658T644 641T656 632L732 695Q734 697 745 697Q758 697 761 692T765 668V627V489V449Q765 428 761 424T741 419H731H724Q705 419 702 422T695 444Q683 520 631 577T495 635Q364 635 295 563Q261 528 247 477T232 343Q232 296 236 260T256 185T296 120T366 76T472 52Q481 51 498 51Q544 51 573 67T607 108Q608 111 608 164V214H464V276H479Q506 273 680 273Q816 273 834 276H845V214H765V113V51Q765 16 763 8T750 0Q742 2 709 16T658 40L648 46Q592 -10 465 -10Z\"><\/path><path id=\"MJX-46-TEX-V-2032\" d=\"M79 43Q73 43 52 49T30 61Q30 68 85 293T146 528Q161 560 198 560Q218 560 240 545T262 501Q262 496 260 486Q259 479 173 263T84 45T79 43Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"msup\"><g data-mml-node=\"mi\"><use data-c=\"1D406\" xlink:href=\"#MJX-46-TEX-B-1D406\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(937,363) scale(0.707)\"><use data-c=\"2032\" xlink:href=\"#MJX-46-TEX-V-2032\"><\/use><\/g><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow data-mjx-texclass=\"ORD\"><msup><mi mathvariant=\"bold\">G<\/mi><mo data-mjx-alternate=\"1\">\u2032<\/mo><\/msup><\/mrow><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">\\mathbf{G'}<\/script><span>.<\/span><\/p><p><span>Since <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"2.673ex\" height=\"1.74ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -759 1181.5 769\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.023ex;\"><defs><path id=\"MJX-46-TEX-B-1D406\" d=\"M465 -10Q281 -10 173 88T64 343Q64 413 85 471T143 568T217 631T298 670Q371 697 449 697Q452 697 459 697T470 696Q502 696 531 690T582 675T618 658T644 641T656 632L732 695Q734 697 745 697Q758 697 761 692T765 668V627V489V449Q765 428 761 424T741 419H731H724Q705 419 702 422T695 444Q683 520 631 577T495 635Q364 635 295 563Q261 528 247 477T232 343Q232 296 236 260T256 185T296 120T366 76T472 52Q481 51 498 51Q544 51 573 67T607 108Q608 111 608 164V214H464V276H479Q506 273 680 273Q816 273 834 276H845V214H765V113V51Q765 16 763 8T750 0Q742 2 709 16T658 40L648 46Q592 -10 465 -10Z\"><\/path><path id=\"MJX-46-TEX-V-2032\" d=\"M79 43Q73 43 52 49T30 61Q30 68 85 293T146 528Q161 560 198 560Q218 560 240 545T262 501Q262 496 260 486Q259 479 173 263T84 45T79 43Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"msup\"><g data-mml-node=\"mi\"><use data-c=\"1D406\" xlink:href=\"#MJX-46-TEX-B-1D406\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(937,363) scale(0.707)\"><use data-c=\"2032\" xlink:href=\"#MJX-46-TEX-V-2032\"><\/use><\/g><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow data-mjx-texclass=\"ORD\"><msup><mi mathvariant=\"bold\">G<\/mi><mo data-mjx-alternate=\"1\">\u2032<\/mo><\/msup><\/mrow><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">\\mathbf{G'}<\/script><span> has rank <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.179ex\" height=\"1.595ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -694 521 705\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.025ex;\"><defs><path id=\"MJX-1790-TEX-I-1D458\" d=\"M121 647Q121 657 125 670T137 683Q138 683 209 688T282 694Q294 694 294 686Q294 679 244 477Q194 279 194 272Q213 282 223 291Q247 309 292 354T362 415Q402 442 438 442Q468 442 485 423T503 369Q503 344 496 327T477 302T456 291T438 288Q418 288 406 299T394 328Q394 353 410 369T442 390L458 393Q446 405 434 405H430Q398 402 367 380T294 316T228 255Q230 254 243 252T267 246T293 238T320 224T342 206T359 180T365 147Q365 130 360 106T354 66Q354 26 381 26Q429 26 459 145Q461 153 479 153H483Q499 153 499 144Q499 139 496 130Q455 -11 378 -11Q333 -11 305 15T277 90Q277 108 280 121T283 145Q283 167 269 183T234 206T200 217T182 220H180Q168 178 159 139T145 81T136 44T129 20T122 7T111 -2Q98 -11 83 -11Q66 -11 57 -1T48 16Q48 26 85 176T158 471L195 616Q196 629 188 632T149 637H144Q134 637 131 637T124 640T121 647Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D458\" xlink:href=\"#MJX-1790-TEX-I-1D458\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>k<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">k<\/script><span>, there are <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.179ex\" height=\"1.595ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -694 521 705\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.025ex;\"><defs><path id=\"MJX-1790-TEX-I-1D458\" d=\"M121 647Q121 657 125 670T137 683Q138 683 209 688T282 694Q294 694 294 686Q294 679 244 477Q194 279 194 272Q213 282 223 291Q247 309 292 354T362 415Q402 442 438 442Q468 442 485 423T503 369Q503 344 496 327T477 302T456 291T438 288Q418 288 406 299T394 328Q394 353 410 369T442 390L458 393Q446 405 434 405H430Q398 402 367 380T294 316T228 255Q230 254 243 252T267 246T293 238T320 224T342 206T359 180T365 147Q365 130 360 106T354 66Q354 26 381 26Q429 26 459 145Q461 153 479 153H483Q499 153 499 144Q499 139 496 130Q455 -11 378 -11Q333 -11 305 15T277 90Q277 108 280 121T283 145Q283 167 269 183T234 206T200 217T182 220H180Q168 178 159 139T145 81T136 44T129 20T122 7T111 -2Q98 -11 83 -11Q66 -11 57 -1T48 16Q48 26 85 176T158 471L195 616Q196 629 188 632T149 637H144Q134 637 131 637T124 640T121 647Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D458\" xlink:href=\"#MJX-1790-TEX-I-1D458\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>k<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">k<\/script><span> pivot variables and <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"5.302ex\" height=\"1.756ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -694 2343.4 776\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.186ex;\"><defs><path id=\"MJX-1297-TEX-I-1D45B\" d=\"M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z\"><\/path><path id=\"MJX-1297-TEX-N-2212\" d=\"M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z\"><\/path><path id=\"MJX-1297-TEX-I-1D458\" d=\"M121 647Q121 657 125 670T137 683Q138 683 209 688T282 694Q294 694 294 686Q294 679 244 477Q194 279 194 272Q213 282 223 291Q247 309 292 354T362 415Q402 442 438 442Q468 442 485 423T503 369Q503 344 496 327T477 302T456 291T438 288Q418 288 406 299T394 328Q394 353 410 369T442 390L458 393Q446 405 434 405H430Q398 402 367 380T294 316T228 255Q230 254 243 252T267 246T293 238T320 224T342 206T359 180T365 147Q365 130 360 106T354 66Q354 26 381 26Q429 26 459 145Q461 153 479 153H483Q499 153 499 144Q499 139 496 130Q455 -11 378 -11Q333 -11 305 15T277 90Q277 108 280 121T283 145Q283 167 269 183T234 206T200 217T182 220H180Q168 178 159 139T145 81T136 44T129 20T122 7T111 -2Q98 -11 83 -11Q66 -11 57 -1T48 16Q48 26 85 176T158 471L195 616Q196 629 188 632T149 637H144Q134 637 131 637T124 640T121 647Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D45B\" xlink:href=\"#MJX-1297-TEX-I-1D45B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(822.2,0)\"><use data-c=\"2212\" xlink:href=\"#MJX-1297-TEX-N-2212\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1822.4,0)\"><use data-c=\"1D458\" xlink:href=\"#MJX-1297-TEX-I-1D458\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>n<\/mi><mo>\u2212<\/mo><mi>k<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">n-k<\/script><span> free variables, so the nullspace has dimension <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"5.302ex\" height=\"1.756ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -694 2343.4 776\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.186ex;\"><defs><path id=\"MJX-1297-TEX-I-1D45B\" d=\"M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z\"><\/path><path id=\"MJX-1297-TEX-N-2212\" d=\"M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z\"><\/path><path id=\"MJX-1297-TEX-I-1D458\" d=\"M121 647Q121 657 125 670T137 683Q138 683 209 688T282 694Q294 694 294 686Q294 679 244 477Q194 279 194 272Q213 282 223 291Q247 309 292 354T362 415Q402 442 438 442Q468 442 485 423T503 369Q503 344 496 327T477 302T456 291T438 288Q418 288 406 299T394 328Q394 353 410 369T442 390L458 393Q446 405 434 405H430Q398 402 367 380T294 316T228 255Q230 254 243 252T267 246T293 238T320 224T342 206T359 180T365 147Q365 130 360 106T354 66Q354 26 381 26Q429 26 459 145Q461 153 479 153H483Q499 153 499 144Q499 139 496 130Q455 -11 378 -11Q333 -11 305 15T277 90Q277 108 280 121T283 145Q283 167 269 183T234 206T200 217T182 220H180Q168 178 159 139T145 81T136 44T129 20T122 7T111 -2Q98 -11 83 -11Q66 -11 57 -1T48 16Q48 26 85 176T158 471L195 616Q196 629 188 632T149 637H144Q134 637 131 637T124 640T121 647Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D45B\" xlink:href=\"#MJX-1297-TEX-I-1D45B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(822.2,0)\"><use data-c=\"2212\" xlink:href=\"#MJX-1297-TEX-N-2212\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1822.4,0)\"><use data-c=\"1D458\" xlink:href=\"#MJX-1297-TEX-I-1D458\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>n<\/mi><mo>\u2212<\/mo><mi>k<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">n-k<\/script><span>. These <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"5.302ex\" height=\"1.756ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -694 2343.4 776\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.186ex;\"><defs><path id=\"MJX-1297-TEX-I-1D45B\" d=\"M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z\"><\/path><path id=\"MJX-1297-TEX-N-2212\" d=\"M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z\"><\/path><path id=\"MJX-1297-TEX-I-1D458\" d=\"M121 647Q121 657 125 670T137 683Q138 683 209 688T282 694Q294 694 294 686Q294 679 244 477Q194 279 194 272Q213 282 223 291Q247 309 292 354T362 415Q402 442 438 442Q468 442 485 423T503 369Q503 344 496 327T477 302T456 291T438 288Q418 288 406 299T394 328Q394 353 410 369T442 390L458 393Q446 405 434 405H430Q398 402 367 380T294 316T228 255Q230 254 243 252T267 246T293 238T320 224T342 206T359 180T365 147Q365 130 360 106T354 66Q354 26 381 26Q429 26 459 145Q461 153 479 153H483Q499 153 499 144Q499 139 496 130Q455 -11 378 -11Q333 -11 305 15T277 90Q277 108 280 121T283 145Q283 167 269 183T234 206T200 217T182 220H180Q168 178 159 139T145 81T136 44T129 20T122 7T111 -2Q98 -11 83 -11Q66 -11 57 -1T48 16Q48 26 85 176T158 471L195 616Q196 629 188 632T149 637H144Q134 637 131 637T124 640T121 647Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D45B\" xlink:href=\"#MJX-1297-TEX-I-1D45B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(822.2,0)\"><use data-c=\"2212\" xlink:href=\"#MJX-1297-TEX-N-2212\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1822.4,0)\"><use data-c=\"1D458\" xlink:href=\"#MJX-1297-TEX-I-1D458\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>n<\/mi><mo>\u2212<\/mo><mi>k<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">n-k<\/script><span> vectors are special solutions to <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"7.737ex\" height=\"1.762ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -697 3419.6 779\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.186ex;\"><defs><path id=\"MJX-52-TEX-B-1D406\" d=\"M465 -10Q281 -10 173 88T64 343Q64 413 85 471T143 568T217 631T298 670Q371 697 449 697Q452 697 459 697T470 696Q502 696 531 690T582 675T618 658T644 641T656 632L732 695Q734 697 745 697Q758 697 761 692T765 668V627V489V449Q765 428 761 424T741 419H731H724Q705 419 702 422T695 444Q683 520 631 577T495 635Q364 635 295 563Q261 528 247 477T232 343Q232 296 236 260T256 185T296 120T366 76T472 52Q481 51 498 51Q544 51 573 67T607 108Q608 111 608 164V214H464V276H479Q506 273 680 273Q816 273 834 276H845V214H765V113V51Q765 16 763 8T750 0Q742 2 709 16T658 40L648 46Q592 -10 465 -10Z\"><\/path><path id=\"MJX-52-TEX-B-1D431\" d=\"M227 0Q212 3 121 3Q40 3 28 0H21V62H117L245 213L109 382H26V444H34Q49 441 143 441Q247 441 265 444H274V382H246L281 339Q315 297 316 297Q320 297 354 341L389 382H352V444H360Q375 441 466 441Q547 441 559 444H566V382H471L355 246L504 63L545 62H586V0H578Q563 3 469 3Q365 3 347 0H338V62H366Q366 63 326 112T285 163L198 63L217 62H235V0H227Z\"><\/path><path id=\"MJX-52-TEX-N-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path id=\"MJX-52-TEX-B-1D7CE\" d=\"M266 654H280H282Q500 654 524 418Q529 370 529 320Q529 125 456 52Q397 -10 287 -10Q110 -10 63 154Q45 212 45 316Q45 504 113 585Q140 618 185 636T266 654ZM374 548Q347 604 286 604Q247 604 218 575Q197 552 193 511T188 311Q188 159 196 116Q202 87 225 64T287 41Q339 41 367 87Q379 107 382 152T386 329Q386 518 374 548Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D406\" xlink:href=\"#MJX-52-TEX-B-1D406\"><\/use><use data-c=\"1D431\" xlink:href=\"#MJX-52-TEX-B-1D431\" transform=\"translate(904,0)\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(1788.8,0)\"><use data-c=\"3D\" xlink:href=\"#MJX-52-TEX-N-3D\"><\/use><\/g><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\" transform=\"translate(2844.6,0)\"><g data-mml-node=\"mn\"><use data-c=\"1D7CE\" xlink:href=\"#MJX-52-TEX-B-1D7CE\"><\/use><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">Gx<\/mi><\/mrow><mo>=<\/mo><mrow data-mjx-texclass=\"ORD\"><mn mathvariant=\"bold\">0<\/mn><\/mrow><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">\\mathbf{Gx}=\\mathbf{0}<\/script><span> and form the basis of the nullspace.<\/span><\/p><\/blockquote><p><span>Left multiplication by an invertible matrix does not change the code (<\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"3.491ex\" height=\"1.602ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -697 1543 708\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.025ex;\"><defs><path id=\"MJX-53-TEX-B-1D412\" d=\"M64 493Q64 582 120 636T264 696H272Q280 697 285 697Q380 697 454 645L480 669Q484 672 488 676T495 683T500 688T504 691T508 693T511 695T514 696T517 697T522 697Q536 697 539 691T542 652V577Q542 557 542 532T543 500Q543 472 540 465T524 458H511H505Q489 458 485 461T479 478Q472 529 449 564T393 614T336 634T287 639Q228 639 203 610T177 544Q177 517 195 493T247 457Q253 454 343 436T475 391Q574 326 574 207V200Q574 163 559 120Q517 12 389 -9Q380 -10 346 -10Q308 -10 275 -5T221 7T184 22T160 35T151 40L126 17Q122 14 118 10T111 3T106 -2T102 -5T98 -7T95 -9T92 -10T89 -11T84 -11Q70 -11 67 -4T64 35V108Q64 128 64 153T63 185Q63 203 63 211T69 223T77 227T94 228H100Q118 228 122 225T126 205Q130 125 193 88T345 51Q408 51 434 82T460 157Q460 196 439 221T388 257Q384 259 305 276T221 295Q155 313 110 366T64 493Z\"><\/path><path id=\"MJX-53-TEX-B-1D406\" d=\"M465 -10Q281 -10 173 88T64 343Q64 413 85 471T143 568T217 631T298 670Q371 697 449 697Q452 697 459 697T470 696Q502 696 531 690T582 675T618 658T644 641T656 632L732 695Q734 697 745 697Q758 697 761 692T765 668V627V489V449Q765 428 761 424T741 419H731H724Q705 419 702 422T695 444Q683 520 631 577T495 635Q364 635 295 563Q261 528 247 477T232 343Q232 296 236 260T256 185T296 120T366 76T472 52Q481 51 498 51Q544 51 573 67T607 108Q608 111 608 164V214H464V276H479Q506 273 680 273Q816 273 834 276H845V214H765V113V51Q765 16 763 8T750 0Q742 2 709 16T658 40L648 46Q592 -10 465 -10Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D412\" xlink:href=\"#MJX-53-TEX-B-1D412\"><\/use><use data-c=\"1D406\" xlink:href=\"#MJX-53-TEX-B-1D406\" transform=\"translate(639,0)\"><\/use><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">SG<\/mi><\/mrow><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">\\mathbf{SG}<\/script><span> or <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"3.482ex\" height=\"1.602ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -697 1539 708\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.025ex;\"><defs><path id=\"MJX-54-TEX-B-1D412\" d=\"M64 493Q64 582 120 636T264 696H272Q280 697 285 697Q380 697 454 645L480 669Q484 672 488 676T495 683T500 688T504 691T508 693T511 695T514 696T517 697T522 697Q536 697 539 691T542 652V577Q542 557 542 532T543 500Q543 472 540 465T524 458H511H505Q489 458 485 461T479 478Q472 529 449 564T393 614T336 634T287 639Q228 639 203 610T177 544Q177 517 195 493T247 457Q253 454 343 436T475 391Q574 326 574 207V200Q574 163 559 120Q517 12 389 -9Q380 -10 346 -10Q308 -10 275 -5T221 7T184 22T160 35T151 40L126 17Q122 14 118 10T111 3T106 -2T102 -5T98 -7T95 -9T92 -10T89 -11T84 -11Q70 -11 67 -4T64 35V108Q64 128 64 153T63 185Q63 203 63 211T69 223T77 227T94 228H100Q118 228 122 225T126 205Q130 125 193 88T345 51Q408 51 434 82T460 157Q460 196 439 221T388 257Q384 259 305 276T221 295Q155 313 110 366T64 493Z\"><\/path><path id=\"MJX-54-TEX-B-1D407\" d=\"M400 0Q376 3 226 3Q75 3 51 0H39V62H147V624H39V686H51Q75 683 226 683Q376 683 400 686H412V624H304V388H595V624H487V686H499Q523 683 673 683Q824 683 848 686H860V624H752V62H860V0H848Q824 3 674 3Q523 3 499 0H487V62H595V326H304V62H412V0H400Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D412\" xlink:href=\"#MJX-54-TEX-B-1D412\"><\/use><use data-c=\"1D407\" xlink:href=\"#MJX-54-TEX-B-1D407\" transform=\"translate(639,0)\"><\/use><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">SH<\/mi><\/mrow><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">\\mathbf{SH}<\/script><span>), it just gives another basis.<\/span><\/p><p><span>If <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"18.382ex\" height=\"2.262ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 8124.8 1000\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-55-TEX-SS-1D5B1\" d=\"M88 0V694H227H259H302Q365 694 399 689T474 663Q528 637 558 595T589 504Q589 482 584 462T569 426T547 396T522 372T495 353T470 338T449 328T434 322L429 320L440 300Q452 280 477 238T523 160L617 1L565 0Q513 0 512 1Q512 2 424 156L337 309H189V0H88ZM492 504Q492 600 367 620Q354 622 271 623H189V385H271Q363 386 388 392Q432 402 462 430T492 504Z\"><\/path><path id=\"MJX-55-TEX-SS-1D5A4\" d=\"M86 0V691H541V611H366L190 612V397H513V321H190V85H372L554 86V0H86Z\"><\/path><path id=\"MJX-55-TEX-SS-1D5A5\" d=\"M86 0V691H526V611H358L190 612V384H485V308H190V0H86Z\"><\/path><path id=\"MJX-55-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-55-TEX-B-1D406\" d=\"M465 -10Q281 -10 173 88T64 343Q64 413 85 471T143 568T217 631T298 670Q371 697 449 697Q452 697 459 697T470 696Q502 696 531 690T582 675T618 658T644 641T656 632L732 695Q734 697 745 697Q758 697 761 692T765 668V627V489V449Q765 428 761 424T741 419H731H724Q705 419 702 422T695 444Q683 520 631 577T495 635Q364 635 295 563Q261 528 247 477T232 343Q232 296 236 260T256 185T296 120T366 76T472 52Q481 51 498 51Q544 51 573 67T607 108Q608 111 608 164V214H464V276H479Q506 273 680 273Q816 273 834 276H845V214H765V113V51Q765 16 763 8T750 0Q742 2 709 16T658 40L648 46Q592 -10 465 -10Z\"><\/path><path id=\"MJX-55-TEX-N-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><path id=\"MJX-55-TEX-N-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path id=\"MJX-55-TEX-N-5B\" d=\"M118 -250V750H255V710H158V-210H255V-250H118Z\"><\/path><path id=\"MJX-55-TEX-B-1D408\" d=\"M397 0Q370 3 218 3Q65 3 38 0H25V62H139V624H25V686H38Q65 683 218 683Q370 683 397 686H410V624H296V62H410V0H397Z\"><\/path><path id=\"MJX-55-TEX-B-1D424\" d=\"M32 686L123 690Q214 694 215 694H221V255L377 382H346V444H355Q370 441 476 441Q544 441 556 444H562V382H476L347 277L515 62H587V0H579Q564 3 476 3Q370 3 352 0H343V62H358L373 63L260 206L237 189L216 172V62H285V0H277Q259 3 157 3Q46 3 37 0H29V62H98V332Q98 387 98 453T99 534Q99 593 97 605T83 620Q69 624 42 624H29V686H32Z\"><\/path><path id=\"MJX-55-TEX-N-7C\" d=\"M139 -249H137Q125 -249 119 -235V251L120 737Q130 750 139 750Q152 750 159 735V-235Q151 -249 141 -249H139Z\"><\/path><path id=\"MJX-55-TEX-B-1D400\" d=\"M296 0Q278 3 164 3Q58 3 49 0H40V62H92Q144 62 144 64Q388 682 397 689Q403 698 434 698Q463 698 471 689Q475 686 538 530T663 218L724 64Q724 62 776 62H828V0H817Q796 3 658 3Q509 3 485 0H472V62H517Q561 62 561 63L517 175H262L240 120Q218 65 217 64Q217 62 261 62H306V0H296ZM390 237L492 238L440 365Q390 491 388 491Q287 239 287 237H390Z\"><\/path><path id=\"MJX-55-TEX-N-5D\" d=\"M22 710V750H159V-250H22V-210H119V710H22Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D5B1\" xlink:href=\"#MJX-55-TEX-SS-1D5B1\"><\/use><use data-c=\"1D5B1\" xlink:href=\"#MJX-55-TEX-SS-1D5B1\" transform=\"translate(646,0)\"><\/use><use data-c=\"1D5A4\" xlink:href=\"#MJX-55-TEX-SS-1D5A4\" transform=\"translate(1292,0)\"><\/use><use data-c=\"1D5A5\" xlink:href=\"#MJX-55-TEX-SS-1D5A5\" transform=\"translate(1889,0)\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(2458,0)\"><use data-c=\"28\" xlink:href=\"#MJX-55-TEX-N-28\"><\/use><\/g><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\" transform=\"translate(2847,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D406\" xlink:href=\"#MJX-55-TEX-B-1D406\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(3751,0)\"><use data-c=\"29\" xlink:href=\"#MJX-55-TEX-N-29\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(4417.8,0)\"><use data-c=\"3D\" xlink:href=\"#MJX-55-TEX-N-3D\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(5473.6,0)\"><use data-c=\"5B\" xlink:href=\"#MJX-55-TEX-N-5B\"><\/use><\/g><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\" transform=\"translate(5751.6,0)\"><g data-mml-node=\"msub\"><g data-mml-node=\"mi\"><use data-c=\"1D408\" xlink:href=\"#MJX-55-TEX-B-1D408\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(469,-150) scale(0.707)\"><use data-c=\"1D424\" xlink:href=\"#MJX-55-TEX-B-1D424\"><\/use><\/g><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(6699.8,0) translate(0 -0.5)\"><use data-c=\"7C\" xlink:href=\"#MJX-55-TEX-N-7C\"><\/use><\/g><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\" transform=\"translate(6977.8,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D400\" xlink:href=\"#MJX-55-TEX-B-1D400\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(7846.8,0)\"><use data-c=\"5D\" xlink:href=\"#MJX-55-TEX-N-5D\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"sans-serif\">RREF<\/mi><\/mrow><mo stretchy=\"false\">(<\/mo><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">G<\/mi><\/mrow><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mo stretchy=\"false\">[<\/mo><mrow data-mjx-texclass=\"ORD\"><msub><mi mathvariant=\"bold\">I<\/mi><mi mathvariant=\"bold\">k<\/mi><\/msub><\/mrow><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">A<\/mi><\/mrow><mo stretchy=\"false\">]<\/mo><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">\\mathsf{RREF}(\\mathbf{G})=[\\mathbf{I_k}|\\mathbf{A}]<\/script><span>, then <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"16.604ex\" height=\"2.262ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 7338.9 1000\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-56-TEX-B-1D407\" d=\"M400 0Q376 3 226 3Q75 3 51 0H39V62H147V624H39V686H51Q75 683 226 683Q376 683 400 686H412V624H304V388H595V624H487V686H499Q523 683 673 683Q824 683 848 686H860V624H752V62H860V0H848Q824 3 674 3Q523 3 499 0H487V62H595V326H304V62H412V0H400Z\"><\/path><path id=\"MJX-56-TEX-N-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path id=\"MJX-56-TEX-N-5B\" d=\"M118 -250V750H255V710H158V-210H255V-250H118Z\"><\/path><path id=\"MJX-56-TEX-B-2212\" d=\"M119 221Q96 230 96 251T116 279Q121 281 448 281H775Q776 280 779 278T785 274T791 269T795 262T797 251Q797 230 775 221H119Z\"><\/path><path id=\"MJX-56-TEX-B-1D400\" d=\"M296 0Q278 3 164 3Q58 3 49 0H40V62H92Q144 62 144 64Q388 682 397 689Q403 698 434 698Q463 698 471 689Q475 686 538 530T663 218L724 64Q724 62 776 62H828V0H817Q796 3 658 3Q509 3 485 0H472V62H517Q561 62 561 63L517 175H262L240 120Q218 65 217 64Q217 62 261 62H306V0H296ZM390 237L492 238L440 365Q390 491 388 491Q287 239 287 237H390Z\"><\/path><path id=\"MJX-56-TEX-N-22BA\" d=\"M318 -182Q302 -212 280 -212H278H275Q249 -212 239 -182L238 84V351H162L87 352Q57 362 57 391T84 429Q89 431 280 431H470L474 429Q477 427 479 426T484 423T490 417T495 410T499 402T500 391Q500 365 470 352L394 351H318V-182Z\"><\/path><path id=\"MJX-56-TEX-N-7C\" d=\"M139 -249H137Q125 -249 119 -235V251L120 737Q130 750 139 750Q152 750 159 735V-235Q151 -249 141 -249H139Z\"><\/path><path id=\"MJX-56-TEX-B-1D408\" d=\"M397 0Q370 3 218 3Q65 3 38 0H25V62H139V624H25V686H38Q65 683 218 683Q370 683 397 686H410V624H296V62H410V0H397Z\"><\/path><path id=\"MJX-56-TEX-B-1D427\" d=\"M40 442Q217 450 218 450H224V407L225 365Q233 378 245 391T289 422T362 448Q374 450 398 450Q428 450 448 447T491 434T529 402T551 346Q553 335 554 198V62H623V0H614Q596 3 489 3Q374 3 365 0H356V62H425V194V275Q425 348 416 373T371 399Q326 399 288 370T238 290Q236 281 235 171V62H304V0H295Q277 3 171 3Q64 3 46 0H37V62H106V210V303Q106 353 104 363T91 376Q77 380 50 380H37V442H40Z\"><\/path><path id=\"MJX-56-TEX-B-1D424\" d=\"M32 686L123 690Q214 694 215 694H221V255L377 382H346V444H355Q370 441 476 441Q544 441 556 444H562V382H476L347 277L515 62H587V0H579Q564 3 476 3Q370 3 352 0H343V62H358L373 63L260 206L237 189L216 172V62H285V0H277Q259 3 157 3Q46 3 37 0H29V62H98V332Q98 387 98 453T99 534Q99 593 97 605T83 620Q69 624 42 624H29V686H32Z\"><\/path><path id=\"MJX-56-TEX-N-5D\" d=\"M22 710V750H159V-250H22V-210H119V710H22Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D407\" xlink:href=\"#MJX-56-TEX-B-1D407\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(1177.8,0)\"><use data-c=\"3D\" xlink:href=\"#MJX-56-TEX-N-3D\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2233.6,0)\"><use data-c=\"5B\" xlink:href=\"#MJX-56-TEX-N-5B\"><\/use><\/g><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\" transform=\"translate(2511.6,0)\"><g data-mml-node=\"mo\"><use data-c=\"2212\" xlink:href=\"#MJX-56-TEX-B-2212\"><\/use><\/g><g data-mml-node=\"msup\" transform=\"translate(894,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D400\" xlink:href=\"#MJX-56-TEX-B-1D400\"><\/use><\/g><g data-mml-node=\"TeXAtom\" transform=\"translate(902,363) scale(0.707)\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mo\"><use data-c=\"22BA\" xlink:href=\"#MJX-56-TEX-N-22BA\"><\/use><\/g><\/g><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(4750.7,0) translate(0 -0.5)\"><use data-c=\"7C\" xlink:href=\"#MJX-56-TEX-N-7C\"><\/use><\/g><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\" transform=\"translate(5028.7,0)\"><g data-mml-node=\"msub\"><g data-mml-node=\"mi\"><use data-c=\"1D408\" xlink:href=\"#MJX-56-TEX-B-1D408\"><\/use><\/g><g data-mml-node=\"TeXAtom\" transform=\"translate(469,-150) scale(0.707)\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D427\" xlink:href=\"#MJX-56-TEX-B-1D427\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(639,0)\"><use data-c=\"2212\" xlink:href=\"#MJX-56-TEX-B-2212\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1533,0)\"><use data-c=\"1D424\" xlink:href=\"#MJX-56-TEX-B-1D424\"><\/use><\/g><\/g><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(7060.9,0)\"><use data-c=\"5D\" xlink:href=\"#MJX-56-TEX-N-5D\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">H<\/mi><\/mrow><mo>=<\/mo><mo stretchy=\"false\">[<\/mo><mrow data-mjx-texclass=\"ORD\"><mo mathvariant=\"bold\">\u2212<\/mo><msup><mi mathvariant=\"bold\">A<\/mi><mrow data-mjx-texclass=\"ORD\"><mo>\u22ba<\/mo><\/mrow><\/msup><\/mrow><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mrow data-mjx-texclass=\"ORD\"><msub><mi mathvariant=\"bold\">I<\/mi><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">n<\/mi><mo mathvariant=\"bold\">\u2212<\/mo><mi mathvariant=\"bold\">k<\/mi><\/mrow><\/msub><\/mrow><mo stretchy=\"false\">]<\/mo><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">\\mathbf{H}=[\\mathbf{-A^{\\intercal}}|\\mathbf{I_{n-k}}]<\/script><span>, and <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"9.477ex\" height=\"1.834ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -728.8 4188.7 810.8\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.186ex;\"><defs><path id=\"MJX-57-TEX-B-1D406\" d=\"M465 -10Q281 -10 173 88T64 343Q64 413 85 471T143 568T217 631T298 670Q371 697 449 697Q452 697 459 697T470 696Q502 696 531 690T582 675T618 658T644 641T656 632L732 695Q734 697 745 697Q758 697 761 692T765 668V627V489V449Q765 428 761 424T741 419H731H724Q705 419 702 422T695 444Q683 520 631 577T495 635Q364 635 295 563Q261 528 247 477T232 343Q232 296 236 260T256 185T296 120T366 76T472 52Q481 51 498 51Q544 51 573 67T607 108Q608 111 608 164V214H464V276H479Q506 273 680 273Q816 273 834 276H845V214H765V113V51Q765 16 763 8T750 0Q742 2 709 16T658 40L648 46Q592 -10 465 -10Z\"><\/path><path id=\"MJX-57-TEX-B-1D407\" d=\"M400 0Q376 3 226 3Q75 3 51 0H39V62H147V624H39V686H51Q75 683 226 683Q376 683 400 686H412V624H304V388H595V624H487V686H499Q523 683 673 683Q824 683 848 686H860V624H752V62H860V0H848Q824 3 674 3Q523 3 499 0H487V62H595V326H304V62H412V0H400Z\"><\/path><path id=\"MJX-57-TEX-N-22BA\" d=\"M318 -182Q302 -212 280 -212H278H275Q249 -212 239 -182L238 84V351H162L87 352Q57 362 57 391T84 429Q89 431 280 431H470L474 429Q477 427 479 426T484 423T490 417T495 410T499 402T500 391Q500 365 470 352L394 351H318V-182Z\"><\/path><path id=\"MJX-57-TEX-N-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path id=\"MJX-57-TEX-B-1D7CE\" d=\"M266 654H280H282Q500 654 524 418Q529 370 529 320Q529 125 456 52Q397 -10 287 -10Q110 -10 63 154Q45 212 45 316Q45 504 113 585Q140 618 185 636T266 654ZM374 548Q347 604 286 604Q247 604 218 575Q197 552 193 511T188 311Q188 159 196 116Q202 87 225 64T287 41Q339 41 367 87Q379 107 382 152T386 329Q386 518 374 548Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"msup\"><g data-mml-node=\"mi\"><use data-c=\"1D406\" xlink:href=\"#MJX-57-TEX-B-1D406\"><\/use><use data-c=\"1D407\" xlink:href=\"#MJX-57-TEX-B-1D407\" transform=\"translate(904,0)\"><\/use><\/g><g data-mml-node=\"TeXAtom\" transform=\"translate(1837,424.1) scale(0.707)\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mo\"><use data-c=\"22BA\" xlink:href=\"#MJX-57-TEX-N-22BA\"><\/use><\/g><\/g><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(2557.9,0)\"><use data-c=\"3D\" xlink:href=\"#MJX-57-TEX-N-3D\"><\/use><\/g><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\" transform=\"translate(3613.7,0)\"><g data-mml-node=\"mn\"><use data-c=\"1D7CE\" xlink:href=\"#MJX-57-TEX-B-1D7CE\"><\/use><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow data-mjx-texclass=\"ORD\"><msup><mi mathvariant=\"bold\">GH<\/mi><mrow data-mjx-texclass=\"ORD\"><mo>\u22ba<\/mo><\/mrow><\/msup><\/mrow><mo>=<\/mo><mrow data-mjx-texclass=\"ORD\"><mn mathvariant=\"bold\">0<\/mn><\/mrow><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">\\mathbf{GH^{\\intercal}}=\\mathbf{0}<\/script><span>.<\/span><\/p><p><span>Let <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"12.859ex\" height=\"3.024ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -1060.7 5683.9 1336.8\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.625ex;\"><defs><path id=\"MJX-551-TEX-B-1D407\" d=\"M400 0Q376 3 226 3Q75 3 51 0H39V62H147V624H39V686H51Q75 683 226 683Q376 683 400 686H412V624H304V388H595V624H487V686H499Q523 683 673 683Q824 683 848 686H860V624H752V62H860V0H848Q824 3 674 3Q523 3 499 0H487V62H595V326H304V62H412V0H400Z\"><\/path><path id=\"MJX-551-TEX-N-2208\" d=\"M84 250Q84 372 166 450T360 539Q361 539 377 539T419 540T469 540H568Q583 532 583 520Q583 511 570 501L466 500Q355 499 329 494Q280 482 242 458T183 409T147 354T129 306T124 272V270H568Q583 262 583 250T568 230H124V228Q124 207 134 177T167 112T231 48T328 7Q355 1 466 0H570Q583 -10 583 -20Q583 -32 568 -40H471Q464 -40 446 -40T417 -41Q262 -41 172 45Q84 127 84 250Z\"><\/path><path id=\"MJX-551-TEX-D-1D53D\" d=\"M584 499Q569 490 566 490Q558 490 552 497T546 515Q546 535 533 559Q526 574 506 593T469 621Q415 648 326 648Q293 648 287 647T275 641Q264 630 263 617Q262 609 260 492V370L275 372Q323 376 350 392T393 441Q409 473 409 506Q409 529 427 529Q437 529 442 519Q444 511 444 362Q444 212 442 206Q436 197 426 197Q409 197 409 217Q409 265 375 299Q346 328 280 335H260V206Q260 70 262 63Q265 46 276 41T326 35Q362 35 366 28Q377 17 366 3L360 -1H24Q12 5 12 16Q12 35 51 35Q92 38 97 52Q102 60 102 341T97 632Q91 645 51 648Q12 648 12 666Q12 675 24 683H573Q576 678 584 670V499ZM137 341Q137 131 136 89T130 37Q129 36 129 35H182Q233 35 233 39Q226 54 225 92T224 346L226 623L231 635L235 648H129Q132 641 133 638T135 603T137 517T137 341ZM549 603V648H495L506 641Q531 621 533 619L549 603ZM409 317V395L400 386Q390 376 375 366L357 355L373 346Q394 331 397 328L409 317Z\"><\/path><path id=\"MJX-551-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-551-TEX-I-1D45B\" d=\"M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z\"><\/path><path id=\"MJX-551-TEX-N-2212\" d=\"M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z\"><\/path><path id=\"MJX-551-TEX-I-1D458\" d=\"M121 647Q121 657 125 670T137 683Q138 683 209 688T282 694Q294 694 294 686Q294 679 244 477Q194 279 194 272Q213 282 223 291Q247 309 292 354T362 415Q402 442 438 442Q468 442 485 423T503 369Q503 344 496 327T477 302T456 291T438 288Q418 288 406 299T394 328Q394 353 410 369T442 390L458 393Q446 405 434 405H430Q398 402 367 380T294 316T228 255Q230 254 243 252T267 246T293 238T320 224T342 206T359 180T365 147Q365 130 360 106T354 66Q354 26 381 26Q429 26 459 145Q461 153 479 153H483Q499 153 499 144Q499 139 496 130Q455 -11 378 -11Q333 -11 305 15T277 90Q277 108 280 121T283 145Q283 167 269 183T234 206T200 217T182 220H180Q168 178 159 139T145 81T136 44T129 20T122 7T111 -2Q98 -11 83 -11Q66 -11 57 -1T48 16Q48 26 85 176T158 471L195 616Q196 629 188 632T149 637H144Q134 637 131 637T124 640T121 647Z\"><\/path><path id=\"MJX-551-TEX-N-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><path id=\"MJX-551-TEX-N-D7\" d=\"M630 29Q630 9 609 9Q604 9 587 25T493 118L389 222L284 117Q178 13 175 11Q171 9 168 9Q160 9 154 15T147 29Q147 36 161 51T255 146L359 250L255 354Q174 435 161 449T147 471Q147 480 153 485T168 490Q173 490 175 489Q178 487 284 383L389 278L493 382Q570 459 587 475T609 491Q630 491 630 471Q630 464 620 453T522 355L418 250L522 145Q606 61 618 48T630 29Z\"><\/path><path id=\"MJX-551-TEX-I-1D45E\" d=\"M33 157Q33 258 109 349T280 441Q340 441 372 389Q373 390 377 395T388 406T404 418Q438 442 450 442Q454 442 457 439T460 434Q460 425 391 149Q320 -135 320 -139Q320 -147 365 -148H390Q396 -156 396 -157T393 -175Q389 -188 383 -194H370Q339 -192 262 -192Q234 -192 211 -192T174 -192T157 -193Q143 -193 143 -185Q143 -182 145 -170Q149 -154 152 -151T172 -148Q220 -148 230 -141Q238 -136 258 -53T279 32Q279 33 272 29Q224 -10 172 -10Q117 -10 75 30T33 157ZM352 326Q329 405 277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q233 26 290 98L298 109L352 326Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D407\" xlink:href=\"#MJX-551-TEX-B-1D407\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(1177.8,0)\"><use data-c=\"2208\" xlink:href=\"#MJX-551-TEX-N-2208\"><\/use><\/g><g data-mml-node=\"msubsup\" transform=\"translate(2122.6,0)\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D53D\" xlink:href=\"#MJX-551-TEX-D-1D53D\"><\/use><\/g><\/g><g data-mml-node=\"TeXAtom\" transform=\"translate(644,530.4) scale(0.707)\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mo\"><use data-c=\"28\" xlink:href=\"#MJX-551-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(389,0)\"><use data-c=\"1D45B\" xlink:href=\"#MJX-551-TEX-I-1D45B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(989,0)\"><use data-c=\"2212\" xlink:href=\"#MJX-551-TEX-N-2212\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1767,0)\"><use data-c=\"1D458\" xlink:href=\"#MJX-551-TEX-I-1D458\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2288,0)\"><use data-c=\"29\" xlink:href=\"#MJX-551-TEX-N-29\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2677,0)\"><use data-c=\"D7\" xlink:href=\"#MJX-551-TEX-N-D7\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(3455,0)\"><use data-c=\"1D45B\" xlink:href=\"#MJX-551-TEX-I-1D45B\"><\/use><\/g><\/g><g data-mml-node=\"mi\" transform=\"translate(644,-138.9) scale(0.707)\"><use data-c=\"1D45E\" xlink:href=\"#MJX-551-TEX-I-1D45E\"><\/use><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">H<\/mi><\/mrow><mo>\u2208<\/mo><msubsup><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"double-struck\">F<\/mi><\/mrow><mi>q<\/mi><mrow data-mjx-texclass=\"ORD\"><mo stretchy=\"false\">(<\/mo><mi>n<\/mi><mo>\u2212<\/mo><mi>k<\/mi><mo stretchy=\"false\">)<\/mo><mo>\u00d7<\/mo><mi>n<\/mi><\/mrow><\/msubsup><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">\\mathbf{H}\\in\\mathbb{F}_q^{(n-k)\\times n}<\/script><span>, the <\/span><strong><span>syndrome<\/span><\/strong><span> of <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.373ex\" height=\"1.457ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -444 607 644\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.452ex;\"><defs><path id=\"MJX-59-TEX-B-1D432\" d=\"M84 -102Q84 -110 87 -119T102 -138T133 -149Q148 -148 162 -143T186 -131T206 -114T222 -95T234 -76T243 -59T249 -45T252 -37L269 0L96 382H26V444H34Q49 441 146 441Q252 441 270 444H279V382H255Q232 382 232 380L337 151L442 382H394V444H401Q413 441 495 441Q568 441 574 444H580V382H510L406 152Q298 -84 297 -87Q269 -139 225 -169T131 -200Q85 -200 54 -172T23 -100Q23 -64 44 -50T87 -35Q111 -35 130 -50T152 -92V-100H84V-102Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D432\" xlink:href=\"#MJX-59-TEX-B-1D432\"><\/use><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">y<\/mi><\/mrow><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">\\mathbf{y}<\/script><span> with respect to <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"2.036ex\" height=\"1.552ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -686 900 686\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: 0px;\"><defs><path id=\"MJX-60-TEX-B-1D407\" d=\"M400 0Q376 3 226 3Q75 3 51 0H39V62H147V624H39V686H51Q75 683 226 683Q376 683 400 686H412V624H304V388H595V624H487V686H499Q523 683 673 683Q824 683 848 686H860V624H752V62H860V0H848Q824 3 674 3Q523 3 499 0H487V62H595V326H304V62H412V0H400Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D407\" xlink:href=\"#MJX-60-TEX-B-1D407\"><\/use><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">H<\/mi><\/mrow><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">\\mathbf{H}<\/script><span> is defined as <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"4.487ex\" height=\"2.077ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -717.8 1983.2 917.8\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.452ex;\"><defs><path id=\"MJX-61-TEX-B-1D407\" d=\"M400 0Q376 3 226 3Q75 3 51 0H39V62H147V624H39V686H51Q75 683 226 683Q376 683 400 686H412V624H304V388H595V624H487V686H499Q523 683 673 683Q824 683 848 686H860V624H752V62H860V0H848Q824 3 674 3Q523 3 499 0H487V62H595V326H304V62H412V0H400Z\"><\/path><path id=\"MJX-61-TEX-B-1D432\" d=\"M84 -102Q84 -110 87 -119T102 -138T133 -149Q148 -148 162 -143T186 -131T206 -114T222 -95T234 -76T243 -59T249 -45T252 -37L269 0L96 382H26V444H34Q49 441 146 441Q252 441 270 444H279V382H255Q232 382 232 380L337 151L442 382H394V444H401Q413 441 495 441Q568 441 574 444H580V382H510L406 152Q298 -84 297 -87Q269 -139 225 -169T131 -200Q85 -200 54 -172T23 -100Q23 -64 44 -50T87 -35Q111 -35 130 -50T152 -92V-100H84V-102Z\"><\/path><path id=\"MJX-61-TEX-N-22BA\" d=\"M318 -182Q302 -212 280 -212H278H275Q249 -212 239 -182L238 84V351H162L87 352Q57 362 57 391T84 429Q89 431 280 431H470L474 429Q477 427 479 426T484 423T490 417T495 410T499 402T500 391Q500 365 470 352L394 351H318V-182Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"msup\"><g data-mml-node=\"mi\"><use data-c=\"1D407\" xlink:href=\"#MJX-61-TEX-B-1D407\"><\/use><use data-c=\"1D432\" xlink:href=\"#MJX-61-TEX-B-1D432\" transform=\"translate(900,0)\"><\/use><\/g><g data-mml-node=\"TeXAtom\" transform=\"translate(1540,413.1) scale(0.707)\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mo\"><use data-c=\"22BA\" xlink:href=\"#MJX-61-TEX-N-22BA\"><\/use><\/g><\/g><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow data-mjx-texclass=\"ORD\"><msup><mi mathvariant=\"bold\">Hy<\/mi><mrow data-mjx-texclass=\"ORD\"><mo>\u22ba<\/mo><\/mrow><\/msup><\/mrow><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">\\mathbf{Hy^{\\intercal}}<\/script><span>. Any element of <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"4.608ex\" height=\"2.801ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -853.7 2036.8 1237.9\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.869ex;\"><defs><path id=\"MJX-561-TEX-D-1D53D\" d=\"M584 499Q569 490 566 490Q558 490 552 497T546 515Q546 535 533 559Q526 574 506 593T469 621Q415 648 326 648Q293 648 287 647T275 641Q264 630 263 617Q262 609 260 492V370L275 372Q323 376 350 392T393 441Q409 473 409 506Q409 529 427 529Q437 529 442 519Q444 511 444 362Q444 212 442 206Q436 197 426 197Q409 197 409 217Q409 265 375 299Q346 328 280 335H260V206Q260 70 262 63Q265 46 276 41T326 35Q362 35 366 28Q377 17 366 3L360 -1H24Q12 5 12 16Q12 35 51 35Q92 38 97 52Q102 60 102 341T97 632Q91 645 51 648Q12 648 12 666Q12 675 24 683H573Q576 678 584 670V499ZM137 341Q137 131 136 89T130 37Q129 36 129 35H182Q233 35 233 39Q226 54 225 92T224 346L226 623L231 635L235 648H129Q132 641 133 638T135 603T137 517T137 341ZM549 603V648H495L506 641Q531 621 533 619L549 603ZM409 317V395L400 386Q390 376 375 366L357 355L373 346Q394 331 397 328L409 317Z\"><\/path><path id=\"MJX-561-TEX-I-1D45B\" d=\"M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z\"><\/path><path id=\"MJX-561-TEX-N-2212\" d=\"M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z\"><\/path><path id=\"MJX-561-TEX-I-1D458\" d=\"M121 647Q121 657 125 670T137 683Q138 683 209 688T282 694Q294 694 294 686Q294 679 244 477Q194 279 194 272Q213 282 223 291Q247 309 292 354T362 415Q402 442 438 442Q468 442 485 423T503 369Q503 344 496 327T477 302T456 291T438 288Q418 288 406 299T394 328Q394 353 410 369T442 390L458 393Q446 405 434 405H430Q398 402 367 380T294 316T228 255Q230 254 243 252T267 246T293 238T320 224T342 206T359 180T365 147Q365 130 360 106T354 66Q354 26 381 26Q429 26 459 145Q461 153 479 153H483Q499 153 499 144Q499 139 496 130Q455 -11 378 -11Q333 -11 305 15T277 90Q277 108 280 121T283 145Q283 167 269 183T234 206T200 217T182 220H180Q168 178 159 139T145 81T136 44T129 20T122 7T111 -2Q98 -11 83 -11Q66 -11 57 -1T48 16Q48 26 85 176T158 471L195 616Q196 629 188 632T149 637H144Q134 637 131 637T124 640T121 647Z\"><\/path><path id=\"MJX-561-TEX-I-1D45E\" d=\"M33 157Q33 258 109 349T280 441Q340 441 372 389Q373 390 377 395T388 406T404 418Q438 442 450 442Q454 442 457 439T460 434Q460 425 391 149Q320 -135 320 -139Q320 -147 365 -148H390Q396 -156 396 -157T393 -175Q389 -188 383 -194H370Q339 -192 262 -192Q234 -192 211 -192T174 -192T157 -193Q143 -193 143 -185Q143 -182 145 -170Q149 -154 152 -151T172 -148Q220 -148 230 -141Q238 -136 258 -53T279 32Q279 33 272 29Q224 -10 172 -10Q117 -10 75 30T33 157ZM352 326Q329 405 277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q233 26 290 98L298 109L352 326Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"msubsup\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D53D\" xlink:href=\"#MJX-561-TEX-D-1D53D\"><\/use><\/g><\/g><g data-mml-node=\"TeXAtom\" transform=\"translate(644,363) scale(0.707)\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D45B\" xlink:href=\"#MJX-561-TEX-I-1D45B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(600,0)\"><use data-c=\"2212\" xlink:href=\"#MJX-561-TEX-N-2212\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1378,0)\"><use data-c=\"1D458\" xlink:href=\"#MJX-561-TEX-I-1D458\"><\/use><\/g><\/g><g data-mml-node=\"mi\" transform=\"translate(644,-247) scale(0.707)\"><use data-c=\"1D45E\" xlink:href=\"#MJX-561-TEX-I-1D45E\"><\/use><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><msubsup><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"double-struck\">F<\/mi><\/mrow><mi>q<\/mi><mrow data-mjx-texclass=\"ORD\"><mi>n<\/mi><mo>\u2212<\/mo><mi>k<\/mi><\/mrow><\/msubsup><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">\\mathbb{F}_q^{n-k}<\/script><span> is called a syndrome.<\/span><\/p><p><strong><span>Syndrome Decoding:<\/span><\/strong><span> Given <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"12.859ex\" height=\"3.024ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -1060.7 5683.9 1336.8\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.625ex;\"><defs><path id=\"MJX-551-TEX-B-1D407\" d=\"M400 0Q376 3 226 3Q75 3 51 0H39V62H147V624H39V686H51Q75 683 226 683Q376 683 400 686H412V624H304V388H595V624H487V686H499Q523 683 673 683Q824 683 848 686H860V624H752V62H860V0H848Q824 3 674 3Q523 3 499 0H487V62H595V326H304V62H412V0H400Z\"><\/path><path id=\"MJX-551-TEX-N-2208\" d=\"M84 250Q84 372 166 450T360 539Q361 539 377 539T419 540T469 540H568Q583 532 583 520Q583 511 570 501L466 500Q355 499 329 494Q280 482 242 458T183 409T147 354T129 306T124 272V270H568Q583 262 583 250T568 230H124V228Q124 207 134 177T167 112T231 48T328 7Q355 1 466 0H570Q583 -10 583 -20Q583 -32 568 -40H471Q464 -40 446 -40T417 -41Q262 -41 172 45Q84 127 84 250Z\"><\/path><path id=\"MJX-551-TEX-D-1D53D\" d=\"M584 499Q569 490 566 490Q558 490 552 497T546 515Q546 535 533 559Q526 574 506 593T469 621Q415 648 326 648Q293 648 287 647T275 641Q264 630 263 617Q262 609 260 492V370L275 372Q323 376 350 392T393 441Q409 473 409 506Q409 529 427 529Q437 529 442 519Q444 511 444 362Q444 212 442 206Q436 197 426 197Q409 197 409 217Q409 265 375 299Q346 328 280 335H260V206Q260 70 262 63Q265 46 276 41T326 35Q362 35 366 28Q377 17 366 3L360 -1H24Q12 5 12 16Q12 35 51 35Q92 38 97 52Q102 60 102 341T97 632Q91 645 51 648Q12 648 12 666Q12 675 24 683H573Q576 678 584 670V499ZM137 341Q137 131 136 89T130 37Q129 36 129 35H182Q233 35 233 39Q226 54 225 92T224 346L226 623L231 635L235 648H129Q132 641 133 638T135 603T137 517T137 341ZM549 603V648H495L506 641Q531 621 533 619L549 603ZM409 317V395L400 386Q390 376 375 366L357 355L373 346Q394 331 397 328L409 317Z\"><\/path><path id=\"MJX-551-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-551-TEX-I-1D45B\" d=\"M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z\"><\/path><path id=\"MJX-551-TEX-N-2212\" d=\"M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z\"><\/path><path id=\"MJX-551-TEX-I-1D458\" d=\"M121 647Q121 657 125 670T137 683Q138 683 209 688T282 694Q294 694 294 686Q294 679 244 477Q194 279 194 272Q213 282 223 291Q247 309 292 354T362 415Q402 442 438 442Q468 442 485 423T503 369Q503 344 496 327T477 302T456 291T438 288Q418 288 406 299T394 328Q394 353 410 369T442 390L458 393Q446 405 434 405H430Q398 402 367 380T294 316T228 255Q230 254 243 252T267 246T293 238T320 224T342 206T359 180T365 147Q365 130 360 106T354 66Q354 26 381 26Q429 26 459 145Q461 153 479 153H483Q499 153 499 144Q499 139 496 130Q455 -11 378 -11Q333 -11 305 15T277 90Q277 108 280 121T283 145Q283 167 269 183T234 206T200 217T182 220H180Q168 178 159 139T145 81T136 44T129 20T122 7T111 -2Q98 -11 83 -11Q66 -11 57 -1T48 16Q48 26 85 176T158 471L195 616Q196 629 188 632T149 637H144Q134 637 131 637T124 640T121 647Z\"><\/path><path id=\"MJX-551-TEX-N-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><path id=\"MJX-551-TEX-N-D7\" d=\"M630 29Q630 9 609 9Q604 9 587 25T493 118L389 222L284 117Q178 13 175 11Q171 9 168 9Q160 9 154 15T147 29Q147 36 161 51T255 146L359 250L255 354Q174 435 161 449T147 471Q147 480 153 485T168 490Q173 490 175 489Q178 487 284 383L389 278L493 382Q570 459 587 475T609 491Q630 491 630 471Q630 464 620 453T522 355L418 250L522 145Q606 61 618 48T630 29Z\"><\/path><path id=\"MJX-551-TEX-I-1D45E\" d=\"M33 157Q33 258 109 349T280 441Q340 441 372 389Q373 390 377 395T388 406T404 418Q438 442 450 442Q454 442 457 439T460 434Q460 425 391 149Q320 -135 320 -139Q320 -147 365 -148H390Q396 -156 396 -157T393 -175Q389 -188 383 -194H370Q339 -192 262 -192Q234 -192 211 -192T174 -192T157 -193Q143 -193 143 -185Q143 -182 145 -170Q149 -154 152 -151T172 -148Q220 -148 230 -141Q238 -136 258 -53T279 32Q279 33 272 29Q224 -10 172 -10Q117 -10 75 30T33 157ZM352 326Q329 405 277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q233 26 290 98L298 109L352 326Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D407\" xlink:href=\"#MJX-551-TEX-B-1D407\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(1177.8,0)\"><use data-c=\"2208\" xlink:href=\"#MJX-551-TEX-N-2208\"><\/use><\/g><g data-mml-node=\"msubsup\" transform=\"translate(2122.6,0)\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D53D\" xlink:href=\"#MJX-551-TEX-D-1D53D\"><\/use><\/g><\/g><g data-mml-node=\"TeXAtom\" transform=\"translate(644,530.4) scale(0.707)\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mo\"><use data-c=\"28\" xlink:href=\"#MJX-551-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(389,0)\"><use data-c=\"1D45B\" xlink:href=\"#MJX-551-TEX-I-1D45B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(989,0)\"><use data-c=\"2212\" xlink:href=\"#MJX-551-TEX-N-2212\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1767,0)\"><use data-c=\"1D458\" xlink:href=\"#MJX-551-TEX-I-1D458\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2288,0)\"><use data-c=\"29\" xlink:href=\"#MJX-551-TEX-N-29\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2677,0)\"><use data-c=\"D7\" xlink:href=\"#MJX-551-TEX-N-D7\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(3455,0)\"><use data-c=\"1D45B\" xlink:href=\"#MJX-551-TEX-I-1D45B\"><\/use><\/g><\/g><g data-mml-node=\"mi\" transform=\"translate(644,-138.9) scale(0.707)\"><use data-c=\"1D45E\" xlink:href=\"#MJX-551-TEX-I-1D45E\"><\/use><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">H<\/mi><\/mrow><mo>\u2208<\/mo><msubsup><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"double-struck\">F<\/mi><\/mrow><mi>q<\/mi><mrow data-mjx-texclass=\"ORD\"><mo stretchy=\"false\">(<\/mo><mi>n<\/mi><mo>\u2212<\/mo><mi>k<\/mi><mo stretchy=\"false\">)<\/mo><mo>\u00d7<\/mo><mi>n<\/mi><\/mrow><\/msubsup><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">\\mathbf{H}\\in\\mathbb{F}_q^{(n-k)\\times n}<\/script><span> of rank <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"5.302ex\" height=\"1.756ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -694 2343.4 776\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.186ex;\"><defs><path id=\"MJX-1297-TEX-I-1D45B\" d=\"M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z\"><\/path><path id=\"MJX-1297-TEX-N-2212\" d=\"M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z\"><\/path><path id=\"MJX-1297-TEX-I-1D458\" d=\"M121 647Q121 657 125 670T137 683Q138 683 209 688T282 694Q294 694 294 686Q294 679 244 477Q194 279 194 272Q213 282 223 291Q247 309 292 354T362 415Q402 442 438 442Q468 442 485 423T503 369Q503 344 496 327T477 302T456 291T438 288Q418 288 406 299T394 328Q394 353 410 369T442 390L458 393Q446 405 434 405H430Q398 402 367 380T294 316T228 255Q230 254 243 252T267 246T293 238T320 224T342 206T359 180T365 147Q365 130 360 106T354 66Q354 26 381 26Q429 26 459 145Q461 153 479 153H483Q499 153 499 144Q499 139 496 130Q455 -11 378 -11Q333 -11 305 15T277 90Q277 108 280 121T283 145Q283 167 269 183T234 206T200 217T182 220H180Q168 178 159 139T145 81T136 44T129 20T122 7T111 -2Q98 -11 83 -11Q66 -11 57 -1T48 16Q48 26 85 176T158 471L195 616Q196 629 188 632T149 637H144Q134 637 131 637T124 640T121 647Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D45B\" xlink:href=\"#MJX-1297-TEX-I-1D45B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(822.2,0)\"><use data-c=\"2212\" xlink:href=\"#MJX-1297-TEX-N-2212\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1822.4,0)\"><use data-c=\"1D458\" xlink:href=\"#MJX-1297-TEX-I-1D458\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>n<\/mi><mo>\u2212<\/mo><mi>k<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">n-k<\/script><span>, <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"9.646ex\" height=\"3.024ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -1060.7 4263.5 1336.8\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.625ex;\"><defs><path id=\"MJX-65-TEX-B-1D42C\" d=\"M38 315Q38 339 45 360T70 404T127 440T223 453Q273 453 320 436L338 445L357 453H366Q380 453 383 447T386 403V387V355Q386 331 383 326T365 321H355H349Q333 321 329 324T324 341Q317 406 224 406H216Q123 406 123 353Q123 334 143 321T188 304T244 294T285 286Q305 281 325 273T373 237T412 172Q414 162 414 142Q414 -6 230 -6Q154 -6 117 22L68 -6H58Q44 -6 41 0T38 42V73Q38 85 38 101T37 122Q37 144 42 148T68 153H75Q87 153 91 151T97 147T103 132Q131 46 220 46H230Q257 46 265 47Q330 58 330 108Q330 127 316 142Q300 156 284 162Q271 168 212 178T122 202Q38 243 38 315Z\"><\/path><path id=\"MJX-65-TEX-N-2208\" d=\"M84 250Q84 372 166 450T360 539Q361 539 377 539T419 540T469 540H568Q583 532 583 520Q583 511 570 501L466 500Q355 499 329 494Q280 482 242 458T183 409T147 354T129 306T124 272V270H568Q583 262 583 250T568 230H124V228Q124 207 134 177T167 112T231 48T328 7Q355 1 466 0H570Q583 -10 583 -20Q583 -32 568 -40H471Q464 -40 446 -40T417 -41Q262 -41 172 45Q84 127 84 250Z\"><\/path><path id=\"MJX-65-TEX-D-1D53D\" d=\"M584 499Q569 490 566 490Q558 490 552 497T546 515Q546 535 533 559Q526 574 506 593T469 621Q415 648 326 648Q293 648 287 647T275 641Q264 630 263 617Q262 609 260 492V370L275 372Q323 376 350 392T393 441Q409 473 409 506Q409 529 427 529Q437 529 442 519Q444 511 444 362Q444 212 442 206Q436 197 426 197Q409 197 409 217Q409 265 375 299Q346 328 280 335H260V206Q260 70 262 63Q265 46 276 41T326 35Q362 35 366 28Q377 17 366 3L360 -1H24Q12 5 12 16Q12 35 51 35Q92 38 97 52Q102 60 102 341T97 632Q91 645 51 648Q12 648 12 666Q12 675 24 683H573Q576 678 584 670V499ZM137 341Q137 131 136 89T130 37Q129 36 129 35H182Q233 35 233 39Q226 54 225 92T224 346L226 623L231 635L235 648H129Q132 641 133 638T135 603T137 517T137 341ZM549 603V648H495L506 641Q531 621 533 619L549 603ZM409 317V395L400 386Q390 376 375 366L357 355L373 346Q394 331 397 328L409 317Z\"><\/path><path id=\"MJX-65-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-65-TEX-I-1D45B\" d=\"M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z\"><\/path><path id=\"MJX-65-TEX-N-2212\" d=\"M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z\"><\/path><path id=\"MJX-65-TEX-I-1D458\" d=\"M121 647Q121 657 125 670T137 683Q138 683 209 688T282 694Q294 694 294 686Q294 679 244 477Q194 279 194 272Q213 282 223 291Q247 309 292 354T362 415Q402 442 438 442Q468 442 485 423T503 369Q503 344 496 327T477 302T456 291T438 288Q418 288 406 299T394 328Q394 353 410 369T442 390L458 393Q446 405 434 405H430Q398 402 367 380T294 316T228 255Q230 254 243 252T267 246T293 238T320 224T342 206T359 180T365 147Q365 130 360 106T354 66Q354 26 381 26Q429 26 459 145Q461 153 479 153H483Q499 153 499 144Q499 139 496 130Q455 -11 378 -11Q333 -11 305 15T277 90Q277 108 280 121T283 145Q283 167 269 183T234 206T200 217T182 220H180Q168 178 159 139T145 81T136 44T129 20T122 7T111 -2Q98 -11 83 -11Q66 -11 57 -1T48 16Q48 26 85 176T158 471L195 616Q196 629 188 632T149 637H144Q134 637 131 637T124 640T121 647Z\"><\/path><path id=\"MJX-65-TEX-N-29\" d=\"M60 749L64 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data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D42C\" xlink:href=\"#MJX-65-TEX-B-1D42C\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(731.8,0)\"><use data-c=\"2208\" xlink:href=\"#MJX-65-TEX-N-2208\"><\/use><\/g><g data-mml-node=\"msubsup\" transform=\"translate(1676.6,0)\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D53D\" xlink:href=\"#MJX-65-TEX-D-1D53D\"><\/use><\/g><\/g><g data-mml-node=\"TeXAtom\" transform=\"translate(644,530.4) scale(0.707)\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mo\"><use data-c=\"28\" xlink:href=\"#MJX-65-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(389,0)\"><use data-c=\"1D45B\" xlink:href=\"#MJX-65-TEX-I-1D45B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(989,0)\"><use data-c=\"2212\" xlink:href=\"#MJX-65-TEX-N-2212\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1767,0)\"><use data-c=\"1D458\" xlink:href=\"#MJX-65-TEX-I-1D458\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2288,0)\"><use data-c=\"29\" xlink:href=\"#MJX-65-TEX-N-29\"><\/use><\/g><\/g><g data-mml-node=\"mi\" transform=\"translate(644,-138.9) scale(0.707)\"><use data-c=\"1D45E\" xlink:href=\"#MJX-65-TEX-I-1D45E\"><\/use><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">s<\/mi><\/mrow><mo>\u2208<\/mo><msubsup><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"double-struck\">F<\/mi><\/mrow><mi>q<\/mi><mrow data-mjx-texclass=\"ORD\"><mo stretchy=\"false\">(<\/mo><mi>n<\/mi><mo>\u2212<\/mo><mi>k<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><\/msubsup><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">\\mathbf{s}\\in\\mathbb{F}_q^{(n-k)}<\/script><span>, <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"13.375ex\" height=\"2.262ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 5911.6 1000\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-73-TEX-I-1D461\" d=\"M26 385Q19 392 19 395Q19 399 22 411T27 425Q29 430 36 430T87 431H140L159 511Q162 522 166 540T173 566T179 586T187 603T197 615T211 624T229 626Q247 625 254 615T261 596Q261 589 252 549T232 470L222 433Q222 431 272 431H323Q330 424 330 420Q330 398 317 385H210L174 240Q135 80 135 68Q135 26 162 26Q197 26 230 60T283 144Q285 150 288 151T303 153H307Q322 153 322 145Q322 142 319 133Q314 117 301 95T267 48T216 6T155 -11Q125 -11 98 4T59 56Q57 64 57 83V101L92 241Q127 382 128 383Q128 385 77 385H26Z\"><\/path><path id=\"MJX-73-TEX-N-2208\" d=\"M84 250Q84 372 166 450T360 539Q361 539 377 539T419 540T469 540H568Q583 532 583 520Q583 511 570 501L466 500Q355 499 329 494Q280 482 242 458T183 409T147 354T129 306T124 272V270H568Q583 262 583 250T568 230H124V228Q124 207 134 177T167 112T231 48T328 7Q355 1 466 0H570Q583 -10 583 -20Q583 -32 568 -40H471Q464 -40 446 -40T417 -41Q262 -41 172 45Q84 127 84 250Z\"><\/path><path id=\"MJX-73-TEX-N-7B\" d=\"M434 -231Q434 -244 428 -250H410Q281 -250 230 -184Q225 -177 222 -172T217 -161T213 -148T211 -133T210 -111T209 -84T209 -47T209 0Q209 21 209 53Q208 142 204 153Q203 154 203 155Q189 191 153 211T82 231Q71 231 68 234T65 250T68 266T82 269Q116 269 152 289T203 345Q208 356 208 377T209 529V579Q209 634 215 656T244 698Q270 724 324 740Q361 748 377 749Q379 749 390 749T408 750H428Q434 744 434 732Q434 719 431 716Q429 713 415 713Q362 710 332 689T296 647Q291 634 291 499V417Q291 370 288 353T271 314Q240 271 184 255L170 250L184 245Q202 239 220 230T262 196T290 137Q291 131 291 1Q291 -134 296 -147Q306 -174 339 -192T415 -213Q429 -213 431 -216Q434 -219 434 -231Z\"><\/path><path id=\"MJX-73-TEX-N-30\" d=\"M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 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442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z\"><\/path><path id=\"MJX-73-TEX-N-7D\" d=\"M65 731Q65 745 68 747T88 750Q171 750 216 725T279 670Q288 649 289 635T291 501Q292 362 293 357Q306 312 345 291T417 269Q428 269 431 266T434 250T431 234T417 231Q380 231 345 210T298 157Q293 143 292 121T291 -28V-79Q291 -134 285 -156T256 -198Q202 -250 89 -250Q71 -250 68 -247T65 -230Q65 -224 65 -223T66 -218T69 -214T77 -213Q91 -213 108 -210T146 -200T183 -177T207 -139Q208 -134 209 3L210 139Q223 196 280 230Q315 247 330 250Q305 257 280 270Q225 304 212 352L210 362L209 498Q208 635 207 640Q195 680 154 696T77 713Q68 713 67 716T65 731Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D461\" xlink:href=\"#MJX-73-TEX-I-1D461\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(638.8,0)\"><use data-c=\"2208\" xlink:href=\"#MJX-73-TEX-N-2208\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(1583.6,0)\"><use data-c=\"7B\" xlink:href=\"#MJX-73-TEX-N-7B\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(2083.6,0)\"><use data-c=\"30\" xlink:href=\"#MJX-73-TEX-N-30\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2583.6,0)\"><use data-c=\"2C\" xlink:href=\"#MJX-73-TEX-N-2C\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(3028.2,0)\"><use data-c=\"2026\" xlink:href=\"#MJX-73-TEX-N-2026\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(4366.9,0)\"><use data-c=\"2C\" xlink:href=\"#MJX-73-TEX-N-2C\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(4811.6,0)\"><use data-c=\"1D45B\" xlink:href=\"#MJX-73-TEX-I-1D45B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(5411.6,0)\"><use data-c=\"7D\" xlink:href=\"#MJX-73-TEX-N-7D\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>t<\/mi><mo>\u2208<\/mo><mo fence=\"false\" stretchy=\"false\">{<\/mo><mn>0<\/mn><mo>,<\/mo><mo>\u2026<\/mo><mo>,<\/mo><mi>n<\/mi><mo fence=\"false\" stretchy=\"false\">}<\/mo><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">t\\in\\{0,\\dots,n\\}<\/script><span>, find <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.192ex\" height=\"1.036ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -452 527 458\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.014ex;\"><defs><path id=\"MJX-74-TEX-B-1D41E\" d=\"M32 225Q32 332 102 392T272 452H283Q382 452 436 401Q494 343 494 243Q494 226 486 222T440 217Q431 217 394 217T327 218H175V209Q175 177 179 154T196 107T236 69T306 50Q312 49 323 49Q376 49 410 85Q421 99 427 111T434 127T442 133T463 135H468Q494 135 494 117Q494 110 489 97T468 66T431 32T373 5T292 -6Q181 -6 107 55T32 225ZM383 276Q377 346 348 374T280 402Q253 402 230 390T195 357Q179 331 176 279V266H383V276Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D41E\" xlink:href=\"#MJX-74-TEX-B-1D41E\"><\/use><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">e<\/mi><\/mrow><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">\\mathbf{e}<\/script><span> with weight <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"0.817ex\" height=\"1.441ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -626 361 637\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.025ex;\"><defs><path id=\"MJX-1284-TEX-I-1D461\" d=\"M26 385Q19 392 19 395Q19 399 22 411T27 425Q29 430 36 430T87 431H140L159 511Q162 522 166 540T173 566T179 586T187 603T197 615T211 624T229 626Q247 625 254 615T261 596Q261 589 252 549T232 470L222 433Q222 431 272 431H323Q330 424 330 420Q330 398 317 385H210L174 240Q135 80 135 68Q135 26 162 26Q197 26 230 60T283 144Q285 150 288 151T303 153H307Q322 153 322 145Q322 142 319 133Q314 117 301 95T267 48T216 6T155 -11Q125 -11 98 4T59 56Q57 64 57 83V101L92 241Q127 382 128 383Q128 385 77 385H26Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D461\" xlink:href=\"#MJX-1284-TEX-I-1D461\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>t<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">t<\/script><span> such that <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"9.427ex\" height=\"1.81ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -717.8 4166.9 799.8\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.186ex;\"><defs><path id=\"MJX-69-TEX-B-1D407\" d=\"M400 0Q376 3 226 3Q75 3 51 0H39V62H147V624H39V686H51Q75 683 226 683Q376 683 400 686H412V624H304V388H595V624H487V686H499Q523 683 673 683Q824 683 848 686H860V624H752V62H860V0H848Q824 3 674 3Q523 3 499 0H487V62H595V326H304V62H412V0H400Z\"><\/path><path id=\"MJX-69-TEX-B-1D41E\" d=\"M32 225Q32 332 102 392T272 452H283Q382 452 436 401Q494 343 494 243Q494 226 486 222T440 217Q431 217 394 217T327 218H175V209Q175 177 179 154T196 107T236 69T306 50Q312 49 323 49Q376 49 410 85Q421 99 427 111T434 127T442 133T463 135H468Q494 135 494 117Q494 110 489 97T468 66T431 32T373 5T292 -6Q181 -6 107 55T32 225ZM383 276Q377 346 348 374T280 402Q253 402 230 390T195 357Q179 331 176 279V266H383V276Z\"><\/path><path id=\"MJX-69-TEX-N-22BA\" d=\"M318 -182Q302 -212 280 -212H278H275Q249 -212 239 -182L238 84V351H162L87 352Q57 362 57 391T84 429Q89 431 280 431H470L474 429Q477 427 479 426T484 423T490 417T495 410T499 402T500 391Q500 365 470 352L394 351H318V-182Z\"><\/path><path id=\"MJX-69-TEX-N-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path id=\"MJX-69-TEX-B-1D42C\" d=\"M38 315Q38 339 45 360T70 404T127 440T223 453Q273 453 320 436L338 445L357 453H366Q380 453 383 447T386 403V387V355Q386 331 383 326T365 321H355H349Q333 321 329 324T324 341Q317 406 224 406H216Q123 406 123 353Q123 334 143 321T188 304T244 294T285 286Q305 281 325 273T373 237T412 172Q414 162 414 142Q414 -6 230 -6Q154 -6 117 22L68 -6H58Q44 -6 41 0T38 42V73Q38 85 38 101T37 122Q37 144 42 148T68 153H75Q87 153 91 151T97 147T103 132Q131 46 220 46H230Q257 46 265 47Q330 58 330 108Q330 127 316 142Q300 156 284 162Q271 168 212 178T122 202Q38 243 38 315Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"msup\"><g data-mml-node=\"mi\"><use data-c=\"1D407\" xlink:href=\"#MJX-69-TEX-B-1D407\"><\/use><use data-c=\"1D41E\" xlink:href=\"#MJX-69-TEX-B-1D41E\" transform=\"translate(900,0)\"><\/use><\/g><g data-mml-node=\"TeXAtom\" transform=\"translate(1460,413.1) scale(0.707)\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mo\"><use data-c=\"22BA\" xlink:href=\"#MJX-69-TEX-N-22BA\"><\/use><\/g><\/g><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(2180.9,0)\"><use data-c=\"3D\" xlink:href=\"#MJX-69-TEX-N-3D\"><\/use><\/g><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\" transform=\"translate(3236.7,0)\"><g data-mml-node=\"msup\"><g data-mml-node=\"mi\"><use data-c=\"1D42C\" xlink:href=\"#MJX-69-TEX-B-1D42C\"><\/use><\/g><g data-mml-node=\"TeXAtom\" transform=\"translate(487,363) scale(0.707)\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mo\"><use data-c=\"22BA\" xlink:href=\"#MJX-69-TEX-N-22BA\"><\/use><\/g><\/g><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow data-mjx-texclass=\"ORD\"><msup><mi mathvariant=\"bold\">He<\/mi><mrow data-mjx-texclass=\"ORD\"><mo>\u22ba<\/mo><\/mrow><\/msup><\/mrow><mo>=<\/mo><mrow data-mjx-texclass=\"ORD\"><msup><mi mathvariant=\"bold\">s<\/mi><mrow data-mjx-texclass=\"ORD\"><mo>\u22ba<\/mo><\/mrow><\/msup><\/mrow><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">\\mathbf{He^{\\intercal}}=\\mathbf{s^{\\intercal}}<\/script><span>. <\/span><\/p><blockquote><p><span>Any solver of syndrome decoding can be turned in polynomial time into an algorithm solving <\/span><strong><span>Noisy Codeword Decoding:<\/span><\/strong><\/p><p><span>Given <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"9.419ex\" height=\"2.801ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -853.7 4163.4 1237.9\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.869ex;\"><defs><path id=\"MJX-70-TEX-B-1D406\" d=\"M465 -10Q281 -10 173 88T64 343Q64 413 85 471T143 568T217 631T298 670Q371 697 449 697Q452 697 459 697T470 696Q502 696 531 690T582 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stretchy=\"false\">}<\/mo><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">t\\in\\{0,\\dots,n\\}<\/script><span>, find <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.192ex\" height=\"1.036ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -452 527 458\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.014ex;\"><defs><path id=\"MJX-74-TEX-B-1D41E\" d=\"M32 225Q32 332 102 392T272 452H283Q382 452 436 401Q494 343 494 243Q494 226 486 222T440 217Q431 217 394 217T327 218H175V209Q175 177 179 154T196 107T236 69T306 50Q312 49 323 49Q376 49 410 85Q421 99 427 111T434 127T442 133T463 135H468Q494 135 494 117Q494 110 489 97T468 66T431 32T373 5T292 -6Q181 -6 107 55T32 225ZM383 276Q377 346 348 374T280 402Q253 402 230 390T195 357Q179 331 176 279V266H383V276Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D41E\" xlink:href=\"#MJX-74-TEX-B-1D41E\"><\/use><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">e<\/mi><\/mrow><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">\\mathbf{e}<\/script><span> with weight <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"0.817ex\" height=\"1.441ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -626 361 637\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.025ex;\"><defs><path id=\"MJX-1284-TEX-I-1D461\" d=\"M26 385Q19 392 19 395Q19 399 22 411T27 425Q29 430 36 430T87 431H140L159 511Q162 522 166 540T173 566T179 586T187 603T197 615T211 624T229 626Q247 625 254 615T261 596Q261 589 252 549T232 470L222 433Q222 431 272 431H323Q330 424 330 420Q330 398 317 385H210L174 240Q135 80 135 68Q135 26 162 26Q197 26 230 60T283 144Q285 150 288 151T303 153H307Q322 153 322 145Q322 142 319 133Q314 117 301 95T267 48T216 6T155 -11Q125 -11 98 4T59 56Q57 64 57 83V101L92 241Q127 382 128 383Q128 385 77 385H26Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D461\" xlink:href=\"#MJX-1284-TEX-I-1D461\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>t<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">t<\/script><span> such that <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"12.561ex\" height=\"2.029ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -697 5552 897\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.452ex;\"><defs><path id=\"MJX-76-TEX-B-1D432\" d=\"M84 -102Q84 -110 87 -119T102 -138T133 -149Q148 -148 162 -143T186 -131T206 -114T222 -95T234 -76T243 -59T249 -45T252 -37L269 0L96 382H26V444H34Q49 441 146 441Q252 441 270 444H279V382H255Q232 382 232 380L337 151L442 382H394V444H401Q413 441 495 441Q568 441 574 444H580V382H510L406 152Q298 -84 297 -87Q269 -139 225 -169T131 -200Q85 -200 54 -172T23 -100Q23 -64 44 -50T87 -35Q111 -35 130 -50T152 -92V-100H84V-102Z\"><\/path><path id=\"MJX-76-TEX-N-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path id=\"MJX-76-TEX-B-1D426\" d=\"M40 442Q217 450 218 450H224V365Q226 367 235 378T254 397T278 416T314 435T362 448Q376 450 400 450H406Q503 450 534 393Q545 376 545 370Q545 368 555 379Q611 450 716 450Q774 450 809 434Q850 414 861 379T873 276V213V198V62H942V0H933Q915 3 809 3Q702 3 684 0H675V62H744V194V275Q744 348 735 373T690 399Q645 399 607 370T557 290Q555 281 554 171V62H623V0H614Q596 3 489 3Q374 3 365 0H356V62H425V194V275Q425 348 416 373T371 399Q326 399 288 370T238 290Q236 281 235 171V62H304V0H295Q277 3 171 3Q64 3 46 0H37V62H106V210V303Q106 353 104 363T91 376Q77 380 50 380H37V442H40Z\"><\/path><path id=\"MJX-76-TEX-B-1D406\" d=\"M465 -10Q281 -10 173 88T64 343Q64 413 85 471T143 568T217 631T298 670Q371 697 449 697Q452 697 459 697T470 696Q502 696 531 690T582 675T618 658T644 641T656 632L732 695Q734 697 745 697Q758 697 761 692T765 668V627V489V449Q765 428 761 424T741 419H731H724Q705 419 702 422T695 444Q683 520 631 577T495 635Q364 635 295 563Q261 528 247 477T232 343Q232 296 236 260T256 185T296 120T366 76T472 52Q481 51 498 51Q544 51 573 67T607 108Q608 111 608 164V214H464V276H479Q506 273 680 273Q816 273 834 276H845V214H765V113V51Q765 16 763 8T750 0Q742 2 709 16T658 40L648 46Q592 -10 465 -10Z\"><\/path><path id=\"MJX-76-TEX-N-2B\" d=\"M56 237T56 250T70 270H369V420L370 570Q380 583 389 583Q402 583 409 568V270H707Q722 262 722 250T707 230H409V-68Q401 -82 391 -82H389H387Q375 -82 369 -68V230H70Q56 237 56 250Z\"><\/path><path id=\"MJX-76-TEX-B-1D41E\" d=\"M32 225Q32 332 102 392T272 452H283Q382 452 436 401Q494 343 494 243Q494 226 486 222T440 217Q431 217 394 217T327 218H175V209Q175 177 179 154T196 107T236 69T306 50Q312 49 323 49Q376 49 410 85Q421 99 427 111T434 127T442 133T463 135H468Q494 135 494 117Q494 110 489 97T468 66T431 32T373 5T292 -6Q181 -6 107 55T32 225ZM383 276Q377 346 348 374T280 402Q253 402 230 390T195 357Q179 331 176 279V266H383V276Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D432\" xlink:href=\"#MJX-76-TEX-B-1D432\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(884.8,0)\"><use data-c=\"3D\" xlink:href=\"#MJX-76-TEX-N-3D\"><\/use><\/g><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\" transform=\"translate(1940.6,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D426\" xlink:href=\"#MJX-76-TEX-B-1D426\"><\/use><use data-c=\"1D406\" xlink:href=\"#MJX-76-TEX-B-1D406\" transform=\"translate(958,0)\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(4024.8,0)\"><use data-c=\"2B\" xlink:href=\"#MJX-76-TEX-N-2B\"><\/use><\/g><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\" transform=\"translate(5025,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D41E\" xlink:href=\"#MJX-76-TEX-B-1D41E\"><\/use><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">y<\/mi><\/mrow><mo>=<\/mo><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">mG<\/mi><\/mrow><mo>+<\/mo><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">e<\/mi><\/mrow><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">\\mathbf{y}=\\mathbf{mG}+\\mathbf{e}<\/script><span> for some <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"7.337ex\" height=\"2.801ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -853.7 3243 1237.9\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.869ex;\"><defs><path id=\"MJX-77-TEX-B-1D426\" d=\"M40 442Q217 450 218 450H224V365Q226 367 235 378T254 397T278 416T314 435T362 448Q376 450 400 450H406Q503 450 534 393Q545 376 545 370Q545 368 555 379Q611 450 716 450Q774 450 809 434Q850 414 861 379T873 276V213V198V62H942V0H933Q915 3 809 3Q702 3 684 0H675V62H744V194V275Q744 348 735 373T690 399Q645 399 607 370T557 290Q555 281 554 171V62H623V0H614Q596 3 489 3Q374 3 365 0H356V62H425V194V275Q425 348 416 373T371 399Q326 399 288 370T238 290Q236 281 235 171V62H304V0H295Q277 3 171 3Q64 3 46 0H37V62H106V210V303Q106 353 104 363T91 376Q77 380 50 380H37V442H40Z\"><\/path><path id=\"MJX-77-TEX-N-2208\" d=\"M84 250Q84 372 166 450T360 539Q361 539 377 539T419 540T469 540H568Q583 532 583 520Q583 511 570 501L466 500Q355 499 329 494Q280 482 242 458T183 409T147 354T129 306T124 272V270H568Q583 262 583 250T568 230H124V228Q124 207 134 177T167 112T231 48T328 7Q355 1 466 0H570Q583 -10 583 -20Q583 -32 568 -40H471Q464 -40 446 -40T417 -41Q262 -41 172 45Q84 127 84 250Z\"><\/path><path id=\"MJX-77-TEX-D-1D53D\" d=\"M584 499Q569 490 566 490Q558 490 552 497T546 515Q546 535 533 559Q526 574 506 593T469 621Q415 648 326 648Q293 648 287 647T275 641Q264 630 263 617Q262 609 260 492V370L275 372Q323 376 350 392T393 441Q409 473 409 506Q409 529 427 529Q437 529 442 519Q444 511 444 362Q444 212 442 206Q436 197 426 197Q409 197 409 217Q409 265 375 299Q346 328 280 335H260V206Q260 70 262 63Q265 46 276 41T326 35Q362 35 366 28Q377 17 366 3L360 -1H24Q12 5 12 16Q12 35 51 35Q92 38 97 52Q102 60 102 341T97 632Q91 645 51 648Q12 648 12 666Q12 675 24 683H573Q576 678 584 670V499ZM137 341Q137 131 136 89T130 37Q129 36 129 35H182Q233 35 233 39Q226 54 225 92T224 346L226 623L231 635L235 648H129Q132 641 133 638T135 603T137 517T137 341ZM549 603V648H495L506 641Q531 621 533 619L549 603ZM409 317V395L400 386Q390 376 375 366L357 355L373 346Q394 331 397 328L409 317Z\"><\/path><path id=\"MJX-77-TEX-I-1D458\" d=\"M121 647Q121 657 125 670T137 683Q138 683 209 688T282 694Q294 694 294 686Q294 679 244 477Q194 279 194 272Q213 282 223 291Q247 309 292 354T362 415Q402 442 438 442Q468 442 485 423T503 369Q503 344 496 327T477 302T456 291T438 288Q418 288 406 299T394 328Q394 353 410 369T442 390L458 393Q446 405 434 405H430Q398 402 367 380T294 316T228 255Q230 254 243 252T267 246T293 238T320 224T342 206T359 180T365 147Q365 130 360 106T354 66Q354 26 381 26Q429 26 459 145Q461 153 479 153H483Q499 153 499 144Q499 139 496 130Q455 -11 378 -11Q333 -11 305 15T277 90Q277 108 280 121T283 145Q283 167 269 183T234 206T200 217T182 220H180Q168 178 159 139T145 81T136 44T129 20T122 7T111 -2Q98 -11 83 -11Q66 -11 57 -1T48 16Q48 26 85 176T158 471L195 616Q196 629 188 632T149 637H144Q134 637 131 637T124 640T121 647Z\"><\/path><path id=\"MJX-77-TEX-I-1D45E\" d=\"M33 157Q33 258 109 349T280 441Q340 441 372 389Q373 390 377 395T388 406T404 418Q438 442 450 442Q454 442 457 439T460 434Q460 425 391 149Q320 -135 320 -139Q320 -147 365 -148H390Q396 -156 396 -157T393 -175Q389 -188 383 -194H370Q339 -192 262 -192Q234 -192 211 -192T174 -192T157 -193Q143 -193 143 -185Q143 -182 145 -170Q149 -154 152 -151T172 -148Q220 -148 230 -141Q238 -136 258 -53T279 32Q279 33 272 29Q224 -10 172 -10Q117 -10 75 30T33 157ZM352 326Q329 405 277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q233 26 290 98L298 109L352 326Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D426\" xlink:href=\"#MJX-77-TEX-B-1D426\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(1235.8,0)\"><use data-c=\"2208\" xlink:href=\"#MJX-77-TEX-N-2208\"><\/use><\/g><g data-mml-node=\"msubsup\" transform=\"translate(2180.6,0)\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D53D\" xlink:href=\"#MJX-77-TEX-D-1D53D\"><\/use><\/g><\/g><g data-mml-node=\"mi\" transform=\"translate(644,363) scale(0.707)\"><use data-c=\"1D458\" xlink:href=\"#MJX-77-TEX-I-1D458\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(644,-247) scale(0.707)\"><use data-c=\"1D45E\" xlink:href=\"#MJX-77-TEX-I-1D45E\"><\/use><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">m<\/mi><\/mrow><mo>\u2208<\/mo><msubsup><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"double-struck\">F<\/mi><\/mrow><mi>q<\/mi><mi>k<\/mi><\/msubsup><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">\\mathbf{m}\\in\\mathbb{F}_q^k<\/script><span>.<\/span><\/p><\/blockquote><p><strong><span>Hardness of Decoding Problem:<\/span><\/strong><span> From telecommunication side, the decoding problem should be easy to solve. From security side, the decoding problem should be hard to solve. All the subtlety lies in the inputs of the problem. There exist some codes and decoding distances where the problem is easy to solve. The existance of codes that are easy to decode is at the <\/span><u><span>foundation of code-based cryptography<\/span><\/u><span>.<\/span><\/p><ul><li><p><span>Decoding problem is hard in the worst case (NP-complete), which means that we cannot hope to solve the decoding problem in polynomial time for <\/span><u><span>all inputs<\/span><\/u><span>. <\/span><\/p><\/li><\/ul><ul><li><p><span>Decoding problem is hard on average, which means that for well chose <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"0.817ex\" height=\"1.441ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -626 361 637\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.025ex;\"><defs><path id=\"MJX-1284-TEX-I-1D461\" d=\"M26 385Q19 392 19 395Q19 399 22 411T27 425Q29 430 36 430T87 431H140L159 511Q162 522 166 540T173 566T179 586T187 603T197 615T211 624T229 626Q247 625 254 615T261 596Q261 589 252 549T232 470L222 433Q222 431 272 431H323Q330 424 330 420Q330 398 317 385H210L174 240Q135 80 135 68Q135 26 162 26Q197 26 230 60T283 144Q285 150 288 151T303 153H307Q322 153 322 145Q322 142 319 133Q314 117 301 95T267 48T216 6T155 -11Q125 -11 98 4T59 56Q57 64 57 83V101L92 241Q127 382 128 383Q128 385 77 385H26Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D461\" xlink:href=\"#MJX-1284-TEX-I-1D461\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>t<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">t<\/script><span>, <\/span><u><span>almost all<\/span><\/u><span> codes are intractable.<\/span><\/p><\/li><\/ul><blockquote><p><span>The NP\u2013completeness of a problem for a cryptographic use is a nice property but it is not the panacea to ensure its hardness, since it is quite possible that the security of a cryptosystem relies on the difficulty to solve an NP\u2013complete problem but at the same time breaking the scheme amounts to solve the problem on a subset of inputs for which make the problem easy to solve.<\/span><\/p><\/blockquote><p><span>Instead of considering the hardness based on its inputs, we focus on the <\/span><strong><span>algorithmic hardness<\/span><\/strong><span> of this decoding problem. Assume a probabilistic algorithm <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.853ex\" height=\"1.76ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -728 819 778\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.113ex;\"><defs><path id=\"MJX-79-TEX-C-41\" d=\"M576 668Q576 688 606 708T660 728Q676 728 675 712V571Q675 409 688 252Q696 122 720 57Q722 53 723 50T728 46T732 43T737 41T743 39L754 45Q788 61 803 61Q819 61 819 47Q818 43 814 35Q799 15 755 -7T675 -30Q659 -30 648 -25T630 -8T621 11T614 34Q603 77 599 106T594 146T591 160V163H460L329 164L316 145Q241 35 196 -7T119 -50T59 -24T30 43Q30 75 46 100T74 125Q81 125 83 120T88 104T96 84Q118 57 151 57Q189 57 277 182Q432 400 542 625L559 659H567Q574 659 575 660T576 668ZM584 249Q579 333 577 386T575 473T574 520V581L563 560Q497 426 412 290L372 228L370 224H371L383 228L393 232H586L584 249Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"41\" xlink:href=\"#MJX-79-TEX-C-41\"><\/use><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow data-mjx-texclass=\"ORD\"><mi data-mjx-variant=\"-tex-calligraphic\" mathvariant=\"script\">A<\/mi><\/mrow><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">\\mathcal{A}<\/script><span> that solves the decoding problem with success probability <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.054ex\" height=\"1.072ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -452 466 474\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.05ex;\"><defs><path id=\"MJX-80-TEX-I-1D700\" d=\"M190 -22Q124 -22 76 11T27 107Q27 174 97 232L107 239L99 248Q76 273 76 304Q76 364 144 408T290 452H302Q360 452 405 421Q428 405 428 392Q428 381 417 369T391 356Q382 356 371 365T338 383T283 392Q217 392 167 368T116 308Q116 289 133 272Q142 263 145 262T157 264Q188 278 238 278H243Q308 278 308 247Q308 206 223 206Q177 206 142 219L132 212Q68 169 68 112Q68 39 201 39Q253 39 286 49T328 72T345 94T362 105Q376 103 376 88Q376 79 365 62T334 26T275 -8T190 -22Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D700\" xlink:href=\"#MJX-80-TEX-I-1D700\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>\u03b5<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">\\varepsilon<\/script><span>, and a single run of this algorithm costs a time <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.593ex\" height=\"1.532ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -677 704 677\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: 0px;\"><defs><path id=\"MJX-81-TEX-I-1D447\" d=\"M40 437Q21 437 21 445Q21 450 37 501T71 602L88 651Q93 669 101 677H569H659Q691 677 697 676T704 667Q704 661 687 553T668 444Q668 437 649 437Q640 437 637 437T631 442L629 445Q629 451 635 490T641 551Q641 586 628 604T573 629Q568 630 515 631Q469 631 457 630T439 622Q438 621 368 343T298 60Q298 48 386 46Q418 46 427 45T436 36Q436 31 433 22Q429 4 424 1L422 0Q419 0 415 0Q410 0 363 1T228 2Q99 2 64 0H49Q43 6 43 9T45 27Q49 40 55 46H83H94Q174 46 189 55Q190 56 191 56Q196 59 201 76T241 233Q258 301 269 344Q339 619 339 625Q339 630 310 630H279Q212 630 191 624Q146 614 121 583T67 467Q60 445 57 441T43 437H40Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D447\" xlink:href=\"#MJX-81-TEX-I-1D447\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>T<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">T<\/script><span>. Then the algorithm solves the decoding problem in average time <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"3.778ex\" height=\"2.262ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 1670 1000\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-82-TEX-I-1D447\" d=\"M40 437Q21 437 21 445Q21 450 37 501T71 602L88 651Q93 669 101 677H569H659Q691 677 697 676T704 667Q704 661 687 553T668 444Q668 437 649 437Q640 437 637 437T631 442L629 445Q629 451 635 490T641 551Q641 586 628 604T573 629Q568 630 515 631Q469 631 457 630T439 622Q438 621 368 343T298 60Q298 48 386 46Q418 46 427 45T436 36Q436 31 433 22Q429 4 424 1L422 0Q419 0 415 0Q410 0 363 1T228 2Q99 2 64 0H49Q43 6 43 9T45 27Q49 40 55 46H83H94Q174 46 189 55Q190 56 191 56Q196 59 201 76T241 233Q258 301 269 344Q339 619 339 625Q339 630 310 630H279Q212 630 191 624Q146 614 121 583T67 467Q60 445 57 441T43 437H40Z\"><\/path><path id=\"MJX-82-TEX-N-2F\" d=\"M423 750Q432 750 438 744T444 730Q444 725 271 248T92 -240Q85 -250 75 -250Q68 -250 62 -245T56 -231Q56 -221 230 257T407 740Q411 750 423 750Z\"><\/path><path id=\"MJX-82-TEX-I-1D700\" d=\"M190 -22Q124 -22 76 11T27 107Q27 174 97 232L107 239L99 248Q76 273 76 304Q76 364 144 408T290 452H302Q360 452 405 421Q428 405 428 392Q428 381 417 369T391 356Q382 356 371 365T338 383T283 392Q217 392 167 368T116 308Q116 289 133 272Q142 263 145 262T157 264Q188 278 238 278H243Q308 278 308 247Q308 206 223 206Q177 206 142 219L132 212Q68 169 68 112Q68 39 201 39Q253 39 286 49T328 72T345 94T362 105Q376 103 376 88Q376 79 365 62T334 26T275 -8T190 -22Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D447\" xlink:href=\"#MJX-82-TEX-I-1D447\"><\/use><\/g><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\" transform=\"translate(704,0)\"><g data-mml-node=\"mo\"><use data-c=\"2F\" xlink:href=\"#MJX-82-TEX-N-2F\"><\/use><\/g><\/g><g data-mml-node=\"mi\" transform=\"translate(1204,0)\"><use data-c=\"1D700\" xlink:href=\"#MJX-82-TEX-I-1D700\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>T<\/mi><mrow data-mjx-texclass=\"ORD\"><mo>\/<\/mo><\/mrow><mi>\u03b5<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">T\/\\varepsilon<\/script><span> (complexity).<\/span><\/p><div contenteditable=\"false\" spellcheck=\"false\" class=\"mathjax-block md-end-block md-math-block md-rawblock\" id=\"mathjax-n169\" cid=\"n169\" mdtype=\"math_block\" data-math-tag-before=\"10\" data-math-tag-after=\"10\" data-math-labels=\"[]\"><div class=\"md-rawblock-container md-math-container\" tabindex=\"-1\"><mjx-container class=\"MathJax\" jax=\"SVG\" display=\"true\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"53.386ex\" height=\"2.262ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 23596.4 1000\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-1987-TEX-I-1D700\" d=\"M190 -22Q124 -22 76 11T27 107Q27 174 97 232L107 239L99 248Q76 273 76 304Q76 364 144 408T290 452H302Q360 452 405 421Q428 405 428 392Q428 381 417 369T391 356Q382 356 371 365T338 383T283 392Q217 392 167 368T116 308Q116 289 133 272Q142 263 145 262T157 264Q188 278 238 278H243Q308 278 308 247Q308 206 223 206Q177 206 142 219L132 212Q68 169 68 112Q68 39 201 39Q253 39 286 49T328 72T345 94T362 105Q376 103 376 88Q376 79 365 62T334 26T275 -8T190 -22Z\"><\/path><path id=\"MJX-1987-TEX-N-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path 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3Q365 3 347 0H338V62H366Q366 63 326 112T285 163L198 63L217 62H235V0H227Z\"><\/path><path id=\"MJX-1987-TEX-N-22BA\" d=\"M318 -182Q302 -212 280 -212H278H275Q249 -212 239 -182L238 84V351H162L87 352Q57 362 57 391T84 429Q89 431 280 431H470L474 429Q477 427 479 426T484 423T490 417T495 410T499 402T500 391Q500 365 470 352L394 351H318V-182Z\"><\/path><path id=\"MJX-1987-TEX-B-1D430\" d=\"M624 444Q636 441 722 441Q797 441 800 444H805V382H741L593 11Q592 10 590 8T586 4T584 2T581 0T579 -2T575 -3T571 -3T567 -4T561 -4T553 -4H542Q525 -4 518 6T490 70Q474 110 463 137L415 257L367 137Q357 111 341 72Q320 17 313 7T289 -4H277Q259 -4 253 -2T238 11L90 382H25V444H32Q47 441 140 441Q243 441 261 444H270V382H222L310 164L382 342L366 382H303V444H310Q322 441 407 441Q508 441 523 444H531V382H506Q481 382 481 380Q482 376 529 259T577 142L674 382H617V444H624Z\"><\/path><path id=\"MJX-1987-TEX-N-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 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549T232 470L222 433Q222 431 272 431H323Q330 424 330 420Q330 398 317 385H210L174 240Q135 80 135 68Q135 26 162 26Q197 26 230 60T283 144Q285 150 288 151T303 153H307Q322 153 322 145Q322 142 319 133Q314 117 301 95T267 48T216 6T155 -11Q125 -11 98 4T59 56Q57 64 57 83V101L92 241Q127 382 128 383Q128 385 77 385H26Z\"><\/path><path id=\"MJX-1987-TEX-N-61\" d=\"M137 305T115 305T78 320T63 359Q63 394 97 421T218 448Q291 448 336 416T396 340Q401 326 401 309T402 194V124Q402 76 407 58T428 40Q443 40 448 56T453 109V145H493V106Q492 66 490 59Q481 29 455 12T400 -6T353 12T329 54V58L327 55Q325 52 322 49T314 40T302 29T287 17T269 6T247 -2T221 -8T190 -11Q130 -11 82 20T34 107Q34 128 41 147T68 188T116 225T194 253T304 268H318V290Q318 324 312 340Q290 411 215 411Q197 411 181 410T156 406T148 403Q170 388 170 359Q170 334 154 320ZM126 106Q126 75 150 51T209 26Q247 26 276 49T315 109Q317 116 318 175Q318 233 317 233Q309 233 296 232T251 223T193 203T147 166T126 106Z\"><\/path><path id=\"MJX-1987-TEX-N-6E\" d=\"M41 46H55Q94 46 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369Q146 341 139 305T131 211Q131 155 138 120T173 59Q203 26 251 26Q322 26 373 103V342Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mtable\"><g data-mml-node=\"mtr\"><g data-mml-node=\"mtd\"><g data-mml-node=\"mi\"><use data-c=\"1D700\" xlink:href=\"#MJX-1987-TEX-I-1D700\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(743.8,0)\"><use data-c=\"3D\" xlink:href=\"#MJX-1987-TEX-N-3D\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1799.6,0)\"><use data-c=\"1D443\" xlink:href=\"#MJX-1987-TEX-I-1D443\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2550.6,0)\"><use data-c=\"28\" xlink:href=\"#MJX-1987-TEX-N-28\"><\/use><\/g><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\" transform=\"translate(2939.6,0)\"><g data-mml-node=\"mi\"><use data-c=\"41\" xlink:href=\"#MJX-1987-TEX-C-41\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(3758.6,0)\"><use data-c=\"28\" xlink:href=\"#MJX-1987-TEX-N-28\"><\/use><\/g><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\" transform=\"translate(4147.6,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D407\" xlink:href=\"#MJX-1987-TEX-B-1D407\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(5047.6,0)\"><use data-c=\"2C\" xlink:href=\"#MJX-1987-TEX-N-2C\"><\/use><\/g><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\" transform=\"translate(5492.2,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D42C\" xlink:href=\"#MJX-1987-TEX-B-1D42C\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(6224,0)\"><use data-c=\"3D\" xlink:href=\"#MJX-1987-TEX-N-3D\"><\/use><\/g><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\" transform=\"translate(7279.8,0)\"><g data-mml-node=\"msup\"><g data-mml-node=\"mi\"><use data-c=\"1D431\" xlink:href=\"#MJX-1987-TEX-B-1D431\"><\/use><use data-c=\"1D407\" xlink:href=\"#MJX-1987-TEX-B-1D407\" 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xlink:href=\"#MJX-1987-TEX-B-1D41E\"><\/use><\/g><\/g><g data-mml-node=\"mtext\" transform=\"translate(12788.2,0)\"><use data-c=\"A0\" xlink:href=\"#MJX-1987-TEX-N-A0\"><\/use><use data-c=\"73\" xlink:href=\"#MJX-1987-TEX-N-73\" transform=\"translate(250,0)\"><\/use><use data-c=\"2E\" xlink:href=\"#MJX-1987-TEX-N-2E\" transform=\"translate(644,0)\"><\/use><use data-c=\"74\" xlink:href=\"#MJX-1987-TEX-N-74\" transform=\"translate(922,0)\"><\/use><use data-c=\"2E\" xlink:href=\"#MJX-1987-TEX-N-2E\" transform=\"translate(1311,0)\"><\/use><use data-c=\"A0\" xlink:href=\"#MJX-1987-TEX-N-A0\" transform=\"translate(1589,0)\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(14627.2,0) translate(0 -0.5)\"><use data-c=\"7C\" xlink:href=\"#MJX-1987-TEX-N-7C\"><\/use><\/g><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\" transform=\"translate(14905.2,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D41E\" xlink:href=\"#MJX-1987-TEX-B-1D41E\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(15432.2,0) translate(0 -0.5)\"><use data-c=\"7C\" xlink:href=\"#MJX-1987-TEX-N-7C\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(15987.9,0)\"><use data-c=\"3D\" xlink:href=\"#MJX-1987-TEX-N-3D\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(17043.7,0)\"><use data-c=\"1D461\" xlink:href=\"#MJX-1987-TEX-I-1D461\"><\/use><\/g><g data-mml-node=\"mtext\" transform=\"translate(17404.7,0)\"><use data-c=\"A0\" xlink:href=\"#MJX-1987-TEX-N-A0\"><\/use><use data-c=\"61\" xlink:href=\"#MJX-1987-TEX-N-61\" transform=\"translate(250,0)\"><\/use><use data-c=\"6E\" xlink:href=\"#MJX-1987-TEX-N-6E\" transform=\"translate(750,0)\"><\/use><use data-c=\"64\" xlink:href=\"#MJX-1987-TEX-N-64\" transform=\"translate(1306,0)\"><\/use><use data-c=\"A0\" xlink:href=\"#MJX-1987-TEX-N-A0\" transform=\"translate(1862,0)\"><\/use><\/g><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\" transform=\"translate(19516.7,0)\"><g data-mml-node=\"msup\"><g data-mml-node=\"mi\"><use data-c=\"1D41E\" xlink:href=\"#MJX-1987-TEX-B-1D41E\"><\/use><use data-c=\"1D407\" xlink:href=\"#MJX-1987-TEX-B-1D407\" transform=\"translate(527,0)\"><\/use><\/g><g data-mml-node=\"TeXAtom\" transform=\"translate(1460,413.1) scale(0.707)\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mo\"><use data-c=\"22BA\" xlink:href=\"#MJX-1987-TEX-N-22BA\"><\/use><\/g><\/g><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(21697.6,0)\"><use data-c=\"3D\" xlink:href=\"#MJX-1987-TEX-N-3D\"><\/use><\/g><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\" transform=\"translate(22753.4,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D42C\" xlink:href=\"#MJX-1987-TEX-B-1D42C\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(23207.4,0)\"><use data-c=\"29\" xlink:href=\"#MJX-1987-TEX-N-29\"><\/use><\/g><\/g><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"block\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable displaystyle=\"true\" columnalign=\"right\" columnspacing=\"\" rowspacing=\"3pt\"><mtr><mtd><mi>\u03b5<\/mi><mo>=<\/mo><mi>P<\/mi><mo stretchy=\"false\">(<\/mo><mrow data-mjx-texclass=\"ORD\"><mi data-mjx-variant=\"-tex-calligraphic\" mathvariant=\"script\">A<\/mi><\/mrow><mo stretchy=\"false\">(<\/mo><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">H<\/mi><\/mrow><mo>,<\/mo><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">s<\/mi><\/mrow><mo>=<\/mo><mrow data-mjx-texclass=\"ORD\"><msup><mi mathvariant=\"bold\">xH<\/mi><mrow data-mjx-texclass=\"ORD\"><mo>\u22ba<\/mo><\/mrow><\/msup><\/mrow><mo>,<\/mo><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">w<\/mi><\/mrow><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">e<\/mi><\/mrow><mtext>&nbsp;s.t.&nbsp;<\/mtext><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">e<\/mi><\/mrow><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mo>=<\/mo><mi>t<\/mi><mtext>&nbsp;and&nbsp;<\/mtext><mrow data-mjx-texclass=\"ORD\"><msup><mi mathvariant=\"bold\">eH<\/mi><mrow data-mjx-texclass=\"ORD\"><mo>\u22ba<\/mo><\/mrow><\/msup><\/mrow><mo>=<\/mo><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">s<\/mi><\/mrow><mo stretchy=\"false\">)<\/mo><\/mtd><\/mtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><p><span>Decoding Problem is a problem parametrized by <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.357ex\" height=\"1.025ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -442 600 453\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.025ex;\"><defs><path id=\"MJX-1143-TEX-I-1D45B\" d=\"M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D45B\" xlink:href=\"#MJX-1143-TEX-I-1D45B\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>n<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">n<\/script><span> and two functions of <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.357ex\" height=\"1.025ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -442 600 453\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.025ex;\"><defs><path id=\"MJX-1143-TEX-I-1D45B\" d=\"M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D45B\" xlink:href=\"#MJX-1143-TEX-I-1D45B\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>n<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">n<\/script><span>: <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"8.402ex\" height=\"2.262ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 3713.6 1000\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-85-TEX-I-1D445\" d=\"M230 637Q203 637 198 638T193 649Q193 676 204 682Q206 683 378 683Q550 682 564 680Q620 672 658 652T712 606T733 563T739 529Q739 484 710 445T643 385T576 351T538 338L545 333Q612 295 612 223Q612 212 607 162T602 80V71Q602 53 603 43T614 25T640 16Q668 16 686 38T712 85Q717 99 720 102T735 105Q755 105 755 93Q755 75 731 36Q693 -21 641 -21H632Q571 -21 531 4T487 82Q487 109 502 166T517 239Q517 290 474 313Q459 320 449 321T378 323H309L277 193Q244 61 244 59Q244 55 245 54T252 50T269 48T302 46H333Q339 38 339 37T336 19Q332 6 326 0H311Q275 2 180 2Q146 2 117 2T71 2T50 1Q33 1 33 10Q33 12 36 24Q41 43 46 45Q50 46 61 46H67Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628Q287 635 230 637ZM630 554Q630 586 609 608T523 636Q521 636 500 636T462 637H440Q393 637 386 627Q385 624 352 494T319 361Q319 360 388 360Q466 361 492 367Q556 377 592 426Q608 449 619 486T630 554Z\"><\/path><path id=\"MJX-85-TEX-N-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path id=\"MJX-85-TEX-I-1D458\" d=\"M121 647Q121 657 125 670T137 683Q138 683 209 688T282 694Q294 694 294 686Q294 679 244 477Q194 279 194 272Q213 282 223 291Q247 309 292 354T362 415Q402 442 438 442Q468 442 485 423T503 369Q503 344 496 327T477 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505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D445\" xlink:href=\"#MJX-85-TEX-I-1D445\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(1036.8,0)\"><use data-c=\"3D\" xlink:href=\"#MJX-85-TEX-N-3D\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(2092.6,0)\"><use data-c=\"1D458\" xlink:href=\"#MJX-85-TEX-I-1D458\"><\/use><\/g><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\" transform=\"translate(2613.6,0)\"><g data-mml-node=\"mo\"><use data-c=\"2F\" xlink:href=\"#MJX-85-TEX-N-2F\"><\/use><\/g><\/g><g data-mml-node=\"mi\" transform=\"translate(3113.6,0)\"><use data-c=\"1D45B\" xlink:href=\"#MJX-85-TEX-I-1D45B\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>R<\/mi><mo>=<\/mo><mi>k<\/mi><mrow data-mjx-texclass=\"ORD\"><mo>\/<\/mo><\/mrow><mi>n<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">R=k\/n<\/script><span> and <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"7.492ex\" height=\"2.262ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 3311.6 1000\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-86-TEX-I-1D70F\" d=\"M39 284Q18 284 18 294Q18 301 45 338T99 398Q134 425 164 429Q170 431 332 431Q492 431 497 429Q517 424 517 402Q517 388 508 376T485 360Q479 358 389 358T299 356Q298 355 283 274T251 109T233 20Q228 5 215 -4T186 -13Q153 -13 153 20V30L203 192Q214 228 227 272T248 336L254 357Q254 358 208 358Q206 358 197 358T183 359Q105 359 61 295Q56 287 53 286T39 284Z\"><\/path><path id=\"MJX-86-TEX-N-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path id=\"MJX-86-TEX-I-1D461\" d=\"M26 385Q19 392 19 395Q19 399 22 411T27 425Q29 430 36 430T87 431H140L159 511Q162 522 166 540T173 566T179 586T187 603T197 615T211 624T229 626Q247 625 254 615T261 596Q261 589 252 549T232 470L222 433Q222 431 272 431H323Q330 424 330 420Q330 398 317 385H210L174 240Q135 80 135 68Q135 26 162 26Q197 26 230 60T283 144Q285 150 288 151T303 153H307Q322 153 322 145Q322 142 319 133Q314 117 301 95T267 48T216 6T155 -11Q125 -11 98 4T59 56Q57 64 57 83V101L92 241Q127 382 128 383Q128 385 77 385H26Z\"><\/path><path id=\"MJX-86-TEX-N-2F\" d=\"M423 750Q432 750 438 744T444 730Q444 725 271 248T92 -240Q85 -250 75 -250Q68 -250 62 -245T56 -231Q56 -221 230 257T407 740Q411 750 423 750Z\"><\/path><path id=\"MJX-86-TEX-I-1D45B\" d=\"M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D70F\" xlink:href=\"#MJX-86-TEX-I-1D70F\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(794.8,0)\"><use data-c=\"3D\" xlink:href=\"#MJX-86-TEX-N-3D\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1850.6,0)\"><use data-c=\"1D461\" xlink:href=\"#MJX-86-TEX-I-1D461\"><\/use><\/g><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\" transform=\"translate(2211.6,0)\"><g data-mml-node=\"mo\"><use data-c=\"2F\" xlink:href=\"#MJX-86-TEX-N-2F\"><\/use><\/g><\/g><g data-mml-node=\"mi\" transform=\"translate(2711.6,0)\"><use data-c=\"1D45B\" xlink:href=\"#MJX-86-TEX-I-1D45B\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>\u03c4<\/mi><mo>=<\/mo><mi>t<\/mi><mrow data-mjx-texclass=\"ORD\"><mo>\/<\/mo><\/mrow><mi>n<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">\\tau=t\/n<\/script><span>, i.e., <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"13.143ex\" height=\"2.262ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 5809 1000\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-1106-TEX-SS-1D5A3\" d=\"M88 0V694H237H258H316Q383 694 425 686T511 648Q578 604 622 525T666 343Q666 190 564 86Q494 18 400 3Q387 1 237 0H88ZM565 341Q565 409 546 463T495 550T429 600T359 621Q348 623 267 623H189V71H267Q272 71 286 71T310 70Q461 70 527 184Q565 251 565 341Z\"><\/path><path id=\"MJX-1106-TEX-SS-1D5AF\" d=\"M88 0V694H230Q347 693 370 692T410 686Q487 667 535 611T583 485Q583 409 527 348T379 276Q369 274 279 274H192V0H88ZM486 485Q486 523 471 551T432 593T391 612T357 621Q350 622 268 623H189V347H268Q350 348 357 349Q370 351 383 354T416 368T450 391T475 429T486 485Z\"><\/path><path id=\"MJX-1106-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 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4T487 82Q487 109 502 166T517 239Q517 290 474 313Q459 320 449 321T378 323H309L277 193Q244 61 244 59Q244 55 245 54T252 50T269 48T302 46H333Q339 38 339 37T336 19Q332 6 326 0H311Q275 2 180 2Q146 2 117 2T71 2T50 1Q33 1 33 10Q33 12 36 24Q41 43 46 45Q50 46 61 46H67Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628Q287 635 230 637ZM630 554Q630 586 609 608T523 636Q521 636 500 636T462 637H440Q393 637 386 627Q385 624 352 494T319 361Q319 360 388 360Q466 361 492 367Q556 377 592 426Q608 449 619 486T630 554Z\"><\/path><path id=\"MJX-1106-TEX-I-1D70F\" d=\"M39 284Q18 284 18 294Q18 301 45 338T99 398Q134 425 164 429Q170 431 332 431Q492 431 497 429Q517 424 517 402Q517 388 508 376T485 360Q479 358 389 358T299 356Q298 355 283 274T251 109T233 20Q228 5 215 -4T186 -13Q153 -13 153 20V30L203 192Q214 228 227 272T248 336L254 357Q254 358 208 358Q206 358 197 358T183 359Q105 359 61 295Q56 287 53 286T39 284Z\"><\/path><path id=\"MJX-1106-TEX-N-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D5A3\" xlink:href=\"#MJX-1106-TEX-SS-1D5A3\"><\/use><use data-c=\"1D5AF\" xlink:href=\"#MJX-1106-TEX-SS-1D5AF\" transform=\"translate(722,0)\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(1361,0)\"><use data-c=\"28\" xlink:href=\"#MJX-1106-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1750,0)\"><use data-c=\"1D45B\" xlink:href=\"#MJX-1106-TEX-I-1D45B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2350,0)\"><use data-c=\"2C\" xlink:href=\"#MJX-1106-TEX-N-2C\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(2794.7,0)\"><use data-c=\"1D45E\" xlink:href=\"#MJX-1106-TEX-I-1D45E\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(3254.7,0)\"><use data-c=\"2C\" xlink:href=\"#MJX-1106-TEX-N-2C\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(3699.3,0)\"><use data-c=\"1D445\" xlink:href=\"#MJX-1106-TEX-I-1D445\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(4458.3,0)\"><use data-c=\"2C\" xlink:href=\"#MJX-1106-TEX-N-2C\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(4903,0)\"><use data-c=\"1D70F\" xlink:href=\"#MJX-1106-TEX-I-1D70F\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(5420,0)\"><use data-c=\"29\" xlink:href=\"#MJX-1106-TEX-N-29\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"sans-serif\">DP<\/mi><\/mrow><mo stretchy=\"false\">(<\/mo><mi>n<\/mi><mo>,<\/mo><mi>q<\/mi><mo>,<\/mo><mi>R<\/mi><mo>,<\/mo><mi>\u03c4<\/mi><mo stretchy=\"false\">)<\/mo><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">\\mathsf{DP}(n,q,R,\\tau)<\/script><span>. <\/span><mark><span>It can be solved in polynomial time as soon as <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"32.983ex\" height=\"3.236ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -948.1 14578.5 1430.3\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -1.091ex;\"><defs><path id=\"MJX-88-TEX-I-1D70F\" d=\"M39 284Q18 284 18 294Q18 301 45 338T99 398Q134 425 164 429Q170 431 332 431Q492 431 497 429Q517 424 517 402Q517 388 508 376T485 360Q479 358 389 358T299 356Q298 355 283 274T251 109T233 20Q228 5 215 -4T186 -13Q153 -13 153 20V30L203 192Q214 228 227 272T248 336L254 357Q254 358 208 358Q206 358 197 358T183 359Q105 359 61 295Q56 287 53 286T39 284Z\"><\/path><path id=\"MJX-88-TEX-N-2208\" d=\"M84 250Q84 372 166 450T360 539Q361 539 377 539T419 540T469 540H568Q583 532 583 520Q583 511 570 501L466 500Q355 499 329 494Q280 482 242 458T183 409T147 354T129 306T124 272V270H568Q583 262 583 250T568 230H124V228Q124 207 134 177T167 112T231 48T328 7Q355 1 466 0H570Q583 -10 583 -20Q583 -32 568 -40H471Q464 -40 446 -40T417 -41Q262 -41 172 45Q84 127 84 250Z\"><\/path><path id=\"MJX-88-TEX-N-5B\" d=\"M118 -250V750H255V710H158V-210H255V-250H118Z\"><\/path><path id=\"MJX-88-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-88-TEX-N-31\" d=\"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 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637H440Q393 637 386 627Q385 624 352 494T319 361Q319 360 388 360Q466 361 492 367Q556 377 592 426Q608 449 619 486T630 554Z\"><\/path><path id=\"MJX-88-TEX-N-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><path id=\"MJX-88-TEX-I-1D45E\" d=\"M33 157Q33 258 109 349T280 441Q340 441 372 389Q373 390 377 395T388 406T404 418Q438 442 450 442Q454 442 457 439T460 434Q460 425 391 149Q320 -135 320 -139Q320 -147 365 -148H390Q396 -156 396 -157T393 -175Q389 -188 383 -194H370Q339 -192 262 -192Q234 -192 211 -192T174 -192T157 -193Q143 -193 143 -185Q143 -182 145 -170Q149 -154 152 -151T172 -148Q220 -148 230 -141Q238 -136 258 -53T279 32Q279 33 272 29Q224 -10 172 -10Q117 -10 75 30T33 157ZM352 326Q329 405 277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q233 26 290 98L298 109L352 326Z\"><\/path><path id=\"MJX-88-TEX-N-2C\" d=\"M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z\"><\/path><path id=\"MJX-88-TEX-N-2B\" d=\"M56 237T56 250T70 270H369V420L370 570Q380 583 389 583Q402 583 409 568V270H707Q722 262 722 250T707 230H409V-68Q401 -82 391 -82H389H387Q375 -82 369 -68V230H70Q56 237 56 250Z\"><\/path><path id=\"MJX-88-TEX-N-5D\" d=\"M22 710V750H159V-250H22V-210H119V710H22Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D70F\" xlink:href=\"#MJX-88-TEX-I-1D70F\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(794.8,0)\"><use data-c=\"2208\" xlink:href=\"#MJX-88-TEX-N-2208\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(1739.6,0)\"><use data-c=\"5B\" xlink:href=\"#MJX-88-TEX-N-5B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2017.6,0)\"><use data-c=\"28\" xlink:href=\"#MJX-88-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(2406.6,0)\"><use data-c=\"31\" xlink:href=\"#MJX-88-TEX-N-31\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(3128.8,0)\"><use data-c=\"2212\" xlink:href=\"#MJX-88-TEX-N-2212\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(4129,0)\"><use data-c=\"1D445\" xlink:href=\"#MJX-88-TEX-I-1D445\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(4888,0)\"><use data-c=\"29\" xlink:href=\"#MJX-88-TEX-N-29\"><\/use><\/g><g data-mml-node=\"mfrac\" transform=\"translate(5277,0)\"><g data-mml-node=\"mrow\" transform=\"translate(220,477.2) scale(0.707)\"><g data-mml-node=\"mi\"><use data-c=\"1D45E\" xlink:href=\"#MJX-88-TEX-I-1D45E\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(460,0)\"><use data-c=\"2212\" xlink:href=\"#MJX-88-TEX-N-2212\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(1238,0)\"><use data-c=\"31\" xlink:href=\"#MJX-88-TEX-N-31\"><\/use><\/g><\/g><g data-mml-node=\"mi\" transform=\"translate(671.8,-345) scale(0.707)\"><use data-c=\"1D45E\" xlink:href=\"#MJX-88-TEX-I-1D45E\"><\/use><\/g><rect width=\"1429\" height=\"60\" x=\"120\" y=\"220\"><\/rect><\/g><g data-mml-node=\"mo\" transform=\"translate(6946,0)\"><use data-c=\"2C\" xlink:href=\"#MJX-88-TEX-N-2C\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(7390.6,0)\"><use data-c=\"1D445\" xlink:href=\"#MJX-88-TEX-I-1D445\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(8371.8,0)\"><use data-c=\"2B\" xlink:href=\"#MJX-88-TEX-N-2B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(9372.1,0)\"><use data-c=\"28\" xlink:href=\"#MJX-88-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(9761.1,0)\"><use data-c=\"31\" xlink:href=\"#MJX-88-TEX-N-31\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(10483.3,0)\"><use data-c=\"2212\" xlink:href=\"#MJX-88-TEX-N-2212\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(11483.5,0)\"><use data-c=\"1D445\" xlink:href=\"#MJX-88-TEX-I-1D445\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(12242.5,0)\"><use data-c=\"29\" xlink:href=\"#MJX-88-TEX-N-29\"><\/use><\/g><g data-mml-node=\"mfrac\" transform=\"translate(12631.5,0)\"><g data-mml-node=\"mrow\" transform=\"translate(220,477.2) scale(0.707)\"><g data-mml-node=\"mi\"><use data-c=\"1D45E\" xlink:href=\"#MJX-88-TEX-I-1D45E\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(460,0)\"><use data-c=\"2212\" xlink:href=\"#MJX-88-TEX-N-2212\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(1238,0)\"><use data-c=\"31\" xlink:href=\"#MJX-88-TEX-N-31\"><\/use><\/g><\/g><g data-mml-node=\"mi\" transform=\"translate(671.8,-345) scale(0.707)\"><use data-c=\"1D45E\" xlink:href=\"#MJX-88-TEX-I-1D45E\"><\/use><\/g><rect width=\"1429\" height=\"60\" x=\"120\" y=\"220\"><\/rect><\/g><g data-mml-node=\"mo\" transform=\"translate(14300.5,0)\"><use data-c=\"5D\" xlink:href=\"#MJX-88-TEX-N-5D\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>\u03c4<\/mi><mo>\u2208<\/mo><mo stretchy=\"false\">[<\/mo><mo stretchy=\"false\">(<\/mo><mn>1<\/mn><mo>\u2212<\/mo><mi>R<\/mi><mo stretchy=\"false\">)<\/mo><mfrac><mrow><mi>q<\/mi><mo>\u2212<\/mo><mn>1<\/mn><\/mrow><mi>q<\/mi><\/mfrac><mo>,<\/mo><mi>R<\/mi><mo>+<\/mo><mo stretchy=\"false\">(<\/mo><mn>1<\/mn><mo>\u2212<\/mo><mi>R<\/mi><mo stretchy=\"false\">)<\/mo><mfrac><mrow><mi>q<\/mi><mo>\u2212<\/mo><mn>1<\/mn><\/mrow><mi>q<\/mi><\/mfrac><mo stretchy=\"false\">]<\/mo><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">\\tau\\in[(1-R)\\frac{q-1}{q},R+(1-R)\\frac{q-1}{q}]<\/script><span>. For other <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.17ex\" height=\"1.005ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -431 517 444\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.029ex;\"><defs><path id=\"MJX-951-TEX-I-1D70F\" d=\"M39 284Q18 284 18 294Q18 301 45 338T99 398Q134 425 164 429Q170 431 332 431Q492 431 497 429Q517 424 517 402Q517 388 508 376T485 360Q479 358 389 358T299 356Q298 355 283 274T251 109T233 20Q228 5 215 -4T186 -13Q153 -13 153 20V30L203 192Q214 228 227 272T248 336L254 357Q254 358 208 358Q206 358 197 358T183 359Q105 359 61 295Q56 287 53 286T39 284Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D70F\" xlink:href=\"#MJX-951-TEX-I-1D70F\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>\u03c4<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">\\tau<\/script><span>, the best algorithms are exponential in <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"2.527ex\" height=\"1.029ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -442 1117 455\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.029ex;\"><defs><path id=\"MJX-90-TEX-I-1D70F\" d=\"M39 284Q18 284 18 294Q18 301 45 338T99 398Q134 425 164 429Q170 431 332 431Q492 431 497 429Q517 424 517 402Q517 388 508 376T485 360Q479 358 389 358T299 356Q298 355 283 274T251 109T233 20Q228 5 215 -4T186 -13Q153 -13 153 20V30L203 192Q214 228 227 272T248 336L254 357Q254 358 208 358Q206 358 197 358T183 359Q105 359 61 295Q56 287 53 286T39 284Z\"><\/path><path id=\"MJX-90-TEX-I-1D45B\" d=\"M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D70F\" xlink:href=\"#MJX-90-TEX-I-1D70F\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(517,0)\"><use data-c=\"1D45B\" xlink:href=\"#MJX-90-TEX-I-1D45B\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>\u03c4<\/mi><mi>n<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">\\tau n<\/script><span>.<\/span><\/mark><span> (proof in 1.3)<\/span><\/p><h2 id='12-random-codes'><span>1.2 Random Codes<\/span><\/h2><p><span>Random codes&#39; parity-check matrix or generator matrix is drawn unifromly at random.<\/span><\/p><p><span>In cryptography, <\/span><strong><span>negligible functions<\/span><\/strong><span> are often used to describe the probability of an event occurring with negligible probability. We say a function <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"4.362ex\" height=\"2.262ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 1928 1000\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-103-TEX-I-1D453\" d=\"M118 -162Q120 -162 124 -164T135 -167T147 -168Q160 -168 171 -155T187 -126Q197 -99 221 27T267 267T289 382V385H242Q195 385 192 387Q188 390 188 397L195 425Q197 430 203 430T250 431Q298 431 298 432Q298 434 307 482T319 540Q356 705 465 705Q502 703 526 683T550 630Q550 594 529 578T487 561Q443 561 443 603Q443 622 454 636T478 657L487 662Q471 668 457 668Q445 668 434 658T419 630Q412 601 403 552T387 469T380 433Q380 431 435 431Q480 431 487 430T498 424Q499 420 496 407T491 391Q489 386 482 386T428 385H372L349 263Q301 15 282 -47Q255 -132 212 -173Q175 -205 139 -205Q107 -205 81 -186T55 -132Q55 -95 76 -78T118 -61Q162 -61 162 -103Q162 -122 151 -136T127 -157L118 -162Z\"><\/path><path id=\"MJX-103-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-103-TEX-I-1D45B\" d=\"M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z\"><\/path><path id=\"MJX-103-TEX-N-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D453\" xlink:href=\"#MJX-103-TEX-I-1D453\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(550,0)\"><use data-c=\"28\" xlink:href=\"#MJX-103-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(939,0)\"><use data-c=\"1D45B\" xlink:href=\"#MJX-103-TEX-I-1D45B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(1539,0)\"><use data-c=\"29\" xlink:href=\"#MJX-103-TEX-N-29\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>f<\/mi><mo stretchy=\"false\">(<\/mo><mi>n<\/mi><mo stretchy=\"false\">)<\/mo><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">f(n)<\/script><span> is negligible, i.e., <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"14.092ex\" height=\"2.262ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 6228.6 1000\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-105-TEX-I-1D453\" d=\"M118 -162Q120 -162 124 -164T135 -167T147 -168Q160 -168 171 -155T187 -126Q197 -99 221 27T267 267T289 382V385H242Q195 385 192 387Q188 390 188 397L195 425Q197 430 203 430T250 431Q298 431 298 432Q298 434 307 482T319 540Q356 705 465 705Q502 703 526 683T550 630Q550 594 529 578T487 561Q443 561 443 603Q443 622 454 636T478 657L487 662Q471 668 457 668Q445 668 434 658T419 630Q412 601 403 552T387 469T380 433Q380 431 435 431Q480 431 487 430T498 424Q499 420 496 407T491 391Q489 386 482 386T428 385H372L349 263Q301 15 282 -47Q255 -132 212 -173Q175 -205 139 -205Q107 -205 81 -186T55 -132Q55 -95 76 -78T118 -61Q162 -61 162 -103Q162 -122 151 -136T127 -157L118 -162Z\"><\/path><path id=\"MJX-105-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-105-TEX-I-1D45B\" d=\"M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z\"><\/path><path id=\"MJX-105-TEX-N-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><path id=\"MJX-105-TEX-N-2208\" d=\"M84 250Q84 372 166 450T360 539Q361 539 377 539T419 540T469 540H568Q583 532 583 520Q583 511 570 501L466 500Q355 499 329 494Q280 482 242 458T183 409T147 354T129 306T124 272V270H568Q583 262 583 250T568 230H124V228Q124 207 134 177T167 112T231 48T328 7Q355 1 466 0H570Q583 -10 583 -20Q583 -32 568 -40H471Q464 -40 446 -40T417 -41Q262 -41 172 45Q84 127 84 250Z\"><\/path><path id=\"MJX-105-TEX-SS-1D5C7\" d=\"M160 392Q214 446 283 454Q285 454 292 454T303 455H306Q417 455 438 354Q442 331 443 164V0H350V157Q349 315 348 320Q334 378 259 378H253Q224 378 204 358Q180 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28 -68T31 -37T40 -12T52 7T64 21T75 31T82 38Q60 68 60 106Q60 145 80 180L86 189L80 199Q55 240 55 286ZM304 233T304 287T279 362T220 383Q189 383 165 361T140 287Q140 243 161 217T220 191Q253 191 278 212ZM250 -134Q298 -134 331 -122T375 -96T387 -69Q387 -21 306 -7Q288 -5 216 -5Q161 -5 153 -7Q146 -9 139 -13T122 -31T113 -66Q113 -75 113 -80T127 -97T166 -121Q203 -134 250 -134Z\"><\/path><path id=\"MJX-105-TEX-SS-1D5C5\" d=\"M74 0V694H164V0H74Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D453\" xlink:href=\"#MJX-105-TEX-I-1D453\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(550,0)\"><use data-c=\"28\" xlink:href=\"#MJX-105-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(939,0)\"><use data-c=\"1D45B\" xlink:href=\"#MJX-105-TEX-I-1D45B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(1539,0)\"><use data-c=\"29\" xlink:href=\"#MJX-105-TEX-N-29\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2205.8,0)\"><use data-c=\"2208\" xlink:href=\"#MJX-105-TEX-N-2208\"><\/use><\/g><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\" transform=\"translate(3150.6,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D5C7\" xlink:href=\"#MJX-105-TEX-SS-1D5C7\"><\/use><use data-c=\"1D5BE\" xlink:href=\"#MJX-105-TEX-SS-1D5BE\" transform=\"translate(517,0)\"><\/use><use data-c=\"1D5C0\" xlink:href=\"#MJX-105-TEX-SS-1D5C0\" transform=\"translate(961,0)\"><\/use><use data-c=\"1D5C5\" xlink:href=\"#MJX-105-TEX-SS-1D5C5\" transform=\"translate(1461,0)\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(4850.6,0)\"><use data-c=\"28\" xlink:href=\"#MJX-105-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(5239.6,0)\"><use data-c=\"1D45B\" xlink:href=\"#MJX-105-TEX-I-1D45B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(5839.6,0)\"><use data-c=\"29\" xlink:href=\"#MJX-105-TEX-N-29\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>f<\/mi><mo stretchy=\"false\">(<\/mo><mi>n<\/mi><mo stretchy=\"false\">)<\/mo><mo>\u2208<\/mo><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"sans-serif\">negl<\/mi><\/mrow><mo stretchy=\"false\">(<\/mo><mi>n<\/mi><mo stretchy=\"false\">)<\/mo><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">f(n)\\in\\mathsf{negl}(n)<\/script><span>, if for all polynomial <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"4.256ex\" height=\"2.262ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 1881 1000\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-113-TEX-I-1D45D\" d=\"M23 287Q24 290 25 295T30 317T40 348T55 381T75 411T101 433T134 442Q209 442 230 378L240 387Q302 442 358 442Q423 442 460 395T497 281Q497 173 421 82T249 -10Q227 -10 210 -4Q199 1 187 11T168 28L161 36Q160 35 139 -51T118 -138Q118 -144 126 -145T163 -148H188Q194 -155 194 -157T191 -175Q188 -187 185 -190T172 -194Q170 -194 161 -194T127 -193T65 -192Q-5 -192 -24 -194H-32Q-39 -187 -39 -183Q-37 -156 -26 -148H-6Q28 -147 33 -136Q36 -130 94 103T155 350Q156 355 156 364Q156 405 131 405Q109 405 94 377T71 316T59 280Q57 278 43 278H29Q23 284 23 287ZM178 102Q200 26 252 26Q282 26 310 49T356 107Q374 141 392 215T411 325V331Q411 405 350 405Q339 405 328 402T306 393T286 380T269 365T254 350T243 336T235 326L232 322Q232 321 229 308T218 264T204 212Q178 106 178 102Z\"><\/path><path id=\"MJX-113-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-113-TEX-I-1D45B\" d=\"M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z\"><\/path><path id=\"MJX-113-TEX-N-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D45D\" xlink:href=\"#MJX-113-TEX-I-1D45D\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(503,0)\"><use data-c=\"28\" xlink:href=\"#MJX-113-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(892,0)\"><use data-c=\"1D45B\" xlink:href=\"#MJX-113-TEX-I-1D45B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(1492,0)\"><use data-c=\"29\" xlink:href=\"#MJX-113-TEX-N-29\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>p<\/mi><mo stretchy=\"false\">(<\/mo><mi>n<\/mi><mo stretchy=\"false\">)<\/mo><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">p(n)<\/script><span>, <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"16.413ex\" height=\"2.262ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 7254.6 1000\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-109-TEX-N-7C\" d=\"M139 -249H137Q125 -249 119 -235V251L120 737Q130 750 139 750Q152 750 159 735V-235Q151 -249 141 -249H139Z\"><\/path><path id=\"MJX-109-TEX-I-1D453\" d=\"M118 -162Q120 -162 124 -164T135 -167T147 -168Q160 -168 171 -155T187 -126Q197 -99 221 27T267 267T289 382V385H242Q195 385 192 387Q188 390 188 397L195 425Q197 430 203 430T250 431Q298 431 298 432Q298 434 307 482T319 540Q356 705 465 705Q502 703 526 683T550 630Q550 594 529 578T487 561Q443 561 443 603Q443 622 454 636T478 657L487 662Q471 668 457 668Q445 668 434 658T419 630Q412 601 403 552T387 469T380 433Q380 431 435 431Q480 431 487 430T498 424Q499 420 496 407T491 391Q489 386 482 386T428 385H372L349 263Q301 15 282 -47Q255 -132 212 -173Q175 -205 139 -205Q107 -205 81 -186T55 -132Q55 -95 76 -78T118 -61Q162 -61 162 -103Q162 -122 151 -136T127 -157L118 -162Z\"><\/path><path id=\"MJX-109-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-109-TEX-I-1D45B\" d=\"M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z\"><\/path><path id=\"MJX-109-TEX-N-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 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transform=\"translate(278,0)\"><use data-c=\"1D453\" xlink:href=\"#MJX-109-TEX-I-1D453\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(828,0)\"><use data-c=\"28\" xlink:href=\"#MJX-109-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1217,0)\"><use data-c=\"1D45B\" xlink:href=\"#MJX-109-TEX-I-1D45B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(1817,0)\"><use data-c=\"29\" xlink:href=\"#MJX-109-TEX-N-29\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2206,0) translate(0 -0.5)\"><use data-c=\"7C\" xlink:href=\"#MJX-109-TEX-N-7C\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2761.8,0)\"><use data-c=\"3C\" xlink:href=\"#MJX-109-TEX-N-3C\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(3817.6,0)\"><use data-c=\"31\" xlink:href=\"#MJX-109-TEX-N-31\"><\/use><\/g><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\" transform=\"translate(4317.6,0)\"><g data-mml-node=\"mo\"><use data-c=\"2F\" xlink:href=\"#MJX-109-TEX-N-2F\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(4817.6,0) translate(0 -0.5)\"><use data-c=\"7C\" xlink:href=\"#MJX-109-TEX-N-7C\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(5095.6,0)\"><use data-c=\"1D45D\" xlink:href=\"#MJX-109-TEX-I-1D45D\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(5598.6,0)\"><use data-c=\"28\" xlink:href=\"#MJX-109-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(5987.6,0)\"><use data-c=\"1D45B\" xlink:href=\"#MJX-109-TEX-I-1D45B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(6587.6,0)\"><use data-c=\"29\" xlink:href=\"#MJX-109-TEX-N-29\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(6976.6,0) translate(0 -0.5)\"><use data-c=\"7C\" xlink:href=\"#MJX-109-TEX-N-7C\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mi>f<\/mi><mo stretchy=\"false\">(<\/mo><mi>n<\/mi><mo stretchy=\"false\">)<\/mo><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mo>&lt;<\/mo><mn>1<\/mn><mrow data-mjx-texclass=\"ORD\"><mo>\/<\/mo><\/mrow><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mi>p<\/mi><mo stretchy=\"false\">(<\/mo><mi>n<\/mi><mo stretchy=\"false\">)<\/mo><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">|f(n)|<1\/|p(n)|<\/script><span> for all sufficiently large <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.357ex\" height=\"1.025ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -442 600 453\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.025ex;\"><defs><path id=\"MJX-1143-TEX-I-1D45B\" d=\"M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D45B\" xlink:href=\"#MJX-1143-TEX-I-1D45B\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>n<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">n<\/script><span>.<\/span><\/p><blockquote><p><span>The introduction of the function <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"4.256ex\" height=\"2.262ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 1881 1000\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-113-TEX-I-1D45D\" d=\"M23 287Q24 290 25 295T30 317T40 348T55 381T75 411T101 433T134 442Q209 442 230 378L240 387Q302 442 358 442Q423 442 460 395T497 281Q497 173 421 82T249 -10Q227 -10 210 -4Q199 1 187 11T168 28L161 36Q160 35 139 -51T118 -138Q118 -144 126 -145T163 -148H188Q194 -155 194 -157T191 -175Q188 -187 185 -190T172 -194Q170 -194 161 -194T127 -193T65 -192Q-5 -192 -24 -194H-32Q-39 -187 -39 -183Q-37 -156 -26 -148H-6Q28 -147 33 -136Q36 -130 94 103T155 350Q156 355 156 364Q156 405 131 405Q109 405 94 377T71 316T59 280Q57 278 43 278H29Q23 284 23 287ZM178 102Q200 26 252 26Q282 26 310 49T356 107Q374 141 392 215T411 325V331Q411 405 350 405Q339 405 328 402T306 393T286 380T269 365T254 350T243 336T235 326L232 322Q232 321 229 308T218 264T204 212Q178 106 178 102Z\"><\/path><path id=\"MJX-113-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-113-TEX-I-1D45B\" d=\"M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z\"><\/path><path id=\"MJX-113-TEX-N-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D45D\" xlink:href=\"#MJX-113-TEX-I-1D45D\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(503,0)\"><use data-c=\"28\" xlink:href=\"#MJX-113-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(892,0)\"><use data-c=\"1D45B\" xlink:href=\"#MJX-113-TEX-I-1D45B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(1492,0)\"><use data-c=\"29\" xlink:href=\"#MJX-113-TEX-N-29\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>p<\/mi><mo stretchy=\"false\">(<\/mo><mi>n<\/mi><mo stretchy=\"false\">)<\/mo><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">p(n)<\/script><span> in the definition of negligible functions serves to formalize the idea that the function being considered becomes increasingly smaller as the input size <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.357ex\" height=\"1.025ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -442 600 453\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.025ex;\"><defs><path id=\"MJX-1143-TEX-I-1D45B\" d=\"M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D45B\" xlink:href=\"#MJX-1143-TEX-I-1D45B\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>n<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">n<\/script><span> grows. It helps establish a <\/span><u><span>quantitative measure<\/span><\/u><span> for the rate at which the function approaches zero.<\/span><\/p><p><span>The use of <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"4.256ex\" height=\"2.262ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 1881 1000\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-113-TEX-I-1D45D\" d=\"M23 287Q24 290 25 295T30 317T40 348T55 381T75 411T101 433T134 442Q209 442 230 378L240 387Q302 442 358 442Q423 442 460 395T497 281Q497 173 421 82T249 -10Q227 -10 210 -4Q199 1 187 11T168 28L161 36Q160 35 139 -51T118 -138Q118 -144 126 -145T163 -148H188Q194 -155 194 -157T191 -175Q188 -187 185 -190T172 -194Q170 -194 161 -194T127 -193T65 -192Q-5 -192 -24 -194H-32Q-39 -187 -39 -183Q-37 -156 -26 -148H-6Q28 -147 33 -136Q36 -130 94 103T155 350Q156 355 156 364Q156 405 131 405Q109 405 94 377T71 316T59 280Q57 278 43 278H29Q23 284 23 287ZM178 102Q200 26 252 26Q282 26 310 49T356 107Q374 141 392 215T411 325V331Q411 405 350 405Q339 405 328 402T306 393T286 380T269 365T254 350T243 336T235 326L232 322Q232 321 229 308T218 264T204 212Q178 106 178 102Z\"><\/path><path id=\"MJX-113-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-113-TEX-I-1D45B\" d=\"M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z\"><\/path><path id=\"MJX-113-TEX-N-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D45D\" xlink:href=\"#MJX-113-TEX-I-1D45D\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(503,0)\"><use data-c=\"28\" xlink:href=\"#MJX-113-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(892,0)\"><use data-c=\"1D45B\" xlink:href=\"#MJX-113-TEX-I-1D45B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(1492,0)\"><use data-c=\"29\" xlink:href=\"#MJX-113-TEX-N-29\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>p<\/mi><mo stretchy=\"false\">(<\/mo><mi>n<\/mi><mo stretchy=\"false\">)<\/mo><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">p(n)<\/script><span> allows for generality in expressing the notion of negligibility. Different contexts and applications may require different rates of decay for a function to be considered negligible. The choice of <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"4.256ex\" height=\"2.262ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 1881 1000\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-113-TEX-I-1D45D\" d=\"M23 287Q24 290 25 295T30 317T40 348T55 381T75 411T101 433T134 442Q209 442 230 378L240 387Q302 442 358 442Q423 442 460 395T497 281Q497 173 421 82T249 -10Q227 -10 210 -4Q199 1 187 11T168 28L161 36Q160 35 139 -51T118 -138Q118 -144 126 -145T163 -148H188Q194 -155 194 -157T191 -175Q188 -187 185 -190T172 -194Q170 -194 161 -194T127 -193T65 -192Q-5 -192 -24 -194H-32Q-39 -187 -39 -183Q-37 -156 -26 -148H-6Q28 -147 33 -136Q36 -130 94 103T155 350Q156 355 156 364Q156 405 131 405Q109 405 94 377T71 316T59 280Q57 278 43 278H29Q23 284 23 287ZM178 102Q200 26 252 26Q282 26 310 49T356 107Q374 141 392 215T411 325V331Q411 405 350 405Q339 405 328 402T306 393T286 380T269 365T254 350T243 336T235 326L232 322Q232 321 229 308T218 264T204 212Q178 106 178 102Z\"><\/path><path id=\"MJX-113-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-113-TEX-I-1D45B\" d=\"M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z\"><\/path><path id=\"MJX-113-TEX-N-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D45D\" xlink:href=\"#MJX-113-TEX-I-1D45D\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(503,0)\"><use data-c=\"28\" xlink:href=\"#MJX-113-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(892,0)\"><use data-c=\"1D45B\" xlink:href=\"#MJX-113-TEX-I-1D45B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(1492,0)\"><use data-c=\"29\" xlink:href=\"#MJX-113-TEX-N-29\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>p<\/mi><mo stretchy=\"false\">(<\/mo><mi>n<\/mi><mo stretchy=\"false\">)<\/mo><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">p(n)<\/script><span> provides a flexible framework for <\/span><u><span>capturing these variations<\/span><\/u><span>.<\/span><\/p><\/blockquote><p><span>As all the parameters are functions of <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.357ex\" height=\"1.025ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -442 600 453\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.025ex;\"><defs><path id=\"MJX-1143-TEX-I-1D45B\" d=\"M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D45B\" xlink:href=\"#MJX-1143-TEX-I-1D45B\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>n<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">n<\/script><span>, our <\/span><strong><span>asymptotic<\/span><\/strong><span> results will always hold for <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"8.899ex\" height=\"1.505ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -583 3933.6 665\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.186ex;\"><defs><path id=\"MJX-116-TEX-I-1D45B\" d=\"M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z\"><\/path><path id=\"MJX-116-TEX-N-2192\" d=\"M56 237T56 250T70 270H835Q719 357 692 493Q692 494 692 496T691 499Q691 511 708 511H711Q720 511 723 510T729 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214Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D45B\" xlink:href=\"#MJX-116-TEX-I-1D45B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(877.8,0)\"><use data-c=\"2192\" xlink:href=\"#MJX-116-TEX-N-2192\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2155.6,0)\"><use data-c=\"2B\" xlink:href=\"#MJX-116-TEX-N-2B\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(2933.6,0)\"><use data-c=\"221E\" xlink:href=\"#MJX-116-TEX-N-221E\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>n<\/mi><mo stretchy=\"false\">\u2192<\/mo><mo>+<\/mo><mi mathvariant=\"normal\">\u221e<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">n\\rightarrow+\\infin<\/script><span>.<\/span><\/p><p><span>The <\/span><strong><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.041ex\" height=\"1.439ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -442 460 636\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.439ex;\"><defs><path id=\"MJX-713-TEX-I-1D45E\" d=\"M33 157Q33 258 109 349T280 441Q340 441 372 389Q373 390 377 395T388 406T404 418Q438 442 450 442Q454 442 457 439T460 434Q460 425 391 149Q320 -135 320 -139Q320 -147 365 -148H390Q396 -156 396 -157T393 -175Q389 -188 383 -194H370Q339 -192 262 -192Q234 -192 211 -192T174 -192T157 -193Q143 -193 143 -185Q143 -182 145 -170Q149 -154 152 -151T172 -148Q220 -148 230 -141Q238 -136 258 -53T279 32Q279 33 272 29Q224 -10 172 -10Q117 -10 75 30T33 157ZM352 326Q329 405 277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q233 26 290 98L298 109L352 326Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D45E\" xlink:href=\"#MJX-713-TEX-I-1D45E\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>q<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">q<\/script><span>-ary entropy<\/span><\/strong><span> is defined as<\/span><\/p><div contenteditable=\"false\" spellcheck=\"false\" class=\"mathjax-block md-end-block md-math-block md-rawblock\" id=\"mathjax-n192\" cid=\"n192\" mdtype=\"math_block\" data-math-tag-before=\"10\" data-math-labels=\"[]\" data-math-tag-after=\"10\"><div class=\"md-rawblock-container md-math-container\" contenteditable=\"false\" tabindex=\"-1\"><mjx-container class=\"MathJax\" jax=\"SVG\" display=\"true\" width=\"full\" style=\"min-width: 56.328ex; position: relative;\"><svg 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transform=\"translate(22907.9,0)\"><use data-c=\"29\" xlink:href=\"#MJX-1988-TEX-N-29\"><\/use><\/g><\/g><\/g><\/g><\/svg><svg data-labels=\"true\" preserveAspectRatio=\"xMaxYMid\" viewBox=\"0 -1249 1 1998\"><g data-labels=\"true\" transform=\"matrix(1 0 0 -1 0 0)\"><g data-mml-node=\"mtd\" id=\"mjx-eqn:\" transform=\"translate(0,131)\"><text data-id-align=\"true\"><\/text><g data-idbox=\"true\" transform=\"translate(0,0)\"><g data-mml-node=\"mrow\"><\/g><\/g><\/g><\/g><\/svg><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"block\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable displaystyle=\"true\"><mlabeledtr><mtd><mrow><\/mrow><\/mtd><mtd><msub><mi>h<\/mi><mi>q<\/mi><\/msub><mo>:<\/mo><mi>x<\/mi><mo>\u2208<\/mo><mo stretchy=\"false\">[<\/mo><mn>0<\/mn><mo>,<\/mo><mn>1<\/mn><mo stretchy=\"false\">]<\/mo><mo stretchy=\"false\">\u2192<\/mo><mo>\u2212<\/mo><mi>x<\/mi><msub><mi>log<\/mi><mi>q<\/mi><\/msub><mo data-mjx-texclass=\"NONE\">\u2061<\/mo><mrow data-mjx-texclass=\"ORD\"><mo minsize=\"1.2em\" maxsize=\"1.2em\">(<\/mo><\/mrow><mfrac><mi>x<\/mi><mrow><mi>q<\/mi><mo>\u2212<\/mo><mn>1<\/mn><\/mrow><\/mfrac><mrow data-mjx-texclass=\"ORD\"><mo minsize=\"1.2em\" maxsize=\"1.2em\">)<\/mo><\/mrow><mo>\u2212<\/mo><mo stretchy=\"false\">(<\/mo><mn>1<\/mn><mo>\u2212<\/mo><mi>x<\/mi><mo stretchy=\"false\">)<\/mo><msub><mi>log<\/mi><mi>q<\/mi><\/msub><mo data-mjx-texclass=\"NONE\">\u2061<\/mo><mo stretchy=\"false\">(<\/mo><mn>1<\/mn><mo>\u2212<\/mo><mi>x<\/mi><mo stretchy=\"false\">)<\/mo><\/mtd><\/mlabeledtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><p><span>It is equal to the entropy of a random variable <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.054ex\" height=\"1.025ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -442 466 453\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.025ex;\"><defs><path id=\"MJX-209-TEX-I-1D452\" d=\"M39 168Q39 225 58 272T107 350T174 402T244 433T307 442H310Q355 442 388 420T421 355Q421 265 310 237Q261 224 176 223Q139 223 138 221Q138 219 132 186T125 128Q125 81 146 54T209 26T302 45T394 111Q403 121 406 121Q410 121 419 112T429 98T420 82T390 55T344 24T281 -1T205 -11Q126 -11 83 42T39 168ZM373 353Q367 405 305 405Q272 405 244 391T199 357T170 316T154 280T149 261Q149 260 169 260Q282 260 327 284T373 353Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D452\" xlink:href=\"#MJX-209-TEX-I-1D452\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>e<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">e<\/script><span> over <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"2.306ex\" height=\"2.195ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -683 1019.3 970.2\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.65ex;\"><defs><path id=\"MJX-210-TEX-D-1D53D\" d=\"M584 499Q569 490 566 490Q558 490 552 497T546 515Q546 535 533 559Q526 574 506 593T469 621Q415 648 326 648Q293 648 287 647T275 641Q264 630 263 617Q262 609 260 492V370L275 372Q323 376 350 392T393 441Q409 473 409 506Q409 529 427 529Q437 529 442 519Q444 511 444 362Q444 212 442 206Q436 197 426 197Q409 197 409 217Q409 265 375 299Q346 328 280 335H260V206Q260 70 262 63Q265 46 276 41T326 35Q362 35 366 28Q377 17 366 3L360 -1H24Q12 5 12 16Q12 35 51 35Q92 38 97 52Q102 60 102 341T97 632Q91 645 51 648Q12 648 12 666Q12 675 24 683H573Q576 678 584 670V499ZM137 341Q137 131 136 89T130 37Q129 36 129 35H182Q233 35 233 39Q226 54 225 92T224 346L226 623L231 635L235 648H129Q132 641 133 638T135 603T137 517T137 341ZM549 603V648H495L506 641Q531 621 533 619L549 603ZM409 317V395L400 386Q390 376 375 366L357 355L373 346Q394 331 397 328L409 317Z\"><\/path><path id=\"MJX-210-TEX-I-1D45E\" d=\"M33 157Q33 258 109 349T280 441Q340 441 372 389Q373 390 377 395T388 406T404 418Q438 442 450 442Q454 442 457 439T460 434Q460 425 391 149Q320 -135 320 -139Q320 -147 365 -148H390Q396 -156 396 -157T393 -175Q389 -188 383 -194H370Q339 -192 262 -192Q234 -192 211 -192T174 -192T157 -193Q143 -193 143 -185Q143 -182 145 -170Q149 -154 152 -151T172 -148Q220 -148 230 -141Q238 -136 258 -53T279 32Q279 33 272 29Q224 -10 172 -10Q117 -10 75 30T33 157ZM352 326Q329 405 277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q233 26 290 98L298 109L352 326Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"msub\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D53D\" xlink:href=\"#MJX-210-TEX-D-1D53D\"><\/use><\/g><\/g><g data-mml-node=\"mi\" transform=\"translate(644,-150) scale(0.707)\"><use data-c=\"1D45E\" xlink:href=\"#MJX-210-TEX-I-1D45E\"><\/use><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><msub><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"double-struck\">F<\/mi><\/mrow><mi>q<\/mi><\/msub><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">\\mathbb{F}_q<\/script><span> distributed like the error for a <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.041ex\" height=\"1.439ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -442 460 636\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.439ex;\"><defs><path id=\"MJX-713-TEX-I-1D45E\" d=\"M33 157Q33 258 109 349T280 441Q340 441 372 389Q373 390 377 395T388 406T404 418Q438 442 450 442Q454 442 457 439T460 434Q460 425 391 149Q320 -135 320 -139Q320 -147 365 -148H390Q396 -156 396 -157T393 -175Q389 -188 383 -194H370Q339 -192 262 -192Q234 -192 211 -192T174 -192T157 -193Q143 -193 143 -185Q143 -182 145 -170Q149 -154 152 -151T172 -148Q220 -148 230 -141Q238 -136 258 -53T279 32Q279 33 272 29Q224 -10 172 -10Q117 -10 75 30T33 157ZM352 326Q329 405 277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q233 26 290 98L298 109L352 326Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D45E\" xlink:href=\"#MJX-713-TEX-I-1D45E\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>q<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">q<\/script><span>-ary symmetric channel of crossprobability <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.294ex\" height=\"1.025ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -442 572 453\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.025ex;\"><defs><path id=\"MJX-212-TEX-I-1D465\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D465\" xlink:href=\"#MJX-212-TEX-I-1D465\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>x<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">x<\/script><span>.<\/span><\/p><p><span>It can be verified that <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"2.227ex\" height=\"2.22ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -694 984.3 981.2\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.65ex;\"><defs><path id=\"MJX-1096-TEX-I-210E\" d=\"M137 683Q138 683 209 688T282 694Q294 694 294 685Q294 674 258 534Q220 386 220 383Q220 381 227 388Q288 442 357 442Q411 442 444 415T478 336Q478 285 440 178T402 50Q403 36 407 31T422 26Q450 26 474 56T513 138Q516 149 519 151T535 153Q555 153 555 145Q555 144 551 130Q535 71 500 33Q466 -10 419 -10H414Q367 -10 346 17T325 74Q325 90 361 192T398 345Q398 404 354 404H349Q266 404 205 306L198 293L164 158Q132 28 127 16Q114 -11 83 -11Q69 -11 59 -2T48 16Q48 30 121 320L195 616Q195 629 188 632T149 637H128Q122 643 122 645T124 664Q129 683 137 683Z\"><\/path><path id=\"MJX-1096-TEX-I-1D45E\" d=\"M33 157Q33 258 109 349T280 441Q340 441 372 389Q373 390 377 395T388 406T404 418Q438 442 450 442Q454 442 457 439T460 434Q460 425 391 149Q320 -135 320 -139Q320 -147 365 -148H390Q396 -156 396 -157T393 -175Q389 -188 383 -194H370Q339 -192 262 -192Q234 -192 211 -192T174 -192T157 -193Q143 -193 143 -185Q143 -182 145 -170Q149 -154 152 -151T172 -148Q220 -148 230 -141Q238 -136 258 -53T279 32Q279 33 272 29Q224 -10 172 -10Q117 -10 75 30T33 157ZM352 326Q329 405 277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q233 26 290 98L298 109L352 326Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"msub\"><g data-mml-node=\"mi\"><use data-c=\"210E\" xlink:href=\"#MJX-1096-TEX-I-210E\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(609,-150) scale(0.707)\"><use data-c=\"1D45E\" xlink:href=\"#MJX-1096-TEX-I-1D45E\"><\/use><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><msub><mi>h<\/mi><mi>q<\/mi><\/msub><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">h_q<\/script><span> is an increasing function over <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"7.8ex\" height=\"3.236ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -948.1 3447.6 1430.3\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -1.091ex;\"><defs><path id=\"MJX-1101-TEX-SO-5B\" d=\"M202 -349V850H394V810H242V-309H394V-349H202Z\"><\/path><path id=\"MJX-1101-TEX-N-30\" d=\"M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z\"><\/path><path id=\"MJX-1101-TEX-N-2C\" d=\"M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z\"><\/path><path id=\"MJX-1101-TEX-I-1D45E\" d=\"M33 157Q33 258 109 349T280 441Q340 441 372 389Q373 390 377 395T388 406T404 418Q438 442 450 442Q454 442 457 439T460 434Q460 425 391 149Q320 -135 320 -139Q320 -147 365 -148H390Q396 -156 396 -157T393 -175Q389 -188 383 -194H370Q339 -192 262 -192Q234 -192 211 -192T174 -192T157 -193Q143 -193 143 -185Q143 -182 145 -170Q149 -154 152 -151T172 -148Q220 -148 230 -141Q238 -136 258 -53T279 32Q279 33 272 29Q224 -10 172 -10Q117 -10 75 30T33 157ZM352 326Q329 405 277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q233 26 290 98L298 109L352 326Z\"><\/path><path id=\"MJX-1101-TEX-N-2212\" d=\"M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z\"><\/path><path id=\"MJX-1101-TEX-N-31\" d=\"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z\"><\/path><path id=\"MJX-1101-TEX-SO-5D\" d=\"M22 810V850H214V-349H22V-309H174V810H22Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mo\" transform=\"translate(0 -0.5)\"><use data-c=\"5B\" xlink:href=\"#MJX-1101-TEX-SO-5B\"><\/use><\/g><\/g><g data-mml-node=\"mn\" transform=\"translate(417,0)\"><use data-c=\"30\" xlink:href=\"#MJX-1101-TEX-N-30\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(917,0)\"><use data-c=\"2C\" xlink:href=\"#MJX-1101-TEX-N-2C\"><\/use><\/g><g data-mml-node=\"mfrac\" transform=\"translate(1361.7,0)\"><g data-mml-node=\"mrow\" transform=\"translate(220,477.2) scale(0.707)\"><g data-mml-node=\"mi\"><use data-c=\"1D45E\" xlink:href=\"#MJX-1101-TEX-I-1D45E\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(460,0)\"><use data-c=\"2212\" xlink:href=\"#MJX-1101-TEX-N-2212\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(1238,0)\"><use data-c=\"31\" xlink:href=\"#MJX-1101-TEX-N-31\"><\/use><\/g><\/g><g data-mml-node=\"mi\" transform=\"translate(671.8,-345) scale(0.707)\"><use data-c=\"1D45E\" xlink:href=\"#MJX-1101-TEX-I-1D45E\"><\/use><\/g><rect width=\"1429\" height=\"60\" x=\"120\" y=\"220\"><\/rect><\/g><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\" transform=\"translate(3030.6,0)\"><g data-mml-node=\"mo\" transform=\"translate(0 -0.5)\"><use data-c=\"5D\" xlink:href=\"#MJX-1101-TEX-SO-5D\"><\/use><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow data-mjx-texclass=\"ORD\"><mo minsize=\"1.2em\" maxsize=\"1.2em\">[<\/mo><\/mrow><mn>0<\/mn><mo>,<\/mo><mfrac><mrow><mi>q<\/mi><mo>\u2212<\/mo><mn>1<\/mn><\/mrow><mi>q<\/mi><\/mfrac><mrow data-mjx-texclass=\"ORD\"><mo minsize=\"1.2em\" maxsize=\"1.2em\">]<\/mo><\/mrow><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">\\big[0,\\frac{q-1}{q}\\big]<\/script><span> and a decreasing function over <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"7.8ex\" height=\"3.236ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -948.1 3447.6 1430.3\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -1.091ex;\"><defs><path id=\"MJX-1102-TEX-SO-5B\" d=\"M202 -349V850H394V810H242V-309H394V-349H202Z\"><\/path><path id=\"MJX-1102-TEX-I-1D45E\" d=\"M33 157Q33 258 109 349T280 441Q340 441 372 389Q373 390 377 395T388 406T404 418Q438 442 450 442Q454 442 457 439T460 434Q460 425 391 149Q320 -135 320 -139Q320 -147 365 -148H390Q396 -156 396 -157T393 -175Q389 -188 383 -194H370Q339 -192 262 -192Q234 -192 211 -192T174 -192T157 -193Q143 -193 143 -185Q143 -182 145 -170Q149 -154 152 -151T172 -148Q220 -148 230 -141Q238 -136 258 -53T279 32Q279 33 272 29Q224 -10 172 -10Q117 -10 75 30T33 157ZM352 326Q329 405 277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q233 26 290 98L298 109L352 326Z\"><\/path><path id=\"MJX-1102-TEX-N-2212\" d=\"M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z\"><\/path><path id=\"MJX-1102-TEX-N-31\" d=\"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z\"><\/path><path id=\"MJX-1102-TEX-N-2C\" d=\"M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z\"><\/path><path 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xlink:href=\"#MJX-1102-TEX-I-1D45E\"><\/use><\/g><rect width=\"1429\" height=\"60\" x=\"120\" y=\"220\"><\/rect><\/g><g data-mml-node=\"mo\" transform=\"translate(2086,0)\"><use data-c=\"2C\" xlink:href=\"#MJX-1102-TEX-N-2C\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(2530.6,0)\"><use data-c=\"31\" xlink:href=\"#MJX-1102-TEX-N-31\"><\/use><\/g><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\" transform=\"translate(3030.6,0)\"><g data-mml-node=\"mo\" transform=\"translate(0 -0.5)\"><use data-c=\"5D\" xlink:href=\"#MJX-1102-TEX-SO-5D\"><\/use><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow data-mjx-texclass=\"ORD\"><mo minsize=\"1.2em\" maxsize=\"1.2em\">[<\/mo><\/mrow><mfrac><mrow><mi>q<\/mi><mo>\u2212<\/mo><mn>1<\/mn><\/mrow><mi>q<\/mi><\/mfrac><mo>,<\/mo><mn>1<\/mn><mrow data-mjx-texclass=\"ORD\"><mo minsize=\"1.2em\" maxsize=\"1.2em\">]<\/mo><\/mrow><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">\\big[\\frac{q-1}{q},1\\big]<\/script><span>.<\/span><\/p><p><span>Let <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"2.136ex\" height=\"1.952ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -705 944.3 862.8\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.357ex;\"><defs><path id=\"MJX-242-TEX-C-53\" d=\"M554 512Q536 512 536 522Q536 525 539 539T542 564Q542 588 528 604Q515 616 482 625T410 635Q374 635 349 624T312 594T295 561T290 532Q290 505 303 482T342 442T378 419T409 404Q435 391 451 383T494 357T535 323T562 282T574 231Q574 133 464 56T220 -22Q138 -22 78 21T18 123Q18 184 61 227T156 274Q178 274 178 263Q178 260 177 258Q172 247 164 239T151 227T136 218L127 213L124 202Q118 186 118 163Q120 124 165 86T292 48Q374 48 423 86T473 186V193Q473 267 347 327Q268 364 239 389Q191 431 191 486Q191 547 242 600T356 679T470 705Q472 705 478 705T489 704Q551 704 596 682T642 610Q642 566 621 545Q592 516 554 512Z\"><\/path><path id=\"MJX-242-TEX-I-1D461\" d=\"M26 385Q19 392 19 395Q19 399 22 411T27 425Q29 430 36 430T87 431H140L159 511Q162 522 166 540T173 566T179 586T187 603T197 615T211 624T229 626Q247 625 254 615T261 596Q261 589 252 549T232 470L222 433Q222 431 272 431H323Q330 424 330 420Q330 398 317 385H210L174 240Q135 80 135 68Q135 26 162 26Q197 26 230 60T283 144Q285 150 288 151T303 153H307Q322 153 322 145Q322 142 319 133Q314 117 301 95T267 48T216 6T155 -11Q125 -11 98 4T59 56Q57 64 57 83V101L92 241Q127 382 128 383Q128 385 77 385H26Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"msub\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"53\" xlink:href=\"#MJX-242-TEX-C-53\"><\/use><\/g><\/g><g data-mml-node=\"mi\" transform=\"translate(639,-150) scale(0.707)\"><use data-c=\"1D461\" xlink:href=\"#MJX-242-TEX-I-1D461\"><\/use><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><msub><mrow data-mjx-texclass=\"ORD\"><mi data-mjx-variant=\"-tex-calligraphic\" mathvariant=\"script\">S<\/mi><\/mrow><mi>t<\/mi><\/msub><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">\\mathcal{S}_t<\/script><span> denote the vectors that have Hamming distance <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"0.817ex\" height=\"1.441ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -626 361 637\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.025ex;\"><defs><path id=\"MJX-1284-TEX-I-1D461\" d=\"M26 385Q19 392 19 395Q19 399 22 411T27 425Q29 430 36 430T87 431H140L159 511Q162 522 166 540T173 566T179 586T187 603T197 615T211 624T229 626Q247 625 254 615T261 596Q261 589 252 549T232 470L222 433Q222 431 272 431H323Q330 424 330 420Q330 398 317 385H210L174 240Q135 80 135 68Q135 26 162 26Q197 26 230 60T283 144Q285 150 288 151T303 153H307Q322 153 322 145Q322 142 319 133Q314 117 301 95T267 48T216 6T155 -11Q125 -11 98 4T59 56Q57 64 57 83V101L92 241Q127 382 128 383Q128 385 77 385H26Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D461\" xlink:href=\"#MJX-1284-TEX-I-1D461\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>t<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">t<\/script><span>, then <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"16.907ex\" height=\"2.72ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -849.5 7472.8 1202.3\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.798ex;\"><defs><path id=\"MJX-288-TEX-N-7C\" d=\"M139 -249H137Q125 -249 119 -235V251L120 737Q130 750 139 750Q152 750 159 735V-235Q151 -249 141 -249H139Z\"><\/path><path id=\"MJX-288-TEX-C-53\" d=\"M554 512Q536 512 536 522Q536 525 539 539T542 564Q542 588 528 604Q515 616 482 625T410 635Q374 635 349 624T312 594T295 561T290 532Q290 505 303 482T342 442T378 419T409 404Q435 391 451 383T494 357T535 323T562 282T574 231Q574 133 464 56T220 -22Q138 -22 78 21T18 123Q18 184 61 227T156 274Q178 274 178 263Q178 260 177 258Q172 247 164 239T151 227T136 218L127 213L124 202Q118 186 118 163Q120 124 165 86T292 48Q374 48 423 86T473 186V193Q473 267 347 327Q268 364 239 389Q191 431 191 486Q191 547 242 600T356 679T470 705Q472 705 478 705T489 704Q551 704 596 682T642 610Q642 566 621 545Q592 516 554 512Z\"><\/path><path id=\"MJX-288-TEX-I-1D461\" d=\"M26 385Q19 392 19 395Q19 399 22 411T27 425Q29 430 36 430T87 431H140L159 511Q162 522 166 540T173 566T179 586T187 603T197 615T211 624T229 626Q247 625 254 615T261 596Q261 589 252 549T232 470L222 433Q222 431 272 431H323Q330 424 330 420Q330 398 317 385H210L174 240Q135 80 135 68Q135 26 162 26Q197 26 230 60T283 144Q285 150 288 151T303 153H307Q322 153 322 145Q322 142 319 133Q314 117 301 95T267 48T216 6T155 -11Q125 -11 98 4T59 56Q57 64 57 83V101L92 241Q127 382 128 383Q128 385 77 385H26Z\"><\/path><path id=\"MJX-288-TEX-N-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path id=\"MJX-288-TEX-SO-28\" d=\"M152 251Q152 646 388 850H416Q422 844 422 841Q422 837 403 816T357 753T302 649T255 482T236 250Q236 124 255 19T301 -147T356 -251T403 -315T422 -340Q422 -343 416 -349H388Q359 -325 332 -296T271 -213T212 -97T170 56T152 251Z\"><\/path><path id=\"MJX-288-TEX-I-1D45B\" d=\"M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z\"><\/path><path id=\"MJX-288-TEX-SO-29\" d=\"M305 251Q305 -145 69 -349H56Q43 -349 39 -347T35 -338Q37 -333 60 -307T108 -239T160 -136T204 27T221 250T204 473T160 636T108 740T60 807T35 839Q35 850 50 850H56H69Q197 743 256 566Q305 425 305 251Z\"><\/path><path id=\"MJX-288-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-288-TEX-I-1D45E\" d=\"M33 157Q33 258 109 349T280 441Q340 441 372 389Q373 390 377 395T388 406T404 418Q438 442 450 442Q454 442 457 439T460 434Q460 425 391 149Q320 -135 320 -139Q320 -147 365 -148H390Q396 -156 396 -157T393 -175Q389 -188 383 -194H370Q339 -192 262 -192Q234 -192 211 -192T174 -192T157 -193Q143 -193 143 -185Q143 -182 145 -170Q149 -154 152 -151T172 -148Q220 -148 230 -141Q238 -136 258 -53T279 32Q279 33 272 29Q224 -10 172 -10Q117 -10 75 30T33 157ZM352 326Q329 405 277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q233 26 290 98L298 109L352 326Z\"><\/path><path id=\"MJX-288-TEX-N-2212\" d=\"M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z\"><\/path><path id=\"MJX-288-TEX-N-31\" d=\"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z\"><\/path><path id=\"MJX-288-TEX-N-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mo\" transform=\"translate(0 -0.5)\"><use data-c=\"7C\" xlink:href=\"#MJX-288-TEX-N-7C\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(278,0)\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"53\" xlink:href=\"#MJX-288-TEX-C-53\"><\/use><\/g><\/g><g data-mml-node=\"mi\" transform=\"translate(639,-150) scale(0.707)\"><use data-c=\"1D461\" xlink:href=\"#MJX-288-TEX-I-1D461\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(1222.3,0) translate(0 -0.5)\"><use data-c=\"7C\" xlink:href=\"#MJX-288-TEX-N-7C\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(1778,0)\"><use data-c=\"3D\" xlink:href=\"#MJX-288-TEX-N-3D\"><\/use><\/g><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\" transform=\"translate(2833.8,0)\"><g data-mml-node=\"mrow\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"OPEN\"><g data-mml-node=\"mo\" transform=\"translate(0 -0.5)\"><use data-c=\"28\" xlink:href=\"#MJX-288-TEX-SO-28\"><\/use><\/g><\/g><g data-mml-node=\"mfrac\" transform=\"translate(458,0)\"><g data-mml-node=\"mi\" transform=\"translate(0,444) scale(0.707)\"><use data-c=\"1D45B\" xlink:href=\"#MJX-288-TEX-I-1D45B\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(84.5,-345) scale(0.707)\"><use data-c=\"1D461\" xlink:href=\"#MJX-288-TEX-I-1D461\"><\/use><\/g><\/g><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"CLOSE\" transform=\"translate(882.3,0)\"><g data-mml-node=\"mo\" transform=\"translate(0 -0.5)\"><use data-c=\"29\" xlink:href=\"#MJX-288-TEX-SO-29\"><\/use><\/g><\/g><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(4174.1,0)\"><use data-c=\"28\" xlink:href=\"#MJX-288-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(4563.1,0)\"><use data-c=\"1D45E\" xlink:href=\"#MJX-288-TEX-I-1D45E\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(5245.3,0)\"><use data-c=\"2212\" xlink:href=\"#MJX-288-TEX-N-2212\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(6245.5,0)\"><use data-c=\"31\" xlink:href=\"#MJX-288-TEX-N-31\"><\/use><\/g><g data-mml-node=\"msup\" transform=\"translate(6745.5,0)\"><g data-mml-node=\"mo\"><use data-c=\"29\" xlink:href=\"#MJX-288-TEX-N-29\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(422,363) scale(0.707)\"><use data-c=\"1D461\" xlink:href=\"#MJX-288-TEX-I-1D461\"><\/use><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><msub><mrow data-mjx-texclass=\"ORD\"><mi data-mjx-variant=\"-tex-calligraphic\" mathvariant=\"script\">S<\/mi><\/mrow><mi>t<\/mi><\/msub><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mo>=<\/mo><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"OPEN\"><mo minsize=\"1.2em\" maxsize=\"1.2em\">(<\/mo><\/mrow><mfrac linethickness=\"0\"><mi>n<\/mi><mi>t<\/mi><\/mfrac><mrow data-mjx-texclass=\"CLOSE\"><mo minsize=\"1.2em\" maxsize=\"1.2em\">)<\/mo><\/mrow><\/mrow><\/mrow><mo stretchy=\"false\">(<\/mo><mi>q<\/mi><mo>\u2212<\/mo><mn>1<\/mn><msup><mo stretchy=\"false\">)<\/mo><mi>t<\/mi><\/msup><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">|\\mathcal{S}_t|={n\\choose t}(q-1)^t<\/script><span>. We also have<\/span><\/p><div contenteditable=\"false\" spellcheck=\"false\" class=\"mathjax-block md-end-block md-math-block md-rawblock\" id=\"mathjax-n208\" cid=\"n208\" mdtype=\"math_block\" data-math-tag-before=\"10\" data-math-labels=\"[]\" data-math-tag-after=\"10\"><div class=\"md-rawblock-container md-math-container\" contenteditable=\"false\" tabindex=\"-1\"><mjx-container class=\"MathJax\" jax=\"SVG\" display=\"true\" width=\"full\" style=\"min-width: 43.475ex; position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"100%\" height=\"5.428ex\" role=\"img\" focusable=\"false\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -2.148ex; min-width: 43.475ex;\"><defs><path id=\"MJX-1989-TEX-I-1D45E\" d=\"M33 157Q33 258 109 349T280 441Q340 441 372 389Q373 390 377 395T388 406T404 418Q438 442 450 442Q454 442 457 439T460 434Q460 425 391 149Q320 -135 320 -139Q320 -147 365 -148H390Q396 -156 396 -157T393 -175Q389 -188 383 -194H370Q339 -192 262 -192Q234 -192 211 -192T174 -192T157 -193Q143 -193 143 -185Q143 -182 145 -170Q149 -154 152 -151T172 -148Q220 -148 230 -141Q238 -136 258 -53T279 32Q279 33 272 29Q224 -10 172 -10Q117 -10 75 30T33 157ZM352 326Q329 405 277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q233 26 290 98L298 109L352 326Z\"><\/path><path id=\"MJX-1989-TEX-I-1D45B\" d=\"M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z\"><\/path><path id=\"MJX-1989-TEX-SO-28\" d=\"M152 251Q152 646 388 850H416Q422 844 422 841Q422 837 403 816T357 753T302 649T255 482T236 250Q236 124 255 19T301 -147T356 -251T403 -315T422 -340Q422 -343 416 -349H388Q359 -325 332 -296T271 -213T212 -97T170 56T152 251Z\"><\/path><path id=\"MJX-1989-TEX-I-210E\" d=\"M137 683Q138 683 209 688T282 694Q294 694 294 685Q294 674 258 534Q220 386 220 383Q220 381 227 388Q288 442 357 442Q411 442 444 415T478 336Q478 285 440 178T402 50Q403 36 407 31T422 26Q450 26 474 56T513 138Q516 149 519 151T535 153Q555 153 555 145Q555 144 551 130Q535 71 500 33Q466 -10 419 -10H414Q367 -10 346 17T325 74Q325 90 361 192T398 345Q398 404 354 404H349Q266 404 205 306L198 293L164 158Q132 28 127 16Q114 -11 83 -11Q69 -11 59 -2T48 16Q48 30 121 320L195 616Q195 629 188 632T149 637H128Q122 643 122 645T124 664Q129 683 137 683Z\"><\/path><path id=\"MJX-1989-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 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transform=\"translate(0,676)\"><use data-c=\"1D45B\" xlink:href=\"#MJX-1989-TEX-I-1D45B\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(119.5,-686)\"><use data-c=\"1D461\" xlink:href=\"#MJX-1989-TEX-I-1D461\"><\/use><\/g><\/g><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"CLOSE\" transform=\"translate(1336,0)\"><g data-mml-node=\"mo\" transform=\"translate(0 -0.5)\"><use data-c=\"29\" xlink:href=\"#MJX-1989-TEX-S3-29\"><\/use><\/g><\/g><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(10354.2,0)\"><use data-c=\"28\" xlink:href=\"#MJX-1989-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(10743.2,0)\"><use data-c=\"1D45E\" xlink:href=\"#MJX-1989-TEX-I-1D45E\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(11425.5,0)\"><use data-c=\"2212\" xlink:href=\"#MJX-1989-TEX-N-2212\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(12425.7,0)\"><use data-c=\"31\" xlink:href=\"#MJX-1989-TEX-N-31\"><\/use><\/g><g data-mml-node=\"msup\" transform=\"translate(12925.7,0)\"><g data-mml-node=\"mo\"><use data-c=\"29\" xlink:href=\"#MJX-1989-TEX-N-29\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(422,413) scale(0.707)\"><use data-c=\"1D461\" xlink:href=\"#MJX-1989-TEX-I-1D461\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(13930.7,0)\"><use data-c=\"2264\" xlink:href=\"#MJX-1989-TEX-N-2264\"><\/use><\/g><g data-mml-node=\"msup\" transform=\"translate(14986.5,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D45E\" xlink:href=\"#MJX-1989-TEX-I-1D45E\"><\/use><\/g><g data-mml-node=\"TeXAtom\" transform=\"translate(543.7,413) scale(0.707)\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D45B\" xlink:href=\"#MJX-1989-TEX-I-1D45B\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(600,0)\"><g data-mml-node=\"mi\"><use data-c=\"210E\" xlink:href=\"#MJX-1989-TEX-I-210E\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(609,-150) scale(0.707)\"><use data-c=\"1D45E\" xlink:href=\"#MJX-1989-TEX-I-1D45E\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(1584.3,0)\"><use data-c=\"28\" xlink:href=\"#MJX-1989-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1973.3,0)\"><use data-c=\"1D70F\" xlink:href=\"#MJX-1989-TEX-I-1D70F\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2490.3,0)\"><use data-c=\"29\" xlink:href=\"#MJX-1989-TEX-N-29\"><\/use><\/g><\/g><\/g><\/g><\/g><\/g><\/svg><svg data-labels=\"true\" preserveAspectRatio=\"xMaxYMid\" viewBox=\"0 -1449.5 1 2399\"><g data-labels=\"true\" transform=\"matrix(1 0 0 -1 0 0)\"><g data-mml-node=\"mtd\" id=\"mjx-eqn:\"><g data-mml-node=\"mrow\"><\/g><\/g><\/g><\/svg><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"block\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable displaystyle=\"true\"><mlabeledtr><mtd><mrow><\/mrow><\/mtd><mtd><msup><mi>q<\/mi><mrow data-mjx-texclass=\"ORD\"><mi>n<\/mi><mrow data-mjx-texclass=\"ORD\"><mo minsize=\"1.2em\" maxsize=\"1.2em\">(<\/mo><\/mrow><msub><mi>h<\/mi><mi>q<\/mi><\/msub><mo stretchy=\"false\">(<\/mo><mi>\u03c4<\/mi><mo stretchy=\"false\">)<\/mo><mo>+<\/mo><mrow data-mjx-texclass=\"ORD\"><mi data-mjx-variant=\"-tex-calligraphic\" mathvariant=\"script\">O<\/mi><\/mrow><mrow data-mjx-texclass=\"ORD\"><mo minsize=\"1.2em\" maxsize=\"1.2em\">(<\/mo><\/mrow><mfrac><mrow><msub><mi>log<\/mi><mi>q<\/mi><\/msub><mo data-mjx-texclass=\"NONE\">\u2061<\/mo><mo stretchy=\"false\">(<\/mo><mi>n<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><mi>n<\/mi><\/mfrac><mrow data-mjx-texclass=\"ORD\"><mo minsize=\"1.2em\" maxsize=\"1.2em\">)<\/mo><\/mrow><mrow data-mjx-texclass=\"ORD\"><mo minsize=\"1.2em\" maxsize=\"1.2em\">)<\/mo><\/mrow><\/mrow><\/msup><mo>\u2264<\/mo><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"OPEN\"><mo minsize=\"2.047em\" maxsize=\"2.047em\">(<\/mo><\/mrow><mfrac linethickness=\"0\"><mi>n<\/mi><mi>t<\/mi><\/mfrac><mrow data-mjx-texclass=\"CLOSE\"><mo minsize=\"2.047em\" maxsize=\"2.047em\">)<\/mo><\/mrow><\/mrow><\/mrow><mo stretchy=\"false\">(<\/mo><mi>q<\/mi><mo>\u2212<\/mo><mn>1<\/mn><msup><mo stretchy=\"false\">)<\/mo><mi>t<\/mi><\/msup><mo>\u2264<\/mo><msup><mi>q<\/mi><mrow data-mjx-texclass=\"ORD\"><mi>n<\/mi><msub><mi>h<\/mi><mi>q<\/mi><\/msub><mo stretchy=\"false\">(<\/mo><mi>\u03c4<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><\/msup><\/mtd><\/mlabeledtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><p><span>Asymptotically, we have<\/span><\/p><div contenteditable=\"false\" spellcheck=\"false\" class=\"mathjax-block md-end-block md-math-block md-rawblock\" id=\"mathjax-n202\" cid=\"n202\" mdtype=\"math_block\" data-math-tag-before=\"10\" data-math-labels=\"[]\" data-math-tag-after=\"10\"><div class=\"md-rawblock-container md-math-container\" contenteditable=\"false\" tabindex=\"-1\"><mjx-container 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stretchy=\"false\">)<\/mo><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">P_X(A)<\/script><span> represents the probability that random variable <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.928ex\" height=\"1.545ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -683 852 683\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: 0px;\"><defs><path id=\"MJX-548-TEX-I-1D44B\" d=\"M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D44B\" xlink:href=\"#MJX-548-TEX-I-1D44B\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>X<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">X<\/script><span> is equal to event <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.697ex\" height=\"1.62ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -716 750 716\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: 0px;\"><defs><path id=\"MJX-550-TEX-I-1D434\" d=\"M208 74Q208 50 254 46Q272 46 272 35Q272 34 270 22Q267 8 264 4T251 0Q249 0 239 0T205 1T141 2Q70 2 50 0H42Q35 7 35 11Q37 38 48 46H62Q132 49 164 96Q170 102 345 401T523 704Q530 716 547 716H555H572Q578 707 578 706L606 383Q634 60 636 57Q641 46 701 46Q726 46 726 36Q726 34 723 22Q720 7 718 4T704 0Q701 0 690 0T651 1T578 2Q484 2 455 0H443Q437 6 437 9T439 27Q443 40 445 43L449 46H469Q523 49 533 63L521 213H283L249 155Q208 86 208 74ZM516 260Q516 271 504 416T490 562L463 519Q447 492 400 412L310 260L413 259Q516 259 516 260Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D434\" xlink:href=\"#MJX-550-TEX-I-1D434\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>A<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">A<\/script><span>. We choose a random code by picking uniformly at random a parity-check matrix <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"12.859ex\" height=\"3.024ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -1060.7 5683.9 1336.8\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.625ex;\"><defs><path id=\"MJX-551-TEX-B-1D407\" d=\"M400 0Q376 3 226 3Q75 3 51 0H39V62H147V624H39V686H51Q75 683 226 683Q376 683 400 686H412V624H304V388H595V624H487V686H499Q523 683 673 683Q824 683 848 686H860V624H752V62H860V0H848Q824 3 674 3Q523 3 499 0H487V62H595V326H304V62H412V0H400Z\"><\/path><path id=\"MJX-551-TEX-N-2208\" d=\"M84 250Q84 372 166 450T360 539Q361 539 377 539T419 540T469 540H568Q583 532 583 520Q583 511 570 501L466 500Q355 499 329 494Q280 482 242 458T183 409T147 354T129 306T124 272V270H568Q583 262 583 250T568 230H124V228Q124 207 134 177T167 112T231 48T328 7Q355 1 466 0H570Q583 -10 583 -20Q583 -32 568 -40H471Q464 -40 446 -40T417 -41Q262 -41 172 45Q84 127 84 250Z\"><\/path><path id=\"MJX-551-TEX-D-1D53D\" d=\"M584 499Q569 490 566 490Q558 490 552 497T546 515Q546 535 533 559Q526 574 506 593T469 621Q415 648 326 648Q293 648 287 647T275 641Q264 630 263 617Q262 609 260 492V370L275 372Q323 376 350 392T393 441Q409 473 409 506Q409 529 427 529Q437 529 442 519Q444 511 444 362Q444 212 442 206Q436 197 426 197Q409 197 409 217Q409 265 375 299Q346 328 280 335H260V206Q260 70 262 63Q265 46 276 41T326 35Q362 35 366 28Q377 17 366 3L360 -1H24Q12 5 12 16Q12 35 51 35Q92 38 97 52Q102 60 102 341T97 632Q91 645 51 648Q12 648 12 666Q12 675 24 683H573Q576 678 584 670V499ZM137 341Q137 131 136 89T130 37Q129 36 129 35H182Q233 35 233 39Q226 54 225 92T224 346L226 623L231 635L235 648H129Q132 641 133 638T135 603T137 517T137 341ZM549 603V648H495L506 641Q531 621 533 619L549 603ZM409 317V395L400 386Q390 376 375 366L357 355L373 346Q394 331 397 328L409 317Z\"><\/path><path id=\"MJX-551-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-551-TEX-I-1D45B\" d=\"M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z\"><\/path><path id=\"MJX-551-TEX-N-2212\" d=\"M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z\"><\/path><path id=\"MJX-551-TEX-I-1D458\" d=\"M121 647Q121 657 125 670T137 683Q138 683 209 688T282 694Q294 694 294 686Q294 679 244 477Q194 279 194 272Q213 282 223 291Q247 309 292 354T362 415Q402 442 438 442Q468 442 485 423T503 369Q503 344 496 327T477 302T456 291T438 288Q418 288 406 299T394 328Q394 353 410 369T442 390L458 393Q446 405 434 405H430Q398 402 367 380T294 316T228 255Q230 254 243 252T267 246T293 238T320 224T342 206T359 180T365 147Q365 130 360 106T354 66Q354 26 381 26Q429 26 459 145Q461 153 479 153H483Q499 153 499 144Q499 139 496 130Q455 -11 378 -11Q333 -11 305 15T277 90Q277 108 280 121T283 145Q283 167 269 183T234 206T200 217T182 220H180Q168 178 159 139T145 81T136 44T129 20T122 7T111 -2Q98 -11 83 -11Q66 -11 57 -1T48 16Q48 26 85 176T158 471L195 616Q196 629 188 632T149 637H144Q134 637 131 637T124 640T121 647Z\"><\/path><path id=\"MJX-551-TEX-N-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><path id=\"MJX-551-TEX-N-D7\" d=\"M630 29Q630 9 609 9Q604 9 587 25T493 118L389 222L284 117Q178 13 175 11Q171 9 168 9Q160 9 154 15T147 29Q147 36 161 51T255 146L359 250L255 354Q174 435 161 449T147 471Q147 480 153 485T168 490Q173 490 175 489Q178 487 284 383L389 278L493 382Q570 459 587 475T609 491Q630 491 630 471Q630 464 620 453T522 355L418 250L522 145Q606 61 618 48T630 29Z\"><\/path><path id=\"MJX-551-TEX-I-1D45E\" d=\"M33 157Q33 258 109 349T280 441Q340 441 372 389Q373 390 377 395T388 406T404 418Q438 442 450 442Q454 442 457 439T460 434Q460 425 391 149Q320 -135 320 -139Q320 -147 365 -148H390Q396 -156 396 -157T393 -175Q389 -188 383 -194H370Q339 -192 262 -192Q234 -192 211 -192T174 -192T157 -193Q143 -193 143 -185Q143 -182 145 -170Q149 -154 152 -151T172 -148Q220 -148 230 -141Q238 -136 258 -53T279 32Q279 33 272 29Q224 -10 172 -10Q117 -10 75 30T33 157ZM352 326Q329 405 277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q233 26 290 98L298 109L352 326Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D407\" xlink:href=\"#MJX-551-TEX-B-1D407\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(1177.8,0)\"><use data-c=\"2208\" xlink:href=\"#MJX-551-TEX-N-2208\"><\/use><\/g><g data-mml-node=\"msubsup\" transform=\"translate(2122.6,0)\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D53D\" xlink:href=\"#MJX-551-TEX-D-1D53D\"><\/use><\/g><\/g><g data-mml-node=\"TeXAtom\" transform=\"translate(644,530.4) scale(0.707)\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mo\"><use data-c=\"28\" xlink:href=\"#MJX-551-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(389,0)\"><use data-c=\"1D45B\" xlink:href=\"#MJX-551-TEX-I-1D45B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(989,0)\"><use data-c=\"2212\" xlink:href=\"#MJX-551-TEX-N-2212\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1767,0)\"><use data-c=\"1D458\" xlink:href=\"#MJX-551-TEX-I-1D458\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2288,0)\"><use data-c=\"29\" xlink:href=\"#MJX-551-TEX-N-29\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2677,0)\"><use data-c=\"D7\" xlink:href=\"#MJX-551-TEX-N-D7\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(3455,0)\"><use data-c=\"1D45B\" xlink:href=\"#MJX-551-TEX-I-1D45B\"><\/use><\/g><\/g><g data-mml-node=\"mi\" transform=\"translate(644,-138.9) scale(0.707)\"><use data-c=\"1D45E\" xlink:href=\"#MJX-551-TEX-I-1D45E\"><\/use><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">H<\/mi><\/mrow><mo>\u2208<\/mo><msubsup><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"double-struck\">F<\/mi><\/mrow><mi>q<\/mi><mrow data-mjx-texclass=\"ORD\"><mo stretchy=\"false\">(<\/mo><mi>n<\/mi><mo>\u2212<\/mo><mi>k<\/mi><mo stretchy=\"false\">)<\/mo><mo>\u00d7<\/mo><mi>n<\/mi><\/mrow><\/msubsup><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">\\mathbf{H}\\in\\mathbb{F}_q^{(n-k)\\times n}<\/script><span>.<\/span><\/p><blockquote><p><span>Defining <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"3.079ex\" height=\"1.57ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -694 1361 694\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: 0px;\"><defs><path id=\"MJX-875-TEX-SS-1D5A3\" d=\"M88 0V694H237H258H316Q383 694 425 686T511 648Q578 604 622 525T666 343Q666 190 564 86Q494 18 400 3Q387 1 237 0H88ZM565 341Q565 409 546 463T495 550T429 600T359 621Q348 623 267 623H189V71H267Q272 71 286 71T310 70Q461 70 527 184Q565 251 565 341Z\"><\/path><path id=\"MJX-875-TEX-SS-1D5AF\" d=\"M88 0V694H230Q347 693 370 692T410 686Q487 667 535 611T583 485Q583 409 527 348T379 276Q369 274 279 274H192V0H88ZM486 485Q486 523 471 551T432 593T391 612T357 621Q350 622 268 623H189V347H268Q350 348 357 349Q370 351 383 354T416 368T450 391T475 429T486 485Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D5A3\" xlink:href=\"#MJX-875-TEX-SS-1D5A3\"><\/use><use data-c=\"1D5AF\" xlink:href=\"#MJX-875-TEX-SS-1D5AF\" transform=\"translate(722,0)\"><\/use><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"sans-serif\">DP<\/mi><\/mrow><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">\\mathsf{DP}<\/script><span> with generator or parity-check matrix is just a matter of personal taste, it does not change the average hardness.<\/span><\/p><\/blockquote><p><strong><span>Lemma 2.2.3.<\/span><\/strong><span> Given <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"8.401ex\" height=\"2.801ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -853.7 3713.4 1237.9\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.869ex;\"><defs><path id=\"MJX-554-TEX-B-1D42C\" d=\"M38 315Q38 339 45 360T70 404T127 440T223 453Q273 453 320 436L338 445L357 453H366Q380 453 383 447T386 403V387V355Q386 331 383 326T365 321H355H349Q333 321 329 324T324 341Q317 406 224 406H216Q123 406 123 353Q123 334 143 321T188 304T244 294T285 286Q305 281 325 273T373 237T412 172Q414 162 414 142Q414 -6 230 -6Q154 -6 117 22L68 -6H58Q44 -6 41 0T38 42V73Q38 85 38 101T37 122Q37 144 42 148T68 153H75Q87 153 91 151T97 147T103 132Q131 46 220 46H230Q257 46 265 47Q330 58 330 108Q330 127 316 142Q300 156 284 162Q271 168 212 178T122 202Q38 243 38 315Z\"><\/path><path id=\"MJX-554-TEX-N-2208\" d=\"M84 250Q84 372 166 450T360 539Q361 539 377 539T419 540T469 540H568Q583 532 583 520Q583 511 570 501L466 500Q355 499 329 494Q280 482 242 458T183 409T147 354T129 306T124 272V270H568Q583 262 583 250T568 230H124V228Q124 207 134 177T167 112T231 48T328 7Q355 1 466 0H570Q583 -10 583 -20Q583 -32 568 -40H471Q464 -40 446 -40T417 -41Q262 -41 172 45Q84 127 84 250Z\"><\/path><path id=\"MJX-554-TEX-D-1D53D\" d=\"M584 499Q569 490 566 490Q558 490 552 497T546 515Q546 535 533 559Q526 574 506 593T469 621Q415 648 326 648Q293 648 287 647T275 641Q264 630 263 617Q262 609 260 492V370L275 372Q323 376 350 392T393 441Q409 473 409 506Q409 529 427 529Q437 529 442 519Q444 511 444 362Q444 212 442 206Q436 197 426 197Q409 197 409 217Q409 265 375 299Q346 328 280 335H260V206Q260 70 262 63Q265 46 276 41T326 35Q362 35 366 28Q377 17 366 3L360 -1H24Q12 5 12 16Q12 35 51 35Q92 38 97 52Q102 60 102 341T97 632Q91 645 51 648Q12 648 12 666Q12 675 24 683H573Q576 678 584 670V499ZM137 341Q137 131 136 89T130 37Q129 36 129 35H182Q233 35 233 39Q226 54 225 92T224 346L226 623L231 635L235 648H129Q132 641 133 638T135 603T137 517T137 341ZM549 603V648H495L506 641Q531 621 533 619L549 603ZM409 317V395L400 386Q390 376 375 366L357 355L373 346Q394 331 397 328L409 317Z\"><\/path><path id=\"MJX-554-TEX-I-1D45B\" d=\"M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z\"><\/path><path id=\"MJX-554-TEX-N-2212\" d=\"M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z\"><\/path><path id=\"MJX-554-TEX-I-1D458\" d=\"M121 647Q121 657 125 670T137 683Q138 683 209 688T282 694Q294 694 294 686Q294 679 244 477Q194 279 194 272Q213 282 223 291Q247 309 292 354T362 415Q402 442 438 442Q468 442 485 423T503 369Q503 344 496 327T477 302T456 291T438 288Q418 288 406 299T394 328Q394 353 410 369T442 390L458 393Q446 405 434 405H430Q398 402 367 380T294 316T228 255Q230 254 243 252T267 246T293 238T320 224T342 206T359 180T365 147Q365 130 360 106T354 66Q354 26 381 26Q429 26 459 145Q461 153 479 153H483Q499 153 499 144Q499 139 496 130Q455 -11 378 -11Q333 -11 305 15T277 90Q277 108 280 121T283 145Q283 167 269 183T234 206T200 217T182 220H180Q168 178 159 139T145 81T136 44T129 20T122 7T111 -2Q98 -11 83 -11Q66 -11 57 -1T48 16Q48 26 85 176T158 471L195 616Q196 629 188 632T149 637H144Q134 637 131 637T124 640T121 647Z\"><\/path><path id=\"MJX-554-TEX-I-1D45E\" d=\"M33 157Q33 258 109 349T280 441Q340 441 372 389Q373 390 377 395T388 406T404 418Q438 442 450 442Q454 442 457 439T460 434Q460 425 391 149Q320 -135 320 -139Q320 -147 365 -148H390Q396 -156 396 -157T393 -175Q389 -188 383 -194H370Q339 -192 262 -192Q234 -192 211 -192T174 -192T157 -193Q143 -193 143 -185Q143 -182 145 -170Q149 -154 152 -151T172 -148Q220 -148 230 -141Q238 -136 258 -53T279 32Q279 33 272 29Q224 -10 172 -10Q117 -10 75 30T33 157ZM352 326Q329 405 277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q233 26 290 98L298 109L352 326Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D42C\" xlink:href=\"#MJX-554-TEX-B-1D42C\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(731.8,0)\"><use data-c=\"2208\" xlink:href=\"#MJX-554-TEX-N-2208\"><\/use><\/g><g data-mml-node=\"msubsup\" transform=\"translate(1676.6,0)\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D53D\" xlink:href=\"#MJX-554-TEX-D-1D53D\"><\/use><\/g><\/g><g data-mml-node=\"TeXAtom\" transform=\"translate(644,363) scale(0.707)\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D45B\" xlink:href=\"#MJX-554-TEX-I-1D45B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(600,0)\"><use data-c=\"2212\" xlink:href=\"#MJX-554-TEX-N-2212\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1378,0)\"><use data-c=\"1D458\" xlink:href=\"#MJX-554-TEX-I-1D458\"><\/use><\/g><\/g><g data-mml-node=\"mi\" transform=\"translate(644,-247) scale(0.707)\"><use data-c=\"1D45E\" xlink:href=\"#MJX-554-TEX-I-1D45E\"><\/use><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">s<\/mi><\/mrow><mo>\u2208<\/mo><msubsup><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"double-struck\">F<\/mi><\/mrow><mi>q<\/mi><mrow data-mjx-texclass=\"ORD\"><mi>n<\/mi><mo>\u2212<\/mo><mi>k<\/mi><\/mrow><\/msubsup><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">\\mathbf s\\in\\mathbb{F}_q^{n-k}<\/script><span> and <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"6.669ex\" height=\"2.414ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -683 2947.8 1067.2\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.869ex;\"><defs><path id=\"MJX-556-TEX-B-1D432\" d=\"M84 -102Q84 -110 87 -119T102 -138T133 -149Q148 -148 162 -143T186 -131T206 -114T222 -95T234 -76T243 -59T249 -45T252 -37L269 0L96 382H26V444H34Q49 441 146 441Q252 441 270 444H279V382H255Q232 382 232 380L337 151L442 382H394V444H401Q413 441 495 441Q568 441 574 444H580V382H510L406 152Q298 -84 297 -87Q269 -139 225 -169T131 -200Q85 -200 54 -172T23 -100Q23 -64 44 -50T87 -35Q111 -35 130 -50T152 -92V-100H84V-102Z\"><\/path><path id=\"MJX-556-TEX-N-2208\" d=\"M84 250Q84 372 166 450T360 539Q361 539 377 539T419 540T469 540H568Q583 532 583 520Q583 511 570 501L466 500Q355 499 329 494Q280 482 242 458T183 409T147 354T129 306T124 272V270H568Q583 262 583 250T568 230H124V228Q124 207 134 177T167 112T231 48T328 7Q355 1 466 0H570Q583 -10 583 -20Q583 -32 568 -40H471Q464 -40 446 -40T417 -41Q262 -41 172 45Q84 127 84 250Z\"><\/path><path id=\"MJX-556-TEX-D-1D53D\" d=\"M584 499Q569 490 566 490Q558 490 552 497T546 515Q546 535 533 559Q526 574 506 593T469 621Q415 648 326 648Q293 648 287 647T275 641Q264 630 263 617Q262 609 260 492V370L275 372Q323 376 350 392T393 441Q409 473 409 506Q409 529 427 529Q437 529 442 519Q444 511 444 362Q444 212 442 206Q436 197 426 197Q409 197 409 217Q409 265 375 299Q346 328 280 335H260V206Q260 70 262 63Q265 46 276 41T326 35Q362 35 366 28Q377 17 366 3L360 -1H24Q12 5 12 16Q12 35 51 35Q92 38 97 52Q102 60 102 341T97 632Q91 645 51 648Q12 648 12 666Q12 675 24 683H573Q576 678 584 670V499ZM137 341Q137 131 136 89T130 37Q129 36 129 35H182Q233 35 233 39Q226 54 225 92T224 346L226 623L231 635L235 648H129Q132 641 133 638T135 603T137 517T137 341ZM549 603V648H495L506 641Q531 621 533 619L549 603ZM409 317V395L400 386Q390 376 375 366L357 355L373 346Q394 331 397 328L409 317Z\"><\/path><path id=\"MJX-556-TEX-I-1D45B\" d=\"M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z\"><\/path><path id=\"MJX-556-TEX-I-1D45E\" d=\"M33 157Q33 258 109 349T280 441Q340 441 372 389Q373 390 377 395T388 406T404 418Q438 442 450 442Q454 442 457 439T460 434Q460 425 391 149Q320 -135 320 -139Q320 -147 365 -148H390Q396 -156 396 -157T393 -175Q389 -188 383 -194H370Q339 -192 262 -192Q234 -192 211 -192T174 -192T157 -193Q143 -193 143 -185Q143 -182 145 -170Q149 -154 152 -151T172 -148Q220 -148 230 -141Q238 -136 258 -53T279 32Q279 33 272 29Q224 -10 172 -10Q117 -10 75 30T33 157ZM352 326Q329 405 277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q233 26 290 98L298 109L352 326Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D432\" xlink:href=\"#MJX-556-TEX-B-1D432\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(884.8,0)\"><use data-c=\"2208\" xlink:href=\"#MJX-556-TEX-N-2208\"><\/use><\/g><g data-mml-node=\"msubsup\" transform=\"translate(1829.6,0)\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D53D\" xlink:href=\"#MJX-556-TEX-D-1D53D\"><\/use><\/g><\/g><g data-mml-node=\"TeXAtom\" transform=\"translate(644,363) scale(0.707)\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D45B\" xlink:href=\"#MJX-556-TEX-I-1D45B\"><\/use><\/g><\/g><g data-mml-node=\"mi\" transform=\"translate(644,-247) scale(0.707)\"><use data-c=\"1D45E\" xlink:href=\"#MJX-556-TEX-I-1D45E\"><\/use><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">y<\/mi><\/mrow><mo>\u2208<\/mo><msubsup><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"double-struck\">F<\/mi><\/mrow><mi>q<\/mi><mrow data-mjx-texclass=\"ORD\"><mi>n<\/mi><\/mrow><\/msubsup><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">\\mathbf y\\in\\mathbb{F}_q^{n}<\/script><span> such that <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"5.691ex\" height=\"2.106ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -716 2515.6 931\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.486ex;\"><defs><path id=\"MJX-558-TEX-B-1D432\" d=\"M84 -102Q84 -110 87 -119T102 -138T133 -149Q148 -148 162 -143T186 -131T206 -114T222 -95T234 -76T243 -59T249 -45T252 -37L269 0L96 382H26V444H34Q49 441 146 441Q252 441 270 444H279V382H255Q232 382 232 380L337 151L442 382H394V444H401Q413 441 495 441Q568 441 574 444H580V382H510L406 152Q298 -84 297 -87Q269 -139 225 -169T131 -200Q85 -200 54 -172T23 -100Q23 -64 44 -50T87 -35Q111 -35 130 -50T152 -92V-100H84V-102Z\"><\/path><path id=\"MJX-558-TEX-N-2260\" d=\"M166 -215T159 -215T147 -212T141 -204T139 -197Q139 -190 144 -183L306 133H70Q56 140 56 153Q56 168 72 173H327L406 327H72Q56 332 56 347Q56 360 70 367H426Q597 702 602 707Q605 716 618 716Q625 716 630 712T636 703T638 696Q638 692 471 367H707Q722 359 722 347Q722 336 708 328L451 327L371 173H708Q722 163 722 153Q722 140 707 133H351Q175 -210 170 -212Q166 -215 159 -215Z\"><\/path><path id=\"MJX-558-TEX-B-1D7CE\" d=\"M266 654H280H282Q500 654 524 418Q529 370 529 320Q529 125 456 52Q397 -10 287 -10Q110 -10 63 154Q45 212 45 316Q45 504 113 585Q140 618 185 636T266 654ZM374 548Q347 604 286 604Q247 604 218 575Q197 552 193 511T188 311Q188 159 196 116Q202 87 225 64T287 41Q339 41 367 87Q379 107 382 152T386 329Q386 518 374 548Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D432\" xlink:href=\"#MJX-558-TEX-B-1D432\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(884.8,0)\"><use data-c=\"2260\" xlink:href=\"#MJX-558-TEX-N-2260\"><\/use><\/g><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\" transform=\"translate(1940.6,0)\"><g data-mml-node=\"mn\"><use data-c=\"1D7CE\" xlink:href=\"#MJX-558-TEX-B-1D7CE\"><\/use><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">y<\/mi><\/mrow><mo>\u2260<\/mo><mrow data-mjx-texclass=\"ORD\"><mn mathvariant=\"bold\">0<\/mn><\/mrow><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">\\mathbf y\\neq\\mathbf 0<\/script><span>, we have for <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"2.036ex\" height=\"1.552ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -686 900 686\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: 0px;\"><defs><path id=\"MJX-560-TEX-B-1D407\" d=\"M400 0Q376 3 226 3Q75 3 51 0H39V62H147V624H39V686H51Q75 683 226 683Q376 683 400 686H412V624H304V388H595V624H487V686H499Q523 683 673 683Q824 683 848 686H860V624H752V62H860V0H848Q824 3 674 3Q523 3 499 0H487V62H595V326H304V62H412V0H400Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D407\" xlink:href=\"#MJX-560-TEX-B-1D407\"><\/use><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">H<\/mi><\/mrow><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">\\mathbf H<\/script><span> being uniformly distributed at random in <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"4.608ex\" height=\"2.801ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -853.7 2036.8 1237.9\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.869ex;\"><defs><path id=\"MJX-561-TEX-D-1D53D\" d=\"M584 499Q569 490 566 490Q558 490 552 497T546 515Q546 535 533 559Q526 574 506 593T469 621Q415 648 326 648Q293 648 287 647T275 641Q264 630 263 617Q262 609 260 492V370L275 372Q323 376 350 392T393 441Q409 473 409 506Q409 529 427 529Q437 529 442 519Q444 511 444 362Q444 212 442 206Q436 197 426 197Q409 197 409 217Q409 265 375 299Q346 328 280 335H260V206Q260 70 262 63Q265 46 276 41T326 35Q362 35 366 28Q377 17 366 3L360 -1H24Q12 5 12 16Q12 35 51 35Q92 38 97 52Q102 60 102 341T97 632Q91 645 51 648Q12 648 12 666Q12 675 24 683H573Q576 678 584 670V499ZM137 341Q137 131 136 89T130 37Q129 36 129 35H182Q233 35 233 39Q226 54 225 92T224 346L226 623L231 635L235 648H129Q132 641 133 638T135 603T137 517T137 341ZM549 603V648H495L506 641Q531 621 533 619L549 603ZM409 317V395L400 386Q390 376 375 366L357 355L373 346Q394 331 397 328L409 317Z\"><\/path><path id=\"MJX-561-TEX-I-1D45B\" d=\"M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z\"><\/path><path id=\"MJX-561-TEX-N-2212\" d=\"M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z\"><\/path><path id=\"MJX-561-TEX-I-1D458\" d=\"M121 647Q121 657 125 670T137 683Q138 683 209 688T282 694Q294 694 294 686Q294 679 244 477Q194 279 194 272Q213 282 223 291Q247 309 292 354T362 415Q402 442 438 442Q468 442 485 423T503 369Q503 344 496 327T477 302T456 291T438 288Q418 288 406 299T394 328Q394 353 410 369T442 390L458 393Q446 405 434 405H430Q398 402 367 380T294 316T228 255Q230 254 243 252T267 246T293 238T320 224T342 206T359 180T365 147Q365 130 360 106T354 66Q354 26 381 26Q429 26 459 145Q461 153 479 153H483Q499 153 499 144Q499 139 496 130Q455 -11 378 -11Q333 -11 305 15T277 90Q277 108 280 121T283 145Q283 167 269 183T234 206T200 217T182 220H180Q168 178 159 139T145 81T136 44T129 20T122 7T111 -2Q98 -11 83 -11Q66 -11 57 -1T48 16Q48 26 85 176T158 471L195 616Q196 629 188 632T149 637H144Q134 637 131 637T124 640T121 647Z\"><\/path><path id=\"MJX-561-TEX-I-1D45E\" d=\"M33 157Q33 258 109 349T280 441Q340 441 372 389Q373 390 377 395T388 406T404 418Q438 442 450 442Q454 442 457 439T460 434Q460 425 391 149Q320 -135 320 -139Q320 -147 365 -148H390Q396 -156 396 -157T393 -175Q389 -188 383 -194H370Q339 -192 262 -192Q234 -192 211 -192T174 -192T157 -193Q143 -193 143 -185Q143 -182 145 -170Q149 -154 152 -151T172 -148Q220 -148 230 -141Q238 -136 258 -53T279 32Q279 33 272 29Q224 -10 172 -10Q117 -10 75 30T33 157ZM352 326Q329 405 277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q233 26 290 98L298 109L352 326Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"msubsup\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D53D\" xlink:href=\"#MJX-561-TEX-D-1D53D\"><\/use><\/g><\/g><g data-mml-node=\"TeXAtom\" transform=\"translate(644,363) scale(0.707)\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D45B\" xlink:href=\"#MJX-561-TEX-I-1D45B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(600,0)\"><use data-c=\"2212\" xlink:href=\"#MJX-561-TEX-N-2212\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1378,0)\"><use data-c=\"1D458\" xlink:href=\"#MJX-561-TEX-I-1D458\"><\/use><\/g><\/g><g data-mml-node=\"mi\" transform=\"translate(644,-247) scale(0.707)\"><use data-c=\"1D45E\" xlink:href=\"#MJX-561-TEX-I-1D45E\"><\/use><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><msubsup><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"double-struck\">F<\/mi><\/mrow><mi>q<\/mi><mrow data-mjx-texclass=\"ORD\"><mi>n<\/mi><mo>\u2212<\/mo><mi>k<\/mi><\/mrow><\/msubsup><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">\\mathbb{F}_q^{n-k}<\/script><span>,<\/span><\/p><div contenteditable=\"false\" spellcheck=\"false\" class=\"mathjax-block md-end-block md-math-block md-rawblock\" id=\"mathjax-n219\" cid=\"n219\" mdtype=\"math_block\" data-math-tag-before=\"10\" data-math-labels=\"[]\" data-math-tag-after=\"10\"><div class=\"md-rawblock-container md-math-container\" contenteditable=\"false\" tabindex=\"-1\"><mjx-container class=\"MathJax\" jax=\"SVG\" display=\"true\" width=\"full\" style=\"min-width: 25.385ex; position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"100%\" height=\"5.149ex\" role=\"img\" focusable=\"false\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -2.009ex; min-width: 25.385ex;\"><defs><path id=\"MJX-1991-TEX-I-1D443\" d=\"M287 628Q287 635 230 637Q206 637 199 638T192 648Q192 649 194 659Q200 679 203 681T397 683Q587 682 600 680Q664 669 707 631T751 530Q751 453 685 389Q616 321 507 303Q500 302 402 301H307L277 182Q247 66 247 59Q247 55 248 54T255 50T272 48T305 46H336Q342 37 342 35Q342 19 335 5Q330 0 319 0Q316 0 282 1T182 2Q120 2 87 2T51 1Q33 1 33 11Q33 13 36 25Q40 41 44 43T67 46Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628ZM645 554Q645 567 643 575T634 597T609 619T560 635Q553 636 480 637Q463 637 445 637T416 636T404 636Q391 635 386 627Q384 621 367 550T332 412T314 344Q314 342 395 342H407H430Q542 342 590 392Q617 419 631 471T645 554Z\"><\/path><path id=\"MJX-1991-TEX-B-1D407\" d=\"M400 0Q376 3 226 3Q75 3 51 0H39V62H147V624H39V686H51Q75 683 226 683Q376 683 400 686H412V624H304V388H595V624H487V686H499Q523 683 673 683Q824 683 848 686H860V624H752V62H860V0H848Q824 3 674 3Q523 3 499 0H487V62H595V326H304V62H412V0H400Z\"><\/path><path id=\"MJX-1991-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 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transform=\"translate(5067.1,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D42C\" xlink:href=\"#MJX-1991-TEX-B-1D42C\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(5521.1,0)\"><use data-c=\"29\" xlink:href=\"#MJX-1991-TEX-N-29\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(6187.9,0)\"><use data-c=\"3D\" xlink:href=\"#MJX-1991-TEX-N-3D\"><\/use><\/g><g data-mml-node=\"mfrac\" transform=\"translate(7243.7,0)\"><g data-mml-node=\"mn\" transform=\"translate(938.2,676)\"><use data-c=\"31\" xlink:href=\"#MJX-1991-TEX-N-31\"><\/use><\/g><g data-mml-node=\"msup\" transform=\"translate(220,-739.7)\"><g data-mml-node=\"mi\"><use data-c=\"1D45E\" xlink:href=\"#MJX-1991-TEX-I-1D45E\"><\/use><\/g><g data-mml-node=\"TeXAtom\" transform=\"translate(543.7,289) scale(0.707)\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D45B\" xlink:href=\"#MJX-1991-TEX-I-1D45B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(600,0)\"><use data-c=\"2212\" xlink:href=\"#MJX-1991-TEX-N-2212\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1378,0)\"><use data-c=\"1D458\" xlink:href=\"#MJX-1991-TEX-I-1D458\"><\/use><\/g><\/g><\/g><rect width=\"2136.5\" height=\"60\" x=\"120\" y=\"220\"><\/rect><\/g><\/g><\/g><\/g><\/svg><svg data-labels=\"true\" preserveAspectRatio=\"xMaxYMid\" viewBox=\"0 -1387.9 1 2275.7\"><g data-labels=\"true\" transform=\"matrix(1 0 0 -1 0 0)\"><g data-mml-node=\"mtd\" id=\"mjx-eqn:\" transform=\"translate(0,45.9)\"><text data-id-align=\"true\"><\/text><g data-idbox=\"true\" transform=\"translate(0,0)\"><g data-mml-node=\"mrow\"><\/g><\/g><\/g><\/g><\/svg><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"block\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable displaystyle=\"true\"><mlabeledtr><mtd><mrow><\/mrow><\/mtd><mtd><msub><mi>P<\/mi><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">H<\/mi><\/mrow><\/mrow><\/msub><mo stretchy=\"false\">(<\/mo><mrow data-mjx-texclass=\"ORD\"><msup><mi mathvariant=\"bold\">yH<\/mi><mo>\u22ba<\/mo><\/msup><\/mrow><mo>=<\/mo><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">s<\/mi><\/mrow><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mfrac><mn>1<\/mn><msup><mi>q<\/mi><mrow data-mjx-texclass=\"ORD\"><mi>n<\/mi><mo>\u2212<\/mo><mi>k<\/mi><\/mrow><\/msup><\/mfrac><\/mtd><\/mlabeledtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><p><span>This lemma gives the probability over all random codes that a fixed non-zero word <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.373ex\" height=\"1.457ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -444 607 644\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.452ex;\"><defs><path id=\"MJX-1600-TEX-B-1D432\" d=\"M84 -102Q84 -110 87 -119T102 -138T133 -149Q148 -148 162 -143T186 -131T206 -114T222 -95T234 -76T243 -59T249 -45T252 -37L269 0L96 382H26V444H34Q49 441 146 441Q252 441 270 444H279V382H255Q232 382 232 380L337 151L442 382H394V444H401Q413 441 495 441Q568 441 574 444H580V382H510L406 152Q298 -84 297 -87Q269 -139 225 -169T131 -200Q85 -200 54 -172T23 -100Q23 -64 44 -50T87 -35Q111 -35 130 -50T152 -92V-100H84V-102Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D432\" xlink:href=\"#MJX-1600-TEX-B-1D432\"><\/use><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">y<\/mi><\/mrow><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">\\mathbf y<\/script><span> reaches some syndrome <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.027ex\" height=\"1.038ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -453 454 459\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.014ex;\"><defs><path id=\"MJX-1800-TEX-B-1D42C\" d=\"M38 315Q38 339 45 360T70 404T127 440T223 453Q273 453 320 436L338 445L357 453H366Q380 453 383 447T386 403V387V355Q386 331 383 326T365 321H355H349Q333 321 329 324T324 341Q317 406 224 406H216Q123 406 123 353Q123 334 143 321T188 304T244 294T285 286Q305 281 325 273T373 237T412 172Q414 162 414 142Q414 -6 230 -6Q154 -6 117 22L68 -6H58Q44 -6 41 0T38 42V73Q38 85 38 101T37 122Q37 144 42 148T68 153H75Q87 153 91 151T97 147T103 132Q131 46 220 46H230Q257 46 265 47Q330 58 330 108Q330 127 316 142Q300 156 284 162Q271 168 212 178T122 202Q38 243 38 315Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D42C\" xlink:href=\"#MJX-1800-TEX-B-1D42C\"><\/use><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">s<\/mi><\/mrow><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">\\mathbf s<\/script><span> according to a certain code.<\/span><\/p><p><em><span>Proof.<\/span><\/em><span> We assume <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"6.245ex\" height=\"1.971ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -666 2760.1 871\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.464ex;\"><defs><path 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0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z\"><\/path><path id=\"MJX-634-TEX-N-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"msub\"><g data-mml-node=\"mi\"><use data-c=\"1D466\" xlink:href=\"#MJX-634-TEX-I-1D466\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(523,-150) scale(0.707)\"><use data-c=\"31\" xlink:href=\"#MJX-634-TEX-N-31\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(1204.3,0)\"><use data-c=\"3D\" xlink:href=\"#MJX-634-TEX-N-3D\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(2260.1,0)\"><use data-c=\"31\" xlink:href=\"#MJX-634-TEX-N-31\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><msub><mi>y<\/mi><mn>1<\/mn><\/msub><mo>=<\/mo><mn>1<\/mn><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">y_1=1<\/script><span>, then we are looking for the probability of the following event<\/span><\/p><div contenteditable=\"false\" spellcheck=\"false\" class=\"mathjax-block md-end-block md-math-block md-rawblock\" id=\"mathjax-n226\" cid=\"n226\" mdtype=\"math_block\" data-math-tag-before=\"10\" data-math-labels=\"[]\" data-math-tag-after=\"10\"><div class=\"md-rawblock-container md-math-container\" contenteditable=\"false\" tabindex=\"-1\"><mjx-container class=\"MathJax\" jax=\"SVG\" display=\"true\" width=\"full\" style=\"min-width: 42.784ex; position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"100%\" height=\"6.549ex\" role=\"img\" focusable=\"false\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -2.709ex; min-width: 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-165Q219 -156 247 -127T290 -65T313 -9T321 21L315 17Q309 13 296 6T270 -6Q250 -11 231 -11Q185 -11 150 11T104 82Q103 89 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"msub\"><g data-mml-node=\"mi\"><use data-c=\"210E\" xlink:href=\"#MJX-712-TEX-I-210E\"><\/use><\/g><g data-mml-node=\"TeXAtom\" transform=\"translate(609,-150) scale(0.707)\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D456\" xlink:href=\"#MJX-712-TEX-I-1D456\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(345,0)\"><use data-c=\"2C\" xlink:href=\"#MJX-712-TEX-N-2C\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(623,0)\"><use data-c=\"31\" xlink:href=\"#MJX-712-TEX-N-31\"><\/use><\/g><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(1730.9,0)\"><use data-c=\"3D\" xlink:href=\"#MJX-712-TEX-N-3D\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(2786.6,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D460\" xlink:href=\"#MJX-712-TEX-I-1D460\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(502,-150) scale(0.707)\"><use data-c=\"1D456\" xlink:href=\"#MJX-712-TEX-I-1D456\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(3804.8,0)\"><use data-c=\"2212\" xlink:href=\"#MJX-712-TEX-N-2212\"><\/use><\/g><g data-mml-node=\"munderover\" transform=\"translate(4805,0)\"><g data-mml-node=\"mo\"><use data-c=\"2211\" xlink:href=\"#MJX-712-TEX-SO-2211\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1089,477.1) scale(0.707)\"><use data-c=\"1D45B\" xlink:href=\"#MJX-712-TEX-I-1D45B\"><\/use><\/g><g data-mml-node=\"TeXAtom\" transform=\"translate(1089,-285.4) scale(0.707)\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D457\" xlink:href=\"#MJX-712-TEX-I-1D457\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(412,0)\"><use data-c=\"3D\" xlink:href=\"#MJX-712-TEX-N-3D\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(1190,0)\"><use data-c=\"32\" xlink:href=\"#MJX-712-TEX-N-32\"><\/use><\/g><\/g><\/g><g data-mml-node=\"msub\" transform=\"translate(7305.7,0)\"><g data-mml-node=\"mi\"><use data-c=\"210E\" xlink:href=\"#MJX-712-TEX-I-210E\"><\/use><\/g><g data-mml-node=\"TeXAtom\" transform=\"translate(609,-150) scale(0.707)\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D456\" xlink:href=\"#MJX-712-TEX-I-1D456\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(345,0)\"><use data-c=\"2C\" xlink:href=\"#MJX-712-TEX-N-2C\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(623,0)\"><use data-c=\"1D457\" xlink:href=\"#MJX-712-TEX-I-1D457\"><\/use><\/g><\/g><\/g><g data-mml-node=\"msub\" transform=\"translate(8696.6,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D466\" xlink:href=\"#MJX-712-TEX-I-1D466\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(523,-150) scale(0.707)\"><use data-c=\"1D457\" xlink:href=\"#MJX-712-TEX-I-1D457\"><\/use><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><msub><mi>h<\/mi><mrow data-mjx-texclass=\"ORD\"><mi>i<\/mi><mo>,<\/mo><mn>1<\/mn><\/mrow><\/msub><mo>=<\/mo><msub><mi>s<\/mi><mi>i<\/mi><\/msub><mo>\u2212<\/mo><munderover><mo data-mjx-texclass=\"OP\">\u2211<\/mo><mrow data-mjx-texclass=\"ORD\"><mi>j<\/mi><mo>=<\/mo><mn>2<\/mn><\/mrow><mi>n<\/mi><\/munderover><msub><mi>h<\/mi><mrow data-mjx-texclass=\"ORD\"><mi>i<\/mi><mo>,<\/mo><mi>j<\/mi><\/mrow><\/msub><msub><mi>y<\/mi><mi>j<\/mi><\/msub><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">h_{i,1}=s_i-\\sum_{j=2}^nh_{i,j}y_j<\/script><span> only have <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.041ex\" height=\"1.439ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -442 460 636\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.439ex;\"><defs><path id=\"MJX-713-TEX-I-1D45E\" d=\"M33 157Q33 258 109 349T280 441Q340 441 372 389Q373 390 377 395T388 406T404 418Q438 442 450 442Q454 442 457 439T460 434Q460 425 391 149Q320 -135 320 -139Q320 -147 365 -148H390Q396 -156 396 -157T393 -175Q389 -188 383 -194H370Q339 -192 262 -192Q234 -192 211 -192T174 -192T157 -193Q143 -193 143 -185Q143 -182 145 -170Q149 -154 152 -151T172 -148Q220 -148 230 -141Q238 -136 258 -53T279 32Q279 33 272 29Q224 -10 172 -10Q117 -10 75 30T33 157ZM352 326Q329 405 277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q233 26 290 98L298 109L352 326Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D45E\" xlink:href=\"#MJX-713-TEX-I-1D45E\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>q<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">q<\/script><span> possible values, and <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"3.147ex\" height=\"2.236ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -694 1390.9 988.2\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.666ex;\"><defs><path id=\"MJX-715-TEX-I-210E\" d=\"M137 683Q138 683 209 688T282 694Q294 694 294 685Q294 674 258 534Q220 386 220 383Q220 381 227 388Q288 442 357 442Q411 442 444 415T478 336Q478 285 440 178T402 50Q403 36 407 31T422 26Q450 26 474 56T513 138Q516 149 519 151T535 153Q555 153 555 145Q555 144 551 130Q535 71 500 33Q466 -10 419 -10H414Q367 -10 346 17T325 74Q325 90 361 192T398 345Q398 404 354 404H349Q266 404 205 306L198 293L164 158Q132 28 127 16Q114 -11 83 -11Q69 -11 59 -2T48 16Q48 30 121 320L195 616Q195 629 188 632T149 637H128Q122 643 122 645T124 664Q129 683 137 683Z\"><\/path><path id=\"MJX-715-TEX-I-1D456\" d=\"M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z\"><\/path><path id=\"MJX-715-TEX-N-2C\" d=\"M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z\"><\/path><path id=\"MJX-715-TEX-I-1D457\" d=\"M297 596Q297 627 318 644T361 661Q378 661 389 651T403 623Q403 595 384 576T340 557Q322 557 310 567T297 596ZM288 376Q288 405 262 405Q240 405 220 393T185 362T161 325T144 293L137 279Q135 278 121 278H107Q101 284 101 286T105 299Q126 348 164 391T252 441Q253 441 260 441T272 442Q296 441 316 432Q341 418 354 401T367 348V332L318 133Q267 -67 264 -75Q246 -125 194 -164T75 -204Q25 -204 7 -183T-12 -137Q-12 -110 7 -91T53 -71Q70 -71 82 -81T95 -112Q95 -148 63 -167Q69 -168 77 -168Q111 -168 139 -140T182 -74L193 -32Q204 11 219 72T251 197T278 308T289 365Q289 372 288 376Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"msub\"><g data-mml-node=\"mi\"><use data-c=\"210E\" xlink:href=\"#MJX-715-TEX-I-210E\"><\/use><\/g><g data-mml-node=\"TeXAtom\" transform=\"translate(609,-150) scale(0.707)\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D456\" xlink:href=\"#MJX-715-TEX-I-1D456\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(345,0)\"><use data-c=\"2C\" xlink:href=\"#MJX-715-TEX-N-2C\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(623,0)\"><use data-c=\"1D457\" xlink:href=\"#MJX-715-TEX-I-1D457\"><\/use><\/g><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><msub><mi>h<\/mi><mrow data-mjx-texclass=\"ORD\"><mi>i<\/mi><mo>,<\/mo><mi>j<\/mi><\/mrow><\/msub><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">h_{i,j}<\/script><span>&#39;s are independent and equidistributed, the above <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"5.302ex\" height=\"1.756ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -694 2343.4 776\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.186ex;\"><defs><path id=\"MJX-1297-TEX-I-1D45B\" d=\"M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z\"><\/path><path id=\"MJX-1297-TEX-N-2212\" d=\"M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z\"><\/path><path id=\"MJX-1297-TEX-I-1D458\" d=\"M121 647Q121 657 125 670T137 683Q138 683 209 688T282 694Q294 694 294 686Q294 679 244 477Q194 279 194 272Q213 282 223 291Q247 309 292 354T362 415Q402 442 438 442Q468 442 485 423T503 369Q503 344 496 327T477 302T456 291T438 288Q418 288 406 299T394 328Q394 353 410 369T442 390L458 393Q446 405 434 405H430Q398 402 367 380T294 316T228 255Q230 254 243 252T267 246T293 238T320 224T342 206T359 180T365 147Q365 130 360 106T354 66Q354 26 381 26Q429 26 459 145Q461 153 479 153H483Q499 153 499 144Q499 139 496 130Q455 -11 378 -11Q333 -11 305 15T277 90Q277 108 280 121T283 145Q283 167 269 183T234 206T200 217T182 220H180Q168 178 159 139T145 81T136 44T129 20T122 7T111 -2Q98 -11 83 -11Q66 -11 57 -1T48 16Q48 26 85 176T158 471L195 616Q196 629 188 632T149 637H144Q134 637 131 637T124 640T121 647Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D45B\" xlink:href=\"#MJX-1297-TEX-I-1D45B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(822.2,0)\"><use data-c=\"2212\" xlink:href=\"#MJX-1297-TEX-N-2212\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1822.4,0)\"><use data-c=\"1D458\" xlink:href=\"#MJX-1297-TEX-I-1D458\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>n<\/mi><mo>\u2212<\/mo><mi>k<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">n-k<\/script><span> equations will be independently true with probability <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"3.303ex\" height=\"2.262ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 1460 1000\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-718-TEX-N-31\" d=\"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z\"><\/path><path id=\"MJX-718-TEX-N-2F\" d=\"M423 750Q432 750 438 744T444 730Q444 725 271 248T92 -240Q85 -250 75 -250Q68 -250 62 -245T56 -231Q56 -221 230 257T407 740Q411 750 423 750Z\"><\/path><path id=\"MJX-718-TEX-I-1D45E\" d=\"M33 157Q33 258 109 349T280 441Q340 441 372 389Q373 390 377 395T388 406T404 418Q438 442 450 442Q454 442 457 439T460 434Q460 425 391 149Q320 -135 320 -139Q320 -147 365 -148H390Q396 -156 396 -157T393 -175Q389 -188 383 -194H370Q339 -192 262 -192Q234 -192 211 -192T174 -192T157 -193Q143 -193 143 -185Q143 -182 145 -170Q149 -154 152 -151T172 -148Q220 -148 230 -141Q238 -136 258 -53T279 32Q279 33 272 29Q224 -10 172 -10Q117 -10 75 30T33 157ZM352 326Q329 405 277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q233 26 290 98L298 109L352 326Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mn\"><use data-c=\"31\" xlink:href=\"#MJX-718-TEX-N-31\"><\/use><\/g><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\" transform=\"translate(500,0)\"><g data-mml-node=\"mo\"><use data-c=\"2F\" xlink:href=\"#MJX-718-TEX-N-2F\"><\/use><\/g><\/g><g data-mml-node=\"mi\" transform=\"translate(1000,0)\"><use data-c=\"1D45E\" xlink:href=\"#MJX-718-TEX-I-1D45E\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mn>1<\/mn><mrow data-mjx-texclass=\"ORD\"><mo>\/<\/mo><\/mrow><mi>q<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">1\/q<\/script><span>. This concludes the proof.<\/span><\/p><p><strong><span>Question:<\/span><\/strong><span> given a random code <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.192ex\" height=\"1.652ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -705 527 730\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.057ex;\"><defs><path id=\"MJX-32-TEX-C-43\" d=\"M201 -25Q167 -25 136 -14T75 23T29 94T12 202Q12 290 50 394T161 574Q227 642 303 673T433 704Q435 705 457 705Q533 701 533 640Q533 606 507 548T464 474Q431 444 396 444Q381 444 381 453Q381 459 388 473T407 513T428 563Q433 580 433 594Q433 636 381 636Q314 636 260 594T175 489T128 363T112 247Q112 157 153 101T273 44Q347 44 398 121Q413 144 437 157T481 171Q496 171 496 160Q496 150 476 123Q426 56 350 16T201 -25Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"43\" xlink:href=\"#MJX-32-TEX-C-43\"><\/use><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow data-mjx-texclass=\"ORD\"><mi data-mjx-variant=\"-tex-calligraphic\" mathvariant=\"script\">C<\/mi><\/mrow><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">\\mathcal{C}<\/script><span> and a fixed vector <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"6.669ex\" height=\"2.414ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -683 2947.8 1067.2\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.869ex;\"><defs><path id=\"MJX-721-TEX-B-1D432\" d=\"M84 -102Q84 -110 87 -119T102 -138T133 -149Q148 -148 162 -143T186 -131T206 -114T222 -95T234 -76T243 -59T249 -45T252 -37L269 0L96 382H26V444H34Q49 441 146 441Q252 441 270 444H279V382H255Q232 382 232 380L337 151L442 382H394V444H401Q413 441 495 441Q568 441 574 444H580V382H510L406 152Q298 -84 297 -87Q269 -139 225 -169T131 -200Q85 -200 54 -172T23 -100Q23 -64 44 -50T87 -35Q111 -35 130 -50T152 -92V-100H84V-102Z\"><\/path><path id=\"MJX-721-TEX-N-2208\" d=\"M84 250Q84 372 166 450T360 539Q361 539 377 539T419 540T469 540H568Q583 532 583 520Q583 511 570 501L466 500Q355 499 329 494Q280 482 242 458T183 409T147 354T129 306T124 272V270H568Q583 262 583 250T568 230H124V228Q124 207 134 177T167 112T231 48T328 7Q355 1 466 0H570Q583 -10 583 -20Q583 -32 568 -40H471Q464 -40 446 -40T417 -41Q262 -41 172 45Q84 127 84 250Z\"><\/path><path id=\"MJX-721-TEX-D-1D53D\" d=\"M584 499Q569 490 566 490Q558 490 552 497T546 515Q546 535 533 559Q526 574 506 593T469 621Q415 648 326 648Q293 648 287 647T275 641Q264 630 263 617Q262 609 260 492V370L275 372Q323 376 350 392T393 441Q409 473 409 506Q409 529 427 529Q437 529 442 519Q444 511 444 362Q444 212 442 206Q436 197 426 197Q409 197 409 217Q409 265 375 299Q346 328 280 335H260V206Q260 70 262 63Q265 46 276 41T326 35Q362 35 366 28Q377 17 366 3L360 -1H24Q12 5 12 16Q12 35 51 35Q92 38 97 52Q102 60 102 341T97 632Q91 645 51 648Q12 648 12 666Q12 675 24 683H573Q576 678 584 670V499ZM137 341Q137 131 136 89T130 37Q129 36 129 35H182Q233 35 233 39Q226 54 225 92T224 346L226 623L231 635L235 648H129Q132 641 133 638T135 603T137 517T137 341ZM549 603V648H495L506 641Q531 621 533 619L549 603ZM409 317V395L400 386Q390 376 375 366L357 355L373 346Q394 331 397 328L409 317Z\"><\/path><path id=\"MJX-721-TEX-I-1D45B\" d=\"M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z\"><\/path><path id=\"MJX-721-TEX-I-1D45E\" d=\"M33 157Q33 258 109 349T280 441Q340 441 372 389Q373 390 377 395T388 406T404 418Q438 442 450 442Q454 442 457 439T460 434Q460 425 391 149Q320 -135 320 -139Q320 -147 365 -148H390Q396 -156 396 -157T393 -175Q389 -188 383 -194H370Q339 -192 262 -192Q234 -192 211 -192T174 -192T157 -193Q143 -193 143 -185Q143 -182 145 -170Q149 -154 152 -151T172 -148Q220 -148 230 -141Q238 -136 258 -53T279 32Q279 33 272 29Q224 -10 172 -10Q117 -10 75 30T33 157ZM352 326Q329 405 277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q233 26 290 98L298 109L352 326Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D432\" xlink:href=\"#MJX-721-TEX-B-1D432\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(884.8,0)\"><use data-c=\"2208\" xlink:href=\"#MJX-721-TEX-N-2208\"><\/use><\/g><g data-mml-node=\"msubsup\" transform=\"translate(1829.6,0)\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D53D\" xlink:href=\"#MJX-721-TEX-D-1D53D\"><\/use><\/g><\/g><g data-mml-node=\"mi\" transform=\"translate(644,363) scale(0.707)\"><use data-c=\"1D45B\" xlink:href=\"#MJX-721-TEX-I-1D45B\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(644,-247) scale(0.707)\"><use data-c=\"1D45E\" xlink:href=\"#MJX-721-TEX-I-1D45E\"><\/use><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">y<\/mi><\/mrow><mo>\u2208<\/mo><msubsup><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"double-struck\">F<\/mi><\/mrow><mi>q<\/mi><mi>n<\/mi><\/msubsup><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">\\mathbf y \\in \\mathbb{F}_q^n<\/script><span>, how many <\/span><u><span>codewords<\/span><\/u><span> <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"5.114ex\" height=\"1.686ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -705 2260.6 745\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.09ex;\"><defs><path id=\"MJX-725-TEX-B-1D41C\" d=\"M447 131H458Q478 131 478 117Q478 112 471 95T439 51T377 9Q330 -6 286 -6Q196 -6 135 35Q39 96 39 222Q39 324 101 384Q169 453 286 453Q359 453 411 431T464 353Q464 319 445 302T395 284Q360 284 343 305T325 353Q325 380 338 396H333Q317 398 295 398H292Q280 398 271 397T245 390T218 373T197 338T183 283Q182 275 182 231Q182 199 184 180T193 132T220 85T270 57Q289 50 317 50H326Q385 50 414 115Q419 127 423 129T447 131Z\"><\/path><path id=\"MJX-725-TEX-N-2208\" d=\"M84 250Q84 372 166 450T360 539Q361 539 377 539T419 540T469 540H568Q583 532 583 520Q583 511 570 501L466 500Q355 499 329 494Q280 482 242 458T183 409T147 354T129 306T124 272V270H568Q583 262 583 250T568 230H124V228Q124 207 134 177T167 112T231 48T328 7Q355 1 466 0H570Q583 -10 583 -20Q583 -32 568 -40H471Q464 -40 446 -40T417 -41Q262 -41 172 45Q84 127 84 250Z\"><\/path><path id=\"MJX-725-TEX-C-43\" d=\"M201 -25Q167 -25 136 -14T75 23T29 94T12 202Q12 290 50 394T161 574Q227 642 303 673T433 704Q435 705 457 705Q533 701 533 640Q533 606 507 548T464 474Q431 444 396 444Q381 444 381 453Q381 459 388 473T407 513T428 563Q433 580 433 594Q433 636 381 636Q314 636 260 594T175 489T128 363T112 247Q112 157 153 101T273 44Q347 44 398 121Q413 144 437 157T481 171Q496 171 496 160Q496 150 476 123Q426 56 350 16T201 -25Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D41C\" xlink:href=\"#MJX-725-TEX-B-1D41C\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(788.8,0)\"><use data-c=\"2208\" xlink:href=\"#MJX-725-TEX-N-2208\"><\/use><\/g><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\" transform=\"translate(1733.6,0)\"><g data-mml-node=\"mi\"><use data-c=\"43\" xlink:href=\"#MJX-725-TEX-C-43\"><\/use><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">c<\/mi><\/mrow><mo>\u2208<\/mo><mrow data-mjx-texclass=\"ORD\"><mi data-mjx-variant=\"-tex-calligraphic\" mathvariant=\"script\">C<\/mi><\/mrow><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">\\mathbf c\\in\\mathcal{C}<\/script><span> do we expect to be at Hamming distance <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"0.817ex\" height=\"1.441ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -626 361 637\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.025ex;\"><defs><path id=\"MJX-1284-TEX-I-1D461\" d=\"M26 385Q19 392 19 395Q19 399 22 411T27 425Q29 430 36 430T87 431H140L159 511Q162 522 166 540T173 566T179 586T187 603T197 615T211 624T229 626Q247 625 254 615T261 596Q261 589 252 549T232 470L222 433Q222 431 272 431H323Q330 424 330 420Q330 398 317 385H210L174 240Q135 80 135 68Q135 26 162 26Q197 26 230 60T283 144Q285 150 288 151T303 153H307Q322 153 322 145Q322 142 319 133Q314 117 301 95T267 48T216 6T155 -11Q125 -11 98 4T59 56Q57 64 57 83V101L92 241Q127 382 128 383Q128 385 77 385H26Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D461\" xlink:href=\"#MJX-1284-TEX-I-1D461\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>t<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">t<\/script><span> from <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.373ex\" height=\"1.457ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -444 607 644\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.452ex;\"><defs><path id=\"MJX-1600-TEX-B-1D432\" d=\"M84 -102Q84 -110 87 -119T102 -138T133 -149Q148 -148 162 -143T186 -131T206 -114T222 -95T234 -76T243 -59T249 -45T252 -37L269 0L96 382H26V444H34Q49 441 146 441Q252 441 270 444H279V382H255Q232 382 232 380L337 151L442 382H394V444H401Q413 441 495 441Q568 441 574 444H580V382H510L406 152Q298 -84 297 -87Q269 -139 225 -169T131 -200Q85 -200 54 -172T23 -100Q23 -64 44 -50T87 -35Q111 -35 130 -50T152 -92V-100H84V-102Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D432\" xlink:href=\"#MJX-1600-TEX-B-1D432\"><\/use><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">y<\/mi><\/mrow><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">\\mathbf y<\/script><span>, i.e., <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"9.505ex\" height=\"1.771ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -583 4201 783\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.452ex;\"><defs><path id=\"MJX-768-TEX-B-1D432\" d=\"M84 -102Q84 -110 87 -119T102 -138T133 -149Q148 -148 162 -143T186 -131T206 -114T222 -95T234 -76T243 -59T249 -45T252 -37L269 0L96 382H26V444H34Q49 441 146 441Q252 441 270 444H279V382H255Q232 382 232 380L337 151L442 382H394V444H401Q413 441 495 441Q568 441 574 444H580V382H510L406 152Q298 -84 297 -87Q269 -139 225 -169T131 -200Q85 -200 54 -172T23 -100Q23 -64 44 -50T87 -35Q111 -35 130 -50T152 -92V-100H84V-102Z\"><\/path><path id=\"MJX-768-TEX-N-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path id=\"MJX-768-TEX-B-1D41C\" d=\"M447 131H458Q478 131 478 117Q478 112 471 95T439 51T377 9Q330 -6 286 -6Q196 -6 135 35Q39 96 39 222Q39 324 101 384Q169 453 286 453Q359 453 411 431T464 353Q464 319 445 302T395 284Q360 284 343 305T325 353Q325 380 338 396H333Q317 398 295 398H292Q280 398 271 397T245 390T218 373T197 338T183 283Q182 275 182 231Q182 199 184 180T193 132T220 85T270 57Q289 50 317 50H326Q385 50 414 115Q419 127 423 129T447 131Z\"><\/path><path id=\"MJX-768-TEX-N-2B\" d=\"M56 237T56 250T70 270H369V420L370 570Q380 583 389 583Q402 583 409 568V270H707Q722 262 722 250T707 230H409V-68Q401 -82 391 -82H389H387Q375 -82 369 -68V230H70Q56 237 56 250Z\"><\/path><path id=\"MJX-768-TEX-B-1D41E\" d=\"M32 225Q32 332 102 392T272 452H283Q382 452 436 401Q494 343 494 243Q494 226 486 222T440 217Q431 217 394 217T327 218H175V209Q175 177 179 154T196 107T236 69T306 50Q312 49 323 49Q376 49 410 85Q421 99 427 111T434 127T442 133T463 135H468Q494 135 494 117Q494 110 489 97T468 66T431 32T373 5T292 -6Q181 -6 107 55T32 225ZM383 276Q377 346 348 374T280 402Q253 402 230 390T195 357Q179 331 176 279V266H383V276Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D432\" xlink:href=\"#MJX-768-TEX-B-1D432\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(884.8,0)\"><use data-c=\"3D\" xlink:href=\"#MJX-768-TEX-N-3D\"><\/use><\/g><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\" transform=\"translate(1940.6,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D41C\" xlink:href=\"#MJX-768-TEX-B-1D41C\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(2673.8,0)\"><use data-c=\"2B\" xlink:href=\"#MJX-768-TEX-N-2B\"><\/use><\/g><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\" transform=\"translate(3674,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D41E\" xlink:href=\"#MJX-768-TEX-B-1D41E\"><\/use><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">y<\/mi><\/mrow><mo>=<\/mo><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">c<\/mi><\/mrow><mo>+<\/mo><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">e<\/mi><\/mrow><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">\\mathbf{y}=\\mathbf{c}+\\mathbf e<\/script><span>? Or equivalently, given a parity-check matrix of our random code <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.192ex\" height=\"1.652ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -705 527 730\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.057ex;\"><defs><path id=\"MJX-32-TEX-C-43\" d=\"M201 -25Q167 -25 136 -14T75 23T29 94T12 202Q12 290 50 394T161 574Q227 642 303 673T433 704Q435 705 457 705Q533 701 533 640Q533 606 507 548T464 474Q431 444 396 444Q381 444 381 453Q381 459 388 473T407 513T428 563Q433 580 433 594Q433 636 381 636Q314 636 260 594T175 489T128 363T112 247Q112 157 153 101T273 44Q347 44 398 121Q413 144 437 157T481 171Q496 171 496 160Q496 150 476 123Q426 56 350 16T201 -25Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"43\" xlink:href=\"#MJX-32-TEX-C-43\"><\/use><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow data-mjx-texclass=\"ORD\"><mi data-mjx-variant=\"-tex-calligraphic\" mathvariant=\"script\">C<\/mi><\/mrow><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">\\mathcal{C}<\/script><span> and a fixed syndrome <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.027ex\" height=\"1.038ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -453 454 459\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.014ex;\"><defs><path id=\"MJX-1800-TEX-B-1D42C\" d=\"M38 315Q38 339 45 360T70 404T127 440T223 453Q273 453 320 436L338 445L357 453H366Q380 453 383 447T386 403V387V355Q386 331 383 326T365 321H355H349Q333 321 329 324T324 341Q317 406 224 406H216Q123 406 123 353Q123 334 143 321T188 304T244 294T285 286Q305 281 325 273T373 237T412 172Q414 162 414 142Q414 -6 230 -6Q154 -6 117 22L68 -6H58Q44 -6 41 0T38 42V73Q38 85 38 101T37 122Q37 144 42 148T68 153H75Q87 153 91 151T97 147T103 132Q131 46 220 46H230Q257 46 265 47Q330 58 330 108Q330 127 316 142Q300 156 284 162Q271 168 212 178T122 202Q38 243 38 315Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D42C\" xlink:href=\"#MJX-1800-TEX-B-1D42C\"><\/use><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">s<\/mi><\/mrow><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">\\mathbf s<\/script><span>, how many <\/span><u><span>error vectors<\/span><\/u><span> <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.192ex\" height=\"1.036ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -452 527 458\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.014ex;\"><defs><path id=\"MJX-1808-TEX-B-1D41E\" d=\"M32 225Q32 332 102 392T272 452H283Q382 452 436 401Q494 343 494 243Q494 226 486 222T440 217Q431 217 394 217T327 218H175V209Q175 177 179 154T196 107T236 69T306 50Q312 49 323 49Q376 49 410 85Q421 99 427 111T434 127T442 133T463 135H468Q494 135 494 117Q494 110 489 97T468 66T431 32T373 5T292 -6Q181 -6 107 55T32 225ZM383 276Q377 346 348 374T280 402Q253 402 230 390T195 357Q179 331 176 279V266H383V276Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D41E\" xlink:href=\"#MJX-1808-TEX-B-1D41E\"><\/use><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">e<\/mi><\/mrow><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">\\mathbf e<\/script><span> of Hamming weight <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"0.817ex\" height=\"1.441ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -626 361 637\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.025ex;\"><defs><path id=\"MJX-1284-TEX-I-1D461\" d=\"M26 385Q19 392 19 395Q19 399 22 411T27 425Q29 430 36 430T87 431H140L159 511Q162 522 166 540T173 566T179 586T187 603T197 615T211 624T229 626Q247 625 254 615T261 596Q261 589 252 549T232 470L222 433Q222 431 272 431H323Q330 424 330 420Q330 398 317 385H210L174 240Q135 80 135 68Q135 26 162 26Q197 26 230 60T283 144Q285 150 288 151T303 153H307Q322 153 322 145Q322 142 319 133Q314 117 301 95T267 48T216 6T155 -11Q125 -11 98 4T59 56Q57 64 57 83V101L92 241Q127 382 128 383Q128 385 77 385H26Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D461\" xlink:href=\"#MJX-1284-TEX-I-1D461\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>t<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">t<\/script><span> do we expect to reach the syndrome <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.027ex\" height=\"1.038ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -453 454 459\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.014ex;\"><defs><path id=\"MJX-1800-TEX-B-1D42C\" d=\"M38 315Q38 339 45 360T70 404T127 440T223 453Q273 453 320 436L338 445L357 453H366Q380 453 383 447T386 403V387V355Q386 331 383 326T365 321H355H349Q333 321 329 324T324 341Q317 406 224 406H216Q123 406 123 353Q123 334 143 321T188 304T244 294T285 286Q305 281 325 273T373 237T412 172Q414 162 414 142Q414 -6 230 -6Q154 -6 117 22L68 -6H58Q44 -6 41 0T38 42V73Q38 85 38 101T37 122Q37 144 42 148T68 153H75Q87 153 91 151T97 147T103 132Q131 46 220 46H230Q257 46 265 47Q330 58 330 108Q330 127 316 142Q300 156 284 162Q271 168 212 178T122 202Q38 243 38 315Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D42C\" xlink:href=\"#MJX-1800-TEX-B-1D42C\"><\/use><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">s<\/mi><\/mrow><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">\\mathbf s<\/script><span> according to <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"2.036ex\" height=\"1.552ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -686 900 686\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: 0px;\"><defs><path id=\"MJX-560-TEX-B-1D407\" d=\"M400 0Q376 3 226 3Q75 3 51 0H39V62H147V624H39V686H51Q75 683 226 683Q376 683 400 686H412V624H304V388H595V624H487V686H499Q523 683 673 683Q824 683 848 686H860V624H752V62H860V0H848Q824 3 674 3Q523 3 499 0H487V62H595V326H304V62H412V0H400Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D407\" xlink:href=\"#MJX-560-TEX-B-1D407\"><\/use><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">H<\/mi><\/mrow><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">\\mathbf H<\/script><span>, i.e., <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"8.35ex\" height=\"1.81ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -717.8 3690.7 799.8\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.186ex;\"><defs><path id=\"MJX-1146-TEX-B-1D41E\" d=\"M32 225Q32 332 102 392T272 452H283Q382 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mathvariant=\"bold\">s<\/mi><\/mrow><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">\\mathbf{eH^{\\intercal}}=\\mathbf s<\/script><span>? <\/span><\/p><p><span>Let <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"30.748ex\" height=\"2.713ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -849.5 13590.6 1199\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.791ex;\"><defs><path id=\"MJX-874-TEX-I-1D441\" d=\"M234 637Q231 637 226 637Q201 637 196 638T191 649Q191 676 202 682Q204 683 299 683Q376 683 387 683T401 677Q612 181 616 168L670 381Q723 592 723 606Q723 633 659 637Q635 637 635 648Q635 650 637 660Q641 676 643 679T653 683Q656 683 684 682T767 680Q817 680 843 681T873 682Q888 682 888 672Q888 650 880 642Q878 637 858 637Q787 633 769 597L620 7Q618 0 599 0Q585 0 582 2Q579 5 453 305L326 604L261 344Q196 88 196 79Q201 46 268 46H278Q284 41 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data-mml-node=\"msup\"><g data-mml-node=\"mi\"><use data-c=\"1D41E\" xlink:href=\"#MJX-874-TEX-B-1D41E\"><\/use><use data-c=\"1D407\" xlink:href=\"#MJX-874-TEX-B-1D407\" transform=\"translate(527,0)\"><\/use><\/g><g data-mml-node=\"TeXAtom\" transform=\"translate(1460,413.1) scale(0.707)\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mo\"><use data-c=\"22BA\" xlink:href=\"#MJX-874-TEX-N-22BA\"><\/use><\/g><\/g><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(11302.8,0)\"><use data-c=\"3D\" xlink:href=\"#MJX-874-TEX-N-3D\"><\/use><\/g><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\" transform=\"translate(12358.6,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D42C\" xlink:href=\"#MJX-874-TEX-B-1D42C\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(12812.6,0)\"><use data-c=\"7D\" xlink:href=\"#MJX-874-TEX-N-7D\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(13312.6,0) translate(0 -0.5)\"><use data-c=\"7C\" xlink:href=\"#MJX-874-TEX-N-7C\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><msub><mi>N<\/mi><mi>t<\/mi><\/msub><mrow data-mjx-texclass=\"ORD\"><mo minsize=\"1.2em\" maxsize=\"1.2em\">(<\/mo><\/mrow><mrow data-mjx-texclass=\"ORD\"><mi data-mjx-variant=\"-tex-calligraphic\" mathvariant=\"script\">C<\/mi><\/mrow><mo>,<\/mo><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">s<\/mi><\/mrow><mrow data-mjx-texclass=\"ORD\"><mo minsize=\"1.2em\" maxsize=\"1.2em\">)<\/mo><\/mrow><mo>=<\/mo><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mo fence=\"false\" stretchy=\"false\">{<\/mo><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">e<\/mi><\/mrow><mo>\u2208<\/mo><msub><mrow data-mjx-texclass=\"ORD\"><mi data-mjx-variant=\"-tex-calligraphic\" mathvariant=\"script\">S<\/mi><\/mrow><mi>t<\/mi><\/msub><mo>:<\/mo><mrow data-mjx-texclass=\"ORD\"><msup><mi mathvariant=\"bold\">eH<\/mi><mrow data-mjx-texclass=\"ORD\"><mo>\u22ba<\/mo><\/mrow><\/msup><\/mrow><mo>=<\/mo><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">s<\/mi><\/mrow><mo fence=\"false\" stretchy=\"false\">}<\/mo><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">N_t\\big(\\mathcal{C},\\mathbf s\\big)=|\\{\\mathbf e\\in\\mathcal{S}_t:\\mathbf{eH^{\\intercal}}=\\mathbf s\\}|<\/script><span>, which is a random variable that gives the <\/span><strong><span>number of solutions<\/span><\/strong><span> of <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"3.079ex\" height=\"1.57ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -694 1361 694\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: 0px;\"><defs><path id=\"MJX-875-TEX-SS-1D5A3\" d=\"M88 0V694H237H258H316Q383 694 425 686T511 648Q578 604 622 525T666 343Q666 190 564 86Q494 18 400 3Q387 1 237 0H88ZM565 341Q565 409 546 463T495 550T429 600T359 621Q348 623 267 623H189V71H267Q272 71 286 71T310 70Q461 70 527 184Q565 251 565 341Z\"><\/path><path id=\"MJX-875-TEX-SS-1D5AF\" d=\"M88 0V694H230Q347 693 370 692T410 686Q487 667 535 611T583 485Q583 409 527 348T379 276Q369 274 279 274H192V0H88ZM486 485Q486 523 471 551T432 593T391 612T357 621Q350 622 268 623H189V347H268Q350 348 357 349Q370 351 383 354T416 368T450 391T475 429T486 485Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D5A3\" xlink:href=\"#MJX-875-TEX-SS-1D5A3\"><\/use><use data-c=\"1D5AF\" xlink:href=\"#MJX-875-TEX-SS-1D5AF\" transform=\"translate(722,0)\"><\/use><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"sans-serif\">DP<\/mi><\/mrow><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">\\mathsf{DP}<\/script><span> with input <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"5.83ex\" height=\"2.262ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 2576.7 1000\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-877-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-877-TEX-B-1D407\" d=\"M400 0Q376 3 226 3Q75 3 51 0H39V62H147V624H39V686H51Q75 683 226 683Q376 683 400 686H412V624H304V388H595V624H487V686H499Q523 683 673 683Q824 683 848 686H860V624H752V62H860V0H848Q824 3 674 3Q523 3 499 0H487V62H595V326H304V62H412V0H400Z\"><\/path><path id=\"MJX-877-TEX-N-2C\" d=\"M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z\"><\/path><path id=\"MJX-877-TEX-B-1D42C\" d=\"M38 315Q38 339 45 360T70 404T127 440T223 453Q273 453 320 436L338 445L357 453H366Q380 453 383 447T386 403V387V355Q386 331 383 326T365 321H355H349Q333 321 329 324T324 341Q317 406 224 406H216Q123 406 123 353Q123 334 143 321T188 304T244 294T285 286Q305 281 325 273T373 237T412 172Q414 162 414 142Q414 -6 230 -6Q154 -6 117 22L68 -6H58Q44 -6 41 0T38 42V73Q38 85 38 101T37 122Q37 144 42 148T68 153H75Q87 153 91 151T97 147T103 132Q131 46 220 46H230Q257 46 265 47Q330 58 330 108Q330 127 316 142Q300 156 284 162Q271 168 212 178T122 202Q38 243 38 315Z\"><\/path><path id=\"MJX-877-TEX-N-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mo\"><use data-c=\"28\" xlink:href=\"#MJX-877-TEX-N-28\"><\/use><\/g><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\" transform=\"translate(389,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D407\" xlink:href=\"#MJX-877-TEX-B-1D407\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(1289,0)\"><use data-c=\"2C\" xlink:href=\"#MJX-877-TEX-N-2C\"><\/use><\/g><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\" transform=\"translate(1733.7,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D42C\" xlink:href=\"#MJX-877-TEX-B-1D42C\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(2187.7,0)\"><use data-c=\"29\" xlink:href=\"#MJX-877-TEX-N-29\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mo stretchy=\"false\">(<\/mo><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">H<\/mi><\/mrow><mo>,<\/mo><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">s<\/mi><\/mrow><mo stretchy=\"false\">)<\/mo><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">(\\mathbf{H},\\mathbf s)<\/script><span>, where <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"8.401ex\" height=\"2.801ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -853.7 3713.4 1237.9\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.869ex;\"><defs><path id=\"MJX-554-TEX-B-1D42C\" d=\"M38 315Q38 339 45 360T70 404T127 440T223 453Q273 453 320 436L338 445L357 453H366Q380 453 383 447T386 403V387V355Q386 331 383 326T365 321H355H349Q333 321 329 324T324 341Q317 406 224 406H216Q123 406 123 353Q123 334 143 321T188 304T244 294T285 286Q305 281 325 273T373 237T412 172Q414 162 414 142Q414 -6 230 -6Q154 -6 117 22L68 -6H58Q44 -6 41 0T38 42V73Q38 85 38 101T37 122Q37 144 42 148T68 153H75Q87 153 91 151T97 147T103 132Q131 46 220 46H230Q257 46 265 47Q330 58 330 108Q330 127 316 142Q300 156 284 162Q271 168 212 178T122 202Q38 243 38 315Z\"><\/path><path id=\"MJX-554-TEX-N-2208\" d=\"M84 250Q84 372 166 450T360 539Q361 539 377 539T419 540T469 540H568Q583 532 583 520Q583 511 570 501L466 500Q355 499 329 494Q280 482 242 458T183 409T147 354T129 306T124 272V270H568Q583 262 583 250T568 230H124V228Q124 207 134 177T167 112T231 48T328 7Q355 1 466 0H570Q583 -10 583 -20Q583 -32 568 -40H471Q464 -40 446 -40T417 -41Q262 -41 172 45Q84 127 84 250Z\"><\/path><path id=\"MJX-554-TEX-D-1D53D\" d=\"M584 499Q569 490 566 490Q558 490 552 497T546 515Q546 535 533 559Q526 574 506 593T469 621Q415 648 326 648Q293 648 287 647T275 641Q264 630 263 617Q262 609 260 492V370L275 372Q323 376 350 392T393 441Q409 473 409 506Q409 529 427 529Q437 529 442 519Q444 511 444 362Q444 212 442 206Q436 197 426 197Q409 197 409 217Q409 265 375 299Q346 328 280 335H260V206Q260 70 262 63Q265 46 276 41T326 35Q362 35 366 28Q377 17 366 3L360 -1H24Q12 5 12 16Q12 35 51 35Q92 38 97 52Q102 60 102 341T97 632Q91 645 51 648Q12 648 12 666Q12 675 24 683H573Q576 678 584 670V499ZM137 341Q137 131 136 89T130 37Q129 36 129 35H182Q233 35 233 39Q226 54 225 92T224 346L226 623L231 635L235 648H129Q132 641 133 638T135 603T137 517T137 341ZM549 603V648H495L506 641Q531 621 533 619L549 603ZM409 317V395L400 386Q390 376 375 366L357 355L373 346Q394 331 397 328L409 317Z\"><\/path><path id=\"MJX-554-TEX-I-1D45B\" d=\"M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z\"><\/path><path id=\"MJX-554-TEX-N-2212\" d=\"M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z\"><\/path><path id=\"MJX-554-TEX-I-1D458\" d=\"M121 647Q121 657 125 670T137 683Q138 683 209 688T282 694Q294 694 294 686Q294 679 244 477Q194 279 194 272Q213 282 223 291Q247 309 292 354T362 415Q402 442 438 442Q468 442 485 423T503 369Q503 344 496 327T477 302T456 291T438 288Q418 288 406 299T394 328Q394 353 410 369T442 390L458 393Q446 405 434 405H430Q398 402 367 380T294 316T228 255Q230 254 243 252T267 246T293 238T320 224T342 206T359 180T365 147Q365 130 360 106T354 66Q354 26 381 26Q429 26 459 145Q461 153 479 153H483Q499 153 499 144Q499 139 496 130Q455 -11 378 -11Q333 -11 305 15T277 90Q277 108 280 121T283 145Q283 167 269 183T234 206T200 217T182 220H180Q168 178 159 139T145 81T136 44T129 20T122 7T111 -2Q98 -11 83 -11Q66 -11 57 -1T48 16Q48 26 85 176T158 471L195 616Q196 629 188 632T149 637H144Q134 637 131 637T124 640T121 647Z\"><\/path><path id=\"MJX-554-TEX-I-1D45E\" d=\"M33 157Q33 258 109 349T280 441Q340 441 372 389Q373 390 377 395T388 406T404 418Q438 442 450 442Q454 442 457 439T460 434Q460 425 391 149Q320 -135 320 -139Q320 -147 365 -148H390Q396 -156 396 -157T393 -175Q389 -188 383 -194H370Q339 -192 262 -192Q234 -192 211 -192T174 -192T157 -193Q143 -193 143 -185Q143 -182 145 -170Q149 -154 152 -151T172 -148Q220 -148 230 -141Q238 -136 258 -53T279 32Q279 33 272 29Q224 -10 172 -10Q117 -10 75 30T33 157ZM352 326Q329 405 277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q233 26 290 98L298 109L352 326Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D42C\" xlink:href=\"#MJX-554-TEX-B-1D42C\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(731.8,0)\"><use data-c=\"2208\" xlink:href=\"#MJX-554-TEX-N-2208\"><\/use><\/g><g data-mml-node=\"msubsup\" transform=\"translate(1676.6,0)\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D53D\" xlink:href=\"#MJX-554-TEX-D-1D53D\"><\/use><\/g><\/g><g data-mml-node=\"TeXAtom\" transform=\"translate(644,363) scale(0.707)\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D45B\" xlink:href=\"#MJX-554-TEX-I-1D45B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(600,0)\"><use data-c=\"2212\" xlink:href=\"#MJX-554-TEX-N-2212\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1378,0)\"><use data-c=\"1D458\" xlink:href=\"#MJX-554-TEX-I-1D458\"><\/use><\/g><\/g><g data-mml-node=\"mi\" transform=\"translate(644,-247) scale(0.707)\"><use data-c=\"1D45E\" xlink:href=\"#MJX-554-TEX-I-1D45E\"><\/use><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">s<\/mi><\/mrow><mo>\u2208<\/mo><msubsup><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"double-struck\">F<\/mi><\/mrow><mi>q<\/mi><mrow data-mjx-texclass=\"ORD\"><mi>n<\/mi><mo>\u2212<\/mo><mi>k<\/mi><\/mrow><\/msubsup><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">\\mathbf s\\in\\mathbb{F}_q^{n-k}<\/script><span> is fixed. On the other hand, <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"8.154ex\" height=\"2.713ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -849.5 3603.9 1199\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.791ex;\"><defs><path id=\"MJX-898-TEX-I-1D441\" d=\"M234 637Q231 637 226 637Q201 637 196 638T191 649Q191 676 202 682Q204 683 299 683Q376 683 387 683T401 677Q612 181 616 168L670 381Q723 592 723 606Q723 633 659 637Q635 637 635 648Q635 650 637 660Q641 676 643 679T653 683Q656 683 684 682T767 680Q817 680 843 681T873 682Q888 682 888 672Q888 650 880 642Q878 637 858 637Q787 633 769 597L620 7Q618 0 599 0Q585 0 582 2Q579 5 453 305L326 604L261 344Q196 88 196 79Q201 46 268 46H278Q284 41 284 38T282 19Q278 6 272 0H259Q228 2 151 2Q123 2 100 2T63 2T46 1Q31 1 31 10Q31 14 34 26T39 40Q41 46 62 46Q130 49 150 85Q154 91 221 362L289 634Q287 635 234 637Z\"><\/path><path id=\"MJX-898-TEX-I-1D461\" d=\"M26 385Q19 392 19 395Q19 399 22 411T27 425Q29 430 36 430T87 431H140L159 511Q162 522 166 540T173 566T179 586T187 603T197 615T211 624T229 626Q247 625 254 615T261 596Q261 589 252 549T232 470L222 433Q222 431 272 431H323Q330 424 330 420Q330 398 317 385H210L174 240Q135 80 135 68Q135 26 162 26Q197 26 230 60T283 144Q285 150 288 151T303 153H307Q322 153 322 145Q322 142 319 133Q314 117 301 95T267 48T216 6T155 -11Q125 -11 98 4T59 56Q57 64 57 83V101L92 241Q127 382 128 383Q128 385 77 385H26Z\"><\/path><path id=\"MJX-898-TEX-SO-28\" d=\"M152 251Q152 646 388 850H416Q422 844 422 841Q422 837 403 816T357 753T302 649T255 482T236 250Q236 124 255 19T301 -147T356 -251T403 -315T422 -340Q422 -343 416 -349H388Q359 -325 332 -296T271 -213T212 -97T170 56T152 251Z\"><\/path><path id=\"MJX-898-TEX-C-43\" d=\"M201 -25Q167 -25 136 -14T75 23T29 94T12 202Q12 290 50 394T161 574Q227 642 303 673T433 704Q435 705 457 705Q533 701 533 640Q533 606 507 548T464 474Q431 444 396 444Q381 444 381 453Q381 459 388 473T407 513T428 563Q433 580 433 594Q433 636 381 636Q314 636 260 594T175 489T128 363T112 247Q112 157 153 101T273 44Q347 44 398 121Q413 144 437 157T481 171Q496 171 496 160Q496 150 476 123Q426 56 350 16T201 -25Z\"><\/path><path id=\"MJX-898-TEX-N-2C\" d=\"M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z\"><\/path><path id=\"MJX-898-TEX-B-1D7CE\" d=\"M266 654H280H282Q500 654 524 418Q529 370 529 320Q529 125 456 52Q397 -10 287 -10Q110 -10 63 154Q45 212 45 316Q45 504 113 585Q140 618 185 636T266 654ZM374 548Q347 604 286 604Q247 604 218 575Q197 552 193 511T188 311Q188 159 196 116Q202 87 225 64T287 41Q339 41 367 87Q379 107 382 152T386 329Q386 518 374 548Z\"><\/path><path id=\"MJX-898-TEX-SO-29\" d=\"M305 251Q305 -145 69 -349H56Q43 -349 39 -347T35 -338Q37 -333 60 -307T108 -239T160 -136T204 27T221 250T204 473T160 636T108 740T60 807T35 839Q35 850 50 850H56H69Q197 743 256 566Q305 425 305 251Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"msub\"><g data-mml-node=\"mi\"><use data-c=\"1D441\" xlink:href=\"#MJX-898-TEX-I-1D441\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(836,-150) scale(0.707)\"><use data-c=\"1D461\" xlink:href=\"#MJX-898-TEX-I-1D461\"><\/use><\/g><\/g><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\" transform=\"translate(1141.3,0)\"><g data-mml-node=\"mo\" transform=\"translate(0 -0.5)\"><use data-c=\"28\" xlink:href=\"#MJX-898-TEX-SO-28\"><\/use><\/g><\/g><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\" transform=\"translate(1599.3,0)\"><g data-mml-node=\"mi\"><use data-c=\"43\" xlink:href=\"#MJX-898-TEX-C-43\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(2126.3,0)\"><use data-c=\"2C\" xlink:href=\"#MJX-898-TEX-N-2C\"><\/use><\/g><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\" transform=\"translate(2570.9,0)\"><g data-mml-node=\"mn\"><use data-c=\"1D7CE\" xlink:href=\"#MJX-898-TEX-B-1D7CE\"><\/use><\/g><\/g><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\" transform=\"translate(3145.9,0)\"><g data-mml-node=\"mo\" transform=\"translate(0 -0.5)\"><use data-c=\"29\" xlink:href=\"#MJX-898-TEX-SO-29\"><\/use><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><msub><mi>N<\/mi><mi>t<\/mi><\/msub><mrow data-mjx-texclass=\"ORD\"><mo minsize=\"1.2em\" maxsize=\"1.2em\">(<\/mo><\/mrow><mrow data-mjx-texclass=\"ORD\"><mi data-mjx-variant=\"-tex-calligraphic\" mathvariant=\"script\">C<\/mi><\/mrow><mo>,<\/mo><mrow data-mjx-texclass=\"ORD\"><mn mathvariant=\"bold\">0<\/mn><\/mrow><mrow data-mjx-texclass=\"ORD\"><mo minsize=\"1.2em\" maxsize=\"1.2em\">)<\/mo><\/mrow><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">N_t\\big(\\mathcal{C},\\mathbf 0\\big)<\/script><span> gives the number of codewords <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"5.114ex\" height=\"1.686ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -705 2260.6 745\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.09ex;\"><defs><path id=\"MJX-1283-TEX-B-1D41C\" d=\"M447 131H458Q478 131 478 117Q478 112 471 95T439 51T377 9Q330 -6 286 -6Q196 -6 135 35Q39 96 39 222Q39 324 101 384Q169 453 286 453Q359 453 411 431T464 353Q464 319 445 302T395 284Q360 284 343 305T325 353Q325 380 338 396H333Q317 398 295 398H292Q280 398 271 397T245 390T218 373T197 338T183 283Q182 275 182 231Q182 199 184 180T193 132T220 85T270 57Q289 50 317 50H326Q385 50 414 115Q419 127 423 129T447 131Z\"><\/path><path id=\"MJX-1283-TEX-N-2208\" d=\"M84 250Q84 372 166 450T360 539Q361 539 377 539T419 540T469 540H568Q583 532 583 520Q583 511 570 501L466 500Q355 499 329 494Q280 482 242 458T183 409T147 354T129 306T124 272V270H568Q583 262 583 250T568 230H124V228Q124 207 134 177T167 112T231 48T328 7Q355 1 466 0H570Q583 -10 583 -20Q583 -32 568 -40H471Q464 -40 446 -40T417 -41Q262 -41 172 45Q84 127 84 250Z\"><\/path><path id=\"MJX-1283-TEX-C-43\" d=\"M201 -25Q167 -25 136 -14T75 23T29 94T12 202Q12 290 50 394T161 574Q227 642 303 673T433 704Q435 705 457 705Q533 701 533 640Q533 606 507 548T464 474Q431 444 396 444Q381 444 381 453Q381 459 388 473T407 513T428 563Q433 580 433 594Q433 636 381 636Q314 636 260 594T175 489T128 363T112 247Q112 157 153 101T273 44Q347 44 398 121Q413 144 437 157T481 171Q496 171 496 160Q496 150 476 123Q426 56 350 16T201 -25Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D41C\" xlink:href=\"#MJX-1283-TEX-B-1D41C\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(788.8,0)\"><use data-c=\"2208\" xlink:href=\"#MJX-1283-TEX-N-2208\"><\/use><\/g><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\" transform=\"translate(1733.6,0)\"><g data-mml-node=\"mi\"><use data-c=\"43\" xlink:href=\"#MJX-1283-TEX-C-43\"><\/use><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">c<\/mi><\/mrow><mo>\u2208<\/mo><mrow data-mjx-texclass=\"ORD\"><mi data-mjx-variant=\"-tex-calligraphic\" mathvariant=\"script\">C<\/mi><\/mrow><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">\\mathbf c\\in\\mathcal C<\/script><span> of Hamming weight <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"0.817ex\" height=\"1.441ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -626 361 637\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.025ex;\"><defs><path id=\"MJX-1284-TEX-I-1D461\" d=\"M26 385Q19 392 19 395Q19 399 22 411T27 425Q29 430 36 430T87 431H140L159 511Q162 522 166 540T173 566T179 586T187 603T197 615T211 624T229 626Q247 625 254 615T261 596Q261 589 252 549T232 470L222 433Q222 431 272 431H323Q330 424 330 420Q330 398 317 385H210L174 240Q135 80 135 68Q135 26 162 26Q197 26 230 60T283 144Q285 150 288 151T303 153H307Q322 153 322 145Q322 142 319 133Q314 117 301 95T267 48T216 6T155 -11Q125 -11 98 4T59 56Q57 64 57 83V101L92 241Q127 382 128 383Q128 385 77 385H26Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D461\" xlink:href=\"#MJX-1284-TEX-I-1D461\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>t<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">t<\/script><span>.<\/span><\/p><p><span>This number is crucial to know, the reason is as follows.<\/span><\/p><p><span>A trivial solution to solve <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"3.079ex\" height=\"1.57ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -694 1361 694\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: 0px;\"><defs><path id=\"MJX-875-TEX-SS-1D5A3\" d=\"M88 0V694H237H258H316Q383 694 425 686T511 648Q578 604 622 525T666 343Q666 190 564 86Q494 18 400 3Q387 1 237 0H88ZM565 341Q565 409 546 463T495 550T429 600T359 621Q348 623 267 623H189V71H267Q272 71 286 71T310 70Q461 70 527 184Q565 251 565 341Z\"><\/path><path id=\"MJX-875-TEX-SS-1D5AF\" d=\"M88 0V694H230Q347 693 370 692T410 686Q487 667 535 611T583 485Q583 409 527 348T379 276Q369 274 279 274H192V0H88ZM486 485Q486 523 471 551T432 593T391 612T357 621Q350 622 268 623H189V347H268Q350 348 357 349Q370 351 383 354T416 368T450 391T475 429T486 485Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D5A3\" xlink:href=\"#MJX-875-TEX-SS-1D5A3\"><\/use><use data-c=\"1D5AF\" xlink:href=\"#MJX-875-TEX-SS-1D5AF\" transform=\"translate(722,0)\"><\/use><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"sans-serif\">DP<\/mi><\/mrow><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">\\mathsf{DP}<\/script><span> is to pick a ranfom error vector <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"6.095ex\" height=\"1.952ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -705 2693.8 862.8\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.357ex;\"><defs><path id=\"MJX-784-TEX-B-1D41E\" d=\"M32 225Q32 332 102 392T272 452H283Q382 452 436 401Q494 343 494 243Q494 226 486 222T440 217Q431 217 394 217T327 218H175V209Q175 177 179 154T196 107T236 69T306 50Q312 49 323 49Q376 49 410 85Q421 99 427 111T434 127T442 133T463 135H468Q494 135 494 117Q494 110 489 97T468 66T431 32T373 5T292 -6Q181 -6 107 55T32 225ZM383 276Q377 346 348 374T280 402Q253 402 230 390T195 357Q179 331 176 279V266H383V276Z\"><\/path><path id=\"MJX-784-TEX-N-2208\" d=\"M84 250Q84 372 166 450T360 539Q361 539 377 539T419 540T469 540H568Q583 532 583 520Q583 511 570 501L466 500Q355 499 329 494Q280 482 242 458T183 409T147 354T129 306T124 272V270H568Q583 262 583 250T568 230H124V228Q124 207 134 177T167 112T231 48T328 7Q355 1 466 0H570Q583 -10 583 -20Q583 -32 568 -40H471Q464 -40 446 -40T417 -41Q262 -41 172 45Q84 127 84 250Z\"><\/path><path id=\"MJX-784-TEX-C-53\" d=\"M554 512Q536 512 536 522Q536 525 539 539T542 564Q542 588 528 604Q515 616 482 625T410 635Q374 635 349 624T312 594T295 561T290 532Q290 505 303 482T342 442T378 419T409 404Q435 391 451 383T494 357T535 323T562 282T574 231Q574 133 464 56T220 -22Q138 -22 78 21T18 123Q18 184 61 227T156 274Q178 274 178 263Q178 260 177 258Q172 247 164 239T151 227T136 218L127 213L124 202Q118 186 118 163Q120 124 165 86T292 48Q374 48 423 86T473 186V193Q473 267 347 327Q268 364 239 389Q191 431 191 486Q191 547 242 600T356 679T470 705Q472 705 478 705T489 704Q551 704 596 682T642 610Q642 566 621 545Q592 516 554 512Z\"><\/path><path id=\"MJX-784-TEX-I-1D461\" d=\"M26 385Q19 392 19 395Q19 399 22 411T27 425Q29 430 36 430T87 431H140L159 511Q162 522 166 540T173 566T179 586T187 603T197 615T211 624T229 626Q247 625 254 615T261 596Q261 589 252 549T232 470L222 433Q222 431 272 431H323Q330 424 330 420Q330 398 317 385H210L174 240Q135 80 135 68Q135 26 162 26Q197 26 230 60T283 144Q285 150 288 151T303 153H307Q322 153 322 145Q322 142 319 133Q314 117 301 95T267 48T216 6T155 -11Q125 -11 98 4T59 56Q57 64 57 83V101L92 241Q127 382 128 383Q128 385 77 385H26Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D41E\" xlink:href=\"#MJX-784-TEX-B-1D41E\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(804.8,0)\"><use data-c=\"2208\" xlink:href=\"#MJX-784-TEX-N-2208\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(1749.6,0)\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"53\" xlink:href=\"#MJX-784-TEX-C-53\"><\/use><\/g><\/g><g data-mml-node=\"mi\" transform=\"translate(639,-150) scale(0.707)\"><use data-c=\"1D461\" xlink:href=\"#MJX-784-TEX-I-1D461\"><\/use><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">e<\/mi><\/mrow><mo>\u2208<\/mo><msub><mrow data-mjx-texclass=\"ORD\"><mi data-mjx-variant=\"-tex-calligraphic\" mathvariant=\"script\">S<\/mi><\/mrow><mi>t<\/mi><\/msub><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">\\mathbf e\\in\\mathcal{S}_t<\/script><span>, <\/span><\/p><ul><li><p><span>If there is only one solution <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.192ex\" height=\"1.036ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -452 527 458\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.014ex;\"><defs><path id=\"MJX-1808-TEX-B-1D41E\" d=\"M32 225Q32 332 102 392T272 452H283Q382 452 436 401Q494 343 494 243Q494 226 486 222T440 217Q431 217 394 217T327 218H175V209Q175 177 179 154T196 107T236 69T306 50Q312 49 323 49Q376 49 410 85Q421 99 427 111T434 127T442 133T463 135H468Q494 135 494 117Q494 110 489 97T468 66T431 32T373 5T292 -6Q181 -6 107 55T32 225ZM383 276Q377 346 348 374T280 402Q253 402 230 390T195 357Q179 331 176 279V266H383V276Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D41E\" xlink:href=\"#MJX-1808-TEX-B-1D41E\"><\/use><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">e<\/mi><\/mrow><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">\\mathbf e<\/script><span>, then our success probability is <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"7.706ex\" height=\"3.518ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -864.9 3406 1555.1\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -1.561ex;\"><defs><path id=\"MJX-798-TEX-N-31\" d=\"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z\"><\/path><path id=\"MJX-798-TEX-SO-28\" d=\"M152 251Q152 646 388 850H416Q422 844 422 841Q422 837 403 816T357 753T302 649T255 482T236 250Q236 124 255 19T301 -147T356 -251T403 -315T422 -340Q422 -343 416 -349H388Q359 -325 332 -296T271 -213T212 -97T170 56T152 251Z\"><\/path><path id=\"MJX-798-TEX-I-1D45B\" d=\"M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z\"><\/path><path id=\"MJX-798-TEX-I-1D461\" d=\"M26 385Q19 392 19 395Q19 399 22 411T27 425Q29 430 36 430T87 431H140L159 511Q162 522 166 540T173 566T179 586T187 603T197 615T211 624T229 626Q247 625 254 615T261 596Q261 589 252 549T232 470L222 433Q222 431 272 431H323Q330 424 330 420Q330 398 317 385H210L174 240Q135 80 135 68Q135 26 162 26Q197 26 230 60T283 144Q285 150 288 151T303 153H307Q322 153 322 145Q322 142 319 133Q314 117 301 95T267 48T216 6T155 -11Q125 -11 98 4T59 56Q57 64 57 83V101L92 241Q127 382 128 383Q128 385 77 385H26Z\"><\/path><path id=\"MJX-798-TEX-SO-29\" d=\"M305 251Q305 -145 69 -349H56Q43 -349 39 -347T35 -338Q37 -333 60 -307T108 -239T160 -136T204 27T221 250T204 473T160 636T108 740T60 807T35 839Q35 850 50 850H56H69Q197 743 256 566Q305 425 305 251Z\"><\/path><path id=\"MJX-798-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-798-TEX-I-1D45E\" d=\"M33 157Q33 258 109 349T280 441Q340 441 372 389Q373 390 377 395T388 406T404 418Q438 442 450 442Q454 442 457 439T460 434Q460 425 391 149Q320 -135 320 -139Q320 -147 365 -148H390Q396 -156 396 -157T393 -175Q389 -188 383 -194H370Q339 -192 262 -192Q234 -192 211 -192T174 -192T157 -193Q143 -193 143 -185Q143 -182 145 -170Q149 -154 152 -151T172 -148Q220 -148 230 -141Q238 -136 258 -53T279 32Q279 33 272 29Q224 -10 172 -10Q117 -10 75 30T33 157ZM352 326Q329 405 277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q233 26 290 98L298 109L352 326Z\"><\/path><path id=\"MJX-798-TEX-N-2212\" d=\"M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z\"><\/path><path id=\"MJX-798-TEX-N-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mfrac\"><g data-mml-node=\"mn\" transform=\"translate(1526.2,394) scale(0.707)\"><use data-c=\"31\" xlink:href=\"#MJX-798-TEX-N-31\"><\/use><\/g><g data-mml-node=\"mrow\" transform=\"translate(220,-440.7) scale(0.707)\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mrow\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"OPEN\"><g data-mml-node=\"mo\" transform=\"translate(0 -0.5)\"><use data-c=\"28\" xlink:href=\"#MJX-798-TEX-SO-28\"><\/use><\/g><\/g><g data-mml-node=\"mfrac\" transform=\"translate(458,0)\"><g data-mml-node=\"mi\" transform=\"translate(0,444) scale(0.707)\"><use data-c=\"1D45B\" xlink:href=\"#MJX-798-TEX-I-1D45B\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(84.5,-345) scale(0.707)\"><use data-c=\"1D461\" xlink:href=\"#MJX-798-TEX-I-1D461\"><\/use><\/g><\/g><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"CLOSE\" transform=\"translate(882.3,0)\"><g data-mml-node=\"mo\" transform=\"translate(0 -0.5)\"><use data-c=\"29\" xlink:href=\"#MJX-798-TEX-SO-29\"><\/use><\/g><\/g><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(1340.3,0)\"><use data-c=\"28\" xlink:href=\"#MJX-798-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1729.3,0)\"><use data-c=\"1D45E\" xlink:href=\"#MJX-798-TEX-I-1D45E\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2189.3,0)\"><use data-c=\"2212\" xlink:href=\"#MJX-798-TEX-N-2212\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(2967.3,0)\"><use data-c=\"31\" xlink:href=\"#MJX-798-TEX-N-31\"><\/use><\/g><g data-mml-node=\"msup\" transform=\"translate(3467.3,0)\"><g data-mml-node=\"mo\"><use data-c=\"29\" xlink:href=\"#MJX-798-TEX-N-29\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(422,289) scale(0.707)\"><use data-c=\"1D461\" xlink:href=\"#MJX-798-TEX-I-1D461\"><\/use><\/g><\/g><\/g><rect width=\"3166\" height=\"60\" x=\"120\" y=\"220\"><\/rect><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mfrac><mn>1<\/mn><mrow><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"OPEN\"><mo minsize=\"1.2em\" maxsize=\"1.2em\">(<\/mo><\/mrow><mfrac linethickness=\"0\"><mi>n<\/mi><mi>t<\/mi><\/mfrac><mrow data-mjx-texclass=\"CLOSE\"><mo minsize=\"1.2em\" maxsize=\"1.2em\">)<\/mo><\/mrow><\/mrow><\/mrow><mo stretchy=\"false\">(<\/mo><mi>q<\/mi><mo>\u2212<\/mo><mn>1<\/mn><msup><mo stretchy=\"false\">)<\/mo><mi>t<\/mi><\/msup><\/mrow><\/mfrac><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">\\frac{1}{{n\\choose t}(q-1)^t}<\/script><span>.<\/span><\/p><\/li><li><p><span>If there are <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"2.009ex\" height=\"1.545ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -683 888 683\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: 0px;\"><defs><path id=\"MJX-802-TEX-I-1D441\" d=\"M234 637Q231 637 226 637Q201 637 196 638T191 649Q191 676 202 682Q204 683 299 683Q376 683 387 683T401 677Q612 181 616 168L670 381Q723 592 723 606Q723 633 659 637Q635 637 635 648Q635 650 637 660Q641 676 643 679T653 683Q656 683 684 682T767 680Q817 680 843 681T873 682Q888 682 888 672Q888 650 880 642Q878 637 858 637Q787 633 769 597L620 7Q618 0 599 0Q585 0 582 2Q579 5 453 305L326 604L261 344Q196 88 196 79Q201 46 268 46H278Q284 41 284 38T282 19Q278 6 272 0H259Q228 2 151 2Q123 2 100 2T63 2T46 1Q31 1 31 10Q31 14 34 26T39 40Q41 46 62 46Q130 49 150 85Q154 91 221 362L289 634Q287 635 234 637Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D441\" xlink:href=\"#MJX-802-TEX-I-1D441\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>N<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">N<\/script><span> solutions <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.192ex\" height=\"1.036ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -452 527 458\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.014ex;\"><defs><path id=\"MJX-1808-TEX-B-1D41E\" d=\"M32 225Q32 332 102 392T272 452H283Q382 452 436 401Q494 343 494 243Q494 226 486 222T440 217Q431 217 394 217T327 218H175V209Q175 177 179 154T196 107T236 69T306 50Q312 49 323 49Q376 49 410 85Q421 99 427 111T434 127T442 133T463 135H468Q494 135 494 117Q494 110 489 97T468 66T431 32T373 5T292 -6Q181 -6 107 55T32 225ZM383 276Q377 346 348 374T280 402Q253 402 230 390T195 357Q179 331 176 279V266H383V276Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D41E\" xlink:href=\"#MJX-1808-TEX-B-1D41E\"><\/use><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">e<\/mi><\/mrow><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">\\mathbf e<\/script><span>, then the success probability is <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"7.706ex\" height=\"3.545ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -877 3406 1567.1\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -1.561ex;\"><defs><path id=\"MJX-801-TEX-I-1D441\" d=\"M234 637Q231 637 226 637Q201 637 196 638T191 649Q191 676 202 682Q204 683 299 683Q376 683 387 683T401 677Q612 181 616 168L670 381Q723 592 723 606Q723 633 659 637Q635 637 635 648Q635 650 637 660Q641 676 643 679T653 683Q656 683 684 682T767 680Q817 680 843 681T873 682Q888 682 888 672Q888 650 880 642Q878 637 858 637Q787 633 769 597L620 7Q618 0 599 0Q585 0 582 2Q579 5 453 305L326 604L261 344Q196 88 196 79Q201 46 268 46H278Q284 41 284 38T282 19Q278 6 272 0H259Q228 2 151 2Q123 2 100 2T63 2T46 1Q31 1 31 10Q31 14 34 26T39 40Q41 46 62 46Q130 49 150 85Q154 91 221 362L289 634Q287 635 234 637Z\"><\/path><path id=\"MJX-801-TEX-SO-28\" d=\"M152 251Q152 646 388 850H416Q422 844 422 841Q422 837 403 816T357 753T302 649T255 482T236 250Q236 124 255 19T301 -147T356 -251T403 -315T422 -340Q422 -343 416 -349H388Q359 -325 332 -296T271 -213T212 -97T170 56T152 251Z\"><\/path><path id=\"MJX-801-TEX-I-1D45B\" d=\"M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z\"><\/path><path id=\"MJX-801-TEX-I-1D461\" d=\"M26 385Q19 392 19 395Q19 399 22 411T27 425Q29 430 36 430T87 431H140L159 511Q162 522 166 540T173 566T179 586T187 603T197 615T211 624T229 626Q247 625 254 615T261 596Q261 589 252 549T232 470L222 433Q222 431 272 431H323Q330 424 330 420Q330 398 317 385H210L174 240Q135 80 135 68Q135 26 162 26Q197 26 230 60T283 144Q285 150 288 151T303 153H307Q322 153 322 145Q322 142 319 133Q314 117 301 95T267 48T216 6T155 -11Q125 -11 98 4T59 56Q57 64 57 83V101L92 241Q127 382 128 383Q128 385 77 385H26Z\"><\/path><path id=\"MJX-801-TEX-SO-29\" d=\"M305 251Q305 -145 69 -349H56Q43 -349 39 -347T35 -338Q37 -333 60 -307T108 -239T160 -136T204 27T221 250T204 473T160 636T108 740T60 807T35 839Q35 850 50 850H56H69Q197 743 256 566Q305 425 305 251Z\"><\/path><path id=\"MJX-801-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-801-TEX-I-1D45E\" d=\"M33 157Q33 258 109 349T280 441Q340 441 372 389Q373 390 377 395T388 406T404 418Q438 442 450 442Q454 442 457 439T460 434Q460 425 391 149Q320 -135 320 -139Q320 -147 365 -148H390Q396 -156 396 -157T393 -175Q389 -188 383 -194H370Q339 -192 262 -192Q234 -192 211 -192T174 -192T157 -193Q143 -193 143 -185Q143 -182 145 -170Q149 -154 152 -151T172 -148Q220 -148 230 -141Q238 -136 258 -53T279 32Q279 33 272 29Q224 -10 172 -10Q117 -10 75 30T33 157ZM352 326Q329 405 277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q233 26 290 98L298 109L352 326Z\"><\/path><path id=\"MJX-801-TEX-N-2212\" d=\"M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z\"><\/path><path id=\"MJX-801-TEX-N-31\" d=\"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z\"><\/path><path id=\"MJX-801-TEX-N-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mfrac\"><g data-mml-node=\"mi\" transform=\"translate(1389,394) scale(0.707)\"><use data-c=\"1D441\" xlink:href=\"#MJX-801-TEX-I-1D441\"><\/use><\/g><g data-mml-node=\"mrow\" transform=\"translate(220,-440.7) scale(0.707)\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mrow\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"OPEN\"><g data-mml-node=\"mo\" transform=\"translate(0 -0.5)\"><use data-c=\"28\" xlink:href=\"#MJX-801-TEX-SO-28\"><\/use><\/g><\/g><g data-mml-node=\"mfrac\" transform=\"translate(458,0)\"><g data-mml-node=\"mi\" transform=\"translate(0,444) scale(0.707)\"><use data-c=\"1D45B\" xlink:href=\"#MJX-801-TEX-I-1D45B\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(84.5,-345) scale(0.707)\"><use data-c=\"1D461\" xlink:href=\"#MJX-801-TEX-I-1D461\"><\/use><\/g><\/g><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"CLOSE\" transform=\"translate(882.3,0)\"><g data-mml-node=\"mo\" transform=\"translate(0 -0.5)\"><use data-c=\"29\" xlink:href=\"#MJX-801-TEX-SO-29\"><\/use><\/g><\/g><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(1340.3,0)\"><use data-c=\"28\" xlink:href=\"#MJX-801-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1729.3,0)\"><use data-c=\"1D45E\" xlink:href=\"#MJX-801-TEX-I-1D45E\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2189.3,0)\"><use data-c=\"2212\" xlink:href=\"#MJX-801-TEX-N-2212\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(2967.3,0)\"><use data-c=\"31\" xlink:href=\"#MJX-801-TEX-N-31\"><\/use><\/g><g data-mml-node=\"msup\" transform=\"translate(3467.3,0)\"><g data-mml-node=\"mo\"><use data-c=\"29\" xlink:href=\"#MJX-801-TEX-N-29\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(422,289) scale(0.707)\"><use data-c=\"1D461\" xlink:href=\"#MJX-801-TEX-I-1D461\"><\/use><\/g><\/g><\/g><rect width=\"3166\" height=\"60\" x=\"120\" y=\"220\"><\/rect><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mfrac><mi>N<\/mi><mrow><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"OPEN\"><mo minsize=\"1.2em\" maxsize=\"1.2em\">(<\/mo><\/mrow><mfrac linethickness=\"0\"><mi>n<\/mi><mi>t<\/mi><\/mfrac><mrow data-mjx-texclass=\"CLOSE\"><mo minsize=\"1.2em\" maxsize=\"1.2em\">)<\/mo><\/mrow><\/mrow><\/mrow><mo stretchy=\"false\">(<\/mo><mi>q<\/mi><mo>\u2212<\/mo><mn>1<\/mn><msup><mo stretchy=\"false\">)<\/mo><mi>t<\/mi><\/msup><\/mrow><\/mfrac><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">\\frac{N}{{n\\choose t}(q-1)^t}<\/script><span>.<\/span><\/p><\/li><\/ul><p><span>It is therefore crucial to know the value of <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"2.009ex\" height=\"1.545ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -683 888 683\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: 0px;\"><defs><path id=\"MJX-802-TEX-I-1D441\" d=\"M234 637Q231 637 226 637Q201 637 196 638T191 649Q191 676 202 682Q204 683 299 683Q376 683 387 683T401 677Q612 181 616 168L670 381Q723 592 723 606Q723 633 659 637Q635 637 635 648Q635 650 637 660Q641 676 643 679T653 683Q656 683 684 682T767 680Q817 680 843 681T873 682Q888 682 888 672Q888 650 880 642Q878 637 858 637Q787 633 769 597L620 7Q618 0 599 0Q585 0 582 2Q579 5 453 305L326 604L261 344Q196 88 196 79Q201 46 268 46H278Q284 41 284 38T282 19Q278 6 272 0H259Q228 2 151 2Q123 2 100 2T63 2T46 1Q31 1 31 10Q31 14 34 26T39 40Q41 46 62 46Q130 49 150 85Q154 91 221 362L289 634Q287 635 234 637Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D441\" xlink:href=\"#MJX-802-TEX-I-1D441\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>N<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">N<\/script><span> to be able to predict the <\/span><strong><span>running time<\/span><\/strong><span> of our algorithm.<\/span><\/p><p><strong><span>Solution:<\/span><\/strong><span> <\/span><\/p><p><span>The expectation of <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"7.88ex\" height=\"2.713ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -849.5 3482.9 1199\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.791ex;\"><defs><path id=\"MJX-1105-TEX-I-1D441\" d=\"M234 637Q231 637 226 637Q201 637 196 638T191 649Q191 676 202 682Q204 683 299 683Q376 683 387 683T401 677Q612 181 616 168L670 381Q723 592 723 606Q723 633 659 637Q635 637 635 648Q635 650 637 660Q641 676 643 679T653 683Q656 683 684 682T767 680Q817 680 843 681T873 682Q888 682 888 672Q888 650 880 642Q878 637 858 637Q787 633 769 597L620 7Q618 0 599 0Q585 0 582 2Q579 5 453 305L326 604L261 344Q196 88 196 79Q201 46 268 46H278Q284 41 284 38T282 19Q278 6 272 0H259Q228 2 151 2Q123 2 100 2T63 2T46 1Q31 1 31 10Q31 14 34 26T39 40Q41 46 62 46Q130 49 150 85Q154 91 221 362L289 634Q287 635 234 637Z\"><\/path><path id=\"MJX-1105-TEX-I-1D461\" d=\"M26 385Q19 392 19 395Q19 399 22 411T27 425Q29 430 36 430T87 431H140L159 511Q162 522 166 540T173 566T179 586T187 603T197 615T211 624T229 626Q247 625 254 615T261 596Q261 589 252 549T232 470L222 433Q222 431 272 431H323Q330 424 330 420Q330 398 317 385H210L174 240Q135 80 135 68Q135 26 162 26Q197 26 230 60T283 144Q285 150 288 151T303 153H307Q322 153 322 145Q322 142 319 133Q314 117 301 95T267 48T216 6T155 -11Q125 -11 98 4T59 56Q57 64 57 83V101L92 241Q127 382 128 383Q128 385 77 385H26Z\"><\/path><path id=\"MJX-1105-TEX-SO-28\" d=\"M152 251Q152 646 388 850H416Q422 844 422 841Q422 837 403 816T357 753T302 649T255 482T236 250Q236 124 255 19T301 -147T356 -251T403 -315T422 -340Q422 -343 416 -349H388Q359 -325 332 -296T271 -213T212 -97T170 56T152 251Z\"><\/path><path id=\"MJX-1105-TEX-C-43\" d=\"M201 -25Q167 -25 136 -14T75 23T29 94T12 202Q12 290 50 394T161 574Q227 642 303 673T433 704Q435 705 457 705Q533 701 533 640Q533 606 507 548T464 474Q431 444 396 444Q381 444 381 453Q381 459 388 473T407 513T428 563Q433 580 433 594Q433 636 381 636Q314 636 260 594T175 489T128 363T112 247Q112 157 153 101T273 44Q347 44 398 121Q413 144 437 157T481 171Q496 171 496 160Q496 150 476 123Q426 56 350 16T201 -25Z\"><\/path><path id=\"MJX-1105-TEX-N-2C\" d=\"M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z\"><\/path><path id=\"MJX-1105-TEX-B-1D42C\" d=\"M38 315Q38 339 45 360T70 404T127 440T223 453Q273 453 320 436L338 445L357 453H366Q380 453 383 447T386 403V387V355Q386 331 383 326T365 321H355H349Q333 321 329 324T324 341Q317 406 224 406H216Q123 406 123 353Q123 334 143 321T188 304T244 294T285 286Q305 281 325 273T373 237T412 172Q414 162 414 142Q414 -6 230 -6Q154 -6 117 22L68 -6H58Q44 -6 41 0T38 42V73Q38 85 38 101T37 122Q37 144 42 148T68 153H75Q87 153 91 151T97 147T103 132Q131 46 220 46H230Q257 46 265 47Q330 58 330 108Q330 127 316 142Q300 156 284 162Q271 168 212 178T122 202Q38 243 38 315Z\"><\/path><path id=\"MJX-1105-TEX-SO-29\" d=\"M305 251Q305 -145 69 -349H56Q43 -349 39 -347T35 -338Q37 -333 60 -307T108 -239T160 -136T204 27T221 250T204 473T160 636T108 740T60 807T35 839Q35 850 50 850H56H69Q197 743 256 566Q305 425 305 251Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"msub\"><g data-mml-node=\"mi\"><use data-c=\"1D441\" xlink:href=\"#MJX-1105-TEX-I-1D441\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(836,-150) scale(0.707)\"><use data-c=\"1D461\" xlink:href=\"#MJX-1105-TEX-I-1D461\"><\/use><\/g><\/g><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\" transform=\"translate(1141.3,0)\"><g data-mml-node=\"mo\" transform=\"translate(0 -0.5)\"><use data-c=\"28\" xlink:href=\"#MJX-1105-TEX-SO-28\"><\/use><\/g><\/g><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\" transform=\"translate(1599.3,0)\"><g data-mml-node=\"mi\"><use data-c=\"43\" xlink:href=\"#MJX-1105-TEX-C-43\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(2126.3,0)\"><use data-c=\"2C\" xlink:href=\"#MJX-1105-TEX-N-2C\"><\/use><\/g><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\" transform=\"translate(2570.9,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D42C\" xlink:href=\"#MJX-1105-TEX-B-1D42C\"><\/use><\/g><\/g><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\" transform=\"translate(3024.9,0)\"><g data-mml-node=\"mo\" transform=\"translate(0 -0.5)\"><use data-c=\"29\" xlink:href=\"#MJX-1105-TEX-SO-29\"><\/use><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><msub><mi>N<\/mi><mi>t<\/mi><\/msub><mrow data-mjx-texclass=\"ORD\"><mo minsize=\"1.2em\" maxsize=\"1.2em\">(<\/mo><\/mrow><mrow data-mjx-texclass=\"ORD\"><mi data-mjx-variant=\"-tex-calligraphic\" mathvariant=\"script\">C<\/mi><\/mrow><mo>,<\/mo><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">s<\/mi><\/mrow><mrow data-mjx-texclass=\"ORD\"><mo minsize=\"1.2em\" maxsize=\"1.2em\">)<\/mo><\/mrow><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">N_t\\big(\\mathcal{C},\\mathbf s\\big)<\/script><span> is <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"16.008ex\" height=\"2.73ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -853.7 7075.5 1206.5\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.798ex;\"><defs><path id=\"MJX-916-TEX-SO-28\" d=\"M152 251Q152 646 388 850H416Q422 844 422 841Q422 837 403 816T357 753T302 649T255 482T236 250Q236 124 255 19T301 -147T356 -251T403 -315T422 -340Q422 -343 416 -349H388Q359 -325 332 -296T271 -213T212 -97T170 56T152 251Z\"><\/path><path id=\"MJX-916-TEX-I-1D45B\" d=\"M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z\"><\/path><path 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147Q365 130 360 106T354 66Q354 26 381 26Q429 26 459 145Q461 153 479 153H483Q499 153 499 144Q499 139 496 130Q455 -11 378 -11Q333 -11 305 15T277 90Q277 108 280 121T283 145Q283 167 269 183T234 206T200 217T182 220H180Q168 178 159 139T145 81T136 44T129 20T122 7T111 -2Q98 -11 83 -11Q66 -11 57 -1T48 16Q48 26 85 176T158 471L195 616Q196 629 188 632T149 637H144Q134 637 131 637T124 640T121 647Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mrow\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"OPEN\"><g data-mml-node=\"mo\" transform=\"translate(0 -0.5)\"><use data-c=\"28\" xlink:href=\"#MJX-916-TEX-SO-28\"><\/use><\/g><\/g><g data-mml-node=\"mfrac\" transform=\"translate(458,0)\"><g data-mml-node=\"mi\" transform=\"translate(0,444) scale(0.707)\"><use data-c=\"1D45B\" xlink:href=\"#MJX-916-TEX-I-1D45B\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(84.5,-345) scale(0.707)\"><use data-c=\"1D461\" xlink:href=\"#MJX-916-TEX-I-1D461\"><\/use><\/g><\/g><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"CLOSE\" transform=\"translate(882.3,0)\"><g data-mml-node=\"mo\" transform=\"translate(0 -0.5)\"><use data-c=\"29\" xlink:href=\"#MJX-916-TEX-SO-29\"><\/use><\/g><\/g><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(1340.3,0)\"><use data-c=\"28\" xlink:href=\"#MJX-916-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1729.3,0)\"><use data-c=\"1D45E\" xlink:href=\"#MJX-916-TEX-I-1D45E\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2411.5,0)\"><use data-c=\"2212\" xlink:href=\"#MJX-916-TEX-N-2212\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(3411.7,0)\"><use data-c=\"31\" xlink:href=\"#MJX-916-TEX-N-31\"><\/use><\/g><g data-mml-node=\"msup\" transform=\"translate(3911.7,0)\"><g data-mml-node=\"mo\"><use data-c=\"29\" xlink:href=\"#MJX-916-TEX-N-29\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(422,363) scale(0.707)\"><use data-c=\"1D461\" xlink:href=\"#MJX-916-TEX-I-1D461\"><\/use><\/g><\/g><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\" transform=\"translate(4639,0)\"><g data-mml-node=\"mo\"><use data-c=\"2F\" xlink:href=\"#MJX-916-TEX-N-2F\"><\/use><\/g><\/g><g data-mml-node=\"msup\" transform=\"translate(5139,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D45E\" xlink:href=\"#MJX-916-TEX-I-1D45E\"><\/use><\/g><g data-mml-node=\"TeXAtom\" transform=\"translate(543.7,363) scale(0.707)\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D45B\" xlink:href=\"#MJX-916-TEX-I-1D45B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(600,0)\"><use data-c=\"2212\" xlink:href=\"#MJX-916-TEX-N-2212\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1378,0)\"><use data-c=\"1D458\" xlink:href=\"#MJX-916-TEX-I-1D458\"><\/use><\/g><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"OPEN\"><mo minsize=\"1.2em\" maxsize=\"1.2em\">(<\/mo><\/mrow><mfrac linethickness=\"0\"><mi>n<\/mi><mi>t<\/mi><\/mfrac><mrow data-mjx-texclass=\"CLOSE\"><mo minsize=\"1.2em\" maxsize=\"1.2em\">)<\/mo><\/mrow><\/mrow><\/mrow><mo stretchy=\"false\">(<\/mo><mi>q<\/mi><mo>\u2212<\/mo><mn>1<\/mn><msup><mo stretchy=\"false\">)<\/mo><mi>t<\/mi><\/msup><mrow data-mjx-texclass=\"ORD\"><mo>\/<\/mo><\/mrow><msup><mi>q<\/mi><mrow data-mjx-texclass=\"ORD\"><mi>n<\/mi><mo>\u2212<\/mo><mi>k<\/mi><\/mrow><\/msup><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">{n\\choose t}(q-1)^t\/q^{n-k}<\/script><span>. 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data-c=\"1D45B\" xlink:href=\"#MJX-1993-TEX-I-1D45B\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(119.5,-686)\"><use data-c=\"1D461\" xlink:href=\"#MJX-1993-TEX-I-1D461\"><\/use><\/g><\/g><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"CLOSE\" transform=\"translate(1336,0)\"><g data-mml-node=\"mo\" transform=\"translate(0 -0.5)\"><use data-c=\"29\" xlink:href=\"#MJX-1993-TEX-S3-29\"><\/use><\/g><\/g><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(5561.9,0)\"><use data-c=\"28\" xlink:href=\"#MJX-1993-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(5950.9,0)\"><use data-c=\"1D45E\" xlink:href=\"#MJX-1993-TEX-I-1D45E\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(6633.2,0)\"><use data-c=\"2212\" xlink:href=\"#MJX-1993-TEX-N-2212\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(7633.4,0)\"><use data-c=\"31\" xlink:href=\"#MJX-1993-TEX-N-31\"><\/use><\/g><g data-mml-node=\"msup\" transform=\"translate(8133.4,0)\"><g 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xlink:href=\"#MJX-1993-TEX-I-1D458\"><\/use><\/g><\/g><\/g><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\" transform=\"translate(11297.1,0)\"><g data-mml-node=\"mo\" transform=\"translate(0 -0.5)\"><use data-c=\"29\" xlink:href=\"#MJX-1993-TEX-LO-29\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(12171.9,0)\"><use data-c=\"3D\" xlink:href=\"#MJX-1993-TEX-N-3D\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(13227.7,0)\"><g data-mml-node=\"mi\"><use data-c=\"210E\" xlink:href=\"#MJX-1993-TEX-I-210E\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(609,-150) scale(0.707)\"><use data-c=\"1D45E\" xlink:href=\"#MJX-1993-TEX-I-1D45E\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(14212,0)\"><use data-c=\"28\" xlink:href=\"#MJX-1993-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(14601,0)\"><use data-c=\"1D70F\" xlink:href=\"#MJX-1993-TEX-I-1D70F\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(15118,0)\"><use data-c=\"29\" xlink:href=\"#MJX-1993-TEX-N-29\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(15729.2,0)\"><use data-c=\"2212\" xlink:href=\"#MJX-1993-TEX-N-2212\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(16729.4,0)\"><use data-c=\"28\" xlink:href=\"#MJX-1993-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(17118.4,0)\"><use data-c=\"31\" xlink:href=\"#MJX-1993-TEX-N-31\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(17840.6,0)\"><use data-c=\"2212\" xlink:href=\"#MJX-1993-TEX-N-2212\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(18840.9,0)\"><use data-c=\"1D445\" xlink:href=\"#MJX-1993-TEX-I-1D445\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(19599.9,0)\"><use data-c=\"29\" xlink:href=\"#MJX-1993-TEX-N-29\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(20211.1,0)\"><use data-c=\"2B\" xlink:href=\"#MJX-1993-TEX-N-2B\"><\/use><\/g><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\" transform=\"translate(21211.3,0)\"><g data-mml-node=\"mi\"><use data-c=\"4F\" xlink:href=\"#MJX-1993-TEX-C-4F\"><\/use><\/g><\/g><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\" transform=\"translate(22007.3,0)\"><g data-mml-node=\"mo\" transform=\"translate(0 -0.5)\"><use data-c=\"28\" xlink:href=\"#MJX-1993-TEX-SO-28\"><\/use><\/g><\/g><g data-mml-node=\"mfrac\" transform=\"translate(22465.3,0)\"><g data-mml-node=\"mrow\" transform=\"translate(220,838.5)\"><g data-mml-node=\"msub\"><g data-mml-node=\"mi\"><use data-c=\"6C\" xlink:href=\"#MJX-1993-TEX-N-6C\"><\/use><use data-c=\"6F\" xlink:href=\"#MJX-1993-TEX-N-6F\" transform=\"translate(278,0)\"><\/use><use data-c=\"67\" xlink:href=\"#MJX-1993-TEX-N-67\" transform=\"translate(778,0)\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1311,-241.4) scale(0.707)\"><use data-c=\"1D45E\" xlink:href=\"#MJX-1993-TEX-I-1D45E\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(1686.3,0)\"><use data-c=\"2061\" xlink:href=\"#MJX-1993-TEX-N-2061\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1852.9,0)\"><use data-c=\"1D45B\" xlink:href=\"#MJX-1993-TEX-I-1D45B\"><\/use><\/g><\/g><g data-mml-node=\"mi\" transform=\"translate(1146.5,-686)\"><use data-c=\"1D45B\" xlink:href=\"#MJX-1993-TEX-I-1D45B\"><\/use><\/g><rect width=\"2652.9\" height=\"60\" x=\"120\" y=\"220\"><\/rect><\/g><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\" transform=\"translate(25358.2,0)\"><g data-mml-node=\"mo\" transform=\"translate(0 -0.5)\"><use data-c=\"29\" xlink:href=\"#MJX-1993-TEX-SO-29\"><\/use><\/g><\/g><\/g><\/g><\/g><\/svg><svg data-labels=\"true\" preserveAspectRatio=\"xMaxYMid\" viewBox=\"0 -1491 1 2482\"><g data-labels=\"true\" transform=\"matrix(1 0 0 -1 0 0)\"><g data-mml-node=\"mtd\" id=\"mjx-eqn:\" transform=\"translate(0,-41.5)\"><text data-id-align=\"true\"><\/text><g data-idbox=\"true\" transform=\"translate(0,0)\"><g data-mml-node=\"mrow\"><\/g><\/g><\/g><\/g><\/svg><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"block\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable displaystyle=\"true\"><mlabeledtr><mtd><mrow><\/mrow><\/mtd><mtd><mfrac><mn>1<\/mn><mi>n<\/mi><\/mfrac><msub><mi>log<\/mi><mi>q<\/mi><\/msub><mo data-mjx-texclass=\"NONE\">\u2061<\/mo><mrow data-mjx-texclass=\"ORD\"><mo minsize=\"1.623em\" maxsize=\"1.623em\">(<\/mo><\/mrow><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"OPEN\"><mo minsize=\"2.047em\" maxsize=\"2.047em\">(<\/mo><\/mrow><mfrac linethickness=\"0\"><mi>n<\/mi><mi>t<\/mi><\/mfrac><mrow data-mjx-texclass=\"CLOSE\"><mo minsize=\"2.047em\" maxsize=\"2.047em\">)<\/mo><\/mrow><\/mrow><\/mrow><mo stretchy=\"false\">(<\/mo><mi>q<\/mi><mo>\u2212<\/mo><mn>1<\/mn><msup><mo stretchy=\"false\">)<\/mo><mi>t<\/mi><\/msup><mrow data-mjx-texclass=\"ORD\"><mo>\/<\/mo><\/mrow><msup><mi>q<\/mi><mrow data-mjx-texclass=\"ORD\"><mi>n<\/mi><mo>\u2212<\/mo><mi>k<\/mi><\/mrow><\/msup><mrow data-mjx-texclass=\"ORD\"><mo minsize=\"1.623em\" maxsize=\"1.623em\">)<\/mo><\/mrow><mo>=<\/mo><msub><mi>h<\/mi><mi>q<\/mi><\/msub><mo stretchy=\"false\">(<\/mo><mi>\u03c4<\/mi><mo stretchy=\"false\">)<\/mo><mo>\u2212<\/mo><mo stretchy=\"false\">(<\/mo><mn>1<\/mn><mo>\u2212<\/mo><mi>R<\/mi><mo stretchy=\"false\">)<\/mo><mo>+<\/mo><mrow data-mjx-texclass=\"ORD\"><mi data-mjx-variant=\"-tex-calligraphic\" mathvariant=\"script\">O<\/mi><\/mrow><mrow data-mjx-texclass=\"ORD\"><mo minsize=\"1.2em\" maxsize=\"1.2em\">(<\/mo><\/mrow><mfrac><mrow><msub><mi>log<\/mi><mi>q<\/mi><\/msub><mo data-mjx-texclass=\"NONE\">\u2061<\/mo><mi>n<\/mi><\/mrow><mi>n<\/mi><\/mfrac><mrow data-mjx-texclass=\"ORD\"><mo minsize=\"1.2em\" maxsize=\"1.2em\">)<\/mo><\/mrow><\/mtd><\/mlabeledtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><p><span>where <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"12.19ex\" height=\"1.984ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -683 5387.8 877\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.439ex;\"><defs><path id=\"MJX-972-TEX-N-30\" d=\"M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z\"><\/path><path id=\"MJX-972-TEX-N-2264\" d=\"M674 636Q682 636 688 630T694 615T687 601Q686 600 417 472L151 346L399 228Q687 92 691 87Q694 81 694 76Q694 58 676 56H670L382 192Q92 329 90 331Q83 336 83 348Q84 359 96 365Q104 369 382 500T665 634Q669 636 674 636ZM84 -118Q84 -108 99 -98H678Q694 -104 694 -118Q694 -130 679 -138H98Q84 -131 84 -118Z\"><\/path><path id=\"MJX-972-TEX-I-1D70F\" d=\"M39 284Q18 284 18 294Q18 301 45 338T99 398Q134 425 164 429Q170 431 332 431Q492 431 497 429Q517 424 517 402Q517 388 508 376T485 360Q479 358 389 358T299 356Q298 355 283 274T251 109T233 20Q228 5 215 -4T186 -13Q153 -13 153 20V30L203 192Q214 228 227 272T248 336L254 357Q254 358 208 358Q206 358 197 358T183 359Q105 359 61 295Q56 287 53 286T39 284Z\"><\/path><path id=\"MJX-972-TEX-N-2C\" d=\"M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z\"><\/path><path id=\"MJX-972-TEX-I-1D445\" d=\"M230 637Q203 637 198 638T193 649Q193 676 204 682Q206 683 378 683Q550 682 564 680Q620 672 658 652T712 606T733 563T739 529Q739 484 710 445T643 385T576 351T538 338L545 333Q612 295 612 223Q612 212 607 162T602 80V71Q602 53 603 43T614 25T640 16Q668 16 686 38T712 85Q717 99 720 102T735 105Q755 105 755 93Q755 75 731 36Q693 -21 641 -21H632Q571 -21 531 4T487 82Q487 109 502 166T517 239Q517 290 474 313Q459 320 449 321T378 323H309L277 193Q244 61 244 59Q244 55 245 54T252 50T269 48T302 46H333Q339 38 339 37T336 19Q332 6 326 0H311Q275 2 180 2Q146 2 117 2T71 2T50 1Q33 1 33 10Q33 12 36 24Q41 43 46 45Q50 46 61 46H67Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628Q287 635 230 637ZM630 554Q630 586 609 608T523 636Q521 636 500 636T462 637H440Q393 637 386 627Q385 624 352 494T319 361Q319 360 388 360Q466 361 492 367Q556 377 592 426Q608 449 619 486T630 554Z\"><\/path><path id=\"MJX-972-TEX-N-31\" d=\"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mn\"><use data-c=\"30\" xlink:href=\"#MJX-972-TEX-N-30\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(777.8,0)\"><use data-c=\"2264\" xlink:href=\"#MJX-972-TEX-N-2264\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1833.6,0)\"><use data-c=\"1D70F\" xlink:href=\"#MJX-972-TEX-I-1D70F\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2350.6,0)\"><use data-c=\"2C\" xlink:href=\"#MJX-972-TEX-N-2C\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(2795.2,0)\"><use data-c=\"1D445\" xlink:href=\"#MJX-972-TEX-I-1D445\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(3832,0)\"><use data-c=\"2264\" xlink:href=\"#MJX-972-TEX-N-2264\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(4887.8,0)\"><use data-c=\"31\" xlink:href=\"#MJX-972-TEX-N-31\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mn>0<\/mn><mo>\u2264<\/mo><mi>\u03c4<\/mi><mo>,<\/mo><mi>R<\/mi><mo>\u2264<\/mo><mn>1<\/mn><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">0\\le\\tau,R\\le 1<\/script><span>. Since <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"9.267ex\" height=\"2.347ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 4095.8 1037.2\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.65ex;\"><defs><path id=\"MJX-973-TEX-I-210E\" d=\"M137 683Q138 683 209 688T282 694Q294 694 294 685Q294 674 258 534Q220 386 220 383Q220 381 227 388Q288 442 357 442Q411 442 444 415T478 336Q478 285 440 178T402 50Q403 36 407 31T422 26Q450 26 474 56T513 138Q516 149 519 151T535 153Q555 153 555 145Q555 144 551 130Q535 71 500 33Q466 -10 419 -10H414Q367 -10 346 17T325 74Q325 90 361 192T398 345Q398 404 354 404H349Q266 404 205 306L198 293L164 158Q132 28 127 16Q114 -11 83 -11Q69 -11 59 -2T48 16Q48 30 121 320L195 616Q195 629 188 632T149 637H128Q122 643 122 645T124 664Q129 683 137 683Z\"><\/path><path id=\"MJX-973-TEX-I-1D45E\" d=\"M33 157Q33 258 109 349T280 441Q340 441 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transform=\"translate(609,-150) scale(0.707)\"><use data-c=\"1D45E\" xlink:href=\"#MJX-973-TEX-I-1D45E\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(984.3,0)\"><use data-c=\"28\" xlink:href=\"#MJX-973-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(1373.3,0)\"><use data-c=\"30\" xlink:href=\"#MJX-973-TEX-N-30\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(1873.3,0)\"><use data-c=\"29\" xlink:href=\"#MJX-973-TEX-N-29\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2540,0)\"><use data-c=\"3D\" xlink:href=\"#MJX-973-TEX-N-3D\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(3595.8,0)\"><use data-c=\"30\" xlink:href=\"#MJX-973-TEX-N-30\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><msub><mi>h<\/mi><mi>q<\/mi><\/msub><mo stretchy=\"false\">(<\/mo><mn>0<\/mn><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mn>0<\/mn><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">h_q(0)=0<\/script><span> and <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"18.648ex\" height=\"2.553ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 8242.5 1128.5\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.856ex;\"><defs><path id=\"MJX-975-TEX-I-210E\" d=\"M137 683Q138 683 209 688T282 694Q294 694 294 685Q294 674 258 534Q220 386 220 383Q220 381 227 388Q288 442 357 442Q411 442 444 415T478 336Q478 285 440 178T402 50Q403 36 407 31T422 26Q450 26 474 56T513 138Q516 149 519 151T535 153Q555 153 555 145Q555 144 551 130Q535 71 500 33Q466 -10 419 -10H414Q367 -10 346 17T325 74Q325 90 361 192T398 345Q398 404 354 404H349Q266 404 205 306L198 293L164 158Q132 28 127 16Q114 -11 83 -11Q69 -11 59 -2T48 16Q48 30 121 320L195 616Q195 629 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292Q52 355 102 398T223 442Q274 442 318 416L329 409ZM299 343Q294 371 273 387T221 404Q192 404 171 388T145 343Q142 326 142 292Q142 248 149 227T179 192Q196 182 222 182Q244 182 260 189T283 207T294 227T299 242Q302 258 302 292T299 343ZM403 -75Q403 -50 389 -34T348 -11T299 -2T245 0H218Q151 0 138 -6Q118 -15 107 -34T95 -74Q95 -84 101 -97T122 -127T170 -155T250 -167Q319 -167 361 -139T403 -75Z\"><\/path><path id=\"MJX-975-TEX-N-2061\" d=\"\"><\/path><path id=\"MJX-975-TEX-N-2212\" d=\"M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"msub\"><g data-mml-node=\"mi\"><use data-c=\"210E\" xlink:href=\"#MJX-975-TEX-I-210E\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(609,-150) scale(0.707)\"><use data-c=\"1D45E\" xlink:href=\"#MJX-975-TEX-I-1D45E\"><\/use><\/g><\/g><g data-mml-node=\"mo\" 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data-c=\"2061\" xlink:href=\"#MJX-975-TEX-N-2061\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(5282.1,0)\"><use data-c=\"28\" xlink:href=\"#MJX-975-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(5671.1,0)\"><use data-c=\"1D45E\" xlink:href=\"#MJX-975-TEX-I-1D45E\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(6353.3,0)\"><use data-c=\"2212\" xlink:href=\"#MJX-975-TEX-N-2212\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(7353.5,0)\"><use data-c=\"31\" xlink:href=\"#MJX-975-TEX-N-31\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(7853.5,0)\"><use data-c=\"29\" xlink:href=\"#MJX-975-TEX-N-29\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><msub><mi>h<\/mi><mi>q<\/mi><\/msub><mo stretchy=\"false\">(<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><msub><mi>log<\/mi><mi>q<\/mi><\/msub><mo data-mjx-texclass=\"NONE\">\u2061<\/mo><mo stretchy=\"false\">(<\/mo><mi>q<\/mi><mo>\u2212<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">h_q(1)=\\log_q(q-1)<\/script><span>, <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"7.88ex\" height=\"2.713ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -849.5 3482.9 1199\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.791ex;\"><defs><path id=\"MJX-1105-TEX-I-1D441\" d=\"M234 637Q231 637 226 637Q201 637 196 638T191 649Q191 676 202 682Q204 683 299 683Q376 683 387 683T401 677Q612 181 616 168L670 381Q723 592 723 606Q723 633 659 637Q635 637 635 648Q635 650 637 660Q641 676 643 679T653 683Q656 683 684 682T767 680Q817 680 843 681T873 682Q888 682 888 672Q888 650 880 642Q878 637 858 637Q787 633 769 597L620 7Q618 0 599 0Q585 0 582 2Q579 5 453 305L326 604L261 344Q196 88 196 79Q201 46 268 46H278Q284 41 284 38T282 19Q278 6 272 0H259Q228 2 151 2Q123 2 100 2T63 2T46 1Q31 1 31 10Q31 14 34 26T39 40Q41 46 62 46Q130 49 150 85Q154 91 221 362L289 634Q287 635 234 637Z\"><\/path><path id=\"MJX-1105-TEX-I-1D461\" d=\"M26 385Q19 392 19 395Q19 399 22 411T27 425Q29 430 36 430T87 431H140L159 511Q162 522 166 540T173 566T179 586T187 603T197 615T211 624T229 626Q247 625 254 615T261 596Q261 589 252 549T232 470L222 433Q222 431 272 431H323Q330 424 330 420Q330 398 317 385H210L174 240Q135 80 135 68Q135 26 162 26Q197 26 230 60T283 144Q285 150 288 151T303 153H307Q322 153 322 145Q322 142 319 133Q314 117 301 95T267 48T216 6T155 -11Q125 -11 98 4T59 56Q57 64 57 83V101L92 241Q127 382 128 383Q128 385 77 385H26Z\"><\/path><path id=\"MJX-1105-TEX-SO-28\" d=\"M152 251Q152 646 388 850H416Q422 844 422 841Q422 837 403 816T357 753T302 649T255 482T236 250Q236 124 255 19T301 -147T356 -251T403 -315T422 -340Q422 -343 416 -349H388Q359 -325 332 -296T271 -213T212 -97T170 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294T285 286Q305 281 325 273T373 237T412 172Q414 162 414 142Q414 -6 230 -6Q154 -6 117 22L68 -6H58Q44 -6 41 0T38 42V73Q38 85 38 101T37 122Q37 144 42 148T68 153H75Q87 153 91 151T97 147T103 132Q131 46 220 46H230Q257 46 265 47Q330 58 330 108Q330 127 316 142Q300 156 284 162Q271 168 212 178T122 202Q38 243 38 315Z\"><\/path><path id=\"MJX-1105-TEX-SO-29\" d=\"M305 251Q305 -145 69 -349H56Q43 -349 39 -347T35 -338Q37 -333 60 -307T108 -239T160 -136T204 27T221 250T204 473T160 636T108 740T60 807T35 839Q35 850 50 850H56H69Q197 743 256 566Q305 425 305 251Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"msub\"><g data-mml-node=\"mi\"><use data-c=\"1D441\" xlink:href=\"#MJX-1105-TEX-I-1D441\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(836,-150) scale(0.707)\"><use data-c=\"1D461\" xlink:href=\"#MJX-1105-TEX-I-1D461\"><\/use><\/g><\/g><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\" transform=\"translate(1141.3,0)\"><g data-mml-node=\"mo\" transform=\"translate(0 -0.5)\"><use data-c=\"28\" xlink:href=\"#MJX-1105-TEX-SO-28\"><\/use><\/g><\/g><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\" transform=\"translate(1599.3,0)\"><g data-mml-node=\"mi\"><use data-c=\"43\" xlink:href=\"#MJX-1105-TEX-C-43\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(2126.3,0)\"><use data-c=\"2C\" xlink:href=\"#MJX-1105-TEX-N-2C\"><\/use><\/g><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\" transform=\"translate(2570.9,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D42C\" xlink:href=\"#MJX-1105-TEX-B-1D42C\"><\/use><\/g><\/g><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\" transform=\"translate(3024.9,0)\"><g data-mml-node=\"mo\" transform=\"translate(0 -0.5)\"><use data-c=\"29\" xlink:href=\"#MJX-1105-TEX-SO-29\"><\/use><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><msub><mi>N<\/mi><mi>t<\/mi><\/msub><mrow data-mjx-texclass=\"ORD\"><mo minsize=\"1.2em\" maxsize=\"1.2em\">(<\/mo><\/mrow><mrow data-mjx-texclass=\"ORD\"><mi data-mjx-variant=\"-tex-calligraphic\" mathvariant=\"script\">C<\/mi><\/mrow><mo>,<\/mo><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">s<\/mi><\/mrow><mrow data-mjx-texclass=\"ORD\"><mo minsize=\"1.2em\" maxsize=\"1.2em\">)<\/mo><\/mrow><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">N_t\\big(\\mathcal{C},\\mathbf s\\big)<\/script><span> is expected to be exponentially small at one value <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"2.724ex\" height=\"1.783ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -775.2 1204.1 788.2\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.029ex;\"><defs><path id=\"MJX-1103-TEX-I-1D70F\" d=\"M39 284Q18 284 18 294Q18 301 45 338T99 398Q134 425 164 429Q170 431 332 431Q492 431 497 429Q517 424 517 402Q517 388 508 376T485 360Q479 358 389 358T299 356Q298 355 283 274T251 109T233 20Q228 5 215 -4T186 -13Q153 -13 153 20V30L203 192Q214 228 227 272T248 336L254 357Q254 358 208 358Q206 358 197 358T183 359Q105 359 61 295Q56 287 53 286T39 284Z\"><\/path><path id=\"MJX-1103-TEX-N-2212\" d=\"M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"msup\"><g data-mml-node=\"mi\"><use data-c=\"1D70F\" xlink:href=\"#MJX-1103-TEX-I-1D70F\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(604,363) scale(0.707)\"><use data-c=\"2212\" xlink:href=\"#MJX-1103-TEX-N-2212\"><\/use><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><msup><mi>\u03c4<\/mi><mo>\u2212<\/mo><\/msup><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">\\tau^-<\/script><span> and potentially a second value <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"2.724ex\" height=\"1.783ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -775.2 1204.1 788.2\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.029ex;\"><defs><path id=\"MJX-1104-TEX-I-1D70F\" d=\"M39 284Q18 284 18 294Q18 301 45 338T99 398Q134 425 164 429Q170 431 332 431Q492 431 497 429Q517 424 517 402Q517 388 508 376T485 360Q479 358 389 358T299 356Q298 355 283 274T251 109T233 20Q228 5 215 -4T186 -13Q153 -13 153 20V30L203 192Q214 228 227 272T248 336L254 357Q254 358 208 358Q206 358 197 358T183 359Q105 359 61 295Q56 287 53 286T39 284Z\"><\/path><path id=\"MJX-1104-TEX-N-2B\" d=\"M56 237T56 250T70 270H369V420L370 570Q380 583 389 583Q402 583 409 568V270H707Q722 262 722 250T707 230H409V-68Q401 -82 391 -82H389H387Q375 -82 369 -68V230H70Q56 237 56 250Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"msup\"><g data-mml-node=\"mi\"><use data-c=\"1D70F\" xlink:href=\"#MJX-1104-TEX-I-1D70F\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(604,363) scale(0.707)\"><use data-c=\"2B\" xlink:href=\"#MJX-1104-TEX-N-2B\"><\/use><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><msup><mi>\u03c4<\/mi><mo>+<\/mo><\/msup><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">\\tau^+<\/script><span> in the case that <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"20.904ex\" height=\"2.553ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 9239.7 1128.5\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.856ex;\"><defs><path id=\"MJX-983-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-983-TEX-N-31\" d=\"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z\"><\/path><path id=\"MJX-983-TEX-N-2212\" d=\"M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z\"><\/path><path id=\"MJX-983-TEX-I-1D445\" d=\"M230 637Q203 637 198 638T193 649Q193 676 204 682Q206 683 378 683Q550 682 564 680Q620 672 658 652T712 606T733 563T739 529Q739 484 710 445T643 385T576 351T538 338L545 333Q612 295 612 223Q612 212 607 162T602 80V71Q602 53 603 43T614 25T640 16Q668 16 686 38T712 85Q717 99 720 102T735 105Q755 105 755 93Q755 75 731 36Q693 -21 641 -21H632Q571 -21 531 4T487 82Q487 109 502 166T517 239Q517 290 474 313Q459 320 449 321T378 323H309L277 193Q244 61 244 59Q244 55 245 54T252 50T269 48T302 46H333Q339 38 339 37T336 19Q332 6 326 0H311Q275 2 180 2Q146 2 117 2T71 2T50 1Q33 1 33 10Q33 12 36 24Q41 43 46 45Q50 46 61 46H67Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628Q287 635 230 637ZM630 554Q630 586 609 608T523 636Q521 636 500 636T462 637H440Q393 637 386 627Q385 624 352 494T319 361Q319 360 388 360Q466 361 492 367Q556 377 592 426Q608 449 619 486T630 554Z\"><\/path><path id=\"MJX-983-TEX-N-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><path id=\"MJX-983-TEX-N-2265\" d=\"M83 616Q83 624 89 630T99 636Q107 636 253 568T543 431T687 361Q694 356 694 346T687 331Q685 329 395 192L107 56H101Q83 58 83 76Q83 77 83 79Q82 86 98 95Q117 105 248 167Q326 204 378 228L626 346L360 472Q291 505 200 548Q112 589 98 597T83 616ZM84 -118Q84 -108 99 -98H678Q694 -104 694 -118Q694 -130 679 -138H98Q84 -131 84 -118Z\"><\/path><path id=\"MJX-983-TEX-N-6C\" d=\"M42 46H56Q95 46 103 60V68Q103 77 103 91T103 124T104 167T104 217T104 272T104 329Q104 366 104 407T104 482T104 542T103 586T103 603Q100 622 89 628T44 637H26V660Q26 683 28 683L38 684Q48 685 67 686T104 688Q121 689 141 690T171 693T182 694H185V379Q185 62 186 60Q190 52 198 49Q219 46 247 46H263V0H255L232 1Q209 2 183 2T145 3T107 3T57 1L34 0H26V46H42Z\"><\/path><path id=\"MJX-983-TEX-N-6F\" d=\"M28 214Q28 309 93 378T250 448Q340 448 405 380T471 215Q471 120 407 55T250 -10Q153 -10 91 57T28 214ZM250 30Q372 30 372 193V225V250Q372 272 371 288T364 326T348 362T317 390T268 410Q263 411 252 411Q222 411 195 399Q152 377 139 338T126 246V226Q126 130 145 91Q177 30 250 30Z\"><\/path><path id=\"MJX-983-TEX-N-67\" d=\"M329 409Q373 453 429 453Q459 453 472 434T485 396Q485 382 476 371T449 360Q416 360 412 390Q410 404 415 411Q415 412 416 414V415Q388 412 363 393Q355 388 355 386Q355 385 359 381T368 369T379 351T388 325T392 292Q392 230 343 187T222 143Q172 143 123 171Q112 153 112 133Q112 98 138 81Q147 75 155 75T227 73Q311 72 335 67Q396 58 431 26Q470 -13 470 -72Q470 -139 392 -175Q332 -206 250 -206Q167 -206 107 -175Q29 -140 29 -75Q29 -39 50 -15T92 18L103 24Q67 55 67 108Q67 155 96 193Q52 237 52 292Q52 355 102 398T223 442Q274 442 318 416L329 409ZM299 343Q294 371 273 387T221 404Q192 404 171 388T145 343Q142 326 142 292Q142 248 149 227T179 192Q196 182 222 182Q244 182 260 189T283 207T294 227T299 242Q302 258 302 292T299 343ZM403 -75Q403 -50 389 -34T348 -11T299 -2T245 0H218Q151 0 138 -6Q118 -15 107 -34T95 -74Q95 -84 101 -97T122 -127T170 -155T250 -167Q319 -167 361 -139T403 -75Z\"><\/path><path id=\"MJX-983-TEX-I-1D45E\" d=\"M33 157Q33 258 109 349T280 441Q340 441 372 389Q373 390 377 395T388 406T404 418Q438 442 450 442Q454 442 457 439T460 434Q460 425 391 149Q320 -135 320 -139Q320 -147 365 -148H390Q396 -156 396 -157T393 -175Q389 -188 383 -194H370Q339 -192 262 -192Q234 -192 211 -192T174 -192T157 -193Q143 -193 143 -185Q143 -182 145 -170Q149 -154 152 -151T172 -148Q220 -148 230 -141Q238 -136 258 -53T279 32Q279 33 272 29Q224 -10 172 -10Q117 -10 75 30T33 157ZM352 326Q329 405 277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q233 26 290 98L298 109L352 326Z\"><\/path><path id=\"MJX-983-TEX-N-2061\" d=\"\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mo\"><use data-c=\"28\" xlink:href=\"#MJX-983-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(389,0)\"><use data-c=\"31\" xlink:href=\"#MJX-983-TEX-N-31\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(1111.2,0)\"><use data-c=\"2212\" xlink:href=\"#MJX-983-TEX-N-2212\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(2111.4,0)\"><use data-c=\"1D445\" xlink:href=\"#MJX-983-TEX-I-1D445\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2870.4,0)\"><use data-c=\"29\" xlink:href=\"#MJX-983-TEX-N-29\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(3537.2,0)\"><use data-c=\"2265\" xlink:href=\"#MJX-983-TEX-N-2265\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(4593,0)\"><g data-mml-node=\"mi\"><use data-c=\"6C\" xlink:href=\"#MJX-983-TEX-N-6C\"><\/use><use data-c=\"6F\" xlink:href=\"#MJX-983-TEX-N-6F\" transform=\"translate(278,0)\"><\/use><use data-c=\"67\" xlink:href=\"#MJX-983-TEX-N-67\" transform=\"translate(778,0)\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1311,-241.4) scale(0.707)\"><use data-c=\"1D45E\" xlink:href=\"#MJX-983-TEX-I-1D45E\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(6279.3,0)\"><use data-c=\"2061\" xlink:href=\"#MJX-983-TEX-N-2061\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(6279.3,0)\"><use data-c=\"28\" xlink:href=\"#MJX-983-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(6668.3,0)\"><use data-c=\"1D45E\" xlink:href=\"#MJX-983-TEX-I-1D45E\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(7350.5,0)\"><use data-c=\"2212\" xlink:href=\"#MJX-983-TEX-N-2212\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(8350.7,0)\"><use data-c=\"31\" xlink:href=\"#MJX-983-TEX-N-31\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(8850.7,0)\"><use data-c=\"29\" xlink:href=\"#MJX-983-TEX-N-29\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mo stretchy=\"false\">(<\/mo><mn>1<\/mn><mo>\u2212<\/mo><mi>R<\/mi><mo stretchy=\"false\">)<\/mo><mo>\u2265<\/mo><msub><mi>log<\/mi><mi>q<\/mi><\/msub><mo data-mjx-texclass=\"NONE\">\u2061<\/mo><mo stretchy=\"false\">(<\/mo><mi>q<\/mi><mo>\u2212<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">(1-R)\\ge\\log_q(q-1)<\/script><span>. The analytic expression of <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"2.724ex\" height=\"1.783ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -775.2 1204.1 788.2\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.029ex;\"><defs><path id=\"MJX-1103-TEX-I-1D70F\" d=\"M39 284Q18 284 18 294Q18 301 45 338T99 398Q134 425 164 429Q170 431 332 431Q492 431 497 429Q517 424 517 402Q517 388 508 376T485 360Q479 358 389 358T299 356Q298 355 283 274T251 109T233 20Q228 5 215 -4T186 -13Q153 -13 153 20V30L203 192Q214 228 227 272T248 336L254 357Q254 358 208 358Q206 358 197 358T183 359Q105 359 61 295Q56 287 53 286T39 284Z\"><\/path><path id=\"MJX-1103-TEX-N-2212\" d=\"M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"msup\"><g data-mml-node=\"mi\"><use data-c=\"1D70F\" xlink:href=\"#MJX-1103-TEX-I-1D70F\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(604,363) scale(0.707)\"><use data-c=\"2212\" xlink:href=\"#MJX-1103-TEX-N-2212\"><\/use><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><msup><mi>\u03c4<\/mi><mo>\u2212<\/mo><\/msup><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">\\tau^-<\/script><span> and <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"2.724ex\" height=\"1.783ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -775.2 1204.1 788.2\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.029ex;\"><defs><path id=\"MJX-1104-TEX-I-1D70F\" d=\"M39 284Q18 284 18 294Q18 301 45 338T99 398Q134 425 164 429Q170 431 332 431Q492 431 497 429Q517 424 517 402Q517 388 508 376T485 360Q479 358 389 358T299 356Q298 355 283 274T251 109T233 20Q228 5 215 -4T186 -13Q153 -13 153 20V30L203 192Q214 228 227 272T248 336L254 357Q254 358 208 358Q206 358 197 358T183 359Q105 359 61 295Q56 287 53 286T39 284Z\"><\/path><path id=\"MJX-1104-TEX-N-2B\" d=\"M56 237T56 250T70 270H369V420L370 570Q380 583 389 583Q402 583 409 568V270H707Q722 262 722 250T707 230H409V-68Q401 -82 391 -82H389H387Q375 -82 369 -68V230H70Q56 237 56 250Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"msup\"><g data-mml-node=\"mi\"><use data-c=\"1D70F\" xlink:href=\"#MJX-1104-TEX-I-1D70F\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(604,363) scale(0.707)\"><use data-c=\"2B\" xlink:href=\"#MJX-1104-TEX-N-2B\"><\/use><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><msup><mi>\u03c4<\/mi><mo>+<\/mo><\/msup><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">\\tau^+<\/script><span> is<\/span><\/p><div contenteditable=\"false\" spellcheck=\"false\" class=\"mathjax-block md-end-block md-math-block md-rawblock\" id=\"mathjax-n288\" cid=\"n288\" mdtype=\"math_block\" data-math-tag-before=\"10\" data-math-labels=\"[]\" data-math-tag-after=\"10\"><div class=\"md-rawblock-container md-math-container\" contenteditable=\"false\" tabindex=\"-1\"><mjx-container class=\"MathJax\" jax=\"SVG\" display=\"true\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"41.056ex\" height=\"6.151ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -1609.4 18147 2718.8\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -2.51ex;\"><defs><path id=\"MJX-1994-TEX-I-1D70F\" d=\"M39 284Q18 284 18 294Q18 301 45 338T99 398Q134 425 164 429Q170 431 332 431Q492 431 497 429Q517 424 517 402Q517 388 508 376T485 360Q479 358 389 358T299 356Q298 355 283 274T251 109T233 20Q228 5 215 -4T186 -13Q153 -13 153 20V30L203 192Q214 228 227 272T248 336L254 357Q254 358 208 358Q206 358 197 358T183 359Q105 359 61 295Q56 287 53 286T39 284Z\"><\/path><path id=\"MJX-1994-TEX-N-2212\" d=\"M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z\"><\/path><path id=\"MJX-1994-TEX-N-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path id=\"MJX-1994-TEX-I-1D454\" d=\"M311 43Q296 30 267 15T206 0Q143 0 105 45T66 160Q66 265 143 353T314 442Q361 442 401 394L404 398Q406 401 409 404T418 412T431 419T447 422Q461 422 470 413T480 394Q480 379 423 152T363 -80Q345 -134 286 -169T151 -205Q10 -205 10 -137Q10 -111 28 -91T74 -71Q89 -71 102 -80T116 -111Q116 -121 114 -130T107 -144T99 -154T92 -162L90 -164H91Q101 -167 151 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741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-1994-TEX-N-31\" d=\"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z\"><\/path><path id=\"MJX-1994-TEX-I-1D445\" d=\"M230 637Q203 637 198 638T193 649Q193 676 204 682Q206 683 378 683Q550 682 564 680Q620 672 658 652T712 606T733 563T739 529Q739 484 710 445T643 385T576 351T538 338L545 333Q612 295 612 223Q612 212 607 162T602 80V71Q602 53 603 43T614 25T640 16Q668 16 686 38T712 85Q717 99 720 102T735 105Q755 105 755 93Q755 75 731 36Q693 -21 641 -21H632Q571 -21 531 4T487 82Q487 109 502 166T517 239Q517 290 474 313Q459 320 449 321T378 323H309L277 193Q244 61 244 59Q244 55 245 54T252 50T269 48T302 46H333Q339 38 339 37T336 19Q332 6 326 0H311Q275 2 180 2Q146 2 117 2T71 2T50 1Q33 1 33 10Q33 12 36 24Q41 43 46 45Q50 46 61 46H67Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628Q287 635 230 637ZM630 554Q630 586 609 608T523 636Q521 636 500 636T462 637H440Q393 637 386 627Q385 624 352 494T319 361Q319 360 388 360Q466 361 492 367Q556 377 592 426Q608 449 619 486T630 554Z\"><\/path><path id=\"MJX-1994-TEX-N-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><path id=\"MJX-1994-TEX-N-2B\" d=\"M56 237T56 250T70 270H369V420L370 570Q380 583 389 583Q402 583 409 568V270H707Q722 262 722 250T707 230H409V-68Q401 -82 391 -82H389H387Q375 -82 369 -68V230H70Q56 237 56 250Z\"><\/path><path id=\"MJX-1994-TEX-N-A0\" d=\"\"><\/path><path id=\"MJX-1994-TEX-N-77\" d=\"M90 368Q84 378 76 380T40 385H18V431H24L43 430Q62 430 84 429T116 428Q206 428 221 431H229V385H215Q177 383 177 368Q177 367 221 239L265 113L339 328L333 345Q323 374 316 379Q308 384 278 385H258V431H264Q270 428 348 428Q439 428 454 431H461V385H452Q404 385 404 369Q404 366 418 324T449 234T481 143L496 100L537 219Q579 341 579 347Q579 363 564 373T530 385H522V431H529Q541 428 624 428Q692 428 698 431H703V385H697Q696 385 691 385T682 384Q635 377 619 334L559 161Q546 124 528 71Q508 12 503 1T487 -11H479Q460 -11 456 -4Q455 -3 407 133L361 267Q359 263 266 -4Q261 -11 243 -11H238Q225 -11 220 -3L90 368Z\"><\/path><path id=\"MJX-1994-TEX-N-68\" d=\"M41 46H55Q94 46 102 60V68Q102 77 102 91T102 124T102 167T103 217T103 272T103 329Q103 366 103 407T103 482T102 542T102 586T102 603Q99 622 88 628T43 637H25V660Q25 683 27 683L37 684Q47 685 66 686T103 688Q120 689 140 690T170 693T181 694H184V367Q244 442 328 442Q451 442 463 329Q464 322 464 190V104Q464 66 466 59T477 49Q498 46 526 46H542V0H534L510 1Q487 2 460 2T422 3Q319 3 310 0H302V46H318Q379 46 379 62Q380 64 380 200Q379 335 378 343Q372 371 358 385T334 402T308 404Q263 404 229 370Q202 343 195 315T187 232V168V108Q187 78 188 68T191 55T200 49Q221 46 249 46H265V0H257L234 1Q210 2 183 2T145 3Q42 3 33 0H25V46H41Z\"><\/path><path id=\"MJX-1994-TEX-N-65\" d=\"M28 218Q28 273 48 318T98 391T163 433T229 448Q282 448 320 430T378 380T406 316T415 245Q415 238 408 231H126V216Q126 68 226 36Q246 30 270 30Q312 30 342 62Q359 79 369 104L379 128Q382 131 395 131H398Q415 131 415 121Q415 117 412 108Q393 53 349 21T250 -11Q155 -11 92 58T28 218ZM333 275Q322 403 238 411H236Q228 411 220 410T195 402T166 381T143 340T127 274V267H333V275Z\"><\/path><path id=\"MJX-1994-TEX-N-6E\" d=\"M41 46H55Q94 46 102 60V68Q102 77 102 91T102 122T103 161T103 203Q103 234 103 269T102 328V351Q99 370 88 376T43 385H25V408Q25 431 27 431L37 432Q47 433 65 434T102 436Q119 437 138 438T167 441T178 442H181V402Q181 364 182 364T187 369T199 384T218 402T247 421T285 437Q305 442 336 442Q450 438 463 329Q464 322 464 190V104Q464 66 466 59T477 49Q498 46 526 46H542V0H534L510 1Q487 2 460 2T422 3Q319 3 310 0H302V46H318Q379 46 379 62Q380 64 380 200Q379 335 378 343Q372 371 358 385T334 402T308 404Q263 404 229 370Q202 343 195 315T187 232V168V108Q187 78 188 68T191 55T200 49Q221 46 249 46H265V0H257L234 1Q210 2 183 2T145 3Q42 3 33 0H25V46H41Z\"><\/path><path id=\"MJX-1994-TEX-N-2264\" d=\"M674 636Q682 636 688 630T694 615T687 601Q686 600 417 472L151 346L399 228Q687 92 691 87Q694 81 694 76Q694 58 676 56H670L382 192Q92 329 90 331Q83 336 83 348Q84 359 96 365Q104 369 382 500T665 634Q669 636 674 636ZM84 -118Q84 -108 99 -98H678Q694 -104 694 -118Q694 -130 679 -138H98Q84 -131 84 -118Z\"><\/path><path id=\"MJX-1994-TEX-N-6C\" d=\"M42 46H56Q95 46 103 60V68Q103 77 103 91T103 124T104 167T104 217T104 272T104 329Q104 366 104 407T104 482T104 542T103 586T103 603Q100 622 89 628T44 637H26V660Q26 683 28 683L38 684Q48 685 67 686T104 688Q121 689 141 690T171 693T182 694H185V379Q185 62 186 60Q190 52 198 49Q219 46 247 46H263V0H255L232 1Q209 2 183 2T145 3T107 3T57 1L34 0H26V46H42Z\"><\/path><path id=\"MJX-1994-TEX-N-6F\" d=\"M28 214Q28 309 93 378T250 448Q340 448 405 380T471 215Q471 120 407 55T250 -10Q153 -10 91 57T28 214ZM250 30Q372 30 372 193V225V250Q372 272 371 288T364 326T348 362T317 390T268 410Q263 411 252 411Q222 411 195 399Q152 377 139 338T126 246V226Q126 130 145 91Q177 30 250 30Z\"><\/path><path id=\"MJX-1994-TEX-N-67\" d=\"M329 409Q373 453 429 453Q459 453 472 434T485 396Q485 382 476 371T449 360Q416 360 412 390Q410 404 415 411Q415 412 416 414V415Q388 412 363 393Q355 388 355 386Q355 385 359 381T368 369T379 351T388 325T392 292Q392 230 343 187T222 143Q172 143 123 171Q112 153 112 133Q112 98 138 81Q147 75 155 75T227 73Q311 72 335 67Q396 58 431 26Q470 -13 470 -72Q470 -139 392 -175Q332 -206 250 -206Q167 -206 107 -175Q29 -140 29 -75Q29 -39 50 -15T92 18L103 24Q67 55 67 108Q67 155 96 193Q52 237 52 292Q52 355 102 398T223 442Q274 442 318 416L329 409ZM299 343Q294 371 273 387T221 404Q192 404 171 388T145 343Q142 326 142 292Q142 248 149 227T179 192Q196 182 222 182Q244 182 260 189T283 207T294 227T299 242Q302 258 302 292T299 343ZM403 -75Q403 -50 389 -34T348 -11T299 -2T245 0H218Q151 0 138 -6Q118 -15 107 -34T95 -74Q95 -84 101 -97T122 -127T170 -155T250 -167Q319 -167 361 -139T403 -75Z\"><\/path><path id=\"MJX-1994-TEX-N-2061\" d=\"\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mtable\"><g data-mml-node=\"mtr\" transform=\"translate(0,784.2)\"><g data-mml-node=\"mtd\"><g data-mml-node=\"msup\"><g data-mml-node=\"mi\"><use data-c=\"1D70F\" xlink:href=\"#MJX-1994-TEX-I-1D70F\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(604,413) scale(0.707)\"><use data-c=\"2212\" xlink:href=\"#MJX-1994-TEX-N-2212\"><\/use><\/g><\/g><\/g><g data-mml-node=\"mtd\" transform=\"translate(1204.1,0)\"><g data-mml-node=\"mi\"><\/g><g data-mml-node=\"mo\" transform=\"translate(277.8,0)\"><use data-c=\"3D\" xlink:href=\"#MJX-1994-TEX-N-3D\"><\/use><\/g><g data-mml-node=\"msubsup\" transform=\"translate(1333.6,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D454\" xlink:href=\"#MJX-1994-TEX-I-1D454\"><\/use><\/g><g data-mml-node=\"TeXAtom\" transform=\"translate(510,413) scale(0.707)\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mo\"><use data-c=\"2212\" xlink:href=\"#MJX-1994-TEX-N-2212\"><\/use><\/g><\/g><g data-mml-node=\"mi\" transform=\"translate(510,-247) scale(0.707)\"><use data-c=\"1D45E\" xlink:href=\"#MJX-1994-TEX-I-1D45E\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(2443.7,0)\"><use data-c=\"28\" xlink:href=\"#MJX-1994-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(2832.7,0)\"><use data-c=\"31\" xlink:href=\"#MJX-1994-TEX-N-31\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(3554.9,0)\"><use data-c=\"2212\" xlink:href=\"#MJX-1994-TEX-N-2212\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(4555.1,0)\"><use data-c=\"1D445\" xlink:href=\"#MJX-1994-TEX-I-1D445\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(5314.1,0)\"><use data-c=\"29\" xlink:href=\"#MJX-1994-TEX-N-29\"><\/use><\/g><\/g><\/g><g data-mml-node=\"mtr\" transform=\"translate(0,-725.2)\"><g data-mml-node=\"mtd\"><g data-mml-node=\"msup\"><g data-mml-node=\"mi\"><use data-c=\"1D70F\" xlink:href=\"#MJX-1994-TEX-I-1D70F\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(604,413) scale(0.707)\"><use data-c=\"2B\" xlink:href=\"#MJX-1994-TEX-N-2B\"><\/use><\/g><\/g><\/g><g data-mml-node=\"mtd\" transform=\"translate(1204.1,0)\"><g data-mml-node=\"mi\"><\/g><g data-mml-node=\"mo\" transform=\"translate(277.8,0)\"><use data-c=\"3D\" xlink:href=\"#MJX-1994-TEX-N-3D\"><\/use><\/g><g data-mml-node=\"msubsup\" transform=\"translate(1333.6,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D454\" xlink:href=\"#MJX-1994-TEX-I-1D454\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(510,413) scale(0.707)\"><use data-c=\"2B\" xlink:href=\"#MJX-1994-TEX-N-2B\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(510,-247) scale(0.707)\"><use data-c=\"1D45E\" xlink:href=\"#MJX-1994-TEX-I-1D45E\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(2443.7,0)\"><use data-c=\"28\" xlink:href=\"#MJX-1994-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(2832.7,0)\"><use data-c=\"31\" xlink:href=\"#MJX-1994-TEX-N-31\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(3554.9,0)\"><use data-c=\"2212\" xlink:href=\"#MJX-1994-TEX-N-2212\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(4555.1,0)\"><use data-c=\"1D445\" xlink:href=\"#MJX-1994-TEX-I-1D445\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(5314.1,0)\"><use data-c=\"29\" xlink:href=\"#MJX-1994-TEX-N-29\"><\/use><\/g><g data-mml-node=\"mtext\" transform=\"translate(5703.1,0)\"><use data-c=\"A0\" xlink:href=\"#MJX-1994-TEX-N-A0\"><\/use><use data-c=\"77\" xlink:href=\"#MJX-1994-TEX-N-77\" transform=\"translate(250,0)\"><\/use><use data-c=\"68\" xlink:href=\"#MJX-1994-TEX-N-68\" transform=\"translate(972,0)\"><\/use><use data-c=\"65\" xlink:href=\"#MJX-1994-TEX-N-65\" transform=\"translate(1528,0)\"><\/use><use data-c=\"6E\" xlink:href=\"#MJX-1994-TEX-N-6E\" transform=\"translate(1972,0)\"><\/use><use data-c=\"A0\" xlink:href=\"#MJX-1994-TEX-N-A0\" transform=\"translate(2528,0)\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(8481.1,0)\"><use data-c=\"1D445\" xlink:href=\"#MJX-1994-TEX-I-1D445\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(9517.9,0)\"><use data-c=\"2264\" xlink:href=\"#MJX-1994-TEX-N-2264\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(10573.7,0)\"><use data-c=\"31\" xlink:href=\"#MJX-1994-TEX-N-31\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(11295.9,0)\"><use data-c=\"2212\" xlink:href=\"#MJX-1994-TEX-N-2212\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(12296.1,0)\"><g data-mml-node=\"mi\"><use data-c=\"6C\" xlink:href=\"#MJX-1994-TEX-N-6C\"><\/use><use data-c=\"6F\" xlink:href=\"#MJX-1994-TEX-N-6F\" transform=\"translate(278,0)\"><\/use><use data-c=\"67\" xlink:href=\"#MJX-1994-TEX-N-67\" transform=\"translate(778,0)\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1311,-241.4) scale(0.707)\"><use data-c=\"1D45E\" xlink:href=\"#MJX-1994-TEX-I-1D45E\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(13982.4,0)\"><use data-c=\"2061\" xlink:href=\"#MJX-1994-TEX-N-2061\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(13982.4,0)\"><use data-c=\"28\" xlink:href=\"#MJX-1994-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(14371.4,0)\"><use data-c=\"1D45E\" xlink:href=\"#MJX-1994-TEX-I-1D45E\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(15053.6,0)\"><use data-c=\"2212\" xlink:href=\"#MJX-1994-TEX-N-2212\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(16053.8,0)\"><use data-c=\"31\" xlink:href=\"#MJX-1994-TEX-N-31\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(16553.8,0)\"><use data-c=\"29\" xlink:href=\"#MJX-1994-TEX-N-29\"><\/use><\/g><\/g><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"block\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable displaystyle=\"true\" columnalign=\"right left\" columnspacing=\"0em\" rowspacing=\"3pt\"><mtr><mtd><msup><mi>\u03c4<\/mi><mo>\u2212<\/mo><\/msup><\/mtd><mtd><mi><\/mi><mo>=<\/mo><msubsup><mi>g<\/mi><mi>q<\/mi><mrow data-mjx-texclass=\"ORD\"><mo>\u2212<\/mo><\/mrow><\/msubsup><mo stretchy=\"false\">(<\/mo><mn>1<\/mn><mo>\u2212<\/mo><mi>R<\/mi><mo stretchy=\"false\">)<\/mo><\/mtd><\/mtr><mtr><mtd><msup><mi>\u03c4<\/mi><mo>+<\/mo><\/msup><\/mtd><mtd><mi><\/mi><mo>=<\/mo><msubsup><mi>g<\/mi><mi>q<\/mi><mo>+<\/mo><\/msubsup><mo stretchy=\"false\">(<\/mo><mn>1<\/mn><mo>\u2212<\/mo><mi>R<\/mi><mo stretchy=\"false\">)<\/mo><mtext>&nbsp;when&nbsp;<\/mtext><mi>R<\/mi><mo>\u2264<\/mo><mn>1<\/mn><mo>\u2212<\/mo><msub><mi>log<\/mi><mi>q<\/mi><\/msub><mo data-mjx-texclass=\"NONE\">\u2061<\/mo><mo stretchy=\"false\">(<\/mo><mi>q<\/mi><mo>\u2212<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><\/mtd><\/mtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><p><span>where <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"2.512ex\" height=\"2.623ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -775.2 1110.1 1159.4\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.869ex;\"><defs><path id=\"MJX-1094-TEX-I-1D454\" d=\"M311 43Q296 30 267 15T206 0Q143 0 105 45T66 160Q66 265 143 353T314 442Q361 442 401 394L404 398Q406 401 409 404T418 412T431 419T447 422Q461 422 470 413T480 394Q480 379 423 152T363 -80Q345 -134 286 -169T151 -205Q10 -205 10 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293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q233 26 290 98L298 109L352 326Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"msubsup\"><g data-mml-node=\"mi\"><use data-c=\"1D454\" xlink:href=\"#MJX-1094-TEX-I-1D454\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(510,363) scale(0.707)\"><use data-c=\"2212\" xlink:href=\"#MJX-1094-TEX-N-2212\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(510,-247) scale(0.707)\"><use data-c=\"1D45E\" xlink:href=\"#MJX-1094-TEX-I-1D45E\"><\/use><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><msubsup><mi>g<\/mi><mi>q<\/mi><mo>\u2212<\/mo><\/msubsup><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">g_q^-<\/script><span> (resp. <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"2.512ex\" height=\"2.623ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -775.2 1110.1 1159.4\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.869ex;\"><defs><path id=\"MJX-1095-TEX-I-1D454\" d=\"M311 43Q296 30 267 15T206 0Q143 0 105 45T66 160Q66 265 143 353T314 442Q361 442 401 394L404 398Q406 401 409 404T418 412T431 419T447 422Q461 422 470 413T480 394Q480 379 423 152T363 -80Q345 -134 286 -169T151 -205Q10 -205 10 -137Q10 -111 28 -91T74 -71Q89 -71 102 -80T116 -111Q116 -121 114 -130T107 -144T99 -154T92 -162L90 -164H91Q101 -167 151 -167Q189 -167 211 -155Q234 -144 254 -122T282 -75Q288 -56 298 -13Q311 35 311 43ZM384 328L380 339Q377 350 375 354T369 368T359 382T346 393T328 402T306 405Q262 405 221 352Q191 313 171 233T151 117Q151 38 213 38Q269 38 323 108L331 118L384 328Z\"><\/path><path id=\"MJX-1095-TEX-N-2B\" d=\"M56 237T56 250T70 270H369V420L370 570Q380 583 389 583Q402 583 409 568V270H707Q722 262 722 250T707 230H409V-68Q401 -82 391 -82H389H387Q375 -82 369 -68V230H70Q56 237 56 250Z\"><\/path><path id=\"MJX-1095-TEX-I-1D45E\" d=\"M33 157Q33 258 109 349T280 441Q340 441 372 389Q373 390 377 395T388 406T404 418Q438 442 450 442Q454 442 457 439T460 434Q460 425 391 149Q320 -135 320 -139Q320 -147 365 -148H390Q396 -156 396 -157T393 -175Q389 -188 383 -194H370Q339 -192 262 -192Q234 -192 211 -192T174 -192T157 -193Q143 -193 143 -185Q143 -182 145 -170Q149 -154 152 -151T172 -148Q220 -148 230 -141Q238 -136 258 -53T279 32Q279 33 272 29Q224 -10 172 -10Q117 -10 75 30T33 157ZM352 326Q329 405 277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q233 26 290 98L298 109L352 326Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"msubsup\"><g data-mml-node=\"mi\"><use data-c=\"1D454\" xlink:href=\"#MJX-1095-TEX-I-1D454\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(510,363) scale(0.707)\"><use data-c=\"2B\" xlink:href=\"#MJX-1095-TEX-N-2B\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(510,-247) scale(0.707)\"><use data-c=\"1D45E\" xlink:href=\"#MJX-1095-TEX-I-1D45E\"><\/use><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><msubsup><mi>g<\/mi><mi>q<\/mi><mo>+<\/mo><\/msubsup><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">g_q^+<\/script><span>) denotes the inverse of <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"2.227ex\" height=\"2.22ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -694 984.3 981.2\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.65ex;\"><defs><path 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type=\"math\/tex\">\\big[\\frac{q-1}{q},1\\big]<\/script><span>).<\/span><\/p><p><mark><span>Quantaties <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"2.724ex\" height=\"1.783ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -775.2 1204.1 788.2\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.029ex;\"><defs><path id=\"MJX-1103-TEX-I-1D70F\" d=\"M39 284Q18 284 18 294Q18 301 45 338T99 398Q134 425 164 429Q170 431 332 431Q492 431 497 429Q517 424 517 402Q517 388 508 376T485 360Q479 358 389 358T299 356Q298 355 283 274T251 109T233 20Q228 5 215 -4T186 -13Q153 -13 153 20V30L203 192Q214 228 227 272T248 336L254 357Q254 358 208 358Q206 358 197 358T183 359Q105 359 61 295Q56 287 53 286T39 284Z\"><\/path><path id=\"MJX-1103-TEX-N-2212\" d=\"M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"msup\"><g data-mml-node=\"mi\"><use data-c=\"1D70F\" xlink:href=\"#MJX-1103-TEX-I-1D70F\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(604,363) scale(0.707)\"><use data-c=\"2212\" xlink:href=\"#MJX-1103-TEX-N-2212\"><\/use><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><msup><mi>\u03c4<\/mi><mo>\u2212<\/mo><\/msup><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">\\tau^-<\/script><span> and <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"2.724ex\" height=\"1.783ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -775.2 1204.1 788.2\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.029ex;\"><defs><path 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xlink:href=\"#MJX-1105-TEX-SO-29\"><\/use><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><msub><mi>N<\/mi><mi>t<\/mi><\/msub><mrow data-mjx-texclass=\"ORD\"><mo minsize=\"1.2em\" maxsize=\"1.2em\">(<\/mo><\/mrow><mrow data-mjx-texclass=\"ORD\"><mi data-mjx-variant=\"-tex-calligraphic\" mathvariant=\"script\">C<\/mi><\/mrow><mo>,<\/mo><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">s<\/mi><\/mrow><mrow data-mjx-texclass=\"ORD\"><mo minsize=\"1.2em\" maxsize=\"1.2em\">)<\/mo><\/mrow><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">N_t\\big(\\mathcal{C},\\mathbf s\\big)<\/script><span> to be exponentially large.<\/span><\/mark><\/p><p><span>The average number of solutions of <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"13.143ex\" height=\"2.262ex\" role=\"img\" 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372 389Q373 390 377 395T388 406T404 418Q438 442 450 442Q454 442 457 439T460 434Q460 425 391 149Q320 -135 320 -139Q320 -147 365 -148H390Q396 -156 396 -157T393 -175Q389 -188 383 -194H370Q339 -192 262 -192Q234 -192 211 -192T174 -192T157 -193Q143 -193 143 -185Q143 -182 145 -170Q149 -154 152 -151T172 -148Q220 -148 230 -141Q238 -136 258 -53T279 32Q279 33 272 29Q224 -10 172 -10Q117 -10 75 30T33 157ZM352 326Q329 405 277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q233 26 290 98L298 109L352 326Z\"><\/path><path id=\"MJX-1106-TEX-I-1D445\" d=\"M230 637Q203 637 198 638T193 649Q193 676 204 682Q206 683 378 683Q550 682 564 680Q620 672 658 652T712 606T733 563T739 529Q739 484 710 445T643 385T576 351T538 338L545 333Q612 295 612 223Q612 212 607 162T602 80V71Q602 53 603 43T614 25T640 16Q668 16 686 38T712 85Q717 99 720 102T735 105Q755 105 755 93Q755 75 731 36Q693 -21 641 -21H632Q571 -21 531 4T487 82Q487 109 502 166T517 239Q517 290 474 313Q459 320 449 321T378 323H309L277 193Q244 61 244 59Q244 55 245 54T252 50T269 48T302 46H333Q339 38 339 37T336 19Q332 6 326 0H311Q275 2 180 2Q146 2 117 2T71 2T50 1Q33 1 33 10Q33 12 36 24Q41 43 46 45Q50 46 61 46H67Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628Q287 635 230 637ZM630 554Q630 586 609 608T523 636Q521 636 500 636T462 637H440Q393 637 386 627Q385 624 352 494T319 361Q319 360 388 360Q466 361 492 367Q556 377 592 426Q608 449 619 486T630 554Z\"><\/path><path id=\"MJX-1106-TEX-I-1D70F\" d=\"M39 284Q18 284 18 294Q18 301 45 338T99 398Q134 425 164 429Q170 431 332 431Q492 431 497 429Q517 424 517 402Q517 388 508 376T485 360Q479 358 389 358T299 356Q298 355 283 274T251 109T233 20Q228 5 215 -4T186 -13Q153 -13 153 20V30L203 192Q214 228 227 272T248 336L254 357Q254 358 208 358Q206 358 197 358T183 359Q105 359 61 295Q56 287 53 286T39 284Z\"><\/path><path id=\"MJX-1106-TEX-N-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D5A3\" xlink:href=\"#MJX-1106-TEX-SS-1D5A3\"><\/use><use data-c=\"1D5AF\" xlink:href=\"#MJX-1106-TEX-SS-1D5AF\" transform=\"translate(722,0)\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(1361,0)\"><use data-c=\"28\" xlink:href=\"#MJX-1106-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1750,0)\"><use data-c=\"1D45B\" xlink:href=\"#MJX-1106-TEX-I-1D45B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2350,0)\"><use data-c=\"2C\" xlink:href=\"#MJX-1106-TEX-N-2C\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(2794.7,0)\"><use data-c=\"1D45E\" xlink:href=\"#MJX-1106-TEX-I-1D45E\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(3254.7,0)\"><use data-c=\"2C\" xlink:href=\"#MJX-1106-TEX-N-2C\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(3699.3,0)\"><use data-c=\"1D445\" xlink:href=\"#MJX-1106-TEX-I-1D445\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(4458.3,0)\"><use data-c=\"2C\" xlink:href=\"#MJX-1106-TEX-N-2C\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(4903,0)\"><use data-c=\"1D70F\" xlink:href=\"#MJX-1106-TEX-I-1D70F\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(5420,0)\"><use data-c=\"29\" xlink:href=\"#MJX-1106-TEX-N-29\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"sans-serif\">DP<\/mi><\/mrow><mo stretchy=\"false\">(<\/mo><mi>n<\/mi><mo>,<\/mo><mi>q<\/mi><mo>,<\/mo><mi>R<\/mi><mo>,<\/mo><mi>\u03c4<\/mi><mo stretchy=\"false\">)<\/mo><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">\\mathsf{DP}(n,q,R,\\tau)<\/script><span> over the input distribution is given by<\/span><\/p><div contenteditable=\"false\" spellcheck=\"false\" class=\"mathjax-block md-end-block md-math-block md-rawblock\" id=\"mathjax-n302\" cid=\"n302\" mdtype=\"math_block\" data-math-tag-before=\"10\" data-math-labels=\"[]\" data-math-tag-after=\"10\"><div class=\"md-rawblock-container md-math-container\" contenteditable=\"false\" tabindex=\"-1\"><mjx-container class=\"MathJax\" jax=\"SVG\" display=\"true\" width=\"full\" style=\"min-width: 22.905ex; position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"100%\" height=\"5.873ex\" role=\"img\" focusable=\"false\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -2.371ex; min-width: 22.905ex;\"><defs><path id=\"MJX-1995-TEX-N-31\" d=\"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z\"><\/path><path id=\"MJX-1995-TEX-N-2B\" d=\"M56 237T56 250T70 270H369V420L370 570Q380 583 389 583Q402 583 409 568V270H707Q722 262 722 250T707 230H409V-68Q401 -82 391 -82H389H387Q375 -82 369 -68V230H70Q56 237 56 250Z\"><\/path><path id=\"MJX-1995-TEX-SO-28\" d=\"M152 251Q152 646 388 850H416Q422 844 422 841Q422 837 403 816T357 753T302 649T255 482T236 250Q236 124 255 19T301 -147T356 -251T403 -315T422 -340Q422 -343 416 -349H388Q359 -325 332 -296T271 -213T212 -97T170 56T152 251Z\"><\/path><path id=\"MJX-1995-TEX-I-1D45B\" d=\"M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z\"><\/path><path id=\"MJX-1995-TEX-I-1D461\" d=\"M26 385Q19 392 19 395Q19 399 22 411T27 425Q29 430 36 430T87 431H140L159 511Q162 522 166 540T173 566T179 586T187 603T197 615T211 624T229 626Q247 625 254 615T261 596Q261 589 252 549T232 470L222 433Q222 431 272 431H323Q330 424 330 420Q330 398 317 385H210L174 240Q135 80 135 68Q135 26 162 26Q197 26 230 60T283 144Q285 150 288 151T303 153H307Q322 153 322 145Q322 142 319 133Q314 117 301 95T267 48T216 6T155 -11Q125 -11 98 4T59 56Q57 64 57 83V101L92 241Q127 382 128 383Q128 385 77 385H26Z\"><\/path><path id=\"MJX-1995-TEX-SO-29\" d=\"M305 251Q305 -145 69 -349H56Q43 -349 39 -347T35 -338Q37 -333 60 -307T108 -239T160 -136T204 27T221 250T204 473T160 636T108 740T60 807T35 839Q35 850 50 850H56H69Q197 743 256 566Q305 425 305 251Z\"><\/path><path id=\"MJX-1995-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-1995-TEX-I-1D45E\" d=\"M33 157Q33 258 109 349T280 441Q340 441 372 389Q373 390 377 395T388 406T404 418Q438 442 450 442Q454 442 457 439T460 434Q460 425 391 149Q320 -135 320 -139Q320 -147 365 -148H390Q396 -156 396 -157T393 -175Q389 -188 383 -194H370Q339 -192 262 -192Q234 -192 211 -192T174 -192T157 -193Q143 -193 143 -185Q143 -182 145 -170Q149 -154 152 -151T172 -148Q220 -148 230 -141Q238 -136 258 -53T279 32Q279 33 272 29Q224 -10 172 -10Q117 -10 75 30T33 157ZM352 326Q329 405 277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q233 26 290 98L298 109L352 326Z\"><\/path><path id=\"MJX-1995-TEX-N-2212\" d=\"M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z\"><\/path><path id=\"MJX-1995-TEX-N-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><path id=\"MJX-1995-TEX-I-1D458\" d=\"M121 647Q121 657 125 670T137 683Q138 683 209 688T282 694Q294 694 294 686Q294 679 244 477Q194 279 194 272Q213 282 223 291Q247 309 292 354T362 415Q402 442 438 442Q468 442 485 423T503 369Q503 344 496 327T477 302T456 291T438 288Q418 288 406 299T394 328Q394 353 410 369T442 390L458 393Q446 405 434 405H430Q398 402 367 380T294 316T228 255Q230 254 243 252T267 246T293 238T320 224T342 206T359 180T365 147Q365 130 360 106T354 66Q354 26 381 26Q429 26 459 145Q461 153 479 153H483Q499 153 499 144Q499 139 496 130Q455 -11 378 -11Q333 -11 305 15T277 90Q277 108 280 121T283 145Q283 167 269 183T234 206T200 217T182 220H180Q168 178 159 139T145 81T136 44T129 20T122 7T111 -2Q98 -11 83 -11Q66 -11 57 -1T48 16Q48 26 85 176T158 471L195 616Q196 629 188 632T149 637H144Q134 637 131 637T124 640T121 647Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(0.018929,-0.018929) translate(0, -1548)\"><g data-mml-node=\"math\"><g data-mml-node=\"mtable\" transform=\"translate(800,0) translate(-800,0)\"><g transform=\"translate(0 1548) matrix(1 0 0 -1 0 0) scale(52.83)\"><svg data-table=\"true\" preserveAspectRatio=\"xMidYMid\" viewBox=\"4261.9 -1548 1 2596\"><g transform=\"matrix(1 0 0 -1 0 0)\"><g data-mml-node=\"mlabeledtr\" transform=\"translate(0,-114.3)\"><g data-mml-node=\"mtd\"><g data-mml-node=\"mn\"><use data-c=\"31\" xlink:href=\"#MJX-1995-TEX-N-31\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(722.2,0)\"><use data-c=\"2B\" xlink:href=\"#MJX-1995-TEX-N-2B\"><\/use><\/g><g data-mml-node=\"mfrac\" 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data-mml-node=\"msup\" transform=\"translate(2432.5,-739.7)\"><g data-mml-node=\"mi\"><use data-c=\"1D45E\" xlink:href=\"#MJX-1995-TEX-I-1D45E\"><\/use><\/g><g data-mml-node=\"TeXAtom\" transform=\"translate(543.7,289) scale(0.707)\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D45B\" xlink:href=\"#MJX-1995-TEX-I-1D45B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(600,0)\"><use data-c=\"2212\" xlink:href=\"#MJX-1995-TEX-N-2212\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1378,0)\"><use data-c=\"1D458\" xlink:href=\"#MJX-1995-TEX-I-1D458\"><\/use><\/g><\/g><\/g><rect width=\"6561.4\" height=\"60\" x=\"120\" y=\"220\"><\/rect><\/g><\/g><\/g><\/g><\/svg><svg data-labels=\"true\" preserveAspectRatio=\"xMaxYMid\" viewBox=\"0 -1548 1 2596\"><g data-labels=\"true\" transform=\"matrix(1 0 0 -1 0 0)\"><g data-mml-node=\"mtd\" id=\"mjx-eqn:\" transform=\"translate(0,-114.3)\"><text data-id-align=\"true\"><\/text><g data-idbox=\"true\" transform=\"translate(0,0)\"><g data-mml-node=\"mrow\"><\/g><\/g><\/g><\/g><\/svg><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"block\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable displaystyle=\"true\"><mlabeledtr><mtd><mrow><\/mrow><\/mtd><mtd><mn>1<\/mn><mo>+<\/mo><mfrac><mrow><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"OPEN\"><mo minsize=\"1.2em\" maxsize=\"1.2em\">(<\/mo><\/mrow><mfrac linethickness=\"0\"><mi>n<\/mi><mi>t<\/mi><\/mfrac><mrow data-mjx-texclass=\"CLOSE\"><mo minsize=\"1.2em\" maxsize=\"1.2em\">)<\/mo><\/mrow><\/mrow><\/mrow><mo stretchy=\"false\">(<\/mo><mi>q<\/mi><mo>\u2212<\/mo><mn>1<\/mn><msup><mo stretchy=\"false\">)<\/mo><mi>t<\/mi><\/msup><mo>\u2212<\/mo><mn>1<\/mn><\/mrow><msup><mi>q<\/mi><mrow data-mjx-texclass=\"ORD\"><mi>n<\/mi><mo>\u2212<\/mo><mi>k<\/mi><\/mrow><\/msup><\/mfrac><\/mtd><\/mlabeledtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><p><img decoding=\"async\" src=\".\/paper reading.assets\/\u622a\u5c4f2023-11-19 16.52.23.png\" referrerpolicy=\"no-referrer\" alt=\"\u622a\u5c4f2023-11-19 16.52.23\"><\/p><p><span>The <\/span><strong><span>expected minimum distance<\/span><\/strong><span> of a code is given by the so-called Gilbert-Varshamov distance, i.e., <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"10.841ex\" height=\"2.262ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 4791.9 1000\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-1176-TEX-I-1D461\" d=\"M26 385Q19 392 19 395Q19 399 22 411T27 425Q29 430 36 430T87 431H140L159 511Q162 522 166 540T173 566T179 586T187 603T197 615T211 624T229 626Q247 625 254 615T261 596Q261 589 252 549T232 470L222 433Q222 431 272 431H323Q330 424 330 420Q330 398 317 385H210L174 240Q135 80 135 68Q135 26 162 26Q197 26 230 60T283 144Q285 150 288 151T303 153H307Q322 153 322 145Q322 142 319 133Q314 117 301 95T267 48T216 6T155 -11Q125 -11 98 4T59 56Q57 64 57 83V101L92 241Q127 382 128 383Q128 385 77 385H26Z\"><\/path><path id=\"MJX-1176-TEX-I-1D43A\" d=\"M50 252Q50 367 117 473T286 641T490 704Q580 704 633 653Q642 643 648 636T656 626L657 623Q660 623 684 649Q691 655 699 663T715 679T725 690L740 705H746Q760 705 760 698Q760 694 728 561Q692 422 692 421Q690 416 687 415T669 413H653Q647 419 647 422Q647 423 648 429T650 449T651 481Q651 552 619 605T510 659Q492 659 471 656T418 643T357 615T294 567T236 496T189 394T158 260Q156 242 156 221Q156 173 170 136T206 79T256 45T308 28T353 24Q407 24 452 47T514 106Q517 114 529 161T541 214Q541 222 528 224T468 227H431Q425 233 425 235T427 254Q431 267 437 273H454Q494 271 594 271Q634 271 659 271T695 272T707 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xlink:href=\"#MJX-1176-TEX-N-2C\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(3881.9,0)\"><use data-c=\"1D458\" xlink:href=\"#MJX-1176-TEX-I-1D458\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(4402.9,0)\"><use data-c=\"29\" xlink:href=\"#MJX-1176-TEX-N-29\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><msub><mi>t<\/mi><mrow data-mjx-texclass=\"ORD\"><mi>G<\/mi><mi>V<\/mi><\/mrow><\/msub><mo stretchy=\"false\">(<\/mo><mi>q<\/mi><mo>,<\/mo><mi>n<\/mi><mo>,<\/mo><mi>k<\/mi><mo stretchy=\"false\">)<\/mo><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">t_{GV}(q,n,k)<\/script><span>, which is defined as the largest integer such that<\/span><\/p><div contenteditable=\"false\" spellcheck=\"false\" class=\"mathjax-block md-end-block md-math-block md-rawblock\" id=\"mathjax-n312\" cid=\"n312\" mdtype=\"math_block\" data-math-tag-before=\"10\" 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data-mml-node=\"msup\" transform=\"translate(9979.1,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D45E\" xlink:href=\"#MJX-1996-TEX-I-1D45E\"><\/use><\/g><g data-mml-node=\"TeXAtom\" transform=\"translate(543.7,413) scale(0.707)\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D45B\" xlink:href=\"#MJX-1996-TEX-I-1D45B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(600,0)\"><use data-c=\"2212\" xlink:href=\"#MJX-1996-TEX-N-2212\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1378,0)\"><use data-c=\"1D458\" xlink:href=\"#MJX-1996-TEX-I-1D458\"><\/use><\/g><\/g><\/g><\/g><\/g><\/g><\/svg><svg data-labels=\"true\" preserveAspectRatio=\"xMaxYMid\" viewBox=\"0 -1816.9 1 3133.8\"><g data-labels=\"true\" transform=\"matrix(1 0 0 -1 0 0)\"><g data-mml-node=\"mtd\" id=\"mjx-eqn:\" transform=\"translate(0,-51.2)\"><text data-id-align=\"true\"><\/text><g data-idbox=\"true\" transform=\"translate(0,0)\"><g data-mml-node=\"mrow\"><\/g><\/g><\/g><\/g><\/svg><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"block\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable displaystyle=\"true\"><mlabeledtr><mtd><mrow><\/mrow><\/mtd><mtd><munderover><mo data-mjx-texclass=\"OP\">\u2211<\/mo><mrow data-mjx-texclass=\"ORD\"><mi>l<\/mi><mo>=<\/mo><mn>0<\/mn><\/mrow><mrow data-mjx-texclass=\"ORD\"><msub><mi>t<\/mi><mrow data-mjx-texclass=\"ORD\"><mi>G<\/mi><mi>V<\/mi><\/mrow><\/msub><mo stretchy=\"false\">(<\/mo><mi>q<\/mi><mo>,<\/mo><mi>n<\/mi><mo>,<\/mo><mi>k<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><\/munderover><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"OPEN\"><mo minsize=\"2.047em\" maxsize=\"2.047em\">(<\/mo><\/mrow><mfrac linethickness=\"0\"><mi>n<\/mi><mi>l<\/mi><\/mfrac><mrow data-mjx-texclass=\"CLOSE\"><mo minsize=\"2.047em\" maxsize=\"2.047em\">)<\/mo><\/mrow><\/mrow><\/mrow><mo stretchy=\"false\">(<\/mo><mi>q<\/mi><mo>\u2212<\/mo><mn>1<\/mn><msup><mo stretchy=\"false\">)<\/mo><mi>l<\/mi><\/msup><mo>\u2264<\/mo><msup><mi>q<\/mi><mrow data-mjx-texclass=\"ORD\"><mi>n<\/mi><mo>\u2212<\/mo><mi>k<\/mi><\/mrow><\/msup><\/mtd><\/mlabeledtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><p><em><span>Proof.<\/span><\/em><span>  <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"25.994ex\" height=\"3.248ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -1060.7 11489.3 1435.8\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.849ex;\"><defs><path id=\"MJX-1229-TEX-SO-2211\" d=\"M61 748Q64 750 489 750H913L954 640Q965 609 976 579T993 533T999 516H979L959 517Q936 579 886 621T777 682Q724 700 655 705T436 710H319Q183 710 183 709Q186 706 348 484T511 259Q517 250 513 244L490 216Q466 188 420 134T330 27L149 -187Q149 -188 362 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xlink:href=\"#MJX-1229-TEX-I-1D458\"><\/use><\/g><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><munderover><mo data-mjx-texclass=\"OP\">\u2211<\/mo><mrow data-mjx-texclass=\"ORD\"><mi>l<\/mi><mo>=<\/mo><mn>0<\/mn><\/mrow><mrow data-mjx-texclass=\"ORD\"><msub><mi>t<\/mi><mrow data-mjx-texclass=\"ORD\"><mi>G<\/mi><mi>V<\/mi><\/mrow><\/msub><mo stretchy=\"false\">(<\/mo><mi>q<\/mi><mo>,<\/mo><mi>n<\/mi><mo>,<\/mo><mi>k<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><\/munderover><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"OPEN\"><mo minsize=\"1.2em\" maxsize=\"1.2em\">(<\/mo><\/mrow><mfrac linethickness=\"0\"><mi>n<\/mi><mi>l<\/mi><\/mfrac><mrow data-mjx-texclass=\"CLOSE\"><mo minsize=\"1.2em\" maxsize=\"1.2em\">)<\/mo><\/mrow><\/mrow><\/mrow><mo stretchy=\"false\">(<\/mo><mi>q<\/mi><mo>\u2212<\/mo><mn>1<\/mn><msup><mo stretchy=\"false\">)<\/mo><mi>l<\/mi><\/msup><mrow data-mjx-texclass=\"ORD\"><mo>\/<\/mo><\/mrow><msup><mi>q<\/mi><mrow data-mjx-texclass=\"ORD\"><mi>n<\/mi><mo>\u2212<\/mo><mi>k<\/mi><\/mrow><\/msup><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">\\sum_{l=0}^{t_{GV}(q,n,k)}{n\\choose l}(q-1)^l\/ q^{n-k}<\/script><span> is the expected number of codewords in a sphere centered around a  codeword, with a radius of <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"3.492ex\" height=\"1.791ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -626 1543.6 791.6\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.375ex;\"><defs><path id=\"MJX-1235-TEX-I-1D461\" d=\"M26 385Q19 392 19 395Q19 399 22 411T27 425Q29 430 36 430T87 431H140L159 511Q162 522 166 540T173 566T179 586T187 603T197 615T211 624T229 626Q247 625 254 615T261 596Q261 589 252 549T232 470L222 433Q222 431 272 431H323Q330 424 330 420Q330 398 317 385H210L174 240Q135 80 135 68Q135 26 162 26Q197 26 230 60T283 144Q285 150 288 151T303 153H307Q322 153 322 145Q322 142 319 133Q314 117 301 95T267 48T216 6T155 -11Q125 -11 98 4T59 56Q57 64 57 83V101L92 241Q127 382 128 383Q128 385 77 385H26Z\"><\/path><path id=\"MJX-1235-TEX-I-1D43A\" d=\"M50 252Q50 367 117 473T286 641T490 704Q580 704 633 653Q642 643 648 636T656 626L657 623Q660 623 684 649Q691 655 699 663T715 679T725 690L740 705H746Q760 705 760 698Q760 694 728 561Q692 422 692 421Q690 416 687 415T669 413H653Q647 419 647 422Q647 423 648 429T650 449T651 481Q651 552 619 605T510 659Q492 659 471 656T418 643T357 615T294 567T236 496T189 394T158 260Q156 242 156 221Q156 173 170 136T206 79T256 45T308 28T353 24Q407 24 452 47T514 106Q517 114 529 161T541 214Q541 222 528 224T468 227H431Q425 233 425 235T427 254Q431 267 437 273H454Q494 271 594 271Q634 271 659 271T695 272T707 272Q721 272 721 263Q721 261 719 249Q714 230 709 228Q706 227 694 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xlink:href=\"#MJX-1235-TEX-I-1D461\"><\/use><\/g><g data-mml-node=\"TeXAtom\" transform=\"translate(394,-150) scale(0.707)\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D43A\" xlink:href=\"#MJX-1235-TEX-I-1D43A\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(786,0)\"><use data-c=\"1D449\" xlink:href=\"#MJX-1235-TEX-I-1D449\"><\/use><\/g><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><msub><mi>t<\/mi><mrow data-mjx-texclass=\"ORD\"><mi>G<\/mi><mi>V<\/mi><\/mrow><\/msub><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">t_{GV}<\/script><span>.<\/span><\/p><blockquote><p><span>The minimum Hamming weight of a code is defined as the smallest Hamming weight of all non-zero codewords.<\/span><\/p><\/blockquote><p><span>It can be verified that<\/span><\/p><div contenteditable=\"false\" spellcheck=\"false\" class=\"mathjax-block md-end-block 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data-c=\"1D45C\" xlink:href=\"#MJX-1997-TEX-C-1D45C\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(9477,0)\"><use data-c=\"28\" xlink:href=\"#MJX-1997-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(9866,0)\"><use data-c=\"31\" xlink:href=\"#MJX-1997-TEX-N-31\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(10366,0)\"><use data-c=\"29\" xlink:href=\"#MJX-1997-TEX-N-29\"><\/use><\/g><\/g><\/g><\/g><\/svg><svg data-labels=\"true\" preserveAspectRatio=\"xMaxYMid\" viewBox=\"0 -1328.5 1 2157\"><g data-labels=\"true\" transform=\"matrix(1 0 0 -1 0 0)\"><g data-mml-node=\"mtd\" id=\"mjx-eqn:\" transform=\"translate(0,-131.5)\"><text data-id-align=\"true\"><\/text><g data-idbox=\"true\" transform=\"translate(0,0)\"><g data-mml-node=\"mrow\"><\/g><\/g><\/g><\/g><\/svg><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"block\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable displaystyle=\"true\"><mlabeledtr><mtd><mrow><\/mrow><\/mtd><mtd><mfrac><mrow><msub><mi>t<\/mi><mrow data-mjx-texclass=\"ORD\"><mi>G<\/mi><mi>V<\/mi><\/mrow><\/msub><mo stretchy=\"false\">(<\/mo><mi>q<\/mi><mo>,<\/mo><mi>n<\/mi><mo>,<\/mo><mi>k<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><mi>n<\/mi><\/mfrac><mo>=<\/mo><msup><mi>\u03c4<\/mi><mo>\u2212<\/mo><\/msup><mo>+<\/mo><mrow data-mjx-texclass=\"ORD\"><mi data-mjx-variant=\"-tex-calligraphic\" mathvariant=\"script\">o<\/mi><\/mrow><mo stretchy=\"false\">(<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><\/mtd><\/mlabeledtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><h2 id='13-information-set-decoding-algorithms'><span>1.3 Information Set Decoding Algorithms<\/span><\/h2><h3 id='pranges-approach-using-linearity'><span>Prange&#39;s approach: using linearity.<\/span><\/h3><p><span>We are looking for some codeword <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"5.114ex\" height=\"1.686ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -705 2260.6 745\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.09ex;\"><defs><path id=\"MJX-1283-TEX-B-1D41C\" d=\"M447 131H458Q478 131 478 117Q478 112 471 95T439 51T377 9Q330 -6 286 -6Q196 -6 135 35Q39 96 39 222Q39 324 101 384Q169 453 286 453Q359 453 411 431T464 353Q464 319 445 302T395 284Q360 284 343 305T325 353Q325 380 338 396H333Q317 398 295 398H292Q280 398 271 397T245 390T218 373T197 338T183 283Q182 275 182 231Q182 199 184 180T193 132T220 85T270 57Q289 50 317 50H326Q385 50 414 115Q419 127 423 129T447 131Z\"><\/path><path id=\"MJX-1283-TEX-N-2208\" d=\"M84 250Q84 372 166 450T360 539Q361 539 377 539T419 540T469 540H568Q583 532 583 520Q583 511 570 501L466 500Q355 499 329 494Q280 482 242 458T183 409T147 354T129 306T124 272V270H568Q583 262 583 250T568 230H124V228Q124 207 134 177T167 112T231 48T328 7Q355 1 466 0H570Q583 -10 583 -20Q583 -32 568 -40H471Q464 -40 446 -40T417 -41Q262 -41 172 45Q84 127 84 250Z\"><\/path><path id=\"MJX-1283-TEX-C-43\" d=\"M201 -25Q167 -25 136 -14T75 23T29 94T12 202Q12 290 50 394T161 574Q227 642 303 673T433 704Q435 705 457 705Q533 701 533 640Q533 606 507 548T464 474Q431 444 396 444Q381 444 381 453Q381 459 388 473T407 513T428 563Q433 580 433 594Q433 636 381 636Q314 636 260 594T175 489T128 363T112 247Q112 157 153 101T273 44Q347 44 398 121Q413 144 437 157T481 171Q496 171 496 160Q496 150 476 123Q426 56 350 16T201 -25Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D41C\" xlink:href=\"#MJX-1283-TEX-B-1D41C\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(788.8,0)\"><use data-c=\"2208\" xlink:href=\"#MJX-1283-TEX-N-2208\"><\/use><\/g><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\" transform=\"translate(1733.6,0)\"><g data-mml-node=\"mi\"><use data-c=\"43\" xlink:href=\"#MJX-1283-TEX-C-43\"><\/use><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">c<\/mi><\/mrow><mo>\u2208<\/mo><mrow data-mjx-texclass=\"ORD\"><mi data-mjx-variant=\"-tex-calligraphic\" mathvariant=\"script\">C<\/mi><\/mrow><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">\\mathbf c\\in\\mathcal C<\/script><span> at distance <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"0.817ex\" height=\"1.441ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -626 361 637\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.025ex;\"><defs><path id=\"MJX-1284-TEX-I-1D461\" d=\"M26 385Q19 392 19 395Q19 399 22 411T27 425Q29 430 36 430T87 431H140L159 511Q162 522 166 540T173 566T179 586T187 603T197 615T211 624T229 626Q247 625 254 615T261 596Q261 589 252 549T232 470L222 433Q222 431 272 431H323Q330 424 330 420Q330 398 317 385H210L174 240Q135 80 135 68Q135 26 162 26Q197 26 230 60T283 144Q285 150 288 151T303 153H307Q322 153 322 145Q322 142 319 133Q314 117 301 95T267 48T216 6T155 -11Q125 -11 98 4T59 56Q57 64 57 83V101L92 241Q127 382 128 383Q128 385 77 385H26Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D461\" xlink:href=\"#MJX-1284-TEX-I-1D461\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>t<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">t<\/script><span> from <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.373ex\" height=\"1.457ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -444 607 644\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.452ex;\"><defs><path id=\"MJX-1600-TEX-B-1D432\" d=\"M84 -102Q84 -110 87 -119T102 -138T133 -149Q148 -148 162 -143T186 -131T206 -114T222 -95T234 -76T243 -59T249 -45T252 -37L269 0L96 382H26V444H34Q49 441 146 441Q252 441 270 444H279V382H255Q232 382 232 380L337 151L442 382H394V444H401Q413 441 495 441Q568 441 574 444H580V382H510L406 152Q298 -84 297 -87Q269 -139 225 -169T131 -200Q85 -200 54 -172T23 -100Q23 -64 44 -50T87 -35Q111 -35 130 -50T152 -92V-100H84V-102Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D432\" xlink:href=\"#MJX-1600-TEX-B-1D432\"><\/use><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">y<\/mi><\/mrow><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">\\mathbf y<\/script><span>. Since <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.192ex\" height=\"1.652ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -705 527 730\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.057ex;\"><defs><path id=\"MJX-1320-TEX-C-43\" d=\"M201 -25Q167 -25 136 -14T75 23T29 94T12 202Q12 290 50 394T161 574Q227 642 303 673T433 704Q435 705 457 705Q533 701 533 640Q533 606 507 548T464 474Q431 444 396 444Q381 444 381 453Q381 459 388 473T407 513T428 563Q433 580 433 594Q433 636 381 636Q314 636 260 594T175 489T128 363T112 247Q112 157 153 101T273 44Q347 44 398 121Q413 144 437 157T481 171Q496 171 496 160Q496 150 476 123Q426 56 350 16T201 -25Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"43\" xlink:href=\"#MJX-1320-TEX-C-43\"><\/use><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow data-mjx-texclass=\"ORD\"><mi data-mjx-variant=\"-tex-calligraphic\" mathvariant=\"script\">C<\/mi><\/mrow><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">\\mathcal C<\/script><span> is a <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.179ex\" height=\"1.595ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -694 521 705\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.025ex;\"><defs><path id=\"MJX-1790-TEX-I-1D458\" d=\"M121 647Q121 657 125 670T137 683Q138 683 209 688T282 694Q294 694 294 686Q294 679 244 477Q194 279 194 272Q213 282 223 291Q247 309 292 354T362 415Q402 442 438 442Q468 442 485 423T503 369Q503 344 496 327T477 302T456 291T438 288Q418 288 406 299T394 328Q394 353 410 369T442 390L458 393Q446 405 434 405H430Q398 402 367 380T294 316T228 255Q230 254 243 252T267 246T293 238T320 224T342 206T359 180T365 147Q365 130 360 106T354 66Q354 26 381 26Q429 26 459 145Q461 153 479 153H483Q499 153 499 144Q499 139 496 130Q455 -11 378 -11Q333 -11 305 15T277 90Q277 108 280 121T283 145Q283 167 269 183T234 206T200 217T182 220H180Q168 178 159 139T145 81T136 44T129 20T122 7T111 -2Q98 -11 83 -11Q66 -11 57 -1T48 16Q48 26 85 176T158 471L195 616Q196 629 188 632T149 637H144Q134 637 131 637T124 640T121 647Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D458\" xlink:href=\"#MJX-1790-TEX-I-1D458\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>k<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">k<\/script><span>-dimensional subspace in <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"2.53ex\" height=\"2.414ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -683 1118.3 1067.2\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.869ex;\"><defs><path id=\"MJX-1293-TEX-D-1D53D\" d=\"M584 499Q569 490 566 490Q558 490 552 497T546 515Q546 535 533 559Q526 574 506 593T469 621Q415 648 326 648Q293 648 287 647T275 641Q264 630 263 617Q262 609 260 492V370L275 372Q323 376 350 392T393 441Q409 473 409 506Q409 529 427 529Q437 529 442 519Q444 511 444 362Q444 212 442 206Q436 197 426 197Q409 197 409 217Q409 265 375 299Q346 328 280 335H260V206Q260 70 262 63Q265 46 276 41T326 35Q362 35 366 28Q377 17 366 3L360 -1H24Q12 5 12 16Q12 35 51 35Q92 38 97 52Q102 60 102 341T97 632Q91 645 51 648Q12 648 12 666Q12 675 24 683H573Q576 678 584 670V499ZM137 341Q137 131 136 89T130 37Q129 36 129 35H182Q233 35 233 39Q226 54 225 92T224 346L226 623L231 635L235 648H129Q132 641 133 638T135 603T137 517T137 341ZM549 603V648H495L506 641Q531 621 533 619L549 603ZM409 317V395L400 386Q390 376 375 366L357 355L373 346Q394 331 397 328L409 317Z\"><\/path><path id=\"MJX-1293-TEX-I-1D45B\" d=\"M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z\"><\/path><path id=\"MJX-1293-TEX-I-1D45E\" d=\"M33 157Q33 258 109 349T280 441Q340 441 372 389Q373 390 377 395T388 406T404 418Q438 442 450 442Q454 442 457 439T460 434Q460 425 391 149Q320 -135 320 -139Q320 -147 365 -148H390Q396 -156 396 -157T393 -175Q389 -188 383 -194H370Q339 -192 262 -192Q234 -192 211 -192T174 -192T157 -193Q143 -193 143 -185Q143 -182 145 -170Q149 -154 152 -151T172 -148Q220 -148 230 -141Q238 -136 258 -53T279 32Q279 33 272 29Q224 -10 172 -10Q117 -10 75 30T33 157ZM352 326Q329 405 277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q233 26 290 98L298 109L352 326Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"msubsup\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D53D\" xlink:href=\"#MJX-1293-TEX-D-1D53D\"><\/use><\/g><\/g><g data-mml-node=\"mi\" transform=\"translate(644,363) scale(0.707)\"><use data-c=\"1D45B\" xlink:href=\"#MJX-1293-TEX-I-1D45B\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(644,-247) scale(0.707)\"><use data-c=\"1D45E\" xlink:href=\"#MJX-1293-TEX-I-1D45E\"><\/use><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><msubsup><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"double-struck\">F<\/mi><\/mrow><mi>q<\/mi><mi>n<\/mi><\/msubsup><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">\\mathbb F_q^n<\/script><span>, there exists some set of positions of size <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.179ex\" height=\"1.595ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -694 521 705\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.025ex;\"><defs><path id=\"MJX-1790-TEX-I-1D458\" d=\"M121 647Q121 657 125 670T137 683Q138 683 209 688T282 694Q294 694 294 686Q294 679 244 477Q194 279 194 272Q213 282 223 291Q247 309 292 354T362 415Q402 442 438 442Q468 442 485 423T503 369Q503 344 496 327T477 302T456 291T438 288Q418 288 406 299T394 328Q394 353 410 369T442 390L458 393Q446 405 434 405H430Q398 402 367 380T294 316T228 255Q230 254 243 252T267 246T293 238T320 224T342 206T359 180T365 147Q365 130 360 106T354 66Q354 26 381 26Q429 26 459 145Q461 153 479 153H483Q499 153 499 144Q499 139 496 130Q455 -11 378 -11Q333 -11 305 15T277 90Q277 108 280 121T283 145Q283 167 269 183T234 206T200 217T182 220H180Q168 178 159 139T145 81T136 44T129 20T122 7T111 -2Q98 -11 83 -11Q66 -11 57 -1T48 16Q48 26 85 176T158 471L195 616Q196 629 188 632T149 637H144Q134 637 131 637T124 640T121 647Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D458\" xlink:href=\"#MJX-1790-TEX-I-1D458\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>k<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">k<\/script><span>, called an <\/span><strong><span>information set<\/span><\/strong><span>, which uniquely determines every codewords.<\/span><\/p><blockquote><p><span>This idea comes from that in the nullspace (<\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"2.036ex\" height=\"1.552ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -686 900 686\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: 0px;\"><defs><path id=\"MJX-560-TEX-B-1D407\" d=\"M400 0Q376 3 226 3Q75 3 51 0H39V62H147V624H39V686H51Q75 683 226 683Q376 683 400 686H412V624H304V388H595V624H487V686H499Q523 683 673 683Q824 683 848 686H860V624H752V62H860V0H848Q824 3 674 3Q523 3 499 0H487V62H595V326H304V62H412V0H400Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D407\" xlink:href=\"#MJX-560-TEX-B-1D407\"><\/use><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">H<\/mi><\/mrow><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">\\mathbf H<\/script><span>) there are <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"5.302ex\" height=\"1.756ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -694 2343.4 776\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.186ex;\"><defs><path id=\"MJX-1297-TEX-I-1D45B\" d=\"M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z\"><\/path><path id=\"MJX-1297-TEX-N-2212\" d=\"M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z\"><\/path><path id=\"MJX-1297-TEX-I-1D458\" d=\"M121 647Q121 657 125 670T137 683Q138 683 209 688T282 694Q294 694 294 686Q294 679 244 477Q194 279 194 272Q213 282 223 291Q247 309 292 354T362 415Q402 442 438 442Q468 442 485 423T503 369Q503 344 496 327T477 302T456 291T438 288Q418 288 406 299T394 328Q394 353 410 369T442 390L458 393Q446 405 434 405H430Q398 402 367 380T294 316T228 255Q230 254 243 252T267 246T293 238T320 224T342 206T359 180T365 147Q365 130 360 106T354 66Q354 26 381 26Q429 26 459 145Q461 153 479 153H483Q499 153 499 144Q499 139 496 130Q455 -11 378 -11Q333 -11 305 15T277 90Q277 108 280 121T283 145Q283 167 269 183T234 206T200 217T182 220H180Q168 178 159 139T145 81T136 44T129 20T122 7T111 -2Q98 -11 83 -11Q66 -11 57 -1T48 16Q48 26 85 176T158 471L195 616Q196 629 188 632T149 637H144Q134 637 131 637T124 640T121 647Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D45B\" xlink:href=\"#MJX-1297-TEX-I-1D45B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(822.2,0)\"><use data-c=\"2212\" xlink:href=\"#MJX-1297-TEX-N-2212\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1822.4,0)\"><use data-c=\"1D458\" xlink:href=\"#MJX-1297-TEX-I-1D458\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>n<\/mi><mo>\u2212<\/mo><mi>k<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">n-k<\/script><span> free variables, and their positions are fixed.<\/span><\/p><\/blockquote><p><span>Let <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.192ex\" height=\"1.652ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -705 527 730\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.057ex;\"><defs><path id=\"MJX-1320-TEX-C-43\" d=\"M201 -25Q167 -25 136 -14T75 23T29 94T12 202Q12 290 50 394T161 574Q227 642 303 673T433 704Q435 705 457 705Q533 701 533 640Q533 606 507 548T464 474Q431 444 396 444Q381 444 381 453Q381 459 388 473T407 513T428 563Q433 580 433 594Q433 636 381 636Q314 636 260 594T175 489T128 363T112 247Q112 157 153 101T273 44Q347 44 398 121Q413 144 437 157T481 171Q496 171 496 160Q496 150 476 123Q426 56 350 16T201 -25Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"43\" xlink:href=\"#MJX-1320-TEX-C-43\"><\/use><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow data-mjx-texclass=\"ORD\"><mi data-mjx-variant=\"-tex-calligraphic\" mathvariant=\"script\">C<\/mi><\/mrow><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">\\mathcal C<\/script><span> be an <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"4.8ex\" height=\"2.262ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 2121.7 1000\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-1322-TEX-N-5B\" d=\"M118 -250V750H255V710H158V-210H255V-250H118Z\"><\/path><path id=\"MJX-1322-TEX-I-1D45B\" d=\"M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z\"><\/path><path id=\"MJX-1322-TEX-N-2C\" d=\"M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z\"><\/path><path id=\"MJX-1322-TEX-I-1D458\" d=\"M121 647Q121 657 125 670T137 683Q138 683 209 688T282 694Q294 694 294 686Q294 679 244 477Q194 279 194 272Q213 282 223 291Q247 309 292 354T362 415Q402 442 438 442Q468 442 485 423T503 369Q503 344 496 327T477 302T456 291T438 288Q418 288 406 299T394 328Q394 353 410 369T442 390L458 393Q446 405 434 405H430Q398 402 367 380T294 316T228 255Q230 254 243 252T267 246T293 238T320 224T342 206T359 180T365 147Q365 130 360 106T354 66Q354 26 381 26Q429 26 459 145Q461 153 479 153H483Q499 153 499 144Q499 139 496 130Q455 -11 378 -11Q333 -11 305 15T277 90Q277 108 280 121T283 145Q283 167 269 183T234 206T200 217T182 220H180Q168 178 159 139T145 81T136 44T129 20T122 7T111 -2Q98 -11 83 -11Q66 -11 57 -1T48 16Q48 26 85 176T158 471L195 616Q196 629 188 632T149 637H144Q134 637 131 637T124 640T121 647Z\"><\/path><path id=\"MJX-1322-TEX-N-5D\" d=\"M22 710V750H159V-250H22V-210H119V710H22Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mo\"><use data-c=\"5B\" xlink:href=\"#MJX-1322-TEX-N-5B\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(278,0)\"><use data-c=\"1D45B\" xlink:href=\"#MJX-1322-TEX-I-1D45B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(878,0)\"><use data-c=\"2C\" xlink:href=\"#MJX-1322-TEX-N-2C\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1322.7,0)\"><use data-c=\"1D458\" xlink:href=\"#MJX-1322-TEX-I-1D458\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(1843.7,0)\"><use data-c=\"5D\" xlink:href=\"#MJX-1322-TEX-N-5D\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mo stretchy=\"false\">[<\/mo><mi>n<\/mi><mo>,<\/mo><mi>k<\/mi><mo stretchy=\"false\">]<\/mo><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">[n,k]<\/script><span>-code, and <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"14.456ex\" height=\"2.262ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 6389.6 1000\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-1603-TEX-C-4A\" d=\"M148 78Q148 16 189 -17T286 -50Q319 -50 348 -33T396 10T426 59T444 101L471 204Q498 306 521 372Q575 532 649 605L659 614H591Q517 613 494 607Q433 591 400 550T360 477Q353 454 325 437T275 419Q256 419 260 435Q280 523 376 597T583 681Q603 683 713 683H830Q839 674 839 671Q839 654 810 634T754 614Q735 614 721 601Q688 571 654 495T600 351T561 209T541 132Q507 29 412 -45T213 -119Q141 -119 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499V417Q291 370 288 353T271 314Q240 271 184 255L170 250L184 245Q202 239 220 230T262 196T290 137Q291 131 291 1Q291 -134 296 -147Q306 -174 339 -192T415 -213Q429 -213 431 -216Q434 -219 434 -231Z\"><\/path><path id=\"MJX-1603-TEX-N-31\" d=\"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z\"><\/path><path id=\"MJX-1603-TEX-N-2C\" d=\"M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z\"><\/path><path id=\"MJX-1603-TEX-N-2026\" d=\"M78 60Q78 84 95 102T138 120Q162 120 180 104T199 61Q199 36 182 18T139 0T96 17T78 60ZM525 60Q525 84 542 102T585 120Q609 120 627 104T646 61Q646 36 629 18T586 0T543 17T525 60ZM972 60Q972 84 989 102T1032 120Q1056 120 1074 104T1093 61Q1093 36 1076 18T1033 0T990 17T972 60Z\"><\/path><path id=\"MJX-1603-TEX-I-1D45B\" d=\"M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z\"><\/path><path id=\"MJX-1603-TEX-N-7D\" d=\"M65 731Q65 745 68 747T88 750Q171 750 216 725T279 670Q288 649 289 635T291 501Q292 362 293 357Q306 312 345 291T417 269Q428 269 431 266T434 250T431 234T417 231Q380 231 345 210T298 157Q293 143 292 121T291 -28V-79Q291 -134 285 -156T256 -198Q202 -250 89 -250Q71 -250 68 -247T65 -230Q65 -224 65 -223T66 -218T69 -214T77 -213Q91 -213 108 -210T146 -200T183 -177T207 -139Q208 -134 209 3L210 139Q223 196 280 230Q315 247 330 250Q305 257 280 270Q225 304 212 352L210 362L209 498Q208 635 207 640Q195 680 154 696T77 713Q68 713 67 716T65 731Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"4A\" xlink:href=\"#MJX-1603-TEX-C-4A\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(1116.8,0)\"><use data-c=\"2208\" xlink:href=\"#MJX-1603-TEX-N-2208\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2061.6,0)\"><use data-c=\"7B\" xlink:href=\"#MJX-1603-TEX-N-7B\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(2561.6,0)\"><use data-c=\"31\" xlink:href=\"#MJX-1603-TEX-N-31\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(3061.6,0)\"><use data-c=\"2C\" xlink:href=\"#MJX-1603-TEX-N-2C\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(3506.2,0)\"><use data-c=\"2026\" xlink:href=\"#MJX-1603-TEX-N-2026\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(4844.9,0)\"><use data-c=\"2C\" xlink:href=\"#MJX-1603-TEX-N-2C\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(5289.6,0)\"><use data-c=\"1D45B\" xlink:href=\"#MJX-1603-TEX-I-1D45B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(5889.6,0)\"><use data-c=\"7D\" xlink:href=\"#MJX-1603-TEX-N-7D\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow data-mjx-texclass=\"ORD\"><mi data-mjx-variant=\"-tex-calligraphic\" mathvariant=\"script\">J<\/mi><\/mrow><mo>\u2208<\/mo><mo fence=\"false\" stretchy=\"false\">{<\/mo><mn>1<\/mn><mo>,<\/mo><mo>\u2026<\/mo><mo>,<\/mo><mi>n<\/mi><mo fence=\"false\" stretchy=\"false\">}<\/mo><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">\\mathcal J\\in\\{1,\\dots,n\\}<\/script><span> be of size <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.179ex\" height=\"1.595ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -694 521 705\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.025ex;\"><defs><path id=\"MJX-1790-TEX-I-1D458\" d=\"M121 647Q121 657 125 670T137 683Q138 683 209 688T282 694Q294 694 294 686Q294 679 244 477Q194 279 194 272Q213 282 223 291Q247 309 292 354T362 415Q402 442 438 442Q468 442 485 423T503 369Q503 344 496 327T477 302T456 291T438 288Q418 288 406 299T394 328Q394 353 410 369T442 390L458 393Q446 405 434 405H430Q398 402 367 380T294 316T228 255Q230 254 243 252T267 246T293 238T320 224T342 206T359 180T365 147Q365 130 360 106T354 66Q354 26 381 26Q429 26 459 145Q461 153 479 153H483Q499 153 499 144Q499 139 496 130Q455 -11 378 -11Q333 -11 305 15T277 90Q277 108 280 121T283 145Q283 167 269 183T234 206T200 217T182 220H180Q168 178 159 139T145 81T136 44T129 20T122 7T111 -2Q98 -11 83 -11Q66 -11 57 -1T48 16Q48 26 85 176T158 471L195 616Q196 629 188 632T149 637H144Q134 637 131 637T124 640T121 647Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D458\" xlink:href=\"#MJX-1790-TEX-I-1D458\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>k<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">k<\/script><span>, then<\/span><\/p><div contenteditable=\"false\" spellcheck=\"false\" class=\"mathjax-block md-end-block md-math-block md-rawblock\" id=\"mathjax-n343\" cid=\"n343\" mdtype=\"math_block\" data-math-tag-before=\"10\" data-math-labels=\"[]\" data-math-tag-after=\"10\"><div class=\"md-rawblock-container md-math-container\" contenteditable=\"false\" tabindex=\"-1\"><mjx-container class=\"MathJax\" jax=\"SVG\" display=\"true\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"74.886ex\" height=\"5.523ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -1470.5 33099.7 2441\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -2.196ex;\"><defs><path id=\"MJX-1998-TEX-C-4A\" d=\"M148 78Q148 16 189 -17T286 -50Q319 -50 348 -33T396 10T426 59T444 101L471 204Q498 306 521 372Q575 532 649 605L659 614H591Q517 613 494 607Q433 591 400 550T360 477Q353 454 325 437T275 419Q256 419 260 435Q280 523 376 597T583 681Q603 683 713 683H830Q839 674 839 671Q839 654 810 634T754 614Q735 614 721 601Q688 571 654 495T600 351T561 209T541 132Q507 29 412 -45T213 -119Q141 -119 94 -77T47 33Q47 55 50 69T58 90T71 103Q105 131 135 131Q152 131 152 120Q152 119 151 114T149 99T148 78Z\"><\/path><path id=\"MJX-1998-TEX-N-A0\" d=\"\"><\/path><path id=\"MJX-1998-TEX-N-69\" d=\"M69 609Q69 637 87 653T131 669Q154 667 171 652T188 609Q188 579 171 564T129 549Q104 549 87 564T69 609ZM247 0Q232 3 143 3Q132 3 106 3T56 1L34 0H26V46H42Q70 46 91 49Q100 53 102 60T104 102V205V293Q104 345 102 359T88 378Q74 385 41 385H30V408Q30 431 32 431L42 432Q52 433 70 434T106 436Q123 437 142 438T171 441T182 442H185V62Q190 52 197 50T232 46H255V0H247Z\"><\/path><path id=\"MJX-1998-TEX-N-73\" d=\"M295 316Q295 356 268 385T190 414Q154 414 128 401Q98 382 98 349Q97 344 98 336T114 312T157 287Q175 282 201 278T245 269T277 256Q294 248 310 236T342 195T359 133Q359 71 321 31T198 -10H190Q138 -10 94 26L86 19L77 10Q71 4 65 -1L54 -11H46H42Q39 -11 33 -5V74V132Q33 153 35 157T45 162H54Q66 162 70 158T75 146T82 119T101 77Q136 26 198 26Q295 26 295 104Q295 133 277 151Q257 175 194 187T111 210Q75 227 54 256T33 318Q33 357 50 384T93 424T143 442T187 447H198Q238 447 268 432L283 424L292 431Q302 440 314 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transform=\"translate(8287,0)\"><\/use><use data-c=\"A0\" xlink:href=\"#MJX-1998-TEX-N-A0\" transform=\"translate(8593,0)\"><\/use><\/g><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\" transform=\"translate(10303,0)\"><g data-mml-node=\"mi\"><use data-c=\"43\" xlink:href=\"#MJX-1998-TEX-C-43\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(10830,0)\"><use data-c=\"2C\" xlink:href=\"#MJX-1998-TEX-N-2C\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(11274.7,0)\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D406\" xlink:href=\"#MJX-1998-TEX-B-1D406\"><\/use><\/g><\/g><g data-mml-node=\"TeXAtom\" transform=\"translate(937,-150) scale(0.707)\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"4A\" xlink:href=\"#MJX-1998-TEX-C-4A\"><\/use><\/g><\/g><\/g><\/g><g data-mml-node=\"mtext\" transform=\"translate(12854.9,0)\"><use data-c=\"A0\" xlink:href=\"#MJX-1998-TEX-N-A0\"><\/use><use data-c=\"69\" xlink:href=\"#MJX-1998-TEX-N-69\" transform=\"translate(250,0)\"><\/use><use data-c=\"73\" xlink:href=\"#MJX-1998-TEX-N-73\" transform=\"translate(528,0)\"><\/use><use data-c=\"20\" xlink:href=\"#MJX-1998-TEX-N-20\" transform=\"translate(922,0)\"><\/use><use data-c=\"69\" xlink:href=\"#MJX-1998-TEX-N-69\" transform=\"translate(1172,0)\"><\/use><use data-c=\"6E\" xlink:href=\"#MJX-1998-TEX-N-6E\" transform=\"translate(1450,0)\"><\/use><use data-c=\"76\" xlink:href=\"#MJX-1998-TEX-N-76\" transform=\"translate(2006,0)\"><\/use><use data-c=\"65\" xlink:href=\"#MJX-1998-TEX-N-65\" transform=\"translate(2534,0)\"><\/use><use data-c=\"72\" xlink:href=\"#MJX-1998-TEX-N-72\" transform=\"translate(2978,0)\"><\/use><use data-c=\"74\" xlink:href=\"#MJX-1998-TEX-N-74\" transform=\"translate(3370,0)\"><\/use><use data-c=\"69\" xlink:href=\"#MJX-1998-TEX-N-69\" transform=\"translate(3759,0)\"><\/use><use data-c=\"62\" xlink:href=\"#MJX-1998-TEX-N-62\" transform=\"translate(4037,0)\"><\/use><use data-c=\"6C\" xlink:href=\"#MJX-1998-TEX-N-6C\" transform=\"translate(4593,0)\"><\/use><use data-c=\"65\" xlink:href=\"#MJX-1998-TEX-N-65\" transform=\"translate(4871,0)\"><\/use><use data-c=\"2E\" xlink:href=\"#MJX-1998-TEX-N-2E\" transform=\"translate(5315,0)\"><\/use><\/g><\/g><\/g><g data-mml-node=\"mtr\" transform=\"translate(0,-579.5)\"><g data-mml-node=\"mtd\" transform=\"translate(12384.8,0)\"><g data-mml-node=\"mo\"><use data-c=\"21D4\" xlink:href=\"#MJX-1998-TEX-N-21D4\"><\/use><\/g><\/g><g data-mml-node=\"mtd\" transform=\"translate(13384.8,0)\"><g data-mml-node=\"mi\"><use data-c=\"2200\" xlink:href=\"#MJX-1998-TEX-N-2200\"><\/use><\/g><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\" transform=\"translate(556,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D407\" xlink:href=\"#MJX-1998-TEX-B-1D407\"><\/use><\/g><\/g><g data-mml-node=\"mtext\" transform=\"translate(1456,0)\"><use data-c=\"A0\" xlink:href=\"#MJX-1998-TEX-N-A0\"><\/use><use data-c=\"70\" xlink:href=\"#MJX-1998-TEX-N-70\" transform=\"translate(250,0)\"><\/use><use data-c=\"61\" xlink:href=\"#MJX-1998-TEX-N-61\" transform=\"translate(806,0)\"><\/use><use data-c=\"72\" xlink:href=\"#MJX-1998-TEX-N-72\" transform=\"translate(1306,0)\"><\/use><use data-c=\"69\" xlink:href=\"#MJX-1998-TEX-N-69\" transform=\"translate(1698,0)\"><\/use><use data-c=\"74\" xlink:href=\"#MJX-1998-TEX-N-74\" transform=\"translate(1976,0)\"><\/use><use data-c=\"79\" xlink:href=\"#MJX-1998-TEX-N-79\" transform=\"translate(2365,0)\"><\/use><use data-c=\"2D\" xlink:href=\"#MJX-1998-TEX-N-2D\" transform=\"translate(2893,0)\"><\/use><use data-c=\"63\" xlink:href=\"#MJX-1998-TEX-N-63\" transform=\"translate(3226,0)\"><\/use><use data-c=\"68\" xlink:href=\"#MJX-1998-TEX-N-68\" transform=\"translate(3670,0)\"><\/use><use data-c=\"65\" xlink:href=\"#MJX-1998-TEX-N-65\" transform=\"translate(4226,0)\"><\/use><use data-c=\"63\" xlink:href=\"#MJX-1998-TEX-N-63\" transform=\"translate(4670,0)\"><\/use><use data-c=\"6B\" xlink:href=\"#MJX-1998-TEX-N-6B\" transform=\"translate(5114,0)\"><\/use><use data-c=\"20\" xlink:href=\"#MJX-1998-TEX-N-20\" transform=\"translate(5642,0)\"><\/use><use data-c=\"6D\" xlink:href=\"#MJX-1998-TEX-N-6D\" transform=\"translate(5892,0)\"><\/use><use data-c=\"61\" xlink:href=\"#MJX-1998-TEX-N-61\" transform=\"translate(6725,0)\"><\/use><use data-c=\"74\" xlink:href=\"#MJX-1998-TEX-N-74\" transform=\"translate(7225,0)\"><\/use><use data-c=\"72\" xlink:href=\"#MJX-1998-TEX-N-72\" transform=\"translate(7614,0)\"><\/use><use data-c=\"69\" xlink:href=\"#MJX-1998-TEX-N-69\" transform=\"translate(8006,0)\"><\/use><use data-c=\"78\" xlink:href=\"#MJX-1998-TEX-N-78\" transform=\"translate(8284,0)\"><\/use><use data-c=\"20\" xlink:href=\"#MJX-1998-TEX-N-20\" transform=\"translate(8812,0)\"><\/use><use data-c=\"6F\" xlink:href=\"#MJX-1998-TEX-N-6F\" transform=\"translate(9062,0)\"><\/use><use data-c=\"66\" xlink:href=\"#MJX-1998-TEX-N-66\" transform=\"translate(9562,0)\"><\/use><use data-c=\"A0\" xlink:href=\"#MJX-1998-TEX-N-A0\" transform=\"translate(9868,0)\"><\/use><\/g><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\" transform=\"translate(11574,0)\"><g data-mml-node=\"mi\"><use data-c=\"43\" xlink:href=\"#MJX-1998-TEX-C-43\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(12101,0)\"><use data-c=\"2C\" xlink:href=\"#MJX-1998-TEX-N-2C\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(12545.7,0)\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D407\" xlink:href=\"#MJX-1998-TEX-B-1D407\"><\/use><\/g><\/g><g data-mml-node=\"TeXAtom\" transform=\"translate(933,-306.8) scale(0.707)\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mover\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"4A\" xlink:href=\"#MJX-1998-TEX-C-4A\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(626.8,244) translate(-250 0)\"><use data-c=\"AF\" xlink:href=\"#MJX-1998-TEX-N-AF\"><\/use><\/g><\/g><\/g><\/g><\/g><g data-mml-node=\"mtext\" transform=\"translate(14121.9,0)\"><use data-c=\"A0\" xlink:href=\"#MJX-1998-TEX-N-A0\"><\/use><use data-c=\"69\" xlink:href=\"#MJX-1998-TEX-N-69\" transform=\"translate(250,0)\"><\/use><use data-c=\"73\" xlink:href=\"#MJX-1998-TEX-N-73\" transform=\"translate(528,0)\"><\/use><use data-c=\"20\" xlink:href=\"#MJX-1998-TEX-N-20\" transform=\"translate(922,0)\"><\/use><use data-c=\"69\" xlink:href=\"#MJX-1998-TEX-N-69\" transform=\"translate(1172,0)\"><\/use><use data-c=\"6E\" xlink:href=\"#MJX-1998-TEX-N-6E\" transform=\"translate(1450,0)\"><\/use><use data-c=\"76\" xlink:href=\"#MJX-1998-TEX-N-76\" transform=\"translate(2006,0)\"><\/use><use data-c=\"65\" xlink:href=\"#MJX-1998-TEX-N-65\" transform=\"translate(2534,0)\"><\/use><use data-c=\"72\" xlink:href=\"#MJX-1998-TEX-N-72\" transform=\"translate(2978,0)\"><\/use><use data-c=\"74\" xlink:href=\"#MJX-1998-TEX-N-74\" transform=\"translate(3370,0)\"><\/use><use data-c=\"69\" xlink:href=\"#MJX-1998-TEX-N-69\" transform=\"translate(3759,0)\"><\/use><use data-c=\"62\" xlink:href=\"#MJX-1998-TEX-N-62\" transform=\"translate(4037,0)\"><\/use><use data-c=\"6C\" xlink:href=\"#MJX-1998-TEX-N-6C\" transform=\"translate(4593,0)\"><\/use><use data-c=\"65\" xlink:href=\"#MJX-1998-TEX-N-65\" transform=\"translate(4871,0)\"><\/use><use data-c=\"2E\" xlink:href=\"#MJX-1998-TEX-N-2E\" transform=\"translate(5315,0)\"><\/use><\/g><\/g><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"block\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable displaystyle=\"true\" columnalign=\"right left\" columnspacing=\"0em\" rowspacing=\"3pt\"><mtr><mtd><mrow data-mjx-texclass=\"ORD\"><mi data-mjx-variant=\"-tex-calligraphic\" mathvariant=\"script\">J<\/mi><\/mrow><mtext>&nbsp;is an information set for&nbsp;<\/mtext><mrow data-mjx-texclass=\"ORD\"><mi data-mjx-variant=\"-tex-calligraphic\" mathvariant=\"script\">C<\/mi><\/mrow><mo stretchy=\"false\">\u21d4<\/mo><\/mtd><mtd><mi mathvariant=\"normal\">\u2200<\/mi><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">G<\/mi><\/mrow><mtext>&nbsp;generator matrix of&nbsp;<\/mtext><mrow data-mjx-texclass=\"ORD\"><mi data-mjx-variant=\"-tex-calligraphic\" mathvariant=\"script\">C<\/mi><\/mrow><mo>,<\/mo><msub><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">G<\/mi><\/mrow><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mi data-mjx-variant=\"-tex-calligraphic\" mathvariant=\"script\">J<\/mi><\/mrow><\/mrow><\/msub><mtext>&nbsp;is invertible.<\/mtext><\/mtd><\/mtr><mtr><mtd><mo stretchy=\"false\">\u21d4<\/mo><\/mtd><mtd><mi mathvariant=\"normal\">\u2200<\/mi><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">H<\/mi><\/mrow><mtext>&nbsp;parity-check matrix of&nbsp;<\/mtext><mrow data-mjx-texclass=\"ORD\"><mi data-mjx-variant=\"-tex-calligraphic\" mathvariant=\"script\">C<\/mi><\/mrow><mo>,<\/mo><msub><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">H<\/mi><\/mrow><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mover><mrow data-mjx-texclass=\"ORD\"><mi data-mjx-variant=\"-tex-calligraphic\" mathvariant=\"script\">J<\/mi><\/mrow><mo stretchy=\"false\">\u00af<\/mo><\/mover><\/mrow><\/mrow><\/msub><mtext>&nbsp;is invertible.<\/mtext><\/mtd><\/mtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><blockquote><p><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.898ex\" height=\"1.814ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -683 839 802\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.269ex;\"><defs><path id=\"MJX-1634-TEX-C-4A\" d=\"M148 78Q148 16 189 -17T286 -50Q319 -50 348 -33T396 10T426 59T444 101L471 204Q498 306 521 372Q575 532 649 605L659 614H591Q517 613 494 607Q433 591 400 550T360 477Q353 454 325 437T275 419Q256 419 260 435Q280 523 376 597T583 681Q603 683 713 683H830Q839 674 839 671Q839 654 810 634T754 614Q735 614 721 601Q688 571 654 495T600 351T561 209T541 132Q507 29 412 -45T213 -119Q141 -119 94 -77T47 33Q47 55 50 69T58 90T71 103Q105 131 135 131Q152 131 152 120Q152 119 151 114T149 99T148 78Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"4A\" xlink:href=\"#MJX-1634-TEX-C-4A\"><\/use><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow data-mjx-texclass=\"ORD\"><mi data-mjx-variant=\"-tex-calligraphic\" mathvariant=\"script\">J<\/mi><\/mrow><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">\\mathcal J<\/script><span> is an information set means that positions in <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.898ex\" height=\"1.814ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -683 839 802\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.269ex;\"><defs><path id=\"MJX-1634-TEX-C-4A\" d=\"M148 78Q148 16 189 -17T286 -50Q319 -50 348 -33T396 10T426 59T444 101L471 204Q498 306 521 372Q575 532 649 605L659 614H591Q517 613 494 607Q433 591 400 550T360 477Q353 454 325 437T275 419Q256 419 260 435Q280 523 376 597T583 681Q603 683 713 683H830Q839 674 839 671Q839 654 810 634T754 614Q735 614 721 601Q688 571 654 495T600 351T561 209T541 132Q507 29 412 -45T213 -119Q141 -119 94 -77T47 33Q47 55 50 69T58 90T71 103Q105 131 135 131Q152 131 152 120Q152 119 151 114T149 99T148 78Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"4A\" xlink:href=\"#MJX-1634-TEX-C-4A\"><\/use><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow data-mjx-texclass=\"ORD\"><mi data-mjx-variant=\"-tex-calligraphic\" mathvariant=\"script\">J<\/mi><\/mrow><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">\\mathcal J<\/script><span> are pivot variables, and those in <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.898ex\" height=\"2.382ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -934 839 1053\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.269ex;\"><defs><path id=\"MJX-1599-TEX-C-4A\" d=\"M148 78Q148 16 189 -17T286 -50Q319 -50 348 -33T396 10T426 59T444 101L471 204Q498 306 521 372Q575 532 649 605L659 614H591Q517 613 494 607Q433 591 400 550T360 477Q353 454 325 437T275 419Q256 419 260 435Q280 523 376 597T583 681Q603 683 713 683H830Q839 674 839 671Q839 654 810 634T754 614Q735 614 721 601Q688 571 654 495T600 351T561 209T541 132Q507 29 412 -45T213 -119Q141 -119 94 -77T47 33Q47 55 50 69T58 90T71 103Q105 131 135 131Q152 131 152 120Q152 119 151 114T149 99T148 78Z\"><\/path><path id=\"MJX-1599-TEX-N-AF\" d=\"M69 544V590H430V544H69Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mover\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"4A\" xlink:href=\"#MJX-1599-TEX-C-4A\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(626.8,244) translate(-250 0)\"><use data-c=\"AF\" xlink:href=\"#MJX-1599-TEX-N-AF\"><\/use><\/g><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow data-mjx-texclass=\"ORD\"><mover><mrow data-mjx-texclass=\"ORD\"><mi data-mjx-variant=\"-tex-calligraphic\" mathvariant=\"script\">J<\/mi><\/mrow><mo stretchy=\"false\">\u00af<\/mo><\/mover><\/mrow><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">\\bar{\\mathcal J}<\/script><span> are free variables.<\/span><\/p><\/blockquote><p><span>Prange&#39;s idea is to pick some ranfom information set, and compute the unique codeword based on those coordinates, and then check whether the calculated codeword has distance <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"0.817ex\" height=\"1.441ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -626 361 637\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.025ex;\"><defs><path id=\"MJX-1284-TEX-I-1D461\" d=\"M26 385Q19 392 19 395Q19 399 22 411T27 425Q29 430 36 430T87 431H140L159 511Q162 522 166 540T173 566T179 586T187 603T197 615T211 624T229 626Q247 625 254 615T261 596Q261 589 252 549T232 470L222 433Q222 431 272 431H323Q330 424 330 420Q330 398 317 385H210L174 240Q135 80 135 68Q135 26 162 26Q197 26 230 60T283 144Q285 150 288 151T303 153H307Q322 153 322 145Q322 142 319 133Q314 117 301 95T267 48T216 6T155 -11Q125 -11 98 4T59 56Q57 64 57 83V101L92 241Q127 382 128 383Q128 385 77 385H26Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D461\" xlink:href=\"#MJX-1284-TEX-I-1D461\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>t<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">t<\/script><span> with <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.373ex\" height=\"1.457ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -444 607 644\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.452ex;\"><defs><path id=\"MJX-1600-TEX-B-1D432\" d=\"M84 -102Q84 -110 87 -119T102 -138T133 -149Q148 -148 162 -143T186 -131T206 -114T222 -95T234 -76T243 -59T249 -45T252 -37L269 0L96 382H26V444H34Q49 441 146 441Q252 441 270 444H279V382H255Q232 382 232 380L337 151L442 382H394V444H401Q413 441 495 441Q568 441 574 444H580V382H510L406 152Q298 -84 297 -87Q269 -139 225 -169T131 -200Q85 -200 54 -172T23 -100Q23 -64 44 -50T87 -35Q111 -35 130 -50T152 -92V-100H84V-102Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D432\" xlink:href=\"#MJX-1600-TEX-B-1D432\"><\/use><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">y<\/mi><\/mrow><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">\\mathbf y<\/script><span>.<\/span><\/p><p><strong><span>The algorithm:<\/span><\/strong><span> <\/span><\/p><ol><li><p><span>Pick the information set: Let <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"14.456ex\" height=\"2.262ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 6389.6 1000\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-1603-TEX-C-4A\" d=\"M148 78Q148 16 189 -17T286 -50Q319 -50 348 -33T396 10T426 59T444 101L471 204Q498 306 521 372Q575 532 649 605L659 614H591Q517 613 494 607Q433 591 400 550T360 477Q353 454 325 437T275 419Q256 419 260 435Q280 523 376 597T583 681Q603 683 713 683H830Q839 674 839 671Q839 654 810 634T754 614Q735 614 721 601Q688 571 654 495T600 351T561 209T541 132Q507 29 412 -45T213 -119Q141 -119 94 -77T47 33Q47 55 50 69T58 90T71 103Q105 131 135 131Q152 131 152 120Q152 119 151 114T149 99T148 78Z\"><\/path><path id=\"MJX-1603-TEX-N-2208\" d=\"M84 250Q84 372 166 450T360 539Q361 539 377 539T419 540T469 540H568Q583 532 583 520Q583 511 570 501L466 500Q355 499 329 494Q280 482 242 458T183 409T147 354T129 306T124 272V270H568Q583 262 583 250T568 230H124V228Q124 207 134 177T167 112T231 48T328 7Q355 1 466 0H570Q583 -10 583 -20Q583 -32 568 -40H471Q464 -40 446 -40T417 -41Q262 -41 172 45Q84 127 84 250Z\"><\/path><path id=\"MJX-1603-TEX-N-7B\" d=\"M434 -231Q434 -244 428 -250H410Q281 -250 230 -184Q225 -177 222 -172T217 -161T213 -148T211 -133T210 -111T209 -84T209 -47T209 0Q209 21 209 53Q208 142 204 153Q203 154 203 155Q189 191 153 211T82 231Q71 231 68 234T65 250T68 266T82 269Q116 269 152 289T203 345Q208 356 208 377T209 529V579Q209 634 215 656T244 698Q270 724 324 740Q361 748 377 749Q379 749 390 749T408 750H428Q434 744 434 732Q434 719 431 716Q429 713 415 713Q362 710 332 689T296 647Q291 634 291 499V417Q291 370 288 353T271 314Q240 271 184 255L170 250L184 245Q202 239 220 230T262 196T290 137Q291 131 291 1Q291 -134 296 -147Q306 -174 339 -192T415 -213Q429 -213 431 -216Q434 -219 434 -231Z\"><\/path><path id=\"MJX-1603-TEX-N-31\" d=\"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z\"><\/path><path id=\"MJX-1603-TEX-N-2C\" d=\"M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z\"><\/path><path id=\"MJX-1603-TEX-N-2026\" d=\"M78 60Q78 84 95 102T138 120Q162 120 180 104T199 61Q199 36 182 18T139 0T96 17T78 60ZM525 60Q525 84 542 102T585 120Q609 120 627 104T646 61Q646 36 629 18T586 0T543 17T525 60ZM972 60Q972 84 989 102T1032 120Q1056 120 1074 104T1093 61Q1093 36 1076 18T1033 0T990 17T972 60Z\"><\/path><path id=\"MJX-1603-TEX-I-1D45B\" d=\"M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z\"><\/path><path id=\"MJX-1603-TEX-N-7D\" d=\"M65 731Q65 745 68 747T88 750Q171 750 216 725T279 670Q288 649 289 635T291 501Q292 362 293 357Q306 312 345 291T417 269Q428 269 431 266T434 250T431 234T417 231Q380 231 345 210T298 157Q293 143 292 121T291 -28V-79Q291 -134 285 -156T256 -198Q202 -250 89 -250Q71 -250 68 -247T65 -230Q65 -224 65 -223T66 -218T69 -214T77 -213Q91 -213 108 -210T146 -200T183 -177T207 -139Q208 -134 209 3L210 139Q223 196 280 230Q315 247 330 250Q305 257 280 270Q225 304 212 352L210 362L209 498Q208 635 207 640Q195 680 154 696T77 713Q68 713 67 716T65 731Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"4A\" xlink:href=\"#MJX-1603-TEX-C-4A\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(1116.8,0)\"><use data-c=\"2208\" xlink:href=\"#MJX-1603-TEX-N-2208\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2061.6,0)\"><use data-c=\"7B\" xlink:href=\"#MJX-1603-TEX-N-7B\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(2561.6,0)\"><use data-c=\"31\" xlink:href=\"#MJX-1603-TEX-N-31\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(3061.6,0)\"><use data-c=\"2C\" xlink:href=\"#MJX-1603-TEX-N-2C\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(3506.2,0)\"><use data-c=\"2026\" xlink:href=\"#MJX-1603-TEX-N-2026\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(4844.9,0)\"><use data-c=\"2C\" xlink:href=\"#MJX-1603-TEX-N-2C\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(5289.6,0)\"><use data-c=\"1D45B\" xlink:href=\"#MJX-1603-TEX-I-1D45B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(5889.6,0)\"><use data-c=\"7D\" xlink:href=\"#MJX-1603-TEX-N-7D\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow data-mjx-texclass=\"ORD\"><mi data-mjx-variant=\"-tex-calligraphic\" mathvariant=\"script\">J<\/mi><\/mrow><mo>\u2208<\/mo><mo fence=\"false\" stretchy=\"false\">{<\/mo><mn>1<\/mn><mo>,<\/mo><mo>\u2026<\/mo><mo>,<\/mo><mi>n<\/mi><mo fence=\"false\" stretchy=\"false\">}<\/mo><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">\\mathcal J\\in\\{1,\\dots,n\\}<\/script><span> be a random set of size <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.179ex\" height=\"1.595ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -694 521 705\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.025ex;\"><defs><path id=\"MJX-1790-TEX-I-1D458\" d=\"M121 647Q121 657 125 670T137 683Q138 683 209 688T282 694Q294 694 294 686Q294 679 244 477Q194 279 194 272Q213 282 223 291Q247 309 292 354T362 415Q402 442 438 442Q468 442 485 423T503 369Q503 344 496 327T477 302T456 291T438 288Q418 288 406 299T394 328Q394 353 410 369T442 390L458 393Q446 405 434 405H430Q398 402 367 380T294 316T228 255Q230 254 243 252T267 246T293 238T320 224T342 206T359 180T365 147Q365 130 360 106T354 66Q354 26 381 26Q429 26 459 145Q461 153 479 153H483Q499 153 499 144Q499 139 496 130Q455 -11 378 -11Q333 -11 305 15T277 90Q277 108 280 121T283 145Q283 167 269 183T234 206T200 217T182 220H180Q168 178 159 139T145 81T136 44T129 20T122 7T111 -2Q98 -11 83 -11Q66 -11 57 -1T48 16Q48 26 85 176T158 471L195 616Q196 629 188 632T149 637H144Q134 637 131 637T124 640T121 647Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D458\" xlink:href=\"#MJX-1790-TEX-I-1D458\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>k<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">k<\/script><span>. If <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"17.712ex\" height=\"3.284ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -1060.7 7828.8 1451.7\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.885ex;\"><defs><path id=\"MJX-1633-TEX-B-1D407\" d=\"M400 0Q376 3 226 3Q75 3 51 0H39V62H147V624H39V686H51Q75 683 226 683Q376 683 400 686H412V624H304V388H595V624H487V686H499Q523 683 673 683Q824 683 848 686H860V624H752V62H860V0H848Q824 3 674 3Q523 3 499 0H487V62H595V326H304V62H412V0H400Z\"><\/path><path id=\"MJX-1633-TEX-C-4A\" d=\"M148 78Q148 16 189 -17T286 -50Q319 -50 348 -33T396 10T426 59T444 101L471 204Q498 306 521 372Q575 532 649 605L659 614H591Q517 613 494 607Q433 591 400 550T360 477Q353 454 325 437T275 419Q256 419 260 435Q280 523 376 597T583 681Q603 683 713 683H830Q839 674 839 671Q839 654 810 634T754 614Q735 614 721 601Q688 571 654 495T600 351T561 209T541 132Q507 29 412 -45T213 -119Q141 -119 94 -77T47 33Q47 55 50 69T58 90T71 103Q105 131 135 131Q152 131 152 120Q152 119 151 114T149 99T148 78Z\"><\/path><path id=\"MJX-1633-TEX-N-AF\" d=\"M69 544V590H430V544H69Z\"><\/path><path id=\"MJX-1633-TEX-N-2208\" d=\"M84 250Q84 372 166 450T360 539Q361 539 377 539T419 540T469 540H568Q583 532 583 520Q583 511 570 501L466 500Q355 499 329 494Q280 482 242 458T183 409T147 354T129 306T124 272V270H568Q583 262 583 250T568 230H124V228Q124 207 134 177T167 112T231 48T328 7Q355 1 466 0H570Q583 -10 583 -20Q583 -32 568 -40H471Q464 -40 446 -40T417 -41Q262 -41 172 45Q84 127 84 250Z\"><\/path><path id=\"MJX-1633-TEX-D-1D53D\" d=\"M584 499Q569 490 566 490Q558 490 552 497T546 515Q546 535 533 559Q526 574 506 593T469 621Q415 648 326 648Q293 648 287 647T275 641Q264 630 263 617Q262 609 260 492V370L275 372Q323 376 350 392T393 441Q409 473 409 506Q409 529 427 529Q437 529 442 519Q444 511 444 362Q444 212 442 206Q436 197 426 197Q409 197 409 217Q409 265 375 299Q346 328 280 335H260V206Q260 70 262 63Q265 46 276 41T326 35Q362 35 366 28Q377 17 366 3L360 -1H24Q12 5 12 16Q12 35 51 35Q92 38 97 52Q102 60 102 341T97 632Q91 645 51 648Q12 648 12 666Q12 675 24 683H573Q576 678 584 670V499ZM137 341Q137 131 136 89T130 37Q129 36 129 35H182Q233 35 233 39Q226 54 225 92T224 346L226 623L231 635L235 648H129Q132 641 133 638T135 603T137 517T137 341ZM549 603V648H495L506 641Q531 621 533 619L549 603ZM409 317V395L400 386Q390 376 375 366L357 355L373 346Q394 331 397 328L409 317Z\"><\/path><path id=\"MJX-1633-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-1633-TEX-I-1D45B\" d=\"M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z\"><\/path><path id=\"MJX-1633-TEX-N-2212\" d=\"M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z\"><\/path><path id=\"MJX-1633-TEX-I-1D458\" d=\"M121 647Q121 657 125 670T137 683Q138 683 209 688T282 694Q294 694 294 686Q294 679 244 477Q194 279 194 272Q213 282 223 291Q247 309 292 354T362 415Q402 442 438 442Q468 442 485 423T503 369Q503 344 496 327T477 302T456 291T438 288Q418 288 406 299T394 328Q394 353 410 369T442 390L458 393Q446 405 434 405H430Q398 402 367 380T294 316T228 255Q230 254 243 252T267 246T293 238T320 224T342 206T359 180T365 147Q365 130 360 106T354 66Q354 26 381 26Q429 26 459 145Q461 153 479 153H483Q499 153 499 144Q499 139 496 130Q455 -11 378 -11Q333 -11 305 15T277 90Q277 108 280 121T283 145Q283 167 269 183T234 206T200 217T182 220H180Q168 178 159 139T145 81T136 44T129 20T122 7T111 -2Q98 -11 83 -11Q66 -11 57 -1T48 16Q48 26 85 176T158 471L195 616Q196 629 188 632T149 637H144Q134 637 131 637T124 640T121 647Z\"><\/path><path id=\"MJX-1633-TEX-N-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><path id=\"MJX-1633-TEX-N-D7\" d=\"M630 29Q630 9 609 9Q604 9 587 25T493 118L389 222L284 117Q178 13 175 11Q171 9 168 9Q160 9 154 15T147 29Q147 36 161 51T255 146L359 250L255 354Q174 435 161 449T147 471Q147 480 153 485T168 490Q173 490 175 489Q178 487 284 383L389 278L493 382Q570 459 587 475T609 491Q630 491 630 471Q630 464 620 453T522 355L418 250L522 145Q606 61 618 48T630 29Z\"><\/path><path id=\"MJX-1633-TEX-I-1D45E\" d=\"M33 157Q33 258 109 349T280 441Q340 441 372 389Q373 390 377 395T388 406T404 418Q438 442 450 442Q454 442 457 439T460 434Q460 425 391 149Q320 -135 320 -139Q320 -147 365 -148H390Q396 -156 396 -157T393 -175Q389 -188 383 -194H370Q339 -192 262 -192Q234 -192 211 -192T174 -192T157 -193Q143 -193 143 -185Q143 -182 145 -170Q149 -154 152 -151T172 -148Q220 -148 230 -141Q238 -136 258 -53T279 32Q279 33 272 29Q224 -10 172 -10Q117 -10 75 30T33 157ZM352 326Q329 405 277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q233 26 290 98L298 109L352 326Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"msub\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D407\" xlink:href=\"#MJX-1633-TEX-B-1D407\"><\/use><\/g><\/g><g data-mml-node=\"TeXAtom\" transform=\"translate(933,-306.8) scale(0.707)\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mover\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"4A\" xlink:href=\"#MJX-1633-TEX-C-4A\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(626.8,244) translate(-250 0)\"><use data-c=\"AF\" xlink:href=\"#MJX-1633-TEX-N-AF\"><\/use><\/g><\/g><\/g><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(1854,0)\"><use data-c=\"2208\" xlink:href=\"#MJX-1633-TEX-N-2208\"><\/use><\/g><g data-mml-node=\"msubsup\" transform=\"translate(2798.8,0)\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D53D\" xlink:href=\"#MJX-1633-TEX-D-1D53D\"><\/use><\/g><\/g><g data-mml-node=\"TeXAtom\" transform=\"translate(644,530.4) scale(0.707)\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mo\"><use data-c=\"28\" xlink:href=\"#MJX-1633-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(389,0)\"><use data-c=\"1D45B\" xlink:href=\"#MJX-1633-TEX-I-1D45B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(989,0)\"><use data-c=\"2212\" xlink:href=\"#MJX-1633-TEX-N-2212\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1767,0)\"><use data-c=\"1D458\" xlink:href=\"#MJX-1633-TEX-I-1D458\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2288,0)\"><use data-c=\"29\" xlink:href=\"#MJX-1633-TEX-N-29\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2677,0)\"><use data-c=\"D7\" xlink:href=\"#MJX-1633-TEX-N-D7\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(3455,0)\"><use data-c=\"28\" xlink:href=\"#MJX-1633-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(3844,0)\"><use data-c=\"1D45B\" xlink:href=\"#MJX-1633-TEX-I-1D45B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(4444,0)\"><use data-c=\"2212\" xlink:href=\"#MJX-1633-TEX-N-2212\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(5222,0)\"><use data-c=\"1D458\" xlink:href=\"#MJX-1633-TEX-I-1D458\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(5743,0)\"><use data-c=\"29\" xlink:href=\"#MJX-1633-TEX-N-29\"><\/use><\/g><\/g><g data-mml-node=\"mi\" transform=\"translate(644,-138.9) scale(0.707)\"><use data-c=\"1D45E\" xlink:href=\"#MJX-1633-TEX-I-1D45E\"><\/use><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><msub><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">H<\/mi><\/mrow><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mover><mrow data-mjx-texclass=\"ORD\"><mi data-mjx-variant=\"-tex-calligraphic\" mathvariant=\"script\">J<\/mi><\/mrow><mo stretchy=\"false\">\u00af<\/mo><\/mover><\/mrow><\/mrow><\/msub><mo>\u2208<\/mo><msubsup><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"double-struck\">F<\/mi><\/mrow><mi>q<\/mi><mrow data-mjx-texclass=\"ORD\"><mo stretchy=\"false\">(<\/mo><mi>n<\/mi><mo>\u2212<\/mo><mi>k<\/mi><mo stretchy=\"false\">)<\/mo><mo>\u00d7<\/mo><mo stretchy=\"false\">(<\/mo><mi>n<\/mi><mo>\u2212<\/mo><mi>k<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><\/msubsup><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">\\mathbf H_{\\bar{\\mathcal J}}\\in\\mathbb{F}_q^{(n-k)\\times(n-k)}<\/script><span> is not full-rank, pick another set <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.898ex\" height=\"1.814ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -683 839 802\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.269ex;\"><defs><path id=\"MJX-1634-TEX-C-4A\" d=\"M148 78Q148 16 189 -17T286 -50Q319 -50 348 -33T396 10T426 59T444 101L471 204Q498 306 521 372Q575 532 649 605L659 614H591Q517 613 494 607Q433 591 400 550T360 477Q353 454 325 437T275 419Q256 419 260 435Q280 523 376 597T583 681Q603 683 713 683H830Q839 674 839 671Q839 654 810 634T754 614Q735 614 721 601Q688 571 654 495T600 351T561 209T541 132Q507 29 412 -45T213 -119Q141 -119 94 -77T47 33Q47 55 50 69T58 90T71 103Q105 131 135 131Q152 131 152 120Q152 119 151 114T149 99T148 78Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"4A\" xlink:href=\"#MJX-1634-TEX-C-4A\"><\/use><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow data-mjx-texclass=\"ORD\"><mi data-mjx-variant=\"-tex-calligraphic\" mathvariant=\"script\">J<\/mi><\/mrow><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">\\mathcal J<\/script><span>.<\/span><\/p><\/li><li><p><span>Linear Algebra. Calculate <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"3.834ex\" height=\"3.288ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -882.5 1694.7 1453.5\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -1.292ex;\"><defs><path id=\"MJX-1640-TEX-SS-1D5A7\" d=\"M86 0V694H190V399H517V694H621V0H517V323H190V0H86Z\"><\/path><path id=\"MJX-1640-TEX-N-2212\" d=\"M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z\"><\/path><path id=\"MJX-1640-TEX-N-31\" d=\"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z\"><\/path><path id=\"MJX-1640-TEX-C-4A\" d=\"M148 78Q148 16 189 -17T286 -50Q319 -50 348 -33T396 10T426 59T444 101L471 204Q498 306 521 372Q575 532 649 605L659 614H591Q517 613 494 607Q433 591 400 550T360 477Q353 454 325 437T275 419Q256 419 260 435Q280 523 376 597T583 681Q603 683 713 683H830Q839 674 839 671Q839 654 810 634T754 614Q735 614 721 601Q688 571 654 495T600 351T561 209T541 132Q507 29 412 -45T213 -119Q141 -119 94 -77T47 33Q47 55 50 69T58 90T71 103Q105 131 135 131Q152 131 152 120Q152 119 151 114T149 99T148 78Z\"><\/path><path id=\"MJX-1640-TEX-N-AF\" d=\"M69 544V590H430V544H69Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"msubsup\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D5A7\" xlink:href=\"#MJX-1640-TEX-SS-1D5A7\"><\/use><\/g><\/g><g data-mml-node=\"TeXAtom\" transform=\"translate(741,411.6) scale(0.707)\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mo\"><use data-c=\"2212\" xlink:href=\"#MJX-1640-TEX-N-2212\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(778,0)\"><use data-c=\"31\" xlink:href=\"#MJX-1640-TEX-N-31\"><\/use><\/g><\/g><g data-mml-node=\"TeXAtom\" transform=\"translate(741,-486.8) scale(0.707)\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mover\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"4A\" xlink:href=\"#MJX-1640-TEX-C-4A\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(626.8,244) translate(-250 0)\"><use data-c=\"AF\" xlink:href=\"#MJX-1640-TEX-N-AF\"><\/use><\/g><\/g><\/g><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><msubsup><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"sans-serif\">H<\/mi><\/mrow><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mover><mrow data-mjx-texclass=\"ORD\"><mi data-mjx-variant=\"-tex-calligraphic\" mathvariant=\"script\">J<\/mi><\/mrow><mo stretchy=\"false\">\u00af<\/mo><\/mover><\/mrow><\/mrow><mrow data-mjx-texclass=\"ORD\"><mo>\u2212<\/mo><mn>1<\/mn><\/mrow><\/msubsup><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">\\mathsf H_{\\bar{\\mathcal J}}^{-1}<\/script><span>.<\/span><\/p><\/li><li><p><span>Test step: Pick <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"12.282ex\" height=\"2.801ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -853.7 5428.8 1237.9\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.869ex;\"><defs><path id=\"MJX-1768-TEX-B-1D41E\" d=\"M32 225Q32 332 102 392T272 452H283Q382 452 436 401Q494 343 494 243Q494 226 486 222T440 217Q431 217 394 217T327 218H175V209Q175 177 179 154T196 107T236 69T306 50Q312 49 323 49Q376 49 410 85Q421 99 427 111T434 127T442 133T463 135H468Q494 135 494 117Q494 110 489 97T468 66T431 32T373 5T292 -6Q181 -6 107 55T32 225ZM383 276Q377 346 348 374T280 402Q253 402 230 390T195 357Q179 331 176 279V266H383V276Z\"><\/path><path id=\"MJX-1768-TEX-C-4A\" d=\"M148 78Q148 16 189 -17T286 -50Q319 -50 348 -33T396 10T426 59T444 101L471 204Q498 306 521 372Q575 532 649 605L659 614H591Q517 613 494 607Q433 591 400 550T360 477Q353 454 325 437T275 419Q256 419 260 435Q280 523 376 597T583 681Q603 683 713 683H830Q839 674 839 671Q839 654 810 634T754 614Q735 614 721 601Q688 571 654 495T600 351T561 209T541 132Q507 29 412 -45T213 -119Q141 -119 94 -77T47 33Q47 55 50 69T58 90T71 103Q105 131 135 131Q152 131 152 120Q152 119 151 114T149 99T148 78Z\"><\/path><path id=\"MJX-1768-TEX-N-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path id=\"MJX-1768-TEX-B-1D431\" d=\"M227 0Q212 3 121 3Q40 3 28 0H21V62H117L245 213L109 382H26V444H34Q49 441 143 441Q247 441 265 444H274V382H246L281 339Q315 297 316 297Q320 297 354 341L389 382H352V444H360Q375 441 466 441Q547 441 559 444H566V382H471L355 246L504 63L545 62H586V0H578Q563 3 469 3Q365 3 347 0H338V62H366Q366 63 326 112T285 163L198 63L217 62H235V0H227Z\"><\/path><path id=\"MJX-1768-TEX-N-2208\" d=\"M84 250Q84 372 166 450T360 539Q361 539 377 539T419 540T469 540H568Q583 532 583 520Q583 511 570 501L466 500Q355 499 329 494Q280 482 242 458T183 409T147 354T129 306T124 272V270H568Q583 262 583 250T568 230H124V228Q124 207 134 177T167 112T231 48T328 7Q355 1 466 0H570Q583 -10 583 -20Q583 -32 568 -40H471Q464 -40 446 -40T417 -41Q262 -41 172 45Q84 127 84 250Z\"><\/path><path id=\"MJX-1768-TEX-D-1D53D\" d=\"M584 499Q569 490 566 490Q558 490 552 497T546 515Q546 535 533 559Q526 574 506 593T469 621Q415 648 326 648Q293 648 287 647T275 641Q264 630 263 617Q262 609 260 492V370L275 372Q323 376 350 392T393 441Q409 473 409 506Q409 529 427 529Q437 529 442 519Q444 511 444 362Q444 212 442 206Q436 197 426 197Q409 197 409 217Q409 265 375 299Q346 328 280 335H260V206Q260 70 262 63Q265 46 276 41T326 35Q362 35 366 28Q377 17 366 3L360 -1H24Q12 5 12 16Q12 35 51 35Q92 38 97 52Q102 60 102 341T97 632Q91 645 51 648Q12 648 12 666Q12 675 24 683H573Q576 678 584 670V499ZM137 341Q137 131 136 89T130 37Q129 36 129 35H182Q233 35 233 39Q226 54 225 92T224 346L226 623L231 635L235 648H129Q132 641 133 638T135 603T137 517T137 341ZM549 603V648H495L506 641Q531 621 533 619L549 603ZM409 317V395L400 386Q390 376 375 366L357 355L373 346Q394 331 397 328L409 317Z\"><\/path><path id=\"MJX-1768-TEX-I-1D458\" d=\"M121 647Q121 657 125 670T137 683Q138 683 209 688T282 694Q294 694 294 686Q294 679 244 477Q194 279 194 272Q213 282 223 291Q247 309 292 354T362 415Q402 442 438 442Q468 442 485 423T503 369Q503 344 496 327T477 302T456 291T438 288Q418 288 406 299T394 328Q394 353 410 369T442 390L458 393Q446 405 434 405H430Q398 402 367 380T294 316T228 255Q230 254 243 252T267 246T293 238T320 224T342 206T359 180T365 147Q365 130 360 106T354 66Q354 26 381 26Q429 26 459 145Q461 153 479 153H483Q499 153 499 144Q499 139 496 130Q455 -11 378 -11Q333 -11 305 15T277 90Q277 108 280 121T283 145Q283 167 269 183T234 206T200 217T182 220H180Q168 178 159 139T145 81T136 44T129 20T122 7T111 -2Q98 -11 83 -11Q66 -11 57 -1T48 16Q48 26 85 176T158 471L195 616Q196 629 188 632T149 637H144Q134 637 131 637T124 640T121 647Z\"><\/path><path id=\"MJX-1768-TEX-I-1D45E\" d=\"M33 157Q33 258 109 349T280 441Q340 441 372 389Q373 390 377 395T388 406T404 418Q438 442 450 442Q454 442 457 439T460 434Q460 425 391 149Q320 -135 320 -139Q320 -147 365 -148H390Q396 -156 396 -157T393 -175Q389 -188 383 -194H370Q339 -192 262 -192Q234 -192 211 -192T174 -192T157 -193Q143 -193 143 -185Q143 -182 145 -170Q149 -154 152 -151T172 -148Q220 -148 230 -141Q238 -136 258 -53T279 32Q279 33 272 29Q224 -10 172 -10Q117 -10 75 30T33 157ZM352 326Q329 405 277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q233 26 290 98L298 109L352 326Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"msub\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D41E\" xlink:href=\"#MJX-1768-TEX-B-1D41E\"><\/use><\/g><\/g><g data-mml-node=\"TeXAtom\" transform=\"translate(560,-150) scale(0.707)\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"4A\" xlink:href=\"#MJX-1768-TEX-C-4A\"><\/use><\/g><\/g><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(1481,0)\"><use data-c=\"3D\" xlink:href=\"#MJX-1768-TEX-N-3D\"><\/use><\/g><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\" transform=\"translate(2536.8,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D431\" xlink:href=\"#MJX-1768-TEX-B-1D431\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(3421.6,0)\"><use data-c=\"2208\" xlink:href=\"#MJX-1768-TEX-N-2208\"><\/use><\/g><g data-mml-node=\"msubsup\" transform=\"translate(4366.4,0)\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D53D\" xlink:href=\"#MJX-1768-TEX-D-1D53D\"><\/use><\/g><\/g><g data-mml-node=\"mi\" transform=\"translate(644,363) scale(0.707)\"><use data-c=\"1D458\" xlink:href=\"#MJX-1768-TEX-I-1D458\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(644,-247) scale(0.707)\"><use data-c=\"1D45E\" xlink:href=\"#MJX-1768-TEX-I-1D45E\"><\/use><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><msub><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">e<\/mi><\/mrow><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mi data-mjx-variant=\"-tex-calligraphic\" mathvariant=\"script\">J<\/mi><\/mrow><\/mrow><\/msub><mo>=<\/mo><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">x<\/mi><\/mrow><mo>\u2208<\/mo><msubsup><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"double-struck\">F<\/mi><\/mrow><mi>q<\/mi><mi>k<\/mi><\/msubsup><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">\\mathbf e_{\\mathcal J}=\\mathbf x\\in\\mathbb F_q^k<\/script><span> according to distribution <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"2.51ex\" height=\"1.902ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -683 1109.3 840.8\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.357ex;\"><defs><path id=\"MJX-1926-TEX-C-44\" d=\"M37 475Q19 475 19 487Q19 536 103 604T327 682H356Q386 683 408 683H419Q475 683 506 681T582 668T667 633Q766 571 766 450Q766 365 723 287T611 152T455 57T279 6Q248 1 160 0Q148 0 131 0T108 -1Q72 -1 72 11Q72 24 90 40T133 64L144 68L152 88Q247 328 272 587Q275 613 272 613Q272 613 269 613Q225 610 195 602T149 579T129 556T119 532Q118 530 116 525T113 518Q102 502 80 490T37 475ZM665 407Q665 596 412 613Q403 614 383 614Q370 614 370 612Q370 598 363 542T323 357T242 103L228 69H265Q391 73 481 119Q536 148 575 188T633 268T658 338T665 392V407Z\"><\/path><path id=\"MJX-1926-TEX-I-1D461\" d=\"M26 385Q19 392 19 395Q19 399 22 411T27 425Q29 430 36 430T87 431H140L159 511Q162 522 166 540T173 566T179 586T187 603T197 615T211 624T229 626Q247 625 254 615T261 596Q261 589 252 549T232 470L222 433Q222 431 272 431H323Q330 424 330 420Q330 398 317 385H210L174 240Q135 80 135 68Q135 26 162 26Q197 26 230 60T283 144Q285 150 288 151T303 153H307Q322 153 322 145Q322 142 319 133Q314 117 301 95T267 48T216 6T155 -11Q125 -11 98 4T59 56Q57 64 57 83V101L92 241Q127 382 128 383Q128 385 77 385H26Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"msub\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"44\" xlink:href=\"#MJX-1926-TEX-C-44\"><\/use><\/g><\/g><g data-mml-node=\"mi\" transform=\"translate(804,-150) scale(0.707)\"><use data-c=\"1D461\" xlink:href=\"#MJX-1926-TEX-I-1D461\"><\/use><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><msub><mrow data-mjx-texclass=\"ORD\"><mi data-mjx-variant=\"-tex-calligraphic\" mathvariant=\"script\">D<\/mi><\/mrow><mi>t<\/mi><\/msub><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">\\mathcal D_t<\/script><span>, and let <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"6.488ex\" height=\"2.414ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -683 2867.8 1067.2\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.869ex;\"><defs><path id=\"MJX-1646-TEX-B-1D41E\" d=\"M32 225Q32 332 102 392T272 452H283Q382 452 436 401Q494 343 494 243Q494 226 486 222T440 217Q431 217 394 217T327 218H175V209Q175 177 179 154T196 107T236 69T306 50Q312 49 323 49Q376 49 410 85Q421 99 427 111T434 127T442 133T463 135H468Q494 135 494 117Q494 110 489 97T468 66T431 32T373 5T292 -6Q181 -6 107 55T32 225ZM383 276Q377 346 348 374T280 402Q253 402 230 390T195 357Q179 331 176 279V266H383V276Z\"><\/path><path id=\"MJX-1646-TEX-N-2208\" d=\"M84 250Q84 372 166 450T360 539Q361 539 377 539T419 540T469 540H568Q583 532 583 520Q583 511 570 501L466 500Q355 499 329 494Q280 482 242 458T183 409T147 354T129 306T124 272V270H568Q583 262 583 250T568 230H124V228Q124 207 134 177T167 112T231 48T328 7Q355 1 466 0H570Q583 -10 583 -20Q583 -32 568 -40H471Q464 -40 446 -40T417 -41Q262 -41 172 45Q84 127 84 250Z\"><\/path><path id=\"MJX-1646-TEX-D-1D53D\" d=\"M584 499Q569 490 566 490Q558 490 552 497T546 515Q546 535 533 559Q526 574 506 593T469 621Q415 648 326 648Q293 648 287 647T275 641Q264 630 263 617Q262 609 260 492V370L275 372Q323 376 350 392T393 441Q409 473 409 506Q409 529 427 529Q437 529 442 519Q444 511 444 362Q444 212 442 206Q436 197 426 197Q409 197 409 217Q409 265 375 299Q346 328 280 335H260V206Q260 70 262 63Q265 46 276 41T326 35Q362 35 366 28Q377 17 366 3L360 -1H24Q12 5 12 16Q12 35 51 35Q92 38 97 52Q102 60 102 341T97 632Q91 645 51 648Q12 648 12 666Q12 675 24 683H573Q576 678 584 670V499ZM137 341Q137 131 136 89T130 37Q129 36 129 35H182Q233 35 233 39Q226 54 225 92T224 346L226 623L231 635L235 648H129Q132 641 133 638T135 603T137 517T137 341ZM549 603V648H495L506 641Q531 621 533 619L549 603ZM409 317V395L400 386Q390 376 375 366L357 355L373 346Q394 331 397 328L409 317Z\"><\/path><path id=\"MJX-1646-TEX-I-1D45B\" d=\"M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z\"><\/path><path id=\"MJX-1646-TEX-I-1D45E\" d=\"M33 157Q33 258 109 349T280 441Q340 441 372 389Q373 390 377 395T388 406T404 418Q438 442 450 442Q454 442 457 439T460 434Q460 425 391 149Q320 -135 320 -139Q320 -147 365 -148H390Q396 -156 396 -157T393 -175Q389 -188 383 -194H370Q339 -192 262 -192Q234 -192 211 -192T174 -192T157 -193Q143 -193 143 -185Q143 -182 145 -170Q149 -154 152 -151T172 -148Q220 -148 230 -141Q238 -136 258 -53T279 32Q279 33 272 29Q224 -10 172 -10Q117 -10 75 30T33 157ZM352 326Q329 405 277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q233 26 290 98L298 109L352 326Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g 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mathvariant=\"bold\">e<\/mi><\/mrow><mo>\u2208<\/mo><msubsup><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"double-struck\">F<\/mi><\/mrow><mi>q<\/mi><mi>n<\/mi><\/msubsup><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">\\mathbf e\\in\\mathbb F_q^n<\/script><span> be such that<\/span><\/p><\/li><\/ol><div contenteditable=\"false\" spellcheck=\"false\" class=\"mathjax-block md-end-block md-math-block md-rawblock\" id=\"mathjax-n373\" cid=\"n373\" mdtype=\"math_block\" data-math-tag-before=\"10\" data-math-labels=\"[]\" data-math-tag-after=\"10\"><div class=\"md-rawblock-container md-math-container\" contenteditable=\"false\" tabindex=\"-1\"><mjx-container class=\"MathJax\" jax=\"SVG\" display=\"true\" width=\"full\" style=\"min-width: 23.776ex; position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"100%\" height=\"3.364ex\" role=\"img\" focusable=\"false\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" 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data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mover\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"4A\" xlink:href=\"#MJX-1999-TEX-C-4A\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(626.8,244) translate(-250 0)\"><use data-c=\"AF\" xlink:href=\"#MJX-1999-TEX-N-AF\"><\/use><\/g><\/g><\/g><\/g><\/g><\/g><\/g><\/g><\/svg><svg data-labels=\"true\" preserveAspectRatio=\"xMaxYMid\" viewBox=\"0 -993.4 1 1486.7\"><g data-labels=\"true\" transform=\"matrix(1 0 0 -1 0 0)\"><g data-mml-node=\"mtd\" id=\"mjx-eqn:\" transform=\"translate(0,77.6)\"><text data-id-align=\"true\"><\/text><g data-idbox=\"true\" transform=\"translate(0,0)\"><g data-mml-node=\"mrow\"><\/g><\/g><\/g><\/g><\/svg><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"block\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable displaystyle=\"true\"><mlabeledtr><mtd><mrow><\/mrow><\/mtd><mtd><msub><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">e<\/mi><\/mrow><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mover><mrow data-mjx-texclass=\"ORD\"><mi data-mjx-variant=\"-tex-calligraphic\" mathvariant=\"script\">J<\/mi><\/mrow><mo stretchy=\"false\">\u00af<\/mo><\/mover><\/mrow><\/mrow><\/msub><mo>=<\/mo><mo stretchy=\"false\">(<\/mo><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">s<\/mi><\/mrow><mo>\u2212<\/mo><mrow data-mjx-texclass=\"ORD\"><msubsup><mi mathvariant=\"bold\">xH<\/mi><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mi data-mjx-variant=\"-tex-calligraphic\" mathvariant=\"script\">J<\/mi><\/mrow><\/mrow><mrow data-mjx-texclass=\"ORD\"><mo>\u22ba<\/mo><\/mrow><\/msubsup><\/mrow><mo stretchy=\"false\">)<\/mo><msubsup><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"sans-serif\">H<\/mi><\/mrow><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mover><mrow data-mjx-texclass=\"ORD\"><mi data-mjx-variant=\"-tex-calligraphic\" mathvariant=\"script\">J<\/mi><\/mrow><mo stretchy=\"false\">\u00af<\/mo><\/mover><\/mrow><\/mrow><mrow data-mjx-texclass=\"ORD\"><mo>\u2212<\/mo><mo>\u22ba<\/mo><\/mrow><\/msubsup><\/mtd><\/mlabeledtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><p><span>If <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"6.284ex\" height=\"2.26ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -749.5 2777.6 999\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.564ex;\"><defs><path id=\"MJX-1774-TEX-N-7C\" d=\"M139 -249H137Q125 -249 119 -235V251L120 737Q130 750 139 750Q152 750 159 735V-235Q151 -249 141 -249H139Z\"><\/path><path id=\"MJX-1774-TEX-B-1D41E\" d=\"M32 225Q32 332 102 392T272 452H283Q382 452 436 401Q494 343 494 243Q494 226 486 222T440 217Q431 217 394 217T327 218H175V209Q175 177 179 154T196 107T236 69T306 50Q312 49 323 49Q376 49 410 85Q421 99 427 111T434 127T442 133T463 135H468Q494 135 494 117Q494 110 489 97T468 66T431 32T373 5T292 -6Q181 -6 107 55T32 225ZM383 276Q377 346 348 374T280 402Q253 402 230 390T195 357Q179 331 176 279V266H383V276Z\"><\/path><path id=\"MJX-1774-TEX-N-2260\" d=\"M166 -215T159 -215T147 -212T141 -204T139 -197Q139 -190 144 -183L306 133H70Q56 140 56 153Q56 168 72 173H327L406 327H72Q56 332 56 347Q56 360 70 367H426Q597 702 602 707Q605 716 618 716Q625 716 630 712T636 703T638 696Q638 692 471 367H707Q722 359 722 347Q722 336 708 328L451 327L371 173H708Q722 163 722 153Q722 140 707 133H351Q175 -210 170 -212Q166 -215 159 -215Z\"><\/path><path id=\"MJX-1774-TEX-I-1D461\" d=\"M26 385Q19 392 19 395Q19 399 22 411T27 425Q29 430 36 430T87 431H140L159 511Q162 522 166 540T173 566T179 586T187 603T197 615T211 624T229 626Q247 625 254 615T261 596Q261 589 252 549T232 470L222 433Q222 431 272 431H323Q330 424 330 420Q330 398 317 385H210L174 240Q135 80 135 68Q135 26 162 26Q197 26 230 60T283 144Q285 150 288 151T303 153H307Q322 153 322 145Q322 142 319 133Q314 117 301 95T267 48T216 6T155 -11Q125 -11 98 4T59 56Q57 64 57 83V101L92 241Q127 382 128 383Q128 385 77 385H26Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mo\" transform=\"translate(0 -0.5)\"><use data-c=\"7C\" xlink:href=\"#MJX-1774-TEX-N-7C\"><\/use><\/g><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\" transform=\"translate(278,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D41E\" xlink:href=\"#MJX-1774-TEX-B-1D41E\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(805,0) translate(0 -0.5)\"><use data-c=\"7C\" xlink:href=\"#MJX-1774-TEX-N-7C\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(1360.8,0)\"><use data-c=\"2260\" xlink:href=\"#MJX-1774-TEX-N-2260\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(2416.6,0)\"><use data-c=\"1D461\" xlink:href=\"#MJX-1774-TEX-I-1D461\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">e<\/mi><\/mrow><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mo>\u2260<\/mo><mi>t<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">|\\mathbf e|\\neq t<\/script><span> go back to step <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.131ex\" height=\"1.507ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -666 500 666\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: 0px;\"><defs><path id=\"MJX-1775-TEX-N-31\" d=\"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mn\"><use data-c=\"31\" xlink:href=\"#MJX-1775-TEX-N-31\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mn>1<\/mn><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">1<\/script><span>, otherwise it is a solution.<\/span><\/p><blockquote><p><span>Picking <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.373ex\" height=\"1.005ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -444 607 444\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: 0px;\"><defs><path id=\"MJX-1797-TEX-B-1D431\" d=\"M227 0Q212 3 121 3Q40 3 28 0H21V62H117L245 213L109 382H26V444H34Q49 441 143 441Q247 441 265 444H274V382H246L281 339Q315 297 316 297Q320 297 354 341L389 382H352V444H360Q375 441 466 441Q547 441 559 444H566V382H471L355 246L504 63L545 62H586V0H578Q563 3 469 3Q365 3 347 0H338V62H366Q366 63 326 112T285 163L198 63L217 62H235V0H227Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D431\" xlink:href=\"#MJX-1797-TEX-B-1D431\"><\/use><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">x<\/mi><\/mrow><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">\\mathbf x<\/script><span> in step <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.131ex\" height=\"1.554ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -665 500 687\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.05ex;\"><defs><path id=\"MJX-1789-TEX-N-33\" d=\"M127 463Q100 463 85 480T69 524Q69 579 117 622T233 665Q268 665 277 664Q351 652 390 611T430 522Q430 470 396 421T302 350L299 348Q299 347 308 345T337 336T375 315Q457 262 457 175Q457 96 395 37T238 -22Q158 -22 100 21T42 130Q42 158 60 175T105 193Q133 193 151 175T169 130Q169 119 166 110T159 94T148 82T136 74T126 70T118 67L114 66Q165 21 238 21Q293 21 321 74Q338 107 338 175V195Q338 290 274 322Q259 328 213 329L171 330L168 332Q166 335 166 348Q166 366 174 366Q202 366 232 371Q266 376 294 413T322 525V533Q322 590 287 612Q265 626 240 626Q208 626 181 615T143 592T132 580H135Q138 579 143 578T153 573T165 566T175 555T183 540T186 520Q186 498 172 481T127 463Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mn\"><use data-c=\"33\" xlink:href=\"#MJX-1789-TEX-N-33\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mn>3<\/mn><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">3<\/script><span> is like fixing <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.179ex\" height=\"1.595ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -694 521 705\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.025ex;\"><defs><path id=\"MJX-1790-TEX-I-1D458\" d=\"M121 647Q121 657 125 670T137 683Q138 683 209 688T282 694Q294 694 294 686Q294 679 244 477Q194 279 194 272Q213 282 223 291Q247 309 292 354T362 415Q402 442 438 442Q468 442 485 423T503 369Q503 344 496 327T477 302T456 291T438 288Q418 288 406 299T394 328Q394 353 410 369T442 390L458 393Q446 405 434 405H430Q398 402 367 380T294 316T228 255Q230 254 243 252T267 246T293 238T320 224T342 206T359 180T365 147Q365 130 360 106T354 66Q354 26 381 26Q429 26 459 145Q461 153 479 153H483Q499 153 499 144Q499 139 496 130Q455 -11 378 -11Q333 -11 305 15T277 90Q277 108 280 121T283 145Q283 167 269 183T234 206T200 217T182 220H180Q168 178 159 139T145 81T136 44T129 20T122 7T111 -2Q98 -11 83 -11Q66 -11 57 -1T48 16Q48 26 85 176T158 471L195 616Q196 629 188 632T149 637H144Q134 637 131 637T124 640T121 647Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D458\" xlink:href=\"#MJX-1790-TEX-I-1D458\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>k<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">k<\/script><span> unknowns to the value that we want. Suppose we want to find a solution <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.192ex\" height=\"1.036ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -452 527 458\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.014ex;\"><defs><path id=\"MJX-1808-TEX-B-1D41E\" d=\"M32 225Q32 332 102 392T272 452H283Q382 452 436 401Q494 343 494 243Q494 226 486 222T440 217Q431 217 394 217T327 218H175V209Q175 177 179 154T196 107T236 69T306 50Q312 49 323 49Q376 49 410 85Q421 99 427 111T434 127T442 133T463 135H468Q494 135 494 117Q494 110 489 97T468 66T431 32T373 5T292 -6Q181 -6 107 55T32 225ZM383 276Q377 346 348 374T280 402Q253 402 230 390T195 357Q179 331 176 279V266H383V276Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D41E\" xlink:href=\"#MJX-1808-TEX-B-1D41E\"><\/use><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">e<\/mi><\/mrow><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">\\mathbf e<\/script><span> of small Hamming weight, fixing <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.373ex\" height=\"1.005ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -444 607 444\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: 0px;\"><defs><path id=\"MJX-1797-TEX-B-1D431\" d=\"M227 0Q212 3 121 3Q40 3 28 0H21V62H117L245 213L109 382H26V444H34Q49 441 143 441Q247 441 265 444H274V382H246L281 339Q315 297 316 297Q320 297 354 341L389 382H352V444H360Q375 441 466 441Q547 441 559 444H566V382H471L355 246L504 63L545 62H586V0H578Q563 3 469 3Q365 3 347 0H338V62H366Q366 63 326 112T285 163L198 63L217 62H235V0H227Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D431\" xlink:href=\"#MJX-1797-TEX-B-1D431\"><\/use><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">x<\/mi><\/mrow><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">\\mathbf x<\/script><span> to zero vector would be helpful. If we want to find <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.192ex\" height=\"1.036ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -452 527 458\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.014ex;\"><defs><path id=\"MJX-1808-TEX-B-1D41E\" d=\"M32 225Q32 332 102 392T272 452H283Q382 452 436 401Q494 343 494 243Q494 226 486 222T440 217Q431 217 394 217T327 218H175V209Q175 177 179 154T196 107T236 69T306 50Q312 49 323 49Q376 49 410 85Q421 99 427 111T434 127T442 133T463 135H468Q494 135 494 117Q494 110 489 97T468 66T431 32T373 5T292 -6Q181 -6 107 55T32 225ZM383 276Q377 346 348 374T280 402Q253 402 230 390T195 357Q179 331 176 279V266H383V276Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D41E\" xlink:href=\"#MJX-1808-TEX-B-1D41E\"><\/use><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">e<\/mi><\/mrow><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">\\mathbf e<\/script><span> with large weight, fixing <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.373ex\" height=\"1.005ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -444 607 444\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: 0px;\"><defs><path id=\"MJX-1797-TEX-B-1D431\" d=\"M227 0Q212 3 121 3Q40 3 28 0H21V62H117L245 213L109 382H26V444H34Q49 441 143 441Q247 441 265 444H274V382H246L281 339Q315 297 316 297Q320 297 354 341L389 382H352V444H360Q375 441 466 441Q547 441 559 444H566V382H471L355 246L504 63L545 62H586V0H578Q563 3 469 3Q365 3 347 0H338V62H366Q366 63 326 112T285 163L198 63L217 62H235V0H227Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D431\" xlink:href=\"#MJX-1797-TEX-B-1D431\"><\/use><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">x<\/mi><\/mrow><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">\\mathbf x<\/script><span> to non-zero vector will improve the success probability. Therefore, we define the distribution <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"2.51ex\" height=\"1.902ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -683 1109.3 840.8\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.357ex;\"><defs><path id=\"MJX-1926-TEX-C-44\" d=\"M37 475Q19 475 19 487Q19 536 103 604T327 682H356Q386 683 408 683H419Q475 683 506 681T582 668T667 633Q766 571 766 450Q766 365 723 287T611 152T455 57T279 6Q248 1 160 0Q148 0 131 0T108 -1Q72 -1 72 11Q72 24 90 40T133 64L144 68L152 88Q247 328 272 587Q275 613 272 613Q272 613 269 613Q225 610 195 602T149 579T129 556T119 532Q118 530 116 525T113 518Q102 502 80 490T37 475ZM665 407Q665 596 412 613Q403 614 383 614Q370 614 370 612Q370 598 363 542T323 357T242 103L228 69H265Q391 73 481 119Q536 148 575 188T633 268T658 338T665 392V407Z\"><\/path><path id=\"MJX-1926-TEX-I-1D461\" d=\"M26 385Q19 392 19 395Q19 399 22 411T27 425Q29 430 36 430T87 431H140L159 511Q162 522 166 540T173 566T179 586T187 603T197 615T211 624T229 626Q247 625 254 615T261 596Q261 589 252 549T232 470L222 433Q222 431 272 431H323Q330 424 330 420Q330 398 317 385H210L174 240Q135 80 135 68Q135 26 162 26Q197 26 230 60T283 144Q285 150 288 151T303 153H307Q322 153 322 145Q322 142 319 133Q314 117 301 95T267 48T216 6T155 -11Q125 -11 98 4T59 56Q57 64 57 83V101L92 241Q127 382 128 383Q128 385 77 385H26Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"msub\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"44\" xlink:href=\"#MJX-1926-TEX-C-44\"><\/use><\/g><\/g><g data-mml-node=\"mi\" transform=\"translate(804,-150) scale(0.707)\"><use data-c=\"1D461\" xlink:href=\"#MJX-1926-TEX-I-1D461\"><\/use><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><msub><mrow data-mjx-texclass=\"ORD\"><mi data-mjx-variant=\"-tex-calligraphic\" mathvariant=\"script\">D<\/mi><\/mrow><mi>t<\/mi><\/msub><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">\\mathcal D_t<\/script><span>.<\/span><\/p><\/blockquote><p><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"2.51ex\" height=\"1.902ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -683 1109.3 840.8\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.357ex;\"><defs><path id=\"MJX-1926-TEX-C-44\" d=\"M37 475Q19 475 19 487Q19 536 103 604T327 682H356Q386 683 408 683H419Q475 683 506 681T582 668T667 633Q766 571 766 450Q766 365 723 287T611 152T455 57T279 6Q248 1 160 0Q148 0 131 0T108 -1Q72 -1 72 11Q72 24 90 40T133 64L144 68L152 88Q247 328 272 587Q275 613 272 613Q272 613 269 613Q225 610 195 602T149 579T129 556T119 532Q118 530 116 525T113 518Q102 502 80 490T37 475ZM665 407Q665 596 412 613Q403 614 383 614Q370 614 370 612Q370 598 363 542T323 357T242 103L228 69H265Q391 73 481 119Q536 148 575 188T633 268T658 338T665 392V407Z\"><\/path><path id=\"MJX-1926-TEX-I-1D461\" d=\"M26 385Q19 392 19 395Q19 399 22 411T27 425Q29 430 36 430T87 431H140L159 511Q162 522 166 540T173 566T179 586T187 603T197 615T211 624T229 626Q247 625 254 615T261 596Q261 589 252 549T232 470L222 433Q222 431 272 431H323Q330 424 330 420Q330 398 317 385H210L174 240Q135 80 135 68Q135 26 162 26Q197 26 230 60T283 144Q285 150 288 151T303 153H307Q322 153 322 145Q322 142 319 133Q314 117 301 95T267 48T216 6T155 -11Q125 -11 98 4T59 56Q57 64 57 83V101L92 241Q127 382 128 383Q128 385 77 385H26Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"msub\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"44\" xlink:href=\"#MJX-1926-TEX-C-44\"><\/use><\/g><\/g><g data-mml-node=\"mi\" transform=\"translate(804,-150) scale(0.707)\"><use data-c=\"1D461\" xlink:href=\"#MJX-1926-TEX-I-1D461\"><\/use><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><msub><mrow data-mjx-texclass=\"ORD\"><mi data-mjx-variant=\"-tex-calligraphic\" mathvariant=\"script\">D<\/mi><\/mrow><mi>t<\/mi><\/msub><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">\\mathcal D_t<\/script><span> is defined as <\/span><\/p><ul><li><p><span>If <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"14.672ex\" height=\"3.236ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -948.1 6485 1430.3\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -1.091ex;\"><defs><path id=\"MJX-1905-TEX-I-1D461\" d=\"M26 385Q19 392 19 395Q19 399 22 411T27 425Q29 430 36 430T87 431H140L159 511Q162 522 166 540T173 566T179 586T187 603T197 615T211 624T229 626Q247 625 254 615T261 596Q261 589 252 549T232 470L222 433Q222 431 272 431H323Q330 424 330 420Q330 398 317 385H210L174 240Q135 80 135 68Q135 26 162 26Q197 26 230 60T283 144Q285 150 288 151T303 153H307Q322 153 322 145Q322 142 319 133Q314 117 301 95T267 48T216 6T155 -11Q125 -11 98 4T59 56Q57 64 57 83V101L92 241Q127 382 128 383Q128 385 77 385H26Z\"><\/path><path id=\"MJX-1905-TEX-N-3C\" d=\"M694 -11T694 -19T688 -33T678 -40Q671 -40 524 29T234 166L90 235Q83 240 83 250Q83 261 91 266Q664 540 678 540Q681 540 687 534T694 519T687 505Q686 504 417 376L151 250L417 124Q686 -4 687 -5Q694 -11 694 -19Z\"><\/path><path id=\"MJX-1905-TEX-I-1D45E\" d=\"M33 157Q33 258 109 349T280 441Q340 441 372 389Q373 390 377 395T388 406T404 418Q438 442 450 442Q454 442 457 439T460 434Q460 425 391 149Q320 -135 320 -139Q320 -147 365 -148H390Q396 -156 396 -157T393 -175Q389 -188 383 -194H370Q339 -192 262 -192Q234 -192 211 -192T174 -192T157 -193Q143 -193 143 -185Q143 -182 145 -170Q149 -154 152 -151T172 -148Q220 -148 230 -141Q238 -136 258 -53T279 32Q279 33 272 29Q224 -10 172 -10Q117 -10 75 30T33 157ZM352 326Q329 405 277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q233 26 290 98L298 109L352 326Z\"><\/path><path id=\"MJX-1905-TEX-N-2212\" d=\"M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z\"><\/path><path id=\"MJX-1905-TEX-N-31\" d=\"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z\"><\/path><path id=\"MJX-1905-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-1905-TEX-I-1D45B\" d=\"M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z\"><\/path><path id=\"MJX-1905-TEX-I-1D458\" d=\"M121 647Q121 657 125 670T137 683Q138 683 209 688T282 694Q294 694 294 686Q294 679 244 477Q194 279 194 272Q213 282 223 291Q247 309 292 354T362 415Q402 442 438 442Q468 442 485 423T503 369Q503 344 496 327T477 302T456 291T438 288Q418 288 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xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><msub><mrow data-mjx-texclass=\"ORD\"><mi data-mjx-variant=\"-tex-calligraphic\" mathvariant=\"script\">D<\/mi><\/mrow><mi>t<\/mi><\/msub><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">\\mathcal D_t<\/script><span> only outputs <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"6.47ex\" height=\"2.801ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -853.7 2860 1237.9\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.869ex;\"><defs><path id=\"MJX-1908-TEX-B-1D7CE\" d=\"M266 654H280H282Q500 654 524 418Q529 370 529 320Q529 125 456 52Q397 -10 287 -10Q110 -10 63 154Q45 212 45 316Q45 504 113 585Q140 618 185 636T266 654ZM374 548Q347 604 286 604Q247 604 218 575Q197 552 193 511T188 311Q188 159 196 116Q202 87 225 64T287 41Q339 41 367 87Q379 107 382 152T386 329Q386 518 374 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stretchy=\"false\">(<\/mo><mfrac><mrow><mi>q<\/mi><mo>\u2212<\/mo><mn>1<\/mn><\/mrow><mi>q<\/mi><\/mfrac><mo stretchy=\"false\">(<\/mo><mi>n<\/mi><mo>\u2212<\/mo><mi>k<\/mi><mo stretchy=\"false\">)<\/mo><mo>,<\/mo><mi>k<\/mi><mo>+<\/mo><mfrac><mrow><mi>q<\/mi><mo>\u2212<\/mo><mn>1<\/mn><\/mrow><mi>q<\/mi><\/mfrac><mo stretchy=\"false\">(<\/mo><mi>n<\/mi><mo>\u2212<\/mo><mi>k<\/mi><mo stretchy=\"false\">)<\/mo><mo stretchy=\"false\">)<\/mo><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">t\\in(\\frac{q-1}{q}(n-k),k+\\frac{q-1}{q}(n-k))<\/script><span>, <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"2.51ex\" height=\"1.902ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -683 1109.3 840.8\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.357ex;\"><defs><path id=\"MJX-1926-TEX-C-44\" d=\"M37 475Q19 475 19 487Q19 536 103 604T327 682H356Q386 683 408 683H419Q475 683 506 681T582 668T667 633Q766 571 766 450Q766 365 723 287T611 152T455 57T279 6Q248 1 160 0Q148 0 131 0T108 -1Q72 -1 72 11Q72 24 90 40T133 64L144 68L152 88Q247 328 272 587Q275 613 272 613Q272 613 269 613Q225 610 195 602T149 579T129 556T119 532Q118 530 116 525T113 518Q102 502 80 490T37 475ZM665 407Q665 596 412 613Q403 614 383 614Q370 614 370 612Q370 598 363 542T323 357T242 103L228 69H265Q391 73 481 119Q536 148 575 188T633 268T658 338T665 392V407Z\"><\/path><path id=\"MJX-1926-TEX-I-1D461\" d=\"M26 385Q19 392 19 395Q19 399 22 411T27 425Q29 430 36 430T87 431H140L159 511Q162 522 166 540T173 566T179 586T187 603T197 615T211 624T229 626Q247 625 254 615T261 596Q261 589 252 549T232 470L222 433Q222 431 272 431H323Q330 424 330 420Q330 398 317 385H210L174 240Q135 80 135 68Q135 26 162 26Q197 26 230 60T283 144Q285 150 288 151T303 153H307Q322 153 322 145Q322 142 319 133Q314 117 301 95T267 48T216 6T155 -11Q125 -11 98 4T59 56Q57 64 57 83V101L92 241Q127 382 128 383Q128 385 77 385H26Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"msub\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"44\" xlink:href=\"#MJX-1926-TEX-C-44\"><\/use><\/g><\/g><g data-mml-node=\"mi\" transform=\"translate(804,-150) scale(0.707)\"><use data-c=\"1D461\" xlink:href=\"#MJX-1926-TEX-I-1D461\"><\/use><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><msub><mrow data-mjx-texclass=\"ORD\"><mi data-mjx-variant=\"-tex-calligraphic\" mathvariant=\"script\">D<\/mi><\/mrow><mi>t<\/mi><\/msub><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">\\mathcal D_t<\/script><span> outputs uniform vectors of weight <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"14.42ex\" height=\"3.236ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -948.1 6373.8 1430.3\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -1.091ex;\"><defs><path id=\"MJX-1948-TEX-I-1D461\" d=\"M26 385Q19 392 19 395Q19 399 22 411T27 425Q29 430 36 430T87 431H140L159 511Q162 522 166 540T173 566T179 586T187 603T197 615T211 624T229 626Q247 625 254 615T261 596Q261 589 252 549T232 470L222 433Q222 431 272 431H323Q330 424 330 420Q330 398 317 385H210L174 240Q135 80 135 68Q135 26 162 26Q197 26 230 60T283 144Q285 150 288 151T303 153H307Q322 153 322 145Q322 142 319 133Q314 117 301 95T267 48T216 6T155 -11Q125 -11 98 4T59 56Q57 64 57 83V101L92 241Q127 382 128 383Q128 385 77 385H26Z\"><\/path><path id=\"MJX-1948-TEX-N-2212\" d=\"M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z\"><\/path><path id=\"MJX-1948-TEX-I-1D45E\" d=\"M33 157Q33 258 109 349T280 441Q340 441 372 389Q373 390 377 395T388 406T404 418Q438 442 450 442Q454 442 457 439T460 434Q460 425 391 149Q320 -135 320 -139Q320 -147 365 -148H390Q396 -156 396 -157T393 -175Q389 -188 383 -194H370Q339 -192 262 -192Q234 -192 211 -192T174 -192T157 -193Q143 -193 143 -185Q143 -182 145 -170Q149 -154 152 -151T172 -148Q220 -148 230 -141Q238 -136 258 -53T279 32Q279 33 272 29Q224 -10 172 -10Q117 -10 75 30T33 157ZM352 326Q329 405 277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q233 26 290 98L298 109L352 326Z\"><\/path><path id=\"MJX-1948-TEX-N-31\" d=\"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z\"><\/path><path id=\"MJX-1948-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 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type=\"math\/tex\">t-\\frac{q-1}{q}(n-k)<\/script><span>.<\/span><\/p><\/li><li><p><span>If <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"18.616ex\" height=\"3.236ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -948.1 8228.4 1430.3\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -1.091ex;\"><defs><path id=\"MJX-1925-TEX-I-1D461\" d=\"M26 385Q19 392 19 395Q19 399 22 411T27 425Q29 430 36 430T87 431H140L159 511Q162 522 166 540T173 566T179 586T187 603T197 615T211 624T229 626Q247 625 254 615T261 596Q261 589 252 549T232 470L222 433Q222 431 272 431H323Q330 424 330 420Q330 398 317 385H210L174 240Q135 80 135 68Q135 26 162 26Q197 26 230 60T283 144Q285 150 288 151T303 153H307Q322 153 322 145Q322 142 319 133Q314 117 301 95T267 48T216 6T155 -11Q125 -11 98 4T59 56Q57 64 57 83V101L92 241Q127 382 128 383Q128 385 77 385H26Z\"><\/path><path id=\"MJX-1925-TEX-N-3E\" 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230H98Q84 237 84 250Z\"><\/path><path id=\"MJX-1925-TEX-N-31\" d=\"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z\"><\/path><path id=\"MJX-1925-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-1925-TEX-I-1D45B\" d=\"M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z\"><\/path><path id=\"MJX-1925-TEX-N-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D461\" xlink:href=\"#MJX-1925-TEX-I-1D461\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(638.8,0)\"><use data-c=\"3E\" xlink:href=\"#MJX-1925-TEX-N-3E\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1694.6,0)\"><use data-c=\"1D458\" xlink:href=\"#MJX-1925-TEX-I-1D458\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2437.8,0)\"><use data-c=\"2B\" xlink:href=\"#MJX-1925-TEX-N-2B\"><\/use><\/g><g data-mml-node=\"mfrac\" transform=\"translate(3438,0)\"><g data-mml-node=\"mrow\" transform=\"translate(220,477.2) scale(0.707)\"><g data-mml-node=\"mi\"><use data-c=\"1D45E\" xlink:href=\"#MJX-1925-TEX-I-1D45E\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(460,0)\"><use data-c=\"2212\" xlink:href=\"#MJX-1925-TEX-N-2212\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(1238,0)\"><use data-c=\"31\" xlink:href=\"#MJX-1925-TEX-N-31\"><\/use><\/g><\/g><g data-mml-node=\"mi\" transform=\"translate(671.8,-345) scale(0.707)\"><use data-c=\"1D45E\" xlink:href=\"#MJX-1925-TEX-I-1D45E\"><\/use><\/g><rect width=\"1429\" height=\"60\" x=\"120\" y=\"220\"><\/rect><\/g><g data-mml-node=\"mo\" transform=\"translate(5107,0)\"><use data-c=\"28\" xlink:href=\"#MJX-1925-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(5496,0)\"><use data-c=\"1D45B\" xlink:href=\"#MJX-1925-TEX-I-1D45B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(6318.2,0)\"><use data-c=\"2212\" xlink:href=\"#MJX-1925-TEX-N-2212\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(7318.4,0)\"><use data-c=\"1D458\" xlink:href=\"#MJX-1925-TEX-I-1D458\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(7839.4,0)\"><use data-c=\"29\" xlink:href=\"#MJX-1925-TEX-N-29\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>t<\/mi><mo>&gt;<\/mo><mi>k<\/mi><mo>+<\/mo><mfrac><mrow><mi>q<\/mi><mo>\u2212<\/mo><mn>1<\/mn><\/mrow><mi>q<\/mi><\/mfrac><mo stretchy=\"false\">(<\/mo><mi>n<\/mi><mo>\u2212<\/mo><mi>k<\/mi><mo stretchy=\"false\">)<\/mo><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">t>k+\\frac{q-1}{q}(n-k)<\/script><span>, <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"2.51ex\" height=\"1.902ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -683 1109.3 840.8\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.357ex;\"><defs><path id=\"MJX-1926-TEX-C-44\" d=\"M37 475Q19 475 19 487Q19 536 103 604T327 682H356Q386 683 408 683H419Q475 683 506 681T582 668T667 633Q766 571 766 450Q766 365 723 287T611 152T455 57T279 6Q248 1 160 0Q148 0 131 0T108 -1Q72 -1 72 11Q72 24 90 40T133 64L144 68L152 88Q247 328 272 587Q275 613 272 613Q272 613 269 613Q225 610 195 602T149 579T129 556T119 532Q118 530 116 525T113 518Q102 502 80 490T37 475ZM665 407Q665 596 412 613Q403 614 383 614Q370 614 370 612Q370 598 363 542T323 357T242 103L228 69H265Q391 73 481 119Q536 148 575 188T633 268T658 338T665 392V407Z\"><\/path><path id=\"MJX-1926-TEX-I-1D461\" d=\"M26 385Q19 392 19 395Q19 399 22 411T27 425Q29 430 36 430T87 431H140L159 511Q162 522 166 540T173 566T179 586T187 603T197 615T211 624T229 626Q247 625 254 615T261 596Q261 589 252 549T232 470L222 433Q222 431 272 431H323Q330 424 330 420Q330 398 317 385H210L174 240Q135 80 135 68Q135 26 162 26Q197 26 230 60T283 144Q285 150 288 151T303 153H307Q322 153 322 145Q322 142 319 133Q314 117 301 95T267 48T216 6T155 -11Q125 -11 98 4T59 56Q57 64 57 83V101L92 241Q127 382 128 383Q128 385 77 385H26Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"msub\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"44\" xlink:href=\"#MJX-1926-TEX-C-44\"><\/use><\/g><\/g><g data-mml-node=\"mi\" transform=\"translate(804,-150) scale(0.707)\"><use data-c=\"1D461\" xlink:href=\"#MJX-1926-TEX-I-1D461\"><\/use><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><msub><mrow data-mjx-texclass=\"ORD\"><mi data-mjx-variant=\"-tex-calligraphic\" mathvariant=\"script\">D<\/mi><\/mrow><mi>t<\/mi><\/msub><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">\\mathcal D_t<\/script><span> outputs uniform vectors of weight <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.179ex\" height=\"1.595ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -694 521 705\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.025ex;\"><defs><path id=\"MJX-1790-TEX-I-1D458\" d=\"M121 647Q121 657 125 670T137 683Q138 683 209 688T282 694Q294 694 294 686Q294 679 244 477Q194 279 194 272Q213 282 223 291Q247 309 292 354T362 415Q402 442 438 442Q468 442 485 423T503 369Q503 344 496 327T477 302T456 291T438 288Q418 288 406 299T394 328Q394 353 410 369T442 390L458 393Q446 405 434 405H430Q398 402 367 380T294 316T228 255Q230 254 243 252T267 246T293 238T320 224T342 206T359 180T365 147Q365 130 360 106T354 66Q354 26 381 26Q429 26 459 145Q461 153 479 153H483Q499 153 499 144Q499 139 496 130Q455 -11 378 -11Q333 -11 305 15T277 90Q277 108 280 121T283 145Q283 167 269 183T234 206T200 217T182 220H180Q168 178 159 139T145 81T136 44T129 20T122 7T111 -2Q98 -11 83 -11Q66 -11 57 -1T48 16Q48 26 85 176T158 471L195 616Q196 629 188 632T149 637H144Q134 637 131 637T124 640T121 647Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D458\" xlink:href=\"#MJX-1790-TEX-I-1D458\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>k<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">k<\/script><span>.<\/span><\/p><\/li><\/ul><p><strong><span>Rough Analysis:<\/span><\/strong><\/p><p><span>Assume <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.027ex\" height=\"1.038ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -453 454 459\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.014ex;\"><defs><path id=\"MJX-1800-TEX-B-1D42C\" d=\"M38 315Q38 339 45 360T70 404T127 440T223 453Q273 453 320 436L338 445L357 453H366Q380 453 383 447T386 403V387V355Q386 331 383 326T365 321H355H349Q333 321 329 324T324 341Q317 406 224 406H216Q123 406 123 353Q123 334 143 321T188 304T244 294T285 286Q305 281 325 273T373 237T412 172Q414 162 414 142Q414 -6 230 -6Q154 -6 117 22L68 -6H58Q44 -6 41 0T38 42V73Q38 85 38 101T37 122Q37 144 42 148T68 153H75Q87 153 91 151T97 147T103 132Q131 46 220 46H230Q257 46 265 47Q330 58 330 108Q330 127 316 142Q300 156 284 162Q271 168 212 178T122 202Q38 243 38 315Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D42C\" xlink:href=\"#MJX-1800-TEX-B-1D42C\"><\/use><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">s<\/mi><\/mrow><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">\\mathbf s<\/script><span> is uniformly distributed over <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"4.608ex\" height=\"2.801ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -853.7 2036.8 1237.9\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.869ex;\"><defs><path id=\"MJX-1802-TEX-D-1D53D\" d=\"M584 499Q569 490 566 490Q558 490 552 497T546 515Q546 535 533 559Q526 574 506 593T469 621Q415 648 326 648Q293 648 287 647T275 641Q264 630 263 617Q262 609 260 492V370L275 372Q323 376 350 392T393 441Q409 473 409 506Q409 529 427 529Q437 529 442 519Q444 511 444 362Q444 212 442 206Q436 197 426 197Q409 197 409 217Q409 265 375 299Q346 328 280 335H260V206Q260 70 262 63Q265 46 276 41T326 35Q362 35 366 28Q377 17 366 3L360 -1H24Q12 5 12 16Q12 35 51 35Q92 38 97 52Q102 60 102 341T97 632Q91 645 51 648Q12 648 12 666Q12 675 24 683H573Q576 678 584 670V499ZM137 341Q137 131 136 89T130 37Q129 36 129 35H182Q233 35 233 39Q226 54 225 92T224 346L226 623L231 635L235 648H129Q132 641 133 638T135 603T137 517T137 341ZM549 603V648H495L506 641Q531 621 533 619L549 603ZM409 317V395L400 386Q390 376 375 366L357 355L373 346Q394 331 397 328L409 317Z\"><\/path><path id=\"MJX-1802-TEX-I-1D45B\" d=\"M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z\"><\/path><path id=\"MJX-1802-TEX-N-2212\" d=\"M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z\"><\/path><path id=\"MJX-1802-TEX-I-1D458\" d=\"M121 647Q121 657 125 670T137 683Q138 683 209 688T282 694Q294 694 294 686Q294 679 244 477Q194 279 194 272Q213 282 223 291Q247 309 292 354T362 415Q402 442 438 442Q468 442 485 423T503 369Q503 344 496 327T477 302T456 291T438 288Q418 288 406 299T394 328Q394 353 410 369T442 390L458 393Q446 405 434 405H430Q398 402 367 380T294 316T228 255Q230 254 243 252T267 246T293 238T320 224T342 206T359 180T365 147Q365 130 360 106T354 66Q354 26 381 26Q429 26 459 145Q461 153 479 153H483Q499 153 499 144Q499 139 496 130Q455 -11 378 -11Q333 -11 305 15T277 90Q277 108 280 121T283 145Q283 167 269 183T234 206T200 217T182 220H180Q168 178 159 139T145 81T136 44T129 20T122 7T111 -2Q98 -11 83 -11Q66 -11 57 -1T48 16Q48 26 85 176T158 471L195 616Q196 629 188 632T149 637H144Q134 637 131 637T124 640T121 647Z\"><\/path><path id=\"MJX-1802-TEX-I-1D45E\" d=\"M33 157Q33 258 109 349T280 441Q340 441 372 389Q373 390 377 395T388 406T404 418Q438 442 450 442Q454 442 457 439T460 434Q460 425 391 149Q320 -135 320 -139Q320 -147 365 -148H390Q396 -156 396 -157T393 -175Q389 -188 383 -194H370Q339 -192 262 -192Q234 -192 211 -192T174 -192T157 -193Q143 -193 143 -185Q143 -182 145 -170Q149 -154 152 -151T172 -148Q220 -148 230 -141Q238 -136 258 -53T279 32Q279 33 272 29Q224 -10 172 -10Q117 -10 75 30T33 157ZM352 326Q329 405 277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q233 26 290 98L298 109L352 326Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"msubsup\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D53D\" xlink:href=\"#MJX-1802-TEX-D-1D53D\"><\/use><\/g><\/g><g data-mml-node=\"TeXAtom\" transform=\"translate(644,363) scale(0.707)\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D45B\" xlink:href=\"#MJX-1802-TEX-I-1D45B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(600,0)\"><use data-c=\"2212\" xlink:href=\"#MJX-1802-TEX-N-2212\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1378,0)\"><use data-c=\"1D458\" xlink:href=\"#MJX-1802-TEX-I-1D458\"><\/use><\/g><\/g><g data-mml-node=\"mi\" transform=\"translate(644,-247) scale(0.707)\"><use data-c=\"1D45E\" xlink:href=\"#MJX-1802-TEX-I-1D45E\"><\/use><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><msubsup><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"double-struck\">F<\/mi><\/mrow><mi>q<\/mi><mrow data-mjx-texclass=\"ORD\"><mi>n<\/mi><mo>\u2212<\/mo><mi>k<\/mi><\/mrow><\/msubsup><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">\\mathbb F_q^{n-k}<\/script><span>, which is equivalent to assuming that <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"3.305ex\" height=\"2.262ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 1461 1000\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-1803-TEX-I-1D461\" d=\"M26 385Q19 392 19 395Q19 399 22 411T27 425Q29 430 36 430T87 431H140L159 511Q162 522 166 540T173 566T179 586T187 603T197 615T211 624T229 626Q247 625 254 615T261 596Q261 589 252 549T232 470L222 433Q222 431 272 431H323Q330 424 330 420Q330 398 317 385H210L174 240Q135 80 135 68Q135 26 162 26Q197 26 230 60T283 144Q285 150 288 151T303 153H307Q322 153 322 145Q322 142 319 133Q314 117 301 95T267 48T216 6T155 -11Q125 -11 98 4T59 56Q57 64 57 83V101L92 241Q127 382 128 383Q128 385 77 385H26Z\"><\/path><path id=\"MJX-1803-TEX-N-2F\" d=\"M423 750Q432 750 438 744T444 730Q444 725 271 248T92 -240Q85 -250 75 -250Q68 -250 62 -245T56 -231Q56 -221 230 257T407 740Q411 750 423 750Z\"><\/path><path id=\"MJX-1803-TEX-I-1D45B\" d=\"M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D461\" xlink:href=\"#MJX-1803-TEX-I-1D461\"><\/use><\/g><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\" transform=\"translate(361,0)\"><g data-mml-node=\"mo\"><use data-c=\"2F\" xlink:href=\"#MJX-1803-TEX-N-2F\"><\/use><\/g><\/g><g data-mml-node=\"mi\" transform=\"translate(861,0)\"><use data-c=\"1D45B\" xlink:href=\"#MJX-1803-TEX-I-1D45B\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>t<\/mi><mrow data-mjx-texclass=\"ORD\"><mo>\/<\/mo><\/mrow><mi>n<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">t\/n<\/script><span> belongs to <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"7.712ex\" height=\"2.32ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -775.2 3408.9 1025.2\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-1807-TEX-N-5B\" d=\"M118 -250V750H255V710H158V-210H255V-250H118Z\"><\/path><path id=\"MJX-1807-TEX-I-1D70F\" d=\"M39 284Q18 284 18 294Q18 301 45 338T99 398Q134 425 164 429Q170 431 332 431Q492 431 497 429Q517 424 517 402Q517 388 508 376T485 360Q479 358 389 358T299 356Q298 355 283 274T251 109T233 20Q228 5 215 -4T186 -13Q153 -13 153 20V30L203 192Q214 228 227 272T248 336L254 357Q254 358 208 358Q206 358 197 358T183 359Q105 359 61 295Q56 287 53 286T39 284Z\"><\/path><path id=\"MJX-1807-TEX-N-2212\" d=\"M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z\"><\/path><path id=\"MJX-1807-TEX-N-2C\" d=\"M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z\"><\/path><path id=\"MJX-1807-TEX-N-2B\" d=\"M56 237T56 250T70 270H369V420L370 570Q380 583 389 583Q402 583 409 568V270H707Q722 262 722 250T707 230H409V-68Q401 -82 391 -82H389H387Q375 -82 369 -68V230H70Q56 237 56 250Z\"><\/path><path id=\"MJX-1807-TEX-N-5D\" d=\"M22 710V750H159V-250H22V-210H119V710H22Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mo\"><use data-c=\"5B\" xlink:href=\"#MJX-1807-TEX-N-5B\"><\/use><\/g><g data-mml-node=\"msup\" transform=\"translate(278,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D70F\" xlink:href=\"#MJX-1807-TEX-I-1D70F\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(604,363) scale(0.707)\"><use data-c=\"2212\" xlink:href=\"#MJX-1807-TEX-N-2212\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(1482.1,0)\"><use data-c=\"2C\" xlink:href=\"#MJX-1807-TEX-N-2C\"><\/use><\/g><g data-mml-node=\"msup\" transform=\"translate(1926.8,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D70F\" xlink:href=\"#MJX-1807-TEX-I-1D70F\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(604,363) scale(0.707)\"><use data-c=\"2B\" xlink:href=\"#MJX-1807-TEX-N-2B\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(3130.9,0)\"><use data-c=\"5D\" xlink:href=\"#MJX-1807-TEX-N-5D\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mo stretchy=\"false\">[<\/mo><msup><mi>\u03c4<\/mi><mo>\u2212<\/mo><\/msup><mo>,<\/mo><msup><mi>\u03c4<\/mi><mo>+<\/mo><\/msup><mo stretchy=\"false\">]<\/mo><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">[\\tau^-,\\tau^+]<\/script><span>, the expected Hamming weight of <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.192ex\" height=\"1.036ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -452 527 458\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.014ex;\"><defs><path id=\"MJX-1808-TEX-B-1D41E\" d=\"M32 225Q32 332 102 392T272 452H283Q382 452 436 401Q494 343 494 243Q494 226 486 222T440 217Q431 217 394 217T327 218H175V209Q175 177 179 154T196 107T236 69T306 50Q312 49 323 49Q376 49 410 85Q421 99 427 111T434 127T442 133T463 135H468Q494 135 494 117Q494 110 489 97T468 66T431 32T373 5T292 -6Q181 -6 107 55T32 225ZM383 276Q377 346 348 374T280 402Q253 402 230 390T195 357Q179 331 176 279V266H383V276Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D41E\" xlink:href=\"#MJX-1808-TEX-B-1D41E\"><\/use><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">e<\/mi><\/mrow><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">\\mathbf e<\/script><span> is given by<\/span><\/p><div contenteditable=\"false\" spellcheck=\"false\" class=\"mathjax-block md-end-block md-math-block md-rawblock\" id=\"mathjax-n396\" cid=\"n396\" mdtype=\"math_block\" data-math-tag-before=\"10\" data-math-labels=\"[]\" data-math-tag-after=\"10\"><div class=\"md-rawblock-container md-math-container\" contenteditable=\"false\" tabindex=\"-1\"><mjx-container class=\"MathJax\" jax=\"SVG\" display=\"true\" width=\"full\" style=\"min-width: 30.59ex; position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"100%\" height=\"5.027ex\" role=\"img\" focusable=\"false\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -1.948ex; min-width: 30.59ex;\"><defs><path id=\"MJX-2000-TEX-SS-1D5A4\" d=\"M86 0V691H541V611H366L190 612V397H513V321H190V85H372L554 86V0H86Z\"><\/path><path id=\"MJX-2000-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-2000-TEX-N-7C\" d=\"M139 -249H137Q125 -249 119 -235V251L120 737Q130 750 139 750Q152 750 159 735V-235Q151 -249 141 -249H139Z\"><\/path><path id=\"MJX-2000-TEX-B-1D41E\" d=\"M32 225Q32 332 102 392T272 452H283Q382 452 436 401Q494 343 494 243Q494 226 486 222T440 217Q431 217 394 217T327 218H175V209Q175 177 179 154T196 107T236 69T306 50Q312 49 323 49Q376 49 410 85Q421 99 427 111T434 127T442 133T463 135H468Q494 135 494 117Q494 110 489 97T468 66T431 32T373 5T292 -6Q181 -6 107 55T32 225ZM383 276Q377 346 348 374T280 402Q253 402 230 390T195 357Q179 331 176 279V266H383V276Z\"><\/path><path id=\"MJX-2000-TEX-N-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><path id=\"MJX-2000-TEX-N-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path id=\"MJX-2000-TEX-B-1D431\" d=\"M227 0Q212 3 121 3Q40 3 28 0H21V62H117L245 213L109 382H26V444H34Q49 441 143 441Q247 441 265 444H274V382H246L281 339Q315 297 316 297Q320 297 354 341L389 382H352V444H360Q375 441 466 441Q547 441 559 444H566V382H471L355 246L504 63L545 62H586V0H578Q563 3 469 3Q365 3 347 0H338V62H366Q366 63 326 112T285 163L198 63L217 62H235V0H227Z\"><\/path><path id=\"MJX-2000-TEX-N-2B\" d=\"M56 237T56 250T70 270H369V420L370 570Q380 583 389 583Q402 583 409 568V270H707Q722 262 722 250T707 230H409V-68Q401 -82 391 -82H389H387Q375 -82 369 -68V230H70Q56 237 56 250Z\"><\/path><path id=\"MJX-2000-TEX-I-1D45E\" d=\"M33 157Q33 258 109 349T280 441Q340 441 372 389Q373 390 377 395T388 406T404 418Q438 442 450 442Q454 442 457 439T460 434Q460 425 391 149Q320 -135 320 -139Q320 -147 365 -148H390Q396 -156 396 -157T393 -175Q389 -188 383 -194H370Q339 -192 262 -192Q234 -192 211 -192T174 -192T157 -193Q143 -193 143 -185Q143 -182 145 -170Q149 -154 152 -151T172 -148Q220 -148 230 -141Q238 -136 258 -53T279 32Q279 33 272 29Q224 -10 172 -10Q117 -10 75 30T33 157ZM352 326Q329 405 277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q233 26 290 98L298 109L352 326Z\"><\/path><path id=\"MJX-2000-TEX-N-2212\" d=\"M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z\"><\/path><path id=\"MJX-2000-TEX-N-31\" d=\"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 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328Q394 353 410 369T442 390L458 393Q446 405 434 405H430Q398 402 367 380T294 316T228 255Q230 254 243 252T267 246T293 238T320 224T342 206T359 180T365 147Q365 130 360 106T354 66Q354 26 381 26Q429 26 459 145Q461 153 479 153H483Q499 153 499 144Q499 139 496 130Q455 -11 378 -11Q333 -11 305 15T277 90Q277 108 280 121T283 145Q283 167 269 183T234 206T200 217T182 220H180Q168 178 159 139T145 81T136 44T129 20T122 7T111 -2Q98 -11 83 -11Q66 -11 57 -1T48 16Q48 26 85 176T158 471L195 616Q196 629 188 632T149 637H144Q134 637 131 637T124 640T121 647Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(0.018929,-0.018929) translate(0, -1361)\"><g data-mml-node=\"math\"><g data-mml-node=\"mtable\" transform=\"translate(800,0) translate(-800,0)\"><g transform=\"translate(0 1361) matrix(1 0 0 -1 0 0) scale(52.83)\"><svg data-table=\"true\" preserveAspectRatio=\"xMidYMid\" viewBox=\"5960.4 -1361 1 2222\"><g transform=\"matrix(1 0 0 -1 0 0)\"><g data-mml-node=\"mlabeledtr\" transform=\"translate(0,19)\"><g data-mml-node=\"mtd\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D5A4\" xlink:href=\"#MJX-2000-TEX-SS-1D5A4\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(597,0)\"><use data-c=\"28\" xlink:href=\"#MJX-2000-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(986,0) translate(0 -0.5)\"><use data-c=\"7C\" xlink:href=\"#MJX-2000-TEX-N-7C\"><\/use><\/g><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\" transform=\"translate(1264,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D41E\" xlink:href=\"#MJX-2000-TEX-B-1D41E\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(1791,0) translate(0 -0.5)\"><use data-c=\"7C\" xlink:href=\"#MJX-2000-TEX-N-7C\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2069,0)\"><use data-c=\"29\" xlink:href=\"#MJX-2000-TEX-N-29\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2735.8,0)\"><use data-c=\"3D\" xlink:href=\"#MJX-2000-TEX-N-3D\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(3791.6,0) translate(0 -0.5)\"><use data-c=\"7C\" xlink:href=\"#MJX-2000-TEX-N-7C\"><\/use><\/g><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\" transform=\"translate(4069.6,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D431\" xlink:href=\"#MJX-2000-TEX-B-1D431\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(4676.6,0) translate(0 -0.5)\"><use data-c=\"7C\" xlink:href=\"#MJX-2000-TEX-N-7C\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(5176.8,0)\"><use data-c=\"2B\" xlink:href=\"#MJX-2000-TEX-N-2B\"><\/use><\/g><g data-mml-node=\"mfrac\" transform=\"translate(6177,0)\"><g data-mml-node=\"mrow\" transform=\"translate(220,676)\"><g data-mml-node=\"mi\"><use data-c=\"1D45E\" xlink:href=\"#MJX-2000-TEX-I-1D45E\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(682.2,0)\"><use data-c=\"2212\" xlink:href=\"#MJX-2000-TEX-N-2212\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(1682.4,0)\"><use data-c=\"31\" xlink:href=\"#MJX-2000-TEX-N-31\"><\/use><\/g><\/g><g data-mml-node=\"mi\" transform=\"translate(1081.2,-686)\"><use data-c=\"1D45E\" xlink:href=\"#MJX-2000-TEX-I-1D45E\"><\/use><\/g><rect width=\"2382.4\" height=\"60\" x=\"120\" y=\"220\"><\/rect><\/g><g data-mml-node=\"mo\" transform=\"translate(8799.4,0)\"><use data-c=\"28\" xlink:href=\"#MJX-2000-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(9188.4,0)\"><use data-c=\"1D45B\" xlink:href=\"#MJX-2000-TEX-I-1D45B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(10010.7,0)\"><use data-c=\"2212\" xlink:href=\"#MJX-2000-TEX-N-2212\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(11010.9,0)\"><use data-c=\"1D458\" xlink:href=\"#MJX-2000-TEX-I-1D458\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(11531.9,0)\"><use data-c=\"29\" xlink:href=\"#MJX-2000-TEX-N-29\"><\/use><\/g><\/g><\/g><\/g><\/svg><svg data-labels=\"true\" preserveAspectRatio=\"xMaxYMid\" viewBox=\"0 -1361 1 2222\"><g data-labels=\"true\" transform=\"matrix(1 0 0 -1 0 0)\"><g data-mml-node=\"mtd\" id=\"mjx-eqn:\" transform=\"translate(0,19)\"><text data-id-align=\"true\"><\/text><g data-idbox=\"true\" transform=\"translate(0,0)\"><g data-mml-node=\"mrow\"><\/g><\/g><\/g><\/g><\/svg><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"block\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable displaystyle=\"true\"><mlabeledtr><mtd><mrow><\/mrow><\/mtd><mtd><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"sans-serif\">E<\/mi><\/mrow><mo stretchy=\"false\">(<\/mo><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">e<\/mi><\/mrow><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">x<\/mi><\/mrow><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mo>+<\/mo><mfrac><mrow><mi>q<\/mi><mo>\u2212<\/mo><mn>1<\/mn><\/mrow><mi>q<\/mi><\/mfrac><mo stretchy=\"false\">(<\/mo><mi>n<\/mi><mo>\u2212<\/mo><mi>k<\/mi><mo stretchy=\"false\">)<\/mo><\/mtd><\/mlabeledtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><p><span>where <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"10.838ex\" height=\"3.236ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -948.1 4790.4 1430.3\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -1.091ex;\"><defs><path id=\"MJX-1898-TEX-I-1D45E\" d=\"M33 157Q33 258 109 349T280 441Q340 441 372 389Q373 390 377 395T388 406T404 418Q438 442 450 442Q454 442 457 439T460 434Q460 425 391 149Q320 -135 320 -139Q320 -147 365 -148H390Q396 -156 396 -157T393 -175Q389 -188 383 -194H370Q339 -192 262 -192Q234 -192 211 -192T174 -192T157 -193Q143 -193 143 -185Q143 -182 145 -170Q149 -154 152 -151T172 -148Q220 -148 230 -141Q238 -136 258 -53T279 32Q279 33 272 29Q224 -10 172 -10Q117 -10 75 30T33 157ZM352 326Q329 405 277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q233 26 290 98L298 109L352 326Z\"><\/path><path id=\"MJX-1898-TEX-N-2212\" d=\"M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z\"><\/path><path id=\"MJX-1898-TEX-N-31\" d=\"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z\"><\/path><path id=\"MJX-1898-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 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302T456 291T438 288Q418 288 406 299T394 328Q394 353 410 369T442 390L458 393Q446 405 434 405H430Q398 402 367 380T294 316T228 255Q230 254 243 252T267 246T293 238T320 224T342 206T359 180T365 147Q365 130 360 106T354 66Q354 26 381 26Q429 26 459 145Q461 153 479 153H483Q499 153 499 144Q499 139 496 130Q455 -11 378 -11Q333 -11 305 15T277 90Q277 108 280 121T283 145Q283 167 269 183T234 206T200 217T182 220H180Q168 178 159 139T145 81T136 44T129 20T122 7T111 -2Q98 -11 83 -11Q66 -11 57 -1T48 16Q48 26 85 176T158 471L195 616Q196 629 188 632T149 637H144Q134 637 131 637T124 640T121 647Z\"><\/path><path id=\"MJX-1898-TEX-N-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mfrac\"><g data-mml-node=\"mrow\" transform=\"translate(220,477.2) scale(0.707)\"><g data-mml-node=\"mi\"><use data-c=\"1D45E\" xlink:href=\"#MJX-1898-TEX-I-1D45E\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(460,0)\"><use data-c=\"2212\" xlink:href=\"#MJX-1898-TEX-N-2212\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(1238,0)\"><use data-c=\"31\" xlink:href=\"#MJX-1898-TEX-N-31\"><\/use><\/g><\/g><g data-mml-node=\"mi\" transform=\"translate(671.8,-345) scale(0.707)\"><use data-c=\"1D45E\" xlink:href=\"#MJX-1898-TEX-I-1D45E\"><\/use><\/g><rect width=\"1429\" height=\"60\" x=\"120\" y=\"220\"><\/rect><\/g><g data-mml-node=\"mo\" transform=\"translate(1669,0)\"><use data-c=\"28\" xlink:href=\"#MJX-1898-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(2058,0)\"><use data-c=\"1D45B\" xlink:href=\"#MJX-1898-TEX-I-1D45B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2880.2,0)\"><use data-c=\"2212\" xlink:href=\"#MJX-1898-TEX-N-2212\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(3880.4,0)\"><use data-c=\"1D458\" xlink:href=\"#MJX-1898-TEX-I-1D458\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(4401.4,0)\"><use data-c=\"29\" xlink:href=\"#MJX-1898-TEX-N-29\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mfrac><mrow><mi>q<\/mi><mo>\u2212<\/mo><mn>1<\/mn><\/mrow><mi>q<\/mi><\/mfrac><mo stretchy=\"false\">(<\/mo><mi>n<\/mi><mo>\u2212<\/mo><mi>k<\/mi><mo stretchy=\"false\">)<\/mo><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">\\frac{q-1}{q}(n-k)<\/script><span> is the expected value of distribution <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"13.33ex\" height=\"3.236ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -948.1 5892.1 1430.3\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -1.091ex;\"><defs><path id=\"MJX-1901-TEX-C-42\" d=\"M304 342Q292 342 292 353Q292 372 323 391Q331 396 417 428T533 487Q563 512 563 555V562Q563 575 557 589T530 618T475 636Q429 636 396 613T330 539Q263 446 210 238Q196 183 173 120Q135 31 121 16Q108 1 85 -10T47 -22T32 -10Q32 -5 44 18T77 93T112 206Q135 296 154 395T182 550T191 615Q191 616 190 616Q188 616 179 611T157 601T131 594Q113 594 113 605Q113 623 144 644Q154 650 205 676T267 703Q277 705 279 705Q295 705 295 693Q295 686 288 635T278 575Q278 572 287 582Q336 635 402 669T540 704Q603 704 633 673T664 599Q664 559 638 523T580 462Q553 440 504 413L491 407L504 402Q566 381 596 338T627 244Q627 172 575 110T444 13T284 -22Q208 -22 158 28Q144 42 146 50Q150 67 178 85T230 103Q236 103 246 95T267 75T302 56T357 47Q436 47 486 93Q526 136 526 198V210Q526 228 518 249T491 292T436 330T350 345Q335 345 321 344T304 342Z\"><\/path><path id=\"MJX-1901-TEX-N-28\" d=\"M94 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transform=\"translate(2868.4,0)\"><use data-c=\"1D458\" xlink:href=\"#MJX-1901-TEX-I-1D458\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(3389.4,0)\"><use data-c=\"2C\" xlink:href=\"#MJX-1901-TEX-N-2C\"><\/use><\/g><g data-mml-node=\"mfrac\" transform=\"translate(3834.1,0)\"><g data-mml-node=\"mrow\" transform=\"translate(220,477.2) scale(0.707)\"><g data-mml-node=\"mi\"><use data-c=\"1D45E\" xlink:href=\"#MJX-1901-TEX-I-1D45E\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(460,0)\"><use data-c=\"2212\" xlink:href=\"#MJX-1901-TEX-N-2212\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(1238,0)\"><use data-c=\"31\" xlink:href=\"#MJX-1901-TEX-N-31\"><\/use><\/g><\/g><g data-mml-node=\"mi\" transform=\"translate(671.8,-345) scale(0.707)\"><use data-c=\"1D45E\" xlink:href=\"#MJX-1901-TEX-I-1D45E\"><\/use><\/g><rect width=\"1429\" height=\"60\" x=\"120\" y=\"220\"><\/rect><\/g><g data-mml-node=\"mo\" transform=\"translate(5503.1,0)\"><use data-c=\"29\" xlink:href=\"#MJX-1901-TEX-N-29\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow data-mjx-texclass=\"ORD\"><mi data-mjx-variant=\"-tex-calligraphic\" mathvariant=\"script\">B<\/mi><\/mrow><mo stretchy=\"false\">(<\/mo><mi>n<\/mi><mo>\u2212<\/mo><mi>k<\/mi><mo>,<\/mo><mfrac><mrow><mi>q<\/mi><mo>\u2212<\/mo><mn>1<\/mn><\/mrow><mi>q<\/mi><\/mfrac><mo stretchy=\"false\">)<\/mo><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">\\mathcal B(n-k,\\frac{q-1}{q})<\/script><span>.<\/span><\/p><p><span>Since <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"6.543ex\" height=\"2.801ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -853.7 2892 1237.9\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.869ex;\"><defs><path 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316T228 255Q230 254 243 252T267 246T293 238T320 224T342 206T359 180T365 147Q365 130 360 106T354 66Q354 26 381 26Q429 26 459 145Q461 153 479 153H483Q499 153 499 144Q499 139 496 130Q455 -11 378 -11Q333 -11 305 15T277 90Q277 108 280 121T283 145Q283 167 269 183T234 206T200 217T182 220H180Q168 178 159 139T145 81T136 44T129 20T122 7T111 -2Q98 -11 83 -11Q66 -11 57 -1T48 16Q48 26 85 176T158 471L195 616Q196 629 188 632T149 637H144Q134 637 131 637T124 640T121 647Z\"><\/path><path id=\"MJX-1928-TEX-I-1D45E\" d=\"M33 157Q33 258 109 349T280 441Q340 441 372 389Q373 390 377 395T388 406T404 418Q438 442 450 442Q454 442 457 439T460 434Q460 425 391 149Q320 -135 320 -139Q320 -147 365 -148H390Q396 -156 396 -157T393 -175Q389 -188 383 -194H370Q339 -192 262 -192Q234 -192 211 -192T174 -192T157 -193Q143 -193 143 -185Q143 -182 145 -170Q149 -154 152 -151T172 -148Q220 -148 230 -141Q238 -136 258 -53T279 32Q279 33 272 29Q224 -10 172 -10Q117 -10 75 30T33 157ZM352 326Q329 405 277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q233 26 290 98L298 109L352 326Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D431\" xlink:href=\"#MJX-1928-TEX-B-1D431\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(884.8,0)\"><use data-c=\"2208\" xlink:href=\"#MJX-1928-TEX-N-2208\"><\/use><\/g><g data-mml-node=\"msubsup\" transform=\"translate(1829.6,0)\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D53D\" xlink:href=\"#MJX-1928-TEX-D-1D53D\"><\/use><\/g><\/g><g data-mml-node=\"mi\" transform=\"translate(644,363) scale(0.707)\"><use data-c=\"1D458\" xlink:href=\"#MJX-1928-TEX-I-1D458\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(644,-247) scale(0.707)\"><use data-c=\"1D45E\" xlink:href=\"#MJX-1928-TEX-I-1D45E\"><\/use><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">x<\/mi><\/mrow><mo>\u2208<\/mo><msubsup><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"double-struck\">F<\/mi><\/mrow><mi>q<\/mi><mi>k<\/mi><\/msubsup><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">\\mathbf x\\in\\mathbb F_q^k<\/script><span>, we can easily reach <\/span><u><span>any weight<\/span><\/u><span> in<\/span><\/p><div contenteditable=\"false\" spellcheck=\"false\" class=\"mathjax-block md-end-block md-math-block md-rawblock\" id=\"mathjax-n419\" cid=\"n419\" mdtype=\"math_block\" data-math-tag-before=\"10\" data-math-labels=\"[]\" data-math-tag-after=\"10\"><div class=\"md-rawblock-container md-math-container\" contenteditable=\"false\" tabindex=\"-1\"><mjx-container class=\"MathJax\" jax=\"SVG\" 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data-c=\"1D458\" xlink:href=\"#MJX-2001-TEX-I-1D458\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(7320.8,0)\"><use data-c=\"2B\" xlink:href=\"#MJX-2001-TEX-N-2B\"><\/use><\/g><g data-mml-node=\"mfrac\" transform=\"translate(8321,0)\"><g data-mml-node=\"mrow\" transform=\"translate(220,676)\"><g data-mml-node=\"mi\"><use data-c=\"1D45E\" xlink:href=\"#MJX-2001-TEX-I-1D45E\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(682.2,0)\"><use data-c=\"2212\" xlink:href=\"#MJX-2001-TEX-N-2212\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(1682.4,0)\"><use data-c=\"31\" xlink:href=\"#MJX-2001-TEX-N-31\"><\/use><\/g><\/g><g data-mml-node=\"mi\" transform=\"translate(1081.2,-686)\"><use data-c=\"1D45E\" xlink:href=\"#MJX-2001-TEX-I-1D45E\"><\/use><\/g><rect width=\"2382.4\" height=\"60\" x=\"120\" y=\"220\"><\/rect><\/g><g data-mml-node=\"mo\" transform=\"translate(10943.4,0)\"><use data-c=\"28\" xlink:href=\"#MJX-2001-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(11332.4,0)\"><use data-c=\"1D45B\" xlink:href=\"#MJX-2001-TEX-I-1D45B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(12154.7,0)\"><use data-c=\"2212\" xlink:href=\"#MJX-2001-TEX-N-2212\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(13154.9,0)\"><use data-c=\"1D458\" xlink:href=\"#MJX-2001-TEX-I-1D458\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(13675.9,0)\"><use data-c=\"29\" xlink:href=\"#MJX-2001-TEX-N-29\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(14064.9,0)\"><use data-c=\"29\" xlink:href=\"#MJX-2001-TEX-N-29\"><\/use><\/g><\/g><\/g><\/g><\/svg><svg data-labels=\"true\" preserveAspectRatio=\"xMaxYMid\" viewBox=\"0 -1361 1 2222\"><g data-labels=\"true\" transform=\"matrix(1 0 0 -1 0 0)\"><g data-mml-node=\"mtd\" id=\"mjx-eqn:\" transform=\"translate(0,19)\"><text data-id-align=\"true\"><\/text><g data-idbox=\"true\" transform=\"translate(0,0)\"><g data-mml-node=\"mrow\"><\/g><\/g><\/g><\/g><\/svg><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"block\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable displaystyle=\"true\"><mlabeledtr><mtd><mrow><\/mrow><\/mtd><mtd><mo stretchy=\"false\">(<\/mo><mfrac><mrow><mi>q<\/mi><mo>\u2212<\/mo><mn>1<\/mn><\/mrow><mi>q<\/mi><\/mfrac><mo stretchy=\"false\">(<\/mo><mi>n<\/mi><mo>\u2212<\/mo><mi>k<\/mi><mo stretchy=\"false\">)<\/mo><mo>,<\/mo><mi>k<\/mi><mo>+<\/mo><mfrac><mrow><mi>q<\/mi><mo>\u2212<\/mo><mn>1<\/mn><\/mrow><mi>q<\/mi><\/mfrac><mo stretchy=\"false\">(<\/mo><mi>n<\/mi><mo>\u2212<\/mo><mi>k<\/mi><mo stretchy=\"false\">)<\/mo><mo stretchy=\"false\">)<\/mo><\/mtd><\/mlabeledtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><p><span>by carefully choosing <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"15.01ex\" height=\"2.262ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 6634.6 1000\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-1977-TEX-N-7C\" d=\"M139 -249H137Q125 -249 119 -235V251L120 737Q130 750 139 750Q152 750 159 735V-235Q151 -249 141 -249H139Z\"><\/path><path id=\"MJX-1977-TEX-B-1D431\" d=\"M227 0Q212 3 121 3Q40 3 28 0H21V62H117L245 213L109 382H26V444H34Q49 441 143 441Q247 441 265 444H274V382H246L281 339Q315 297 316 297Q320 297 354 341L389 382H352V444H360Q375 441 466 441Q547 441 559 444H566V382H471L355 246L504 63L545 62H586V0H578Q563 3 469 3Q365 3 347 0H338V62H366Q366 63 326 112T285 163L198 63L217 62H235V0H227Z\"><\/path><path id=\"MJX-1977-TEX-N-2208\" d=\"M84 250Q84 372 166 450T360 539Q361 539 377 539T419 540T469 540H568Q583 532 583 520Q583 511 570 501L466 500Q355 499 329 494Q280 482 242 458T183 409T147 354T129 306T124 272V270H568Q583 262 583 250T568 230H124V228Q124 207 134 177T167 112T231 48T328 7Q355 1 466 0H570Q583 -10 583 -20Q583 -32 568 -40H471Q464 -40 446 -40T417 -41Q262 -41 172 45Q84 127 84 250Z\"><\/path><path id=\"MJX-1977-TEX-N-7B\" d=\"M434 -231Q434 -244 428 -250H410Q281 -250 230 -184Q225 -177 222 -172T217 -161T213 -148T211 -133T210 -111T209 -84T209 -47T209 0Q209 21 209 53Q208 142 204 153Q203 154 203 155Q189 191 153 211T82 231Q71 231 68 234T65 250T68 266T82 269Q116 269 152 289T203 345Q208 356 208 377T209 529V579Q209 634 215 656T244 698Q270 724 324 740Q361 748 377 749Q379 749 390 749T408 750H428Q434 744 434 732Q434 719 431 716Q429 713 415 713Q362 710 332 689T296 647Q291 634 291 499V417Q291 370 288 353T271 314Q240 271 184 255L170 250L184 245Q202 239 220 230T262 196T290 137Q291 131 291 1Q291 -134 296 -147Q306 -174 339 -192T415 -213Q429 -213 431 -216Q434 -219 434 -231Z\"><\/path><path id=\"MJX-1977-TEX-N-30\" d=\"M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z\"><\/path><path id=\"MJX-1977-TEX-N-2C\" d=\"M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z\"><\/path><path id=\"MJX-1977-TEX-N-2026\" d=\"M78 60Q78 84 95 102T138 120Q162 120 180 104T199 61Q199 36 182 18T139 0T96 17T78 60ZM525 60Q525 84 542 102T585 120Q609 120 627 104T646 61Q646 36 629 18T586 0T543 17T525 60ZM972 60Q972 84 989 102T1032 120Q1056 120 1074 104T1093 61Q1093 36 1076 18T1033 0T990 17T972 60Z\"><\/path><path id=\"MJX-1977-TEX-I-1D458\" d=\"M121 647Q121 657 125 670T137 683Q138 683 209 688T282 694Q294 694 294 686Q294 679 244 477Q194 279 194 272Q213 282 223 291Q247 309 292 354T362 415Q402 442 438 442Q468 442 485 423T503 369Q503 344 496 327T477 302T456 291T438 288Q418 288 406 299T394 328Q394 353 410 369T442 390L458 393Q446 405 434 405H430Q398 402 367 380T294 316T228 255Q230 254 243 252T267 246T293 238T320 224T342 206T359 180T365 147Q365 130 360 106T354 66Q354 26 381 26Q429 26 459 145Q461 153 479 153H483Q499 153 499 144Q499 139 496 130Q455 -11 378 -11Q333 -11 305 15T277 90Q277 108 280 121T283 145Q283 167 269 183T234 206T200 217T182 220H180Q168 178 159 139T145 81T136 44T129 20T122 7T111 -2Q98 -11 83 -11Q66 -11 57 -1T48 16Q48 26 85 176T158 471L195 616Q196 629 188 632T149 637H144Q134 637 131 637T124 640T121 647Z\"><\/path><path id=\"MJX-1977-TEX-N-7D\" d=\"M65 731Q65 745 68 747T88 750Q171 750 216 725T279 670Q288 649 289 635T291 501Q292 362 293 357Q306 312 345 291T417 269Q428 269 431 266T434 250T431 234T417 231Q380 231 345 210T298 157Q293 143 292 121T291 -28V-79Q291 -134 285 -156T256 -198Q202 -250 89 -250Q71 -250 68 -247T65 -230Q65 -224 65 -223T66 -218T69 -214T77 -213Q91 -213 108 -210T146 -200T183 -177T207 -139Q208 -134 209 3L210 139Q223 196 280 230Q315 247 330 250Q305 257 280 270Q225 304 212 352L210 362L209 498Q208 635 207 640Q195 680 154 696T77 713Q68 713 67 716T65 731Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mo\" transform=\"translate(0 -0.5)\"><use data-c=\"7C\" xlink:href=\"#MJX-1977-TEX-N-7C\"><\/use><\/g><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\" transform=\"translate(278,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D431\" xlink:href=\"#MJX-1977-TEX-B-1D431\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(885,0) translate(0 -0.5)\"><use data-c=\"7C\" xlink:href=\"#MJX-1977-TEX-N-7C\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(1440.8,0)\"><use data-c=\"2208\" xlink:href=\"#MJX-1977-TEX-N-2208\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2385.6,0)\"><use data-c=\"7B\" xlink:href=\"#MJX-1977-TEX-N-7B\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(2885.6,0)\"><use data-c=\"30\" xlink:href=\"#MJX-1977-TEX-N-30\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(3385.6,0)\"><use data-c=\"2C\" xlink:href=\"#MJX-1977-TEX-N-2C\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(3830.2,0)\"><use data-c=\"2026\" xlink:href=\"#MJX-1977-TEX-N-2026\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(5168.9,0)\"><use data-c=\"2C\" xlink:href=\"#MJX-1977-TEX-N-2C\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(5613.6,0)\"><use data-c=\"1D458\" xlink:href=\"#MJX-1977-TEX-I-1D458\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(6134.6,0)\"><use data-c=\"7D\" xlink:href=\"#MJX-1977-TEX-N-7D\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">x<\/mi><\/mrow><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mo>\u2208<\/mo><mo fence=\"false\" stretchy=\"false\">{<\/mo><mn>0<\/mn><mo>,<\/mo><mo>\u2026<\/mo><mo>,<\/mo><mi>k<\/mi><mo fence=\"false\" stretchy=\"false\">}<\/mo><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">|\\mathbf x|\\in\\{0,\\dots,k\\}<\/script><span>. This explains why <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"3.079ex\" height=\"1.57ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -694 1361 694\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: 0px;\"><defs><path id=\"MJX-875-TEX-SS-1D5A3\" d=\"M88 0V694H237H258H316Q383 694 425 686T511 648Q578 604 622 525T666 343Q666 190 564 86Q494 18 400 3Q387 1 237 0H88ZM565 341Q565 409 546 463T495 550T429 600T359 621Q348 623 267 623H189V71H267Q272 71 286 71T310 70Q461 70 527 184Q565 251 565 341Z\"><\/path><path id=\"MJX-875-TEX-SS-1D5AF\" d=\"M88 0V694H230Q347 693 370 692T410 686Q487 667 535 611T583 485Q583 409 527 348T379 276Q369 274 279 274H192V0H88ZM486 485Q486 523 471 551T432 593T391 612T357 621Q350 622 268 623H189V347H268Q350 348 357 349Q370 351 383 354T416 368T450 391T475 429T486 485Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D5A3\" xlink:href=\"#MJX-875-TEX-SS-1D5A3\"><\/use><use data-c=\"1D5AF\" xlink:href=\"#MJX-875-TEX-SS-1D5AF\" transform=\"translate(722,0)\"><\/use><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"sans-serif\">DP<\/mi><\/mrow><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">\\mathsf{DP}<\/script><span> is easy to solve in this interval.<\/span><\/p><p><strong><span>Precise Analysis:<\/span><\/strong><\/p><p><span>....<\/span><\/p><h3 id='dumers-approach-collision-search'><span>Dumer&#39;s approach: collision search<\/span><\/h3><p><span>Dumer takes advantage of the birthday paradox to improve the search.<\/span><\/p><p><strong><span>Lemma 3.2.1.<\/span><\/strong><span> Let <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"2.549ex\" height=\"1.934ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -705 1126.6 855\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.339ex;\"><defs><path id=\"MJX-1979-TEX-C-4C\" d=\"M62 -22T47 -22T32 -11Q32 -1 56 24T83 55Q113 96 138 172T180 320T234 473T323 609Q364 649 419 677T531 705Q559 705 578 696T604 671T615 645T618 623V611Q618 582 615 571T598 548Q581 531 558 520T518 509Q503 509 503 520Q503 523 505 536T507 560Q507 590 494 610T452 630Q423 630 410 617Q367 578 333 492T271 301T233 170Q211 123 204 112L198 103L224 102Q281 102 369 79T509 52H523Q535 64 544 87T579 128Q616 152 641 152Q656 152 656 142Q656 101 588 40T433 -22Q381 -22 289 1T156 28L141 29L131 20Q111 0 87 -11Z\"><\/path><path id=\"MJX-1979-TEX-N-31\" d=\"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"msub\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"4C\" xlink:href=\"#MJX-1979-TEX-C-4C\"><\/use><\/g><\/g><g data-mml-node=\"mn\" transform=\"translate(723,-150) scale(0.707)\"><use data-c=\"31\" xlink:href=\"#MJX-1979-TEX-N-31\"><\/use><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><msub><mrow data-mjx-texclass=\"ORD\"><mi data-mjx-variant=\"-tex-calligraphic\" mathvariant=\"script\">L<\/mi><\/mrow><mn>1<\/mn><\/msub><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">\\mathcal L_1<\/script><span>, <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"2.549ex\" height=\"1.934ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -705 1126.6 855\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.339ex;\"><defs><path id=\"MJX-1981-TEX-C-4C\" d=\"M62 -22T47 -22T32 -11Q32 -1 56 24T83 55Q113 96 138 172T180 320T234 473T323 609Q364 649 419 677T531 705Q559 705 578 696T604 671T615 645T618 623V611Q618 582 615 571T598 548Q581 531 558 520T518 509Q503 509 503 520Q503 523 505 536T507 560Q507 590 494 610T452 630Q423 630 410 617Q367 578 333 492T271 301T233 170Q211 123 204 112L198 103L224 102Q281 102 369 79T509 52H523Q535 64 544 87T579 128Q616 152 641 152Q656 152 656 142Q656 101 588 40T433 -22Q381 -22 289 1T156 28L141 29L131 20Q111 0 87 -11Z\"><\/path><path id=\"MJX-1981-TEX-N-32\" d=\"M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"msub\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"4C\" xlink:href=\"#MJX-1981-TEX-C-4C\"><\/use><\/g><\/g><g data-mml-node=\"mn\" transform=\"translate(723,-150) scale(0.707)\"><use data-c=\"32\" xlink:href=\"#MJX-1981-TEX-N-32\"><\/use><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><msub><mrow data-mjx-texclass=\"ORD\"><mi data-mjx-variant=\"-tex-calligraphic\" mathvariant=\"script\">L<\/mi><\/mrow><mn>2<\/mn><\/msub><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">\\mathcal L_2<\/script><span> be two lists of <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.541ex\" height=\"1.545ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -683 681 683\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: 0px;\"><defs><path id=\"MJX-1983-TEX-I-1D43F\" d=\"M228 637Q194 637 192 641Q191 643 191 649Q191 673 202 682Q204 683 217 683Q271 680 344 680Q485 680 506 683H518Q524 677 524 674T522 656Q517 641 513 637H475Q406 636 394 628Q387 624 380 600T313 336Q297 271 279 198T252 88L243 52Q243 48 252 48T311 46H328Q360 46 379 47T428 54T478 72T522 106T564 161Q580 191 594 228T611 270Q616 273 628 273H641Q647 264 647 262T627 203T583 83T557 9Q555 4 553 3T537 0T494 -1Q483 -1 418 -1T294 0H116Q32 0 32 10Q32 17 34 24Q39 43 44 45Q48 46 59 46H65Q92 46 125 49Q139 52 144 61Q147 65 216 339T285 628Q285 635 228 637Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D43F\" xlink:href=\"#MJX-1983-TEX-I-1D43F\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>L<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">L<\/script><span> random and independent elements in <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"2.306ex\" height=\"2.414ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -683 1019.3 1067.2\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.869ex;\"><defs><path id=\"MJX-1985-TEX-D-1D53D\" d=\"M584 499Q569 490 566 490Q558 490 552 497T546 515Q546 535 533 559Q526 574 506 593T469 621Q415 648 326 648Q293 648 287 647T275 641Q264 630 263 617Q262 609 260 492V370L275 372Q323 376 350 392T393 441Q409 473 409 506Q409 529 427 529Q437 529 442 519Q444 511 444 362Q444 212 442 206Q436 197 426 197Q409 197 409 217Q409 265 375 299Q346 328 280 335H260V206Q260 70 262 63Q265 46 276 41T326 35Q362 35 366 28Q377 17 366 3L360 -1H24Q12 5 12 16Q12 35 51 35Q92 38 97 52Q102 60 102 341T97 632Q91 645 51 648Q12 648 12 666Q12 675 24 683H573Q576 678 584 670V499ZM137 341Q137 131 136 89T130 37Q129 36 129 35H182Q233 35 233 39Q226 54 225 92T224 346L226 623L231 635L235 648H129Q132 641 133 638T135 603T137 517T137 341ZM549 603V648H495L506 641Q531 621 533 619L549 603ZM409 317V395L400 386Q390 376 375 366L357 355L373 346Q394 331 397 328L409 317Z\"><\/path><path id=\"MJX-1985-TEX-I-1D45F\" d=\"M21 287Q22 290 23 295T28 317T38 348T53 381T73 411T99 433T132 442Q161 442 183 430T214 408T225 388Q227 382 228 382T236 389Q284 441 347 441H350Q398 441 422 400Q430 381 430 363Q430 333 417 315T391 292T366 288Q346 288 334 299T322 328Q322 376 378 392Q356 405 342 405Q286 405 239 331Q229 315 224 298T190 165Q156 25 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 114 189T154 366Q154 405 128 405Q107 405 92 377T68 316T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><path id=\"MJX-1985-TEX-I-1D45E\" d=\"M33 157Q33 258 109 349T280 441Q340 441 372 389Q373 390 377 395T388 406T404 418Q438 442 450 442Q454 442 457 439T460 434Q460 425 391 149Q320 -135 320 -139Q320 -147 365 -148H390Q396 -156 396 -157T393 -175Q389 -188 383 -194H370Q339 -192 262 -192Q234 -192 211 -192T174 -192T157 -193Q143 -193 143 -185Q143 -182 145 -170Q149 -154 152 -151T172 -148Q220 -148 230 -141Q238 -136 258 -53T279 32Q279 33 272 29Q224 -10 172 -10Q117 -10 75 30T33 157ZM352 326Q329 405 277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 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type=\"math\/tex\">\\mathbb F_q^r<\/script><span>, we have<\/span><\/p><div contenteditable=\"false\" spellcheck=\"false\" class=\"mathjax-block md-end-block md-math-block md-rawblock\" id=\"mathjax-n431\" cid=\"n431\" mdtype=\"math_block\" data-math-tag-before=\"10\" data-math-labels=\"[]\" data-math-tag-after=\"10\"><div class=\"md-rawblock-container md-math-container\" contenteditable=\"false\" tabindex=\"-1\"><mjx-container class=\"MathJax\" jax=\"SVG\" display=\"true\" width=\"full\" style=\"min-width: 22.142ex; position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"100%\" height=\"5.407ex\" role=\"img\" focusable=\"false\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -2.138ex; min-width: 22.142ex;\"><defs><path id=\"MJX-2077-TEX-SS-1D5A4\" d=\"M86 0V691H541V611H366L190 612V397H513V321H190V85H372L554 86V0H86Z\"><\/path><path id=\"MJX-2077-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-2077-TEX-N-7C\" d=\"M139 -249H137Q125 -249 119 -235V251L120 737Q130 750 139 750Q152 750 159 735V-235Q151 -249 141 -249H139Z\"><\/path><path id=\"MJX-2077-TEX-C-4C\" d=\"M62 -22T47 -22T32 -11Q32 -1 56 24T83 55Q113 96 138 172T180 320T234 473T323 609Q364 649 419 677T531 705Q559 705 578 696T604 671T615 645T618 623V611Q618 582 615 571T598 548Q581 531 558 520T518 509Q503 509 503 520Q503 523 505 536T507 560Q507 590 494 610T452 630Q423 630 410 617Q367 578 333 492T271 301T233 170Q211 123 204 112L198 103L224 102Q281 102 369 79T509 52H523Q535 64 544 87T579 128Q616 152 641 152Q656 152 656 142Q656 101 588 40T433 -22Q381 -22 289 1T156 28L141 29L131 20Q111 0 87 -11Z\"><\/path><path id=\"MJX-2077-TEX-N-31\" d=\"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 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378 392Q356 405 342 405Q286 405 239 331Q229 315 224 298T190 165Q156 25 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 114 189T154 366Q154 405 128 405Q107 405 92 377T68 316T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(0.018929,-0.018929) translate(0, -1445)\"><g data-mml-node=\"math\"><g data-mml-node=\"mtable\" transform=\"translate(800,0) translate(-800,0)\"><g transform=\"translate(0 1445) matrix(1 0 0 -1 0 0) scale(52.83)\"><svg data-table=\"true\" preserveAspectRatio=\"xMidYMid\" viewBox=\"4093.3 -1445 1 2389.9\"><g transform=\"matrix(1 0 0 -1 0 0)\"><g data-mml-node=\"mlabeledtr\" transform=\"translate(0,-65)\"><g data-mml-node=\"mtd\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D5A4\" xlink:href=\"#MJX-2077-TEX-SS-1D5A4\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(597,0)\"><use data-c=\"28\" 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xlink:href=\"#MJX-2077-TEX-N-32\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(4628.6,0) translate(0 -0.5)\"><use data-c=\"7C\" xlink:href=\"#MJX-2077-TEX-N-7C\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(4906.6,0)\"><use data-c=\"29\" xlink:href=\"#MJX-2077-TEX-N-29\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(5573.3,0)\"><use data-c=\"3D\" xlink:href=\"#MJX-2077-TEX-N-3D\"><\/use><\/g><g data-mml-node=\"mfrac\" transform=\"translate(6629.1,0)\"><g data-mml-node=\"msup\" transform=\"translate(220,676)\"><g data-mml-node=\"mi\"><use data-c=\"1D43F\" xlink:href=\"#MJX-2077-TEX-I-1D43F\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(714,363) scale(0.707)\"><use data-c=\"32\" xlink:href=\"#MJX-2077-TEX-N-32\"><\/use><\/g><\/g><g data-mml-node=\"msup\" transform=\"translate(322.5,-686)\"><g data-mml-node=\"mi\"><use data-c=\"1D45E\" xlink:href=\"#MJX-2077-TEX-I-1D45E\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(543.7,289) scale(0.707)\"><use data-c=\"1D45F\" xlink:href=\"#MJX-2077-TEX-I-1D45F\"><\/use><\/g><\/g><rect width=\"1317.6\" height=\"60\" x=\"120\" y=\"220\"><\/rect><\/g><\/g><\/g><\/g><\/svg><svg data-labels=\"true\" preserveAspectRatio=\"xMaxYMid\" viewBox=\"0 -1445 1 2389.9\"><g data-labels=\"true\" transform=\"matrix(1 0 0 -1 0 0)\"><g data-mml-node=\"mtd\" id=\"mjx-eqn:\" transform=\"translate(0,-65)\"><text data-id-align=\"true\"><\/text><g data-idbox=\"true\" transform=\"translate(0,0)\"><g data-mml-node=\"mrow\"><\/g><\/g><\/g><\/g><\/svg><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"block\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable displaystyle=\"true\"><mlabeledtr><mtd><mrow><\/mrow><\/mtd><mtd><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"sans-serif\">E<\/mi><\/mrow><mo stretchy=\"false\">(<\/mo><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><msub><mrow data-mjx-texclass=\"ORD\"><mi data-mjx-variant=\"-tex-calligraphic\" mathvariant=\"script\">L<\/mi><\/mrow><mn>1<\/mn><\/msub><mo>\u2229<\/mo><msub><mrow data-mjx-texclass=\"ORD\"><mi data-mjx-variant=\"-tex-calligraphic\" mathvariant=\"script\">L<\/mi><\/mrow><mn>2<\/mn><\/msub><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mfrac><msup><mi>L<\/mi><mn>2<\/mn><\/msup><msup><mi>q<\/mi><mi>r<\/mi><\/msup><\/mfrac><\/mtd><\/mlabeledtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><p><span>We expect one element in the intersection of the two lists when the list size is <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"8.552ex\" height=\"2.398ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -831.3 3780.2 1060\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.517ex;\"><defs><path id=\"MJX-2080-TEX-I-1D43F\" d=\"M228 637Q194 637 192 641Q191 643 191 649Q191 673 202 682Q204 683 217 683Q271 680 344 680Q485 680 506 683H518Q524 677 524 674T522 656Q517 641 513 637H475Q406 636 394 628Q387 624 380 600T313 336Q297 271 279 198T252 88L243 52Q243 48 252 48T311 46H328Q360 46 379 47T428 54T478 72T522 106T564 161Q580 191 594 228T611 270Q616 273 628 273H641Q647 264 647 262T627 203T583 83T557 9Q555 4 553 3T537 0T494 -1Q483 -1 418 -1T294 0H116Q32 0 32 10Q32 17 34 24Q39 43 44 45Q48 46 59 46H65Q92 46 125 49Q139 52 144 61Q147 65 216 339T285 628Q285 635 228 637Z\"><\/path><path id=\"MJX-2080-TEX-N-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path id=\"MJX-2080-TEX-N-221A\" d=\"M95 178Q89 178 81 186T72 200T103 230T169 280T207 309Q209 311 212 311H213Q219 311 227 294T281 177Q300 134 312 108L397 -77Q398 -77 501 136T707 565T814 786Q820 800 834 800Q841 800 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288Q346 288 334 299T322 328Q322 376 378 392Q356 405 342 405Q286 405 239 331Q229 315 224 298T190 165Q156 25 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 114 189T154 366Q154 405 128 405Q107 405 92 377T68 316T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D43F\" xlink:href=\"#MJX-2080-TEX-I-1D43F\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(958.8,0)\"><use data-c=\"3D\" xlink:href=\"#MJX-2080-TEX-N-3D\"><\/use><\/g><g data-mml-node=\"msqrt\" transform=\"translate(2014.6,0)\"><g transform=\"translate(853,0)\"><g data-mml-node=\"msup\"><g data-mml-node=\"mi\"><use data-c=\"1D45E\" xlink:href=\"#MJX-2080-TEX-I-1D45E\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(543.7,289) scale(0.707)\"><use data-c=\"1D45F\" xlink:href=\"#MJX-2080-TEX-I-1D45F\"><\/use><\/g><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(0,-28.7)\"><use data-c=\"221A\" xlink:href=\"#MJX-2080-TEX-N-221A\"><\/use><\/g><rect width=\"912.6\" height=\"60\" x=\"853\" y=\"711.3\"><\/rect><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>L<\/mi><mo>=<\/mo><msqrt><msup><mi>q<\/mi><mi>r<\/mi><\/msup><\/msqrt><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">L=\\sqrt{q^r}<\/script><span>.<\/span><\/p><p>&nbsp;<\/p><\/div><\/div>\n<\/body>\n<\/html>\n","protected":false},"excerpt":{"rendered":"<p>paper reading 1. Code-based Cryptography: Lecture Notes<a class=\"more-link\" href=\"https:\/\/blog.mhrooz.xyz\/index.php\/2023\/11\/19\/guan_yu_code-based_cryptography_de_lun_wen_bi_ji\/\">\u7ee7\u7eed\u9605\u8bfb<span class=\"screen-reader-text\">&#8220;\u5173\u4e8ecode-based cryptography\u7684\u8bba\u6587\u7b14\u8bb0&#8221;<\/span><\/a><\/p>\n","protected":false},"author":3,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":[],"categories":[49],"tags":[],"aioseo_notices":[],"_links":{"self":[{"href":"https:\/\/blog.mhrooz.xyz\/index.php\/wp-json\/wp\/v2\/posts\/2914"}],"collection":[{"href":"https:\/\/blog.mhrooz.xyz\/index.php\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/blog.mhrooz.xyz\/index.php\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/blog.mhrooz.xyz\/index.php\/wp-json\/wp\/v2\/users\/3"}],"replies":[{"embeddable":true,"href":"https:\/\/blog.mhrooz.xyz\/index.php\/wp-json\/wp\/v2\/comments?post=2914"}],"version-history":[{"count":1,"href":"https:\/\/blog.mhrooz.xyz\/index.php\/wp-json\/wp\/v2\/posts\/2914\/revisions"}],"predecessor-version":[{"id":2915,"href":"https:\/\/blog.mhrooz.xyz\/index.php\/wp-json\/wp\/v2\/posts\/2914\/revisions\/2915"}],"wp:attachment":[{"href":"https:\/\/blog.mhrooz.xyz\/index.php\/wp-json\/wp\/v2\/media?parent=2914"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/blog.mhrooz.xyz\/index.php\/wp-json\/wp\/v2\/categories?post=2914"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/blog.mhrooz.xyz\/index.php\/wp-json\/wp\/v2\/tags?post=2914"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}