{"id":8740,"date":"2026-04-30T18:30:36","date_gmt":"2026-04-30T09:30:36","guid":{"rendered":"https:\/\/qard.is.tohoku.ac.jp\/T-Wave\/?p=8740"},"modified":"2026-04-30T18:30:37","modified_gmt":"2026-04-30T09:30:37","slug":"bposd-2-bp%e3%83%87%e3%82%b3%e3%83%bc%e3%83%87%e3%82%a3%e3%83%b3%e3%82%b0","status":"publish","type":"post","link":"https:\/\/qard.is.tohoku.ac.jp\/T-Wave\/2026\/04\/30\/bposd-2-bp%e3%83%87%e3%82%b3%e3%83%bc%e3%83%87%e3%82%a3%e3%83%b3%e3%82%b0\/","title":{"rendered":"BP+OSD #2 BP\u30c7\u30b3\u30fc\u30c7\u30a3\u30f3\u30b0"},"content":{"rendered":"\n<script>\nwindow.MathJax = {\n  tex: {\n    inlineMath: [['$', '$'], ['\\\\(', '\\\\)']],\n    packages: {'[+]': ['ams']},\n    tags: 'ams'\n  }\n};\n<\/script>\n\n\n<div class=\"taxonomy-post_tag wp-block-post-terms\"><a href=\"https:\/\/qard.is.tohoku.ac.jp\/T-Wave\/tag\/%e9%87%8f%e5%ad%90%e3%82%b3%e3%83%b3%e3%83%94%e3%83%a5%e3%83%bc%e3%82%bf\/\" rel=\"tag\">\u91cf\u5b50\u30b3\u30f3\u30d4\u30e5\u30fc\u30bf<\/a><span class=\"wp-block-post-terms__separator\">, <\/span><a href=\"https:\/\/qard.is.tohoku.ac.jp\/T-Wave\/tag\/%e9%87%8f%e5%ad%90%e8%aa%a4%e3%82%8a%e8%a8%82%e6%ad%a3\/\" rel=\"tag\">\u91cf\u5b50\u8aa4\u308a\u8a02\u6b63<\/a><\/div>\n\n\n<div id=\"ez-toc-container\" class=\"ez-toc-v2_0_82_2 counter-hierarchy ez-toc-counter ez-toc-white ez-toc-container-direction\">\n<div class=\"ez-toc-title-container\">\n<p class=\"ez-toc-title ez-toc-toggle\" style=\"cursor:pointer\">Table of Contents<\/p>\n<span class=\"ez-toc-title-toggle\"><a href=\"#\" class=\"ez-toc-pull-right ez-toc-btn ez-toc-btn-xs ez-toc-btn-default ez-toc-toggle\" aria-label=\"Toggle Table of Content\"><span class=\"ez-toc-js-icon-con\"><span class=\"\"><span class=\"eztoc-hide\" style=\"display:none;\">Toggle<\/span><span class=\"ez-toc-icon-toggle-span\"><svg style=\"fill: #999;color:#999\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" class=\"list-377408\" width=\"20px\" height=\"20px\" viewBox=\"0 0 24 24\" fill=\"none\"><path d=\"M6 6H4v2h2V6zm14 0H8v2h12V6zM4 11h2v2H4v-2zm16 0H8v2h12v-2zM4 16h2v2H4v-2zm16 0H8v2h12v-2z\" fill=\"currentColor\"><\/path><\/svg><svg style=\"fill: #999;color:#999\" class=\"arrow-unsorted-368013\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"10px\" height=\"10px\" viewBox=\"0 0 24 24\" version=\"1.2\" baseProfile=\"tiny\"><path d=\"M18.2 9.3l-6.2-6.3-6.2 6.3c-.2.2-.3.4-.3.7s.1.5.3.7c.2.2.4.3.7.3h11c.3 0 .5-.1.7-.3.2-.2.3-.5.3-.7s-.1-.5-.3-.7zM5.8 14.7l6.2 6.3 6.2-6.3c.2-.2.3-.5.3-.7s-.1-.5-.3-.7c-.2-.2-.4-.3-.7-.3h-11c-.3 0-.5.1-.7.3-.2.2-.3.5-.3.7s.1.5.3.7z\"\/><\/svg><\/span><\/span><\/span><\/a><\/span><\/div>\n<nav><ul class='ez-toc-list ez-toc-list-level-1 ' ><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-1\" href=\"https:\/\/qard.is.tohoku.ac.jp\/T-Wave\/2026\/04\/30\/bposd-2-bp%e3%83%87%e3%82%b3%e3%83%bc%e3%83%87%e3%82%a3%e3%83%b3%e3%82%b0\/#%E6%96%87%E7%8C%AE%E6%83%85%E5%A0%B1\" >\u6587\u732e\u60c5\u5831<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-2\" href=\"https:\/\/qard.is.tohoku.ac.jp\/T-Wave\/2026\/04\/30\/bposd-2-bp%e3%83%87%e3%82%b3%e3%83%bc%e3%83%87%e3%82%a3%e3%83%b3%e3%82%b0\/#%E6%A6%82%E8%A6%81\" >\u6982\u8981<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-3\" href=\"https:\/\/qard.is.tohoku.ac.jp\/T-Wave\/2026\/04\/30\/bposd-2-bp%e3%83%87%e3%82%b3%e3%83%bc%e3%83%87%e3%82%a3%e3%83%b3%e3%82%b0\/#%E5%BE%A9%E7%BF%92%EF%BC%9A%E9%87%8F%E5%AD%90%E8%AA%A4%E3%82%8A%E8%A8%82%E6%AD%A3%E7%AC%A6%E5%8F%B7%E3%81%AE%E4%BB%95%E7%B5%84%E3%81%BF\" >\u5fa9\u7fd2\uff1a\u91cf\u5b50\u8aa4\u308a\u8a02\u6b63\u7b26\u53f7\u306e\u4ed5\u7d44\u307f<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-4\" href=\"https:\/\/qard.is.tohoku.ac.jp\/T-Wave\/2026\/04\/30\/bposd-2-bp%e3%83%87%e3%82%b3%e3%83%bc%e3%83%87%e3%82%a3%e3%83%b3%e3%82%b0\/#%E3%83%95%E3%82%A1%E3%82%AF%E3%82%BF%E3%83%BC%E3%82%B0%E3%83%A9%E3%83%95%EF%BC%9A%E8%AA%A4%E3%82%8A%E8%A8%82%E6%AD%A3%E7%AC%A6%E5%8F%B7%E3%81%AE%E6%A7%8B%E9%80%A0%E3%82%92%E8%A1%A8%E3%81%99%E3%82%B0%E3%83%A9%E3%83%95\" >\u30d5\u30a1\u30af\u30bf\u30fc\u30b0\u30e9\u30d5\uff1a\u8aa4\u308a\u8a02\u6b63\u7b26\u53f7\u306e\u69cb\u9020\u3092\u8868\u3059\u30b0\u30e9\u30d5<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-5\" href=\"https:\/\/qard.is.tohoku.ac.jp\/T-Wave\/2026\/04\/30\/bposd-2-bp%e3%83%87%e3%82%b3%e3%83%bc%e3%83%87%e3%82%a3%e3%83%b3%e3%82%b0\/#%E3%83%95%E3%82%A1%E3%82%AF%E3%82%BF%E3%83%BC%E3%82%B0%E3%83%A9%E3%83%95%E3%81%A7%E8%A6%8B%E3%82%8B%E3%83%87%E3%82%B3%E3%83%BC%E3%83%80%E3%81%AE%E4%BB%95%E7%B5%84%E3%81%BF\" >\u30d5\u30a1\u30af\u30bf\u30fc\u30b0\u30e9\u30d5\u3067\u898b\u308b\u30c7\u30b3\u30fc\u30c0\u306e\u4ed5\u7d44\u307f<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-6\" href=\"https:\/\/qard.is.tohoku.ac.jp\/T-Wave\/2026\/04\/30\/bposd-2-bp%e3%83%87%e3%82%b3%e3%83%bc%e3%83%87%e3%82%a3%e3%83%b3%e3%82%b0\/#%E7%A2%BA%E7%8E%87%E4%BC%9D%E6%92%AD%E6%B3%95%EF%BC%88BP%EF%BC%89\" >\u78ba\u7387\u4f1d\u64ad\u6cd5\uff08BP\uff09<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-7\" href=\"https:\/\/qard.is.tohoku.ac.jp\/T-Wave\/2026\/04\/30\/bposd-2-bp%e3%83%87%e3%82%b3%e3%83%bc%e3%83%87%e3%82%a3%e3%83%b3%e3%82%b0\/#LLR-MSA%EF%BC%9ABP%E3%83%87%E3%82%B3%E3%83%BC%E3%83%80%E3%81%AE%E5%8A%B9%E7%8E%87%E7%9A%84%E3%81%AA%E5%AE%9F%E8%A3%85\" >LLR-MSA\uff1aBP\u30c7\u30b3\u30fc\u30c0\u306e\u52b9\u7387\u7684\u306a\u5b9f\u88c5<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-8\" href=\"https:\/\/qard.is.tohoku.ac.jp\/T-Wave\/2026\/04\/30\/bposd-2-bp%e3%83%87%e3%82%b3%e3%83%bc%e3%83%87%e3%82%a3%e3%83%b3%e3%82%b0\/#BP%E3%81%A0%E3%81%91%E3%81%A7%E3%81%AF%E3%83%80%E3%83%A1%E3%81%AA%E3%81%AE%E3%81%8B%EF%BC%9F\" >BP\u3060\u3051\u3067\u306f\u30c0\u30e1\u306a\u306e\u304b\uff1f<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-9\" href=\"https:\/\/qard.is.tohoku.ac.jp\/T-Wave\/2026\/04\/30\/bposd-2-bp%e3%83%87%e3%82%b3%e3%83%bc%e3%83%87%e3%82%a3%e3%83%b3%e3%82%b0\/#%E3%81%BE%E3%81%A8%E3%82%81\" >\u307e\u3068\u3081<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-10\" href=\"https:\/\/qard.is.tohoku.ac.jp\/T-Wave\/2026\/04\/30\/bposd-2-bp%e3%83%87%e3%82%b3%e3%83%bc%e3%83%87%e3%82%a3%e3%83%b3%e3%82%b0\/#Appendix_A_Min-Sum%E3%81%AE%E6%9B%B4%E6%96%B0%E5%BC%8F%E3%81%AE%E8%A7%A3%E9%87%88\" >Appendix A: Min-Sum\u306e\u66f4\u65b0\u5f0f\u306e\u89e3\u91c8<\/a><ul class='ez-toc-list-level-3' ><li class='ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-11\" href=\"https:\/\/qard.is.tohoku.ac.jp\/T-Wave\/2026\/04\/30\/bposd-2-bp%e3%83%87%e3%82%b3%e3%83%bc%e3%83%87%e3%82%a3%e3%83%b3%e3%82%b0\/#%E2%91%A0_prod_mathrmsignm_vto_u\" >\u2460 $\\prod \\mathrm{sign}(m_{v&#8217;\\to u})$<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-12\" href=\"https:\/\/qard.is.tohoku.ac.jp\/T-Wave\/2026\/04\/30\/bposd-2-bp%e3%83%87%e3%82%b3%e3%83%bc%e3%83%87%e3%82%a3%e3%83%b3%e3%82%b0\/#%E2%91%A1_-1s_i\" >\u2461 $(-1)^{s_i}$<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-13\" href=\"https:\/\/qard.is.tohoku.ac.jp\/T-Wave\/2026\/04\/30\/bposd-2-bp%e3%83%87%e3%82%b3%e3%83%bc%e3%83%87%e3%82%a3%e3%83%b3%e3%82%b0\/#%E2%91%A2_alpha\" >\u2462 $\\alpha$<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-14\" href=\"https:\/\/qard.is.tohoku.ac.jp\/T-Wave\/2026\/04\/30\/bposd-2-bp%e3%83%87%e3%82%b3%e3%83%bc%e3%83%87%e3%82%a3%e3%83%b3%e3%82%b0\/#%E2%91%A3_minm_vto_u\" >\u2463 $\\min|m_{v\\to u}|$<\/a><\/li><\/ul><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-15\" href=\"https:\/\/qard.is.tohoku.ac.jp\/T-Wave\/2026\/04\/30\/bposd-2-bp%e3%83%87%e3%82%b3%e3%83%bc%e3%83%87%e3%82%a3%e3%83%b3%e3%82%b0\/#%E6%9C%AC%E8%A8%98%E4%BA%8B%E3%81%AE%E6%8B%85%E5%BD%93%E8%80%85\" >\u672c\u8a18\u4e8b\u306e\u62c5\u5f53\u8005<\/a><\/li><\/ul><\/nav><\/div>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"%E6%96%87%E7%8C%AE%E6%83%85%E5%A0%B1\"><\/span>\u6587\u732e\u60c5\u5831<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<ul class=\"wp-block-list\">\n<li>\u30bf\u30a4\u30c8\u30eb\uff1a<a href=\"https:\/\/quantum-journal.org\/papers\/q-2021-11-22-585\/#\">Degenerate Quantum LDPC Codes With Good Finite Length Performance<\/a><\/li>\n\n\n\n<li>\u8457\u8005\uff1aPavel Panteleev&nbsp;and&nbsp;Gleb Kalachev<\/li>\n\n\n\n<li>\u66f8\u8a8c\u60c5\u5831(DOI)\uff1a<a href=\"https:\/\/doi.org\/10.22331\/q-2021-11-22-585\">https:\/\/doi.org\/10.22331\/q-2021-11-22-585<\/a><\/li>\n\n\n\n<li>\u51fa\u7248\u5e74\uff1a2021\u5e7411\u6708<\/li>\n<\/ul>\n\n\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"%E6%A6%82%E8%A6%81\"><\/span>\u6982\u8981<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p>\u524d\u56de\u306b\u5f15\u304d\u7d9a\u304d\u3001qLDPC\u7b26\u53f7\u306e\u30c7\u30b3\u30fc\u30c0\u3067\u3042\u308bBP+OSD\u30c7\u30b3\u30fc\u30c0\u306b\u3064\u3044\u3066\u89e3\u8aac\u3057\u307e\u3059\u3002\u524d\u56de\u306e\u8a18\u4e8b\u3067\u306f\u3001\u91cf\u5b50\u8aa4\u308a\u8a02\u6b63\u7b26\u53f7\u306e\u57fa\u790e\u3092\u7c21\u5358\u306b\u7d39\u4ecb\u3057\u307e\u3057\u305f\u3002\u4e2d\u3067\u3082\u3001\u30b9\u30bf\u30d3\u30e9\u30a4\u30b6\u7b26\u53f7\u3068\u305d\u308c\u3092\u8868\u73fe\u3059\u308b\u30d1\u30ea\u30c6\u30a3\u30c1\u30a7\u30c3\u30af\u884c\u5217\u306f\u4eca\u56de\u306e\u8a18\u4e8b\u3067\u3082\u4f55\u5ea6\u3082\u767b\u5834\u3059\u308b\u6982\u5ff5\u306a\u306e\u3067\u3001\u308f\u304b\u3089\u306a\u304f\u306a\u3063\u305f\u3089\u524d\u56de\u306e\u8a18\u4e8b\u3092\u53c2\u7167\u3057\u3066\u304f\u3060\u3055\u3044\u3002<br>\u4eca\u56de\u306e\u8a18\u4e8b\u3067\u306f\u3001BP+OSD\u30c7\u30b3\u30fc\u30c0\u306e\u5fc3\u81d3\u90e8\u3067\u3042\u308b\u78ba\u7387\u4f1d\u64ad\u6cd5 (Belief Propagation, \u4ee5\u4e0b\u300cBP\u300d)\u3092\u4f7f\u3063\u305f\u5fa9\u53f7\u30a2\u30eb\u30b4\u30ea\u30ba\u30e0\u306b\u3064\u3044\u3066\u89e3\u8aac\u3057\u307e\u3059\u3002\u300cBP+OSD\u300d\u306e\u300cBP\u300d\u306e\u90e8\u5206\u3067\u3059\u3002<\/p>\n\n\n\n<p>\u3000\u30c7\u30b3\u30fc\u30c0\u306b\u3064\u3044\u3066\u8aac\u660e\u3059\u308b\u524d\u306b\u3001\u91cf\u5b50\u8aa4\u308a\u8a02\u6b63\u7b26\u53f7\u304c\u30a8\u30e9\u30fc\u3092\u3069\u306e\u3088\u3046\u306b\u691c\u51fa\u3059\u308b\u306e\u304b\u3092\u5fa9\u7fd2\u3057\u306a\u304c\u3089\u3001\u305d\u306e\u4ed5\u7d44\u307f\u3092\u30b0\u30e9\u30d5\u3092\u4f7f\u3063\u3066\u8868\u73fe\u3059\u308b\u65b9\u6cd5\u306b\u3064\u3044\u3066\u8aac\u660e\u3057\u307e\u3059\u3002<br><\/p>\n\n\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"%E5%BE%A9%E7%BF%92%EF%BC%9A%E9%87%8F%E5%AD%90%E8%AA%A4%E3%82%8A%E8%A8%82%E6%AD%A3%E7%AC%A6%E5%8F%B7%E3%81%AE%E4%BB%95%E7%B5%84%E3%81%BF\"><\/span>\u5fa9\u7fd2\uff1a\u91cf\u5b50\u8aa4\u308a\u8a02\u6b63\u7b26\u53f7\u306e\u4ed5\u7d44\u307f<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p>\u3000\u524d\u56de\u306e\u8a18\u4e8b\u3067\u3001\u91cf\u5b50\u8aa4\u308a\u8a02\u6b63\u7b26\u53f7\u306e\u57fa\u790e\u306b\u3064\u3044\u3066\u7d39\u4ecb\u3057\u307e\u3057\u305f\u3002\u305d\u306e\u4e2d\u304b\u3089\u91cd\u8981\u306a\u90e8\u5206\u3092\u629c\u7c8b\u3057\u3066\u3001\u91cf\u5b50\u8aa4\u308a\u8a02\u6b63\u7b26\u53f7\u304c\u30a8\u30e9\u30fc\u3092\u691c\u51fa\u3059\u308b\u4ed5\u7d44\u307f\u306b\u3064\u3044\u3066\u5fa9\u7fd2\u3057\u307e\u3059\u3002<\/p>\n\n\n\n<p>\u3000\u30b9\u30bf\u30d3\u30e9\u30a4\u30b6\u7b26\u53f7\u306f\u3001\u6700\u3082\u57fa\u672c\u7684\u306a\u91cf\u5b50\u8aa4\u308a\u8a02\u6b63\u7b26\u53f7\u3067\u3059\u3002\u7b26\u53f7\u5316\u3055\u308c\u305f\u72b6\u614b\u306b\u5bfe\u3057\u3001\u30b9\u30bf\u30d3\u30e9\u30a4\u30b6\u3068\u547c\u3070\u308c\u308b\u7269\u7406\u91cf\u306e\u6e2c\u5b9a\u3067\u30b7\u30f3\u30c9\u30ed\u30fc\u30e0\u5024\u3068\u547c\u3070\u308c\u308b\u5024\u3092\u53d6\u5f97\u3059\u308b\u3053\u3068\u3067\u3057\u3001\u305d\u308c\u306b\u3088\u3063\u3066\u30a8\u30e9\u30fc\u3092\u691c\u51fa\u3057\u307e\u3059\u3002\u30a8\u30e9\u30fc\u3068\u30b7\u30f3\u30c9\u30ed\u30fc\u30e0\u5024\u306e\u95a2\u4fc2\u3092\u7aef\u7684\u306b\u8868\u73fe\u3057\u305f\u3082\u306e\u304c\u30d1\u30ea\u30c6\u30a3\u30c1\u30a7\u30c3\u30af\u884c\u5217\u3067\u3057\u305f\u3002<\/p>\n\n\n\n<p>\u3000\u30d1\u30ea\u30c6\u30a3\u30c1\u30a7\u30c3\u30af\u884c\u5217\u306f\u3001\u6b21\u306e\u3088\u3046\u306a\u30d0\u30a4\u30ca\u30ea\u884c\u5217\u3067\u3057\u305f\u3002<br>$$\\begin{align}H:=\\left(\\begin{array}{c|c} 01100 &amp; 10010\\\\ 00110&amp;01001\\\\ 00011&amp;10100\\\\10001&amp;01010 \\end{array}\\right)\\end{align}$$<br>\u30d1\u30ea\u30c6\u30a3\u30c1\u30a7\u30c3\u30af\u884c\u5217\u306e\u5404\u884c\u304c\u30b9\u30bf\u30d3\u30e9\u30a4\u30b6\u306b\u5bfe\u5fdc\u3057\u3066\u3044\u3066\u3001\u5404\u5217\u306f\u30a8\u30e9\u30fc\u306b\u5bfe\u5fdc\u3057\u3066\u3044\u307e\u3059\u3002\u7e26\u7dda\u306e\u5de6\u5074\u304cZ\u30a8\u30e9\u30fc\u3092\u691c\u51fa\u3057\u3001\u53f3\u5074\u304cX\u30a8\u30e9\u30fc\u3092\u691c\u51fa\u3057\u307e\u3059\u3002<\/p>\n\n\n\n<p>\u3000\u30d1\u30ea\u30c6\u30a3\u30c1\u30a7\u30c3\u30af\u884c\u5217\u306b\u5bfe\u3057\u3066\u3001\u30d1\u30a6\u30ea\u30a8\u30e9\u30fc\uff08\u30d1\u30a6\u30ea\u6f14\u7b97\u5b50\u306b\u3088\u3063\u3066\u8868\u73fe\u3055\u308c\u308b\u30a8\u30e9\u30fc\uff09$E$\u3092\u30d0\u30a4\u30ca\u30ea\u30b7\u30f3\u30d7\u30ec\u30af\u30c6\u30a3\u30c3\u30af\u8868\u73fe\u3067\u8868\u3057\u305f\u30d9\u30af\u30c8\u30eb $v_E$ \u3092\u304b\u3051\u308b\u3068\u3001<br>$$ \\begin{align}Hv_E=\\begin{pmatrix}v_{M_1} \\odot v_E \\\\ \\vdots \\\\ v_{M_r} \\odot v_E\\end{pmatrix} =\\begin{pmatrix} s_1 \\\\ \\vdots \\\\ s_r \\end{pmatrix}\\end{align} $$<br>\u306e\u3088\u3046\u306b\u3057\u3066\u30b7\u30f3\u30c9\u30ed\u30fc\u30e0\u5024 $s_i$\u304c\u5f97\u3089\u308c\u308b\u306e\u3067\u3057\u305f\u3002\u3053\u308c\u304c\u3001\u30a8\u30e9\u30fc\u304b\u3089\u30b7\u30f3\u30c9\u30ed\u30fc\u30e0\u5024\u304c\u6c7a\u307e\u308b\u4ed5\u7d44\u307f\u3067\u3059\u3002<\/p>\n\n\n\n<p>\u3000\u30d1\u30ea\u30c6\u30a3\u30c1\u30a7\u30c3\u30af\u884c\u5217\u3092\u7528\u3044\u305f\u8868\u73fe\u3067\u3001\u30a8\u30e9\u30fc\u3068\u30b7\u30f3\u30c9\u30ed\u30fc\u30e0\u5024\u306e\u95a2\u4fc2\u306f\u660e\u78ba\u306b\u306a\u3063\u3066\u3044\u307e\u3059\u304c\u3001\u3088\u308a\u308f\u304b\u308a\u6613\u3044\u8868\u73fe\u304c\u3042\u308a\u307e\u3059\u3002\u305d\u308c\u304c\u3001\u30d5\u30a1\u30af\u30bf\u30fc\u30b0\u30e9\u30d5\u3068\u547c\u3070\u308c\u308b\u30b0\u30e9\u30d5\u3092\u4f7f\u3063\u305f\u8868\u73fe\u3067\u3059\u3002\u30d5\u30a1\u30af\u30bf\u30fc\u30b0\u30e9\u30d5\u306f\u30d1\u30ea\u30c6\u30a3\u30c1\u30a7\u30c3\u30af\u884c\u5217\u3068\u4e00\u5bfe\u4e00\u306b\u5bfe\u5fdc\u3057\u3066\u3044\u3066\u3001\u8aa4\u308a\u8a02\u6b63\u7b26\u53f7\u306e\u69cb\u9020\u3092\u8868\u73fe\u3057\u3066\u3044\u307e\u3059\u3002<\/p>\n\n\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"%E3%83%95%E3%82%A1%E3%82%AF%E3%82%BF%E3%83%BC%E3%82%B0%E3%83%A9%E3%83%95%EF%BC%9A%E8%AA%A4%E3%82%8A%E8%A8%82%E6%AD%A3%E7%AC%A6%E5%8F%B7%E3%81%AE%E6%A7%8B%E9%80%A0%E3%82%92%E8%A1%A8%E3%81%99%E3%82%B0%E3%83%A9%E3%83%95\"><\/span>\u30d5\u30a1\u30af\u30bf\u30fc\u30b0\u30e9\u30d5\uff1a\u8aa4\u308a\u8a02\u6b63\u7b26\u53f7\u306e\u69cb\u9020\u3092\u8868\u3059\u30b0\u30e9\u30d5<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p>\u3000\u30d5\u30a1\u30af\u30bf\u30fc\u30b0\u30e9\u30d5\u306f\u8aa4\u308a\u8a02\u6b63\u7b26\u53f7\u304c\u6301\u3063\u3066\u3044\u308b\u69cb\u9020\u3092\u57cb\u3081\u8fbc\u3093\u3060\u30b0\u30e9\u30d5\u3067\u3059\u3002\u3088\u308a\u6b63\u78ba\u306b\u306f\u3001\u30d1\u30ea\u30c6\u30a3\u30c1\u30a7\u30c3\u30af\u884c\u5217\u3092\u30b0\u30e9\u30d5\u3067\u8868\u73fe\u3057\u305f\u3082\u306e\u3067\u3059\u3002\u30b0\u30e9\u30d5\u7406\u8ad6\u306e\u8a00\u8449\u3092\u4f7f\u3046\u3068\u3001\u30d1\u30ea\u30c6\u30a3\u30c1\u30a7\u30c3\u30af\u884c\u5217\u3092\u96a3\u63a5\u884c\u5217\u3068\u3059\u308b\u30b0\u30e9\u30d5\u306e\u3053\u3068\u3067\u3059\u3002\u3053\u3053\u3067\u306f\u3001\u30b0\u30e9\u30d5\u306b\u99b4\u67d3\u307f\u306e\u306a\u3044\u65b9\u5411\u3051\u306b\u3001\u30d1\u30ea\u30c6\u30a3\u30c1\u30a7\u30c3\u30af\u884c\u5217\u304b\u3089\u30b0\u30e9\u30d5\u304c\u69cb\u6210\u3055\u308c\u308b\u69d8\u5b50\u3092\u89b3\u5bdf\u3057\u307e\u3059\u3002\u4e8c\u90e8\u30b0\u30e9\u30d5\u3084\u96a3\u63a5\u884c\u5217\u306b\u99b4\u67d3\u307f\u304c\u3042\u308b\u65b9\u306f\u3001\u30b0\u30e9\u30d5\u306b\u95a2\u3059\u308b\u8a18\u6cd5\u3060\u3051\u78ba\u8a8d\u3057\u3066\u5148\u306b\u9032\u3093\u3067\u3082\u5927\u4e08\u592b\u3067\u3059\u3002<\/p>\n\n\n\n<p>\u3000\u7c21\u5358\u306a\u5177\u4f53\u4f8b\u3092\u901a\u3057\u3066\u3001\u30d5\u30a1\u30af\u30bf\u30fc\u30b0\u30e9\u30d5\u3092\u69cb\u6210\u3057\u3066\u3044\u304d\u307e\u3059\u3002\u3053\u3053\u3067\u306f\u8a71\u3092\u7c21\u5358\u306b\u3059\u308b\u305f\u3081\u306b\u3001\u30b9\u30bf\u30d3\u30e9\u30a4\u30b6\u751f\u6210\u5143\u306f $I,Z$\u306e\u30c6\u30f3\u30bd\u30eb\u7a4d\u3067\u69cb\u6210\u3055\u308c\u3066\u3044\u308b\u3082\u306e\u3068\u3057\u307e\u3059\u3002\u3053\u306e\u3068\u304d\u3001\u30d1\u30ea\u30c6\u30a3\u30c1\u30a7\u30c3\u30af\u884c\u5217\u306f<br>$$ \\begin{align}H=\\left(\\begin{array}{c|c}O&amp;H_Z\\end{array}\\right)\\end{align} $$<br>\u3068\u3044\u3046\u5f62\u3092\u3057\u3066\u3044\u307e\u3059\u3002\u3053\u306e\u3068\u304d\u306f\u3001$H$\u306e\u6210\u5206\u306e\u5de6\u534a\u5206\u306f\u306a\u306b\u3082\u306a\u3044\u306e\u3067\u3001\u4ee3\u308f\u308a\u306b $H_Z$\u3092\u8003\u3048\u308c\u3070\u5341\u5206\u3067\u3059<sup data-fn=\"3b90fa1b-4bc1-4064-af22-e0d656deb491\" class=\"fn\"><a href=\"#3b90fa1b-4bc1-4064-af22-e0d656deb491\" id=\"3b90fa1b-4bc1-4064-af22-e0d656deb491-link\">1<\/a><\/sup>\u3002<\/p>\n\n\n\n<p>\u3000\u5177\u4f53\u4f8b\u3068\u3057\u3066\u3001\u884c\u5217 $H_Z$\u3092\u6b21\u306e\u3088\u3046\u306b\u5b9a\u3081\u307e\u3059\u3002<br>$$ \\begin{align}H_Z=\\begin{pmatrix}1&amp;1&amp;0\\\\0&amp;1&amp;1\\end{pmatrix}\\end{align} $$<br>\u3053\u308c\u306f\u7e70\u308a\u8fd4\u3057\u7b26\u53f7\u3068\u547c\u3070\u308c\u308b\u7b26\u53f7\u306e\u30d1\u30ea\u30c6\u30a3\u30c1\u30a7\u30c3\u30af\u884c\u5217\u3067\u3001\u6700\u3082\u7c21\u5358\u306a\u7b26\u53f7\u306e\u3072\u3068\u3064\u3067\u3059\u3002\u3053\u306e\u3068\u304d\u3001\u30b9\u30bf\u30d3\u30e9\u30a4\u30b6\u306e\u751f\u6210\u5143\u306f<br>$$ \\begin{align}(0\\;0\\;0\\;|\\;1\\;1\\;0) \\to ZZI \\quad (=M_1) \\\\(0\\;0\\;0\\;|\\;0\\;1\\;1) \\to IZZ \\quad (=M_2) \\end{align} $$<br>\u3068\u306a\u308a\u307e\u3059\u3002<\/p>\n\n\n\n<p>\u3000\u7e70\u308a\u8fd4\u3057\u7b26\u53f7\u304c\u3069\u306e\u3088\u3046\u306b\u30a8\u30e9\u30fc\u3092\u691c\u51fa\u3059\u308b\u306e\u304b\u3001\u9806\u3092\u8ffd\u3063\u3066\u898b\u3066\u3044\u304d\u307e\u3059\u3002\u30d1\u30ea\u30c6\u30a3\u30c1\u30a7\u30c3\u30af\u884c\u5217\u306b\u3064\u3044\u3066\u306e\u8b70\u8ad6\u304b\u3089\u3001\u30a8\u30e9\u30fc $E$\u306b\u3064\u3044\u3066\u306eBS\u8868\u73fe $v_E$\u3068\u30b7\u30f3\u30c9\u30ed\u30fc\u30e0\u5024 $s$\u306b\u3064\u3044\u3066\u3001<br>$$ \\begin{align}Hv_E=\\begin{pmatrix}v_{M_1} \\odot v_E \\\\ v_{M_2} \\odot v_E \\end{pmatrix} = \\begin{pmatrix}s_1\\\\s_2\\end{pmatrix}=s\\end{align} $$<br>\u304c\u6210\u308a\u7acb\u3061\u307e\u3059\u3002\u3053\u3053\u3067\u3001 $v_E = (x_1\\;x_2\\;x_3\\;|\\;z_1\\;z_2\\;z_3) \\in \\mathbb Z_2^3\\times \\mathbb Z_2^3$\u3068\u304a\u304f\u3068\u3001\u30b7\u30f3\u30c9\u30ed\u30fc\u30e0\u5024$s$\u306f<br>$$ \\begin{align}s_1=v_{M_1}\\odot v_E= x_1\\oplus x_2 \\\\ s_2 = v_{M_2}\\odot v_E=x_2\\oplus x_3\\end{align} $$<br>\u3068\u306a\u308a\u307e\u3059\u3002\u3053\u3053\u3067\u3001 $\\oplus$\u306fmod 2\u3067\u306e\u548c\uff08\u6392\u4ed6\u7684\u8ad6\u7406\u548c\u3001XOR\uff09\u3067\u3059\u3002\u3064\u307e\u308a\u3001\u30b7\u30f3\u30c9\u30ed\u30fc\u30e0\u5024 $s_1, s_2$\u306f $x_1,x_2,x_3$\u304b\u3089\u6c42\u307e\u308b\u3068\u3044\u3046\u3053\u3068\u304c\u308f\u304b\u308a\u307e\u3059\u3002\u30a8\u30e9\u30fc\u304c\u306a\u3044\u3068\u304d\u306f\u30b7\u30f3\u30c9\u30ed\u30fc\u30e0\u5024\u304c0\u306b\u306a\u308b\u306e\u3067\u3001$s_1, s_2$\u304c\u4e21\u65b9\u3068\u30820\u3067\u306a\u3051\u308c\u3070\u30a8\u30e9\u30fc\u304c\u691c\u51fa\u3055\u308c\u305f\u3068\u3044\u3046\u3053\u3068\u306b\u306a\u308a\u307e\u3059\u3002<\/p>\n\n\n\n<p>\u3000\u3053\u306e\u3088\u3046\u306a\u5236\u7d04\u5f0f\u306e\u3053\u3068\u3092<strong>\u30d1\u30ea\u30c6\u30a3\u691c\u67fb\u6761\u4ef6<\/strong>\u3068\u547c\u3073\u307e\u3059\u3002\u30d1\u30ea\u30c6\u30a3\u691c\u67fb\u6761\u4ef6\u3053\u305d\u304c\u8aa4\u308a\u8a02\u6b63\u7b26\u53f7\u306e\u69cb\u9020\u3092\u76f4\u63a5\u7684\u306b\u8868\u73fe\u3057\u3066\u3044\u307e\u3059\u3002\u8aa4\u308a\u8a02\u6b63\u7b26\u53f7\u306e\u7a7a\u9593\u306f\u3001\u30d1\u30ea\u30c6\u30a3\u691c\u67fb\u6761\u4ef6\u306e\u89e3\u7a7a\u9593\u3068\u3057\u3066\u5b9a\u7fa9\u3055\u308c\u307e\u3059\u3002<br>\u3000\u30d1\u30ea\u30c6\u30a3\u691c\u67fb\u6761\u4ef6\u3092\u30b0\u30e9\u30d5\u3092\u4f7f\u3063\u3066\u8996\u899a\u5316\u3057\u307e\u3059\u3002\u8996\u899a\u5316\u306e\u65b9\u6cd5\u306f\u30011\u3064\u306e\u5236\u7d04\u5f0f\u30921\u3064\u306e\u56db\u89d2\u3044\u30ce\u30fc\u30c9\u3067\u8868\u3057\u3066\u3001\u5236\u7d04\u5f0f\u306b\u73fe\u308c\u308b\u5909\u6570\u3092\u4e38\u3044\u30ce\u30fc\u30c9\u3067\u8868\u3057\u3066\u3001\u305d\u306e\u9593\u3092\u8fba\u3067\u7d50\u3073\u307e\u3059\u3002\u5177\u4f53\u4f8b\u3068\u3057\u3066\u5f0f(8)\u3092\u8003\u3048\u308b\u3068\u3001\u6b21\u306e\u3088\u3046\u306a\u30b0\u30e9\u30d5\u306b\u306a\u308a\u307e\u3059\u3002<\/p>\n\n\n\n<figure class=\"wp-block-image size-large\"><img loading=\"lazy\" decoding=\"async\" width=\"1024\" height=\"546\" src=\"https:\/\/qard.is.tohoku.ac.jp\/T-Wave\/wp-content\/uploads\/2026\/03\/parity_variable-1024x546.png\" alt=\"\" class=\"wp-image-8763\" srcset=\"https:\/\/qard.is.tohoku.ac.jp\/T-Wave\/wp-content\/uploads\/2026\/03\/parity_variable-1024x546.png 1024w, https:\/\/qard.is.tohoku.ac.jp\/T-Wave\/wp-content\/uploads\/2026\/03\/parity_variable-300x160.png 300w, https:\/\/qard.is.tohoku.ac.jp\/T-Wave\/wp-content\/uploads\/2026\/03\/parity_variable-768x410.png 768w, https:\/\/qard.is.tohoku.ac.jp\/T-Wave\/wp-content\/uploads\/2026\/03\/parity_variable.png 1500w\" sizes=\"auto, (max-width: 1024px) 100vw, 1024px\" \/><figcaption class=\"wp-element-caption\">\u56f31 \u5f0f(8)\u306e\u30d1\u30ea\u30c6\u30a3\u691c\u67fb\u6761\u4ef6<\/figcaption><\/figure>\n\n\n\n<p>\u30d1\u30ea\u30c6\u30a3\u691c\u67fb\u6761\u4ef6\u306b\u5bfe\u5fdc\u3059\u308b\u56db\u89d2\u3044\u30ce\u30fc\u30c9\u3092<strong>\u30d1\u30ea\u30c6\u30a3\u30ce\u30fc\u30c9<\/strong>\u3068\u547c\u3073\u3001\u5909\u6570\u306b\u5bfe\u5fdc\u3059\u308b\u4e38\u3044\u30ce\u30fc\u30c9\u3092<strong>\u5909\u6570\u30ce\u30fc\u30c9<\/strong>\u3068\u547c\u3073\u307e\u3059\u3002\u30b0\u30e9\u30d5\u306e\u8fba\u306f\u30d1\u30ea\u30c6\u30a3\u30ce\u30fc\u30c9\u3068\u5909\u6570\u30ce\u30fc\u30c9\u306e\u9593\u306b\u3060\u3051\u5b58\u5728\u3057\u3066\u3001\u30d1\u30ea\u30c6\u30a3\u691c\u67fb\u6761\u4ef6\u306b\u542b\u307e\u308c\u308b\u5909\u6570\u3068\u63a5\u7d9a\u3057\u3066\u3044\u307e\u3059\u3002<\/p>\n\n\n\n<p>\u3000\u91cf\u5b50\u8aa4\u308a\u8a02\u6b63\u306e\u6587\u8108\u3067\u306f\u3001\u5909\u6570\u30ce\u30fc\u30c9\u3068\u91cf\u5b50\u30d3\u30c3\u30c8\u304c\u4e00\u5bfe\u4e00\u306b\u5bfe\u5fdc\u3057\u3066\u3044\u307e\u3059\u3002\u306a\u305c\u306a\u3089\u3001\u5909\u6570\u30ce\u30fc\u30c9\u306f\u7279\u5b9a\u306e\u4f4d\u7f6e\u306e\u30a8\u30e9\u30fc\u306e\u6709\u7121\u3092\u8868\u3057\u3066\u3044\u3066\u3001\u30a8\u30e9\u30fc\u306e\u4f4d\u7f6e\u3068\u91cf\u5b50\u30d3\u30c3\u30c8\u306f\u4e00\u5bfe\u4e00\u306b\u5bfe\u5fdc\u3059\u308b\u304b\u3089\u3067\u3059<sup data-fn=\"dcd6f7e9-e6f8-4fad-9191-b0535b7e2807\" class=\"fn\"><a href=\"#dcd6f7e9-e6f8-4fad-9191-b0535b7e2807\" id=\"dcd6f7e9-e6f8-4fad-9191-b0535b7e2807-link\">2<\/a><\/sup>\u3002<br><br>\u540c\u3058\u65b9\u6cd5\u3067\u5f0f2\u3064\u76ee\u306e\u5236\u7d04\u5f0f(9)\u3082\u30b0\u30e9\u30d5\u306b\u3057\u3066\u3001\u30d1\u30ea\u30c6\u30a3\u691c\u67fb\u6761\u4ef6\u5168\u4f53\u3092\u30b0\u30e9\u30d5\u306b\u3059\u308b\u3068\u6b21\u306e\u3088\u3046\u306b\u306a\u308a\u307e\u3059\u3002<br><\/p>\n\n\n\n<figure class=\"wp-block-image size-large\"><img loading=\"lazy\" decoding=\"async\" width=\"1024\" height=\"546\" src=\"https:\/\/qard.is.tohoku.ac.jp\/T-Wave\/wp-content\/uploads\/2026\/03\/tanne_graph-1-1024x546.png\" alt=\"\" class=\"wp-image-8765\" srcset=\"https:\/\/qard.is.tohoku.ac.jp\/T-Wave\/wp-content\/uploads\/2026\/03\/tanne_graph-1-1024x546.png 1024w, https:\/\/qard.is.tohoku.ac.jp\/T-Wave\/wp-content\/uploads\/2026\/03\/tanne_graph-1-300x160.png 300w, https:\/\/qard.is.tohoku.ac.jp\/T-Wave\/wp-content\/uploads\/2026\/03\/tanne_graph-1-768x410.png 768w, https:\/\/qard.is.tohoku.ac.jp\/T-Wave\/wp-content\/uploads\/2026\/03\/tanne_graph-1.png 1500w\" sizes=\"auto, (max-width: 1024px) 100vw, 1024px\" \/><figcaption class=\"wp-element-caption\">\u56f32 \u5f0f(8), (9)\u306e\u30d1\u30ea\u30c6\u30a3\u691c\u67fb\u6761\u4ef6<\/figcaption><\/figure>\n\n\n\n<p>\u3053\u306e\u30b0\u30e9\u30d5\u306b\u306f\u30d1\u30ea\u30c6\u30a3\u30c1\u30a7\u30c3\u30af\u884c\u5217 $H$ \u3067\u6307\u5b9a\u3055\u308c\u308b\u7b26\u53f7\u306e\u3059\u3079\u3066\u306e\u30d1\u30ea\u30c6\u30a3\u691c\u67fb\u6761\u4ef6\u304c\u57cb\u3081\u8fbc\u307e\u308c\u3066\u3044\u307e\u3059\u3002\u3053\u306e\u3088\u3046\u306a\u30b0\u30e9\u30d5\u3092\u3001\u30d1\u30ea\u30c6\u30a3\u30c1\u30a7\u30c3\u30af\u884c\u5217 $H$ \u306b\u5bfe\u5fdc\u3059\u308b\u7b26\u53f7\u306e<strong>\u30d5\u30a1\u30af\u30bf\u30fc\u30b0\u30e9\u30d5<\/strong>\u3001\u3042\u308b\u3044\u306f\u30bf\u30ca\u30fc\u30b0\u30e9\u30d5\u3068\u547c\u3073\u307e\u3059\u3002<\/p>\n\n\n\n<p>\u3000\u30bf\u30ca\u30fc\u30b0\u30e9\u30d5\u3092\u4f7f\u3046\u3068\u30a8\u30e9\u30fc\u3068\u30b7\u30f3\u30c9\u30ed\u30fc\u30e0\u5024\u306e\u95a2\u4fc2\u304c\u8996\u899a\u7684\u306b\u308f\u304b\u308b\u3053\u3068\u3092\u78ba\u8a8d\u3057\u307e\u3059\u3002 $i$\u756a\u76ee\u306e\u91cf\u5b50\u30d3\u30c3\u30c8\u306bX\u30a8\u30e9\u30fc\u304c\u8d77\u304d\u305f\u3068\u304d $x_i=1$\u3068\u306a\u308a\u3001\u305d\u3046\u3067\u306a\u3044\u3068\u304d\u306f0\u306b\u306a\u308a\u307e\u3059\u3002\u5f93\u3063\u3066\u3001 2\u756a\u76ee\u30683\u756a\u76ee\u306e\u91cf\u5b50\u30d3\u30c3\u30c8\u306bX\u30a8\u30e9\u30fc\u304c\u8d77\u304d\u305f\u5834\u5408\u3001\u30d5\u30a1\u30af\u30bf\u30fc\u30b0\u30e9\u30d5\u306f\u6b21\u306e\u3088\u3046\u306b\u306a\u308a\u307e\u3059\u3002<br><\/p>\n\n\n\n<figure class=\"wp-block-image size-large\"><img loading=\"lazy\" decoding=\"async\" width=\"1024\" height=\"546\" src=\"https:\/\/qard.is.tohoku.ac.jp\/T-Wave\/wp-content\/uploads\/2026\/03\/tanner_graph_with_error-1024x546.png\" alt=\"\" class=\"wp-image-8766\" srcset=\"https:\/\/qard.is.tohoku.ac.jp\/T-Wave\/wp-content\/uploads\/2026\/03\/tanner_graph_with_error-1024x546.png 1024w, https:\/\/qard.is.tohoku.ac.jp\/T-Wave\/wp-content\/uploads\/2026\/03\/tanner_graph_with_error-300x160.png 300w, https:\/\/qard.is.tohoku.ac.jp\/T-Wave\/wp-content\/uploads\/2026\/03\/tanner_graph_with_error-768x410.png 768w, https:\/\/qard.is.tohoku.ac.jp\/T-Wave\/wp-content\/uploads\/2026\/03\/tanner_graph_with_error-1536x819.png 1536w, https:\/\/qard.is.tohoku.ac.jp\/T-Wave\/wp-content\/uploads\/2026\/03\/tanner_graph_with_error-2048x1092.png 2048w\" sizes=\"auto, (max-width: 1024px) 100vw, 1024px\" \/><figcaption class=\"wp-element-caption\">\u56f33 \u30d1\u30ea\u30c6\u30a3\u691c\u67fb\u306e\u69d8\u5b50<\/figcaption><\/figure>\n\n\n\n<p>\u308f\u304b\u308a\u3084\u3059\u3055\u306e\u305f\u3081\u306b\u3001\u30a8\u30e9\u30fc\u304c\u8d77\u304d\u3066\u3044\u308b\u5909\u6570\u30ce\u30fc\u30c9\u3068\u305d\u3053\u304b\u3089\u4f38\u3073\u308b\u8fba\u3092\u8d64\u304f\u3057\u3066\u3044\u307e\u3059\u3002\u3053\u306e\u30b0\u30e9\u30d5\u3092\u898b\u308b\u3068\u3001\u5909\u6570\u30ce\u30fc\u30c9\u306e\u5024\u304b\u3089\u30d1\u30ea\u30c6\u30a3\u30ce\u30fc\u30c9\u306e\u5024\u3092\u7c21\u5358\u306b\u6c42\u3081\u308b\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u3002\u30d1\u30ea\u30c6\u30a3\u306e\u5024\u306f\u3001\u8fba\u3067\u7e4b\u304c\u3063\u305f\u5909\u6570\u30ce\u30fc\u30c9\u306e\u5024\u306e\u6392\u4ed6\u7684\u8ad6\u7406\u548c\u3092\u53d6\u308c\u3070\u6c42\u307e\u308a\u307e\u3059\u3002\u3088\u308a\u5358\u7d14\u306b\u3001\u7e4b\u304c\u3063\u305f\u5909\u6570\u30ce\u30fc\u30c9\u306e\u4e2d\u30671\u306e\u6570\u3092\u6570\u3048\u3066\u3001\u5076\u6570\u500b\u306a\u3089\u30d1\u30ea\u30c6\u30a3\u306f0\u3001\u5947\u6570\u500b\u306a\u3089\u30d1\u30ea\u30c6\u30a3\u306f1\u306b\u306a\u308a\u307e\u3059\u3002<\/p>\n\n\n\n<p>\u3000\u3053\u306e\u3088\u3046\u306b\u3001\u30d5\u30a1\u30af\u30bf\u30fc\u30b0\u30e9\u30d5\u306b\u3088\u3063\u3066\u30a8\u30e9\u30fc\u304b\u3089\u30b7\u30f3\u30c9\u30ed\u30fc\u30e0\u5024\u304c\u6c7a\u5b9a\u3055\u308c\u308b\u69cb\u9020\u3092\u8996\u899a\u7684\u306b\u8868\u73fe\u3059\u308b\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u3002\u305d\u308c\u3060\u3051\u3067\u306f\u306a\u304f\u3001BP+OSD\u30c7\u30b3\u30fc\u30c0\u306e\u5fc3\u81d3\u90e8\u3067\u3042\u308bBP\u3067\u306f\u3001\u30d5\u30a1\u30af\u30bf\u30fc\u30b0\u30e9\u30d5\u4e0a\u3067\u30a2\u30eb\u30b4\u30ea\u30ba\u30e0\u304c\u5b9f\u884c\u3055\u308c\u307e\u3059\u3002<\/p>\n\n\n\n<p>\u3053\u308c\u4ee5\u964d\u306e\u8aac\u660e\u3067\u306f\u30d5\u30a1\u30af\u30bf\u30fc\u30b0\u30e9\u30d5\u306e\u9802\u70b9\u306b\u3064\u3044\u3066\u6b21\u306e\u3088\u3046\u306a\u8868\u8a18\u3092\u4f7f\u3044\u307e\u3059\u3002<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>\u5909\u6570\u30ce\u30fc\u30c9\u306f $v_i$<\/li>\n\n\n\n<li>\u30d1\u30ea\u30c6\u30a3\u30ce\u30fc\u30c9\u306f $u_j$<\/li>\n\n\n\n<li>\u5909\u6570\u30ce\u30fc\u30c9 $v_i$ \u3068\u63a5\u7d9a\u3057\u3066\u3044\u308b\u30d1\u30ea\u30c6\u30a3\u30ce\u30fc\u30c9\u306e\u96c6\u5408\u3092 $U(v_i)$<\/li>\n\n\n\n<li>\u30d1\u30ea\u30c6\u30a3\u30ce\u30fc\u30c9 $u_j$ \u3068\u63a5\u7d9a\u3057\u3066\u3044\u308b\u5909\u6570\u30ce\u30fc\u30c9\u306e\u96c6\u5408\u3092 $V(u_j)$<\/li>\n<\/ul>\n\n\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"%E3%83%95%E3%82%A1%E3%82%AF%E3%82%BF%E3%83%BC%E3%82%B0%E3%83%A9%E3%83%95%E3%81%A7%E8%A6%8B%E3%82%8B%E3%83%87%E3%82%B3%E3%83%BC%E3%83%80%E3%81%AE%E4%BB%95%E7%B5%84%E3%81%BF\"><\/span>\u30d5\u30a1\u30af\u30bf\u30fc\u30b0\u30e9\u30d5\u3067\u898b\u308b\u30c7\u30b3\u30fc\u30c0\u306e\u4ed5\u7d44\u307f<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p>\u3000\u3053\u306e\u8a18\u4e8b\u306e\u672c\u984c\u306f\u30c7\u30b3\u30fc\u30c0\u3067\u3059\u3002\u91cf\u5b50\u8aa4\u308a\u8a02\u6b63\u7b26\u53f7\u306e\u30c7\u30b3\u30fc\u30c0\u306f\u3069\u3093\u306a\u554f\u984c\u3092\u89e3\u304f\u306e\u304b\u3001\u30d5\u30a1\u30af\u30bf\u30fc\u30b0\u30e9\u30d5\u3092\u4f7f\u3063\u3066\u8996\u899a\u7684\u306b\u6574\u7406\u3057\u307e\u3059\u3002<\/p>\n\n\n\n<p>\u3000\u5148\u7a0b\u306e\u30d5\u30a1\u30af\u30bf\u30fc\u30b0\u30e9\u30d5\u306e\u8aac\u660e\u3067\u3001\u30a8\u30e9\u30fc\u304b\u3089\u30b7\u30f3\u30c9\u30ed\u30fc\u30e0\u5024\u304c\u6c7a\u307e\u308b\u3053\u3068\u3092\u8aac\u660e\u3057\u307e\u3057\u305f\u3002\u3057\u304b\u3057\u3001\u5b9f\u969b\u306b\u306f\u30a8\u30e9\u30fc\u304c\u3069\u3046\u306a\u3063\u3066\u3044\u308b\u304b\u306f\u308f\u304b\u3089\u305a\u3001\u30b7\u30f3\u30c9\u30ed\u30fc\u30e0\u5024\u3060\u3051\u304c\u5f97\u3089\u308c\u307e\u3059\u3002\u3064\u307e\u308a\u3001\u5b9f\u969b\u306b\u77e5\u308b\u3053\u3068\u304c\u3067\u304d\u308b\u306e\u306f\u6b21\u306e\u56f3\u306e\u3088\u3046\u306a\u60c5\u5831\u306e\u307f\u3067\u3059\u3002<br><\/p>\n\n\n\n<figure class=\"wp-block-image size-large\"><img loading=\"lazy\" decoding=\"async\" width=\"1024\" height=\"546\" src=\"https:\/\/qard.is.tohoku.ac.jp\/T-Wave\/wp-content\/uploads\/2026\/03\/tanner_graph_with_only_parity-1024x546.png\" alt=\"\" class=\"wp-image-8767\" srcset=\"https:\/\/qard.is.tohoku.ac.jp\/T-Wave\/wp-content\/uploads\/2026\/03\/tanner_graph_with_only_parity-1024x546.png 1024w, https:\/\/qard.is.tohoku.ac.jp\/T-Wave\/wp-content\/uploads\/2026\/03\/tanner_graph_with_only_parity-300x160.png 300w, https:\/\/qard.is.tohoku.ac.jp\/T-Wave\/wp-content\/uploads\/2026\/03\/tanner_graph_with_only_parity-768x410.png 768w, https:\/\/qard.is.tohoku.ac.jp\/T-Wave\/wp-content\/uploads\/2026\/03\/tanner_graph_with_only_parity-1536x819.png 1536w, https:\/\/qard.is.tohoku.ac.jp\/T-Wave\/wp-content\/uploads\/2026\/03\/tanner_graph_with_only_parity-2048x1092.png 2048w\" sizes=\"auto, (max-width: 1024px) 100vw, 1024px\" \/><figcaption class=\"wp-element-caption\">\u56f34 \u5b9f\u969b\u306b\u5f97\u3089\u308c\u308b\u60c5\u5831\uff08\u4e0a\u90e8\u306e\u30b7\u30f3\u30c9\u30ed\u30fc\u30e0\u5024\u3060\u3051\u304c\u308f\u304b\u308a\u3001\u4e0b\u90e8\u306e\u30a8\u30e9\u30fc\u305d\u306e\u3082\u306e\u306f\u308f\u304b\u3089\u306a\u3044\uff09<\/figcaption><\/figure>\n\n\n\n<p>\u3053\u306e\u60c5\u5831\u304b\u3089\u3001\u30a8\u30e9\u30fc\u304c\u3069\u3046\u306a\u3063\u3066\u3044\u308b\u304b\u3092\u63a8\u5b9a\u3059\u308b\u306e\u304c\u30c7\u30b3\u30fc\u30c0\u3067\u3059\u3002\u3064\u307e\u308a\u3001\u56f3\u306e\u300c\uff1f\u300d\u306e\u90e8\u5206\u3092\u63a8\u5b9a\u3057\u3066\u5f53\u3066\u306b\u884c\u304d\u307e\u3059\u3002<\/p>\n\n\n\n<p>\u3000\u30b7\u30f3\u30c9\u30ed\u30fc\u30e0\u5024\u304b\u3089\u30a8\u30e9\u30fc\u3092\u63a8\u5b9a\u3059\u308b\u3053\u3068\u306f\u7c21\u5358\u3067\u306f\u3042\u308a\u307e\u305b\u3093\u3002\u30a8\u30e9\u30fc\u304b\u3089\u30b7\u30f3\u30c9\u30ed\u30fc\u30e0\u5024\u3092\u8a08\u7b97\u3059\u308b\u3068\u304d\u306f\u5f0f\u306b\u5f53\u3066\u306f\u3081\u308b\u3060\u3051\u3067\u7c21\u5358\u306b\u8a08\u7b97\u3067\u304d\u307e\u3057\u305f\u304c\u3001\u9006\u5411\u304d\u306f\u305d\u3046\u306f\u3044\u304d\u307e\u305b\u3093\u3002\u306a\u305c\u306a\u3089\u3001\u4e00\u822c\u306b\u5909\u6570\u306e\u6570\u306f\u30d1\u30ea\u30c6\u30a3\u306e\u6570\u3088\u308a\u3082\u591a\u304f\u3001\u30b7\u30f3\u30c9\u30ed\u30fc\u30e0\u5024\u306e\u60c5\u5831\u306f\u5909\u6570\u306e\u5024\u3092\u6c7a\u5b9a\u3059\u308b\u306b\u306f\u4e0d\u5341\u5206\u3060\u304b\u3089\u3067\u3059\u3002<\/p>\n\n\n\n<p>\u3000\u305d\u3053\u3067\u3001\u78ba\u7387\u7684\u306a\u63a8\u8ad6\u3092\u3059\u308b\u3053\u3068\u306b\u306a\u308a\u307e\u3059\u3002\u5909\u6570\u306e\u5024\u306e\u78ba\u7387\u5206\u5e03\u3068\u5909\u6570\u3068\u30b7\u30f3\u30c9\u30ed\u30fc\u30e0\u5024\u306e\u9593\u306e\u6761\u4ef6\u4ed8\u304d\u78ba\u7387\u3092\u8003\u3048\u3066\u3001\u30b7\u30f3\u30c9\u30ed\u30fc\u30e0\u5024\u304c\u5f97\u3089\u308c\u305f\u3068\u304d\u306e\u5909\u6570\u306e\u5024\u306e\u6761\u4ef6\u4ed8\u304d\u78ba\u7387\u3092\u6c42\u3081\u307e\u3059\u3002\u3053\u306e\u3088\u3046\u306a\u63a8\u5b9a\u306f\u30d9\u30a4\u30ba\u63a8\u5b9a\u3068\u547c\u3070\u308c\u3001\u30d9\u30a4\u30ba\u306e\u5b9a\u7406\u3068\u547c\u3070\u308c\u308b\u6b21\u306e\u5f0f\u3092\u4f7f\u3063\u3066\u8a08\u7b97\u3057\u307e\u3059\u3002<br>$$ \\begin{align}p(e|s) = \\frac{p(s|e)}{p(s)}p(e)\\end{align} $$<br>\u3053\u3053\u3067\u3001 $s$\u306f\u30b7\u30f3\u30c9\u30ed\u30fc\u30e0\u5024\u3092\u3001 $e$\u306f\u30a8\u30e9\u30fc\u3092\u8868\u3057\u307e\u3059\u3002\u3053\u306e\u5f0f\u3067\u6ce8\u76ee\u3059\u3079\u304d\u306f\u3001\u5de6\u8fba\u304c\u6c42\u3081\u305f\u3044\u78ba\u7387\u5206\u5e03\u3067\u3001\u53f3\u8fba\u304c\u30a8\u30e9\u30fc\u306e\u78ba\u7387\u5206\u5e03 $p(e)$\u3084 \u6761\u4ef6\u4ed8\u304d\u78ba\u7387 $p(s|e)$\u306a\u3069\u6bd4\u8f03\u7684\u308f\u304b\u308a\u3084\u3059\u3044\u78ba\u7387\u5206\u5e03\u306b\u306a\u3063\u3066\u3044\u308b\u3068\u3044\u3046\u70b9\u3067\u3059\u3002\u3053\u306e\u3068\u304d\u306e\u3001 $p(e)$\u3092 $e$\u306e\u4e8b\u524d\u5206\u5e03\u3001 $p(e|s)$\u3092 $e$\u306e $s$\u304c\u5f97\u3089\u308c\u305f\u3068\u304d\u306e\u4e8b\u5f8c\u5206\u5e03\u3068\u547c\u3073\u307e\u3059\u3002\u76f4\u611f\u7684\u306b\u306f\u3001\u4e8b\u524d\u5206\u5e03\u306b\u30b7\u30f3\u30c9\u30ed\u30fc\u30e0\u5024 $s$\u306e\u60c5\u5831\u3092\u52a0\u3048\u308b\u3053\u3068\u3067\u3001\u3088\u308a\u7cbe\u5ea6\u306e\u9ad8\u3044\u30a8\u30e9\u30fc $e$\u306e\u78ba\u7387\u5206\u5e03 $p(e|s)$\u304c\u5f97\u3089\u308c\u308b\u3068\u3044\u3046\u30a4\u30e1\u30fc\u30b8\u3067\u3059\u3002<\/p>\n\n\n\n<p>\u3000\u4e8b\u5f8c\u5206\u5e03\u304c\u5f97\u3089\u308c\u308c\u3070\u3001\u524d\u56de\u306e\u8a18\u4e8b\u3067\u7d39\u4ecb\u3057\u305f\u3088\u3046\u306b\u3001\u4e8b\u5f8c\u5206\u5e03\u306e\u5024\u3092\u6700\u5927\u306b\u3059\u308b\u30a8\u30e9\u30fc\u3092\u63a8\u5b9a\u5024\u3068\u3057\u3066\u63a1\u7528\u3059\u308b\u3053\u3068\u3067\u6700\u3082\u78ba\u304b\u3089\u3057\u3044\u30a8\u30e9\u30fc\u63a8\u5b9a\u304c\u3067\u304d\u308b\u3053\u3068\u306b\u306a\u308a\u307e\u3059\u3002\u3064\u307e\u308a\u3001\u6b21\u306e\u5f0f\u306b\u3088\u3063\u3066\u30a8\u30e9\u30fc\u3092\u6c7a\u5b9a\u3057\u307e\u3059\u3002<br>$$ \\begin{align}E^* = \\arg \\max_{E\\in P_n} \\mathrm{Pr}(E \\mid s)\\end{align} $$<br>\u4ee5\u4e0a\u3092\u307e\u3068\u3081\u308b\u3068\u3001\u30c7\u30b3\u30fc\u30c0\u3068\u306f\u30b7\u30f3\u30c9\u30ed\u30fc\u30e0\u5024\u304b\u3089\u30a8\u30e9\u30fc\u3092\u63a8\u5b9a\u3059\u308b\u30d7\u30ed\u30b0\u30e9\u30e0\u3067\u3001\u305d\u308c\u3092\u5b9f\u73fe\u3059\u308b\u305f\u3081\u306b\u4e8b\u5f8c\u5206\u5e03\u3092\u6c42\u3081\u308b\u3082\u306e\u3067\u3059\u3002\u305f\u3060\u3001\u5f0f(10)\u3092\u4f7f\u3063\u3066\u7c21\u5358\u306b\u3001\u305d\u3057\u3066\u53b3\u5bc6\u306b\u4e8b\u5f8c\u5206\u5e03\u304c\u6c42\u3081\u308b\u304b\u3068\u8a00\u3046\u3068\u3001\u305d\u3093\u306a\u4e8b\u306f\u3042\u308a\u307e\u305b\u3093\u3002\u5b9f\u969b\u306b\u306f\u3001\u53f3\u8fba\u306b\u51fa\u3066\u304f\u308b $p(s|e)$\u3084 $p(s)$\u3092\u76f4\u63a5\u8a08\u7b97\u3059\u308b\u3053\u3068\u306f\u56f0\u96e3\u3067\u3059\u3002<\/p>\n\n\n\n<p>\u3000\u305d\u3053\u3067\u3001\u8fd1\u4f3c\u7684\u306a\u624b\u6cd5\u3092\u4f7f\u3044\u307e\u3059\u3002\u78ba\u7387\u4f1d\u64ad\u6cd5\u3068\u547c\u3070\u308c\u308b\u624b\u6cd5\u306f\u3001\u30d5\u30a1\u30af\u30bf\u30fc\u30b0\u30e9\u30d5\u306e\u3088\u3046\u306a\u30b0\u30e9\u30d5\u3092\u4f7f\u3063\u3066\u8868\u3055\u308c\u308b\u78ba\u7387\u30e2\u30c7\u30eb\u306b\u3064\u3044\u3066\u3001\u4e8b\u5f8c\u5206\u5e03 $p(e|s)$\u3092\u52b9\u7387\u7684\u306b\u6c42\u3081\u308b\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u3002<\/p>\n\n\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"%E7%A2%BA%E7%8E%87%E4%BC%9D%E6%92%AD%E6%B3%95%EF%BC%88BP%EF%BC%89\"><\/span>\u78ba\u7387\u4f1d\u64ad\u6cd5\uff08BP\uff09<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p>\u3000\u78ba\u7387\u4f1d\u64ad\u6cd5\u306f\u3001\u30b0\u30e9\u30d5\u3067\u8868\u73fe\u3055\u308c\u308b\u78ba\u7387\u30e2\u30c7\u30eb\u306b\u3064\u3044\u3066\u4e8b\u5f8c\u5206\u5e03 $p(e|s)$\u3092\u52b9\u7387\u7684\u306b\u6c42\u3081\u308b\u30a2\u30eb\u30b4\u30ea\u30ba\u30e0\u3067\u3059\u3002\u3053\u3053\u3067\u306f\u3001\u4e00\u822c\u7684\u306a\u78ba\u7387\u4f1d\u64ad\u6cd5\u3067\u306f\u306a\u304f\u3001\u8aa4\u308a\u8a02\u6b63\u7b26\u53f7\u306e\u30c7\u30b3\u30fc\u30c9\u306b\u4f7f\u308f\u308c\u308b\u7279\u6b8a\u306a\u78ba\u7387\u4f1d\u64ad\u6cd5\u306b\u3064\u3044\u3066\u8aac\u660e\u3057\u307e\u3059\u3002<\/p>\n\n\n\n<p>\u3000\u78ba\u7387\u4f1d\u64ad\u6cd5\u306f\u3001\u540d\u524d\u306b\u300c\u4f1d\u64ad\u300d\u3068\u3064\u3044\u3066\u3044\u308b\u3088\u3046\u306b\u3001\u60c5\u5831\u3092\u30b0\u30e9\u30d5\u4e0a\u3067\u4f1d\u64ad\u3055\u305b\u307e\u3059\u3002\u3069\u306e\u3088\u3046\u306a\u60c5\u5831\u3092\u4f1d\u64ad\u3055\u305b\u308b\u304b\u3068\u3044\u3046\u3068\u3001\u89b3\u6e2c\u6e08\u307f\u306e\u5909\u6570\u306e\u5024\u306e\u60c5\u5831\u3068\u672a\u89b3\u6e2c\u306e\u5909\u6570\u306e\u4e8b\u524d\u5206\u5e03\u306e\u60c5\u5831\u3067\u3059\u3002\u30c7\u30b3\u30fc\u30c9\u306e\u6587\u8108\u3067\u306f\u3001\u30b7\u30f3\u30c9\u30ed\u30fc\u30e0\u5024\u3068\u30a8\u30e9\u30fc\u306e\u4e8b\u524d\u5206\u5e03\u306b\u5bfe\u5fdc\u3057\u307e\u3059\u3002\u3064\u307e\u308a\u3001\u89b3\u6e2c\u3067\u5f97\u3089\u308c\u305f\u30b7\u30f3\u30c9\u30ed\u30fc\u30e0\u5024\u306e\u5024\u306e\u60c5\u5831\u3068\u30a8\u30e9\u30fc\u306e\u4e8b\u524d\u5206\u5e03\u306e\u60c5\u5831\u3092\u4f1d\u64ad\u3055\u305b\u307e\u3059\u3002<\/p>\n\n\n\n<p>\u3000\u3053\u306e\u3068\u304d\u3001\u30b7\u30f3\u30c9\u30ed\u30fc\u30e0\u5024\u306e\u60c5\u5831\u3068\u4e8b\u524d\u5206\u5e03\u306e\u60c5\u5831\u306f\u4ea4\u4e92\u306b\u4f1d\u64ad\u3055\u305b\u307e\u3059\u3002\u30a4\u30e1\u30fc\u30b8\u3068\u3057\u3066\u306f\u6b21\u306e\u56f3\u306e\u3088\u3046\u306a\u611f\u3058\u3067\u3059\u3002<br><\/p>\n\n\n\n<figure class=\"wp-block-image size-large\"><img loading=\"lazy\" decoding=\"async\" width=\"1024\" height=\"273\" src=\"https:\/\/qard.is.tohoku.ac.jp\/T-Wave\/wp-content\/uploads\/2026\/03\/bp_iteration-1024x273.png\" alt=\"\" class=\"wp-image-8768\" srcset=\"https:\/\/qard.is.tohoku.ac.jp\/T-Wave\/wp-content\/uploads\/2026\/03\/bp_iteration-1024x273.png 1024w, https:\/\/qard.is.tohoku.ac.jp\/T-Wave\/wp-content\/uploads\/2026\/03\/bp_iteration-300x80.png 300w, https:\/\/qard.is.tohoku.ac.jp\/T-Wave\/wp-content\/uploads\/2026\/03\/bp_iteration-768x205.png 768w, https:\/\/qard.is.tohoku.ac.jp\/T-Wave\/wp-content\/uploads\/2026\/03\/bp_iteration-1536x410.png 1536w, https:\/\/qard.is.tohoku.ac.jp\/T-Wave\/wp-content\/uploads\/2026\/03\/bp_iteration-2048x546.png 2048w\" sizes=\"auto, (max-width: 1024px) 100vw, 1024px\" \/><figcaption class=\"wp-element-caption\">\u56f35 \u78ba\u7387\u4f1d\u64ad\u6cd5\uff08BP\uff09\u306e\u30a4\u30e1\u30fc\u30b8<\/figcaption><\/figure>\n\n\n\n<p>\u3000\u5909\u6570\u30ce\u30fc\u30c9\u304b\u3089\u30d1\u30ea\u30c6\u30a3\u30ce\u30fc\u30c9\u3078\u4e8b\u524d\u5206\u5e03\u306e\u60c5\u5831\u3092\u4f1d\u64ad\u3055\u305b\u3001\u30d1\u30ea\u30c6\u30a3\u30ce\u30fc\u30c9\u304b\u3089\u5909\u6570\u30ce\u30fc\u30c9\u3078\u30b7\u30f3\u30c9\u30ed\u30fc\u30e0\u5024\u306e\u60c5\u5831\u3092\u4f1d\u64ad\u3055\u305b\u307e\u3059\u3002\u3053\u308c\u3092\u4ea4\u4e92\u306b\u7e70\u308a\u8fd4\u3059\u3053\u3068\u3067\u3001\u30e1\u30c3\u30bb\u30fc\u30b8\u304b\u3089\u8a08\u7b97\u3055\u308c\u308b\u5909\u6570\u30ce\u30fc\u30c9\u306e\u63a8\u5b9a\u5206\u5e03\u304c\u771f\u306e\u4e8b\u5f8c\u5206\u5e03\u306b\u8fd1\u3065\u3044\u3066\u3044\u304d\u307e\u3059\u3002<\/p>\n\n\n\n<p>\u3000\u4f1d\u64ad\u3055\u305b\u308b\u60c5\u5831\u3092\u3069\u306e\u3088\u3046\u306b\u8a08\u7b97\u3059\u308b\u306e\u304b\u3001\u6700\u7d42\u7684\u306b\u4e8b\u5f8c\u5206\u5e03\u3092\u3069\u306e\u3088\u3046\u306b\u8a08\u7b97\u3059\u308b\u306e\u304b\u306a\u3069\u306e\u5177\u4f53\u7684\u306a\u8a71\u3092\u6b21\u7bc0\u3067\u898b\u3066\u3044\u304d\u307e\u3059\u3002<\/p>\n\n\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"LLR-MSA%EF%BC%9ABP%E3%83%87%E3%82%B3%E3%83%BC%E3%83%80%E3%81%AE%E5%8A%B9%E7%8E%87%E7%9A%84%E3%81%AA%E5%AE%9F%E8%A3%85\"><\/span>LLR-MSA\uff1aBP\u30c7\u30b3\u30fc\u30c0\u306e\u52b9\u7387\u7684\u306a\u5b9f\u88c5<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p>\u3000BP\u3092\u7528\u3044\u305f\u30c7\u30b3\u30fc\u30c0\u306e\u52b9\u7387\u7684\u306a\u5b9f\u88c5\u3068\u3057\u3066\u3001<strong>\u5bfe\u6570\u5c24\u5ea6\u6bd4Min-Sum\u30a2\u30eb\u30b4\u30ea\u30ba\u30e0<\/strong>\uff08Log-likelihood ratio Min-sum algorithm,\u4ee5\u4e0b\u300c LLR-MSA\u300d\uff09\u3068\u547c\u3070\u308c\u308b\u30a2\u30eb\u30b4\u30ea\u30ba\u30e0\u304c\u77e5\u3089\u308c\u3066\u3044\u307e\u3059\u3002<\/p>\n\n\n\n<p>\u3000LLR-MSA\u306f\u3001\u30d5\u30a1\u30af\u30bf\u30fc\u30b0\u30e9\u30d5\u306e\u9802\u70b9\u9593\u306e\u30e1\u30c3\u30bb\u30fc\u30b8\u306e\u3084\u308a\u53d6\u308a\u3092\u7e70\u308a\u8fd4\u3059\u3053\u3068\u306b\u3088\u3063\u3066\u4e8b\u5f8c\u5206\u5e03\u3092\u8a08\u7b97\u3057\u307e\u3059\u3002\u30e1\u30c3\u30bb\u30fc\u30b8\u3068\u306f\u3001\u8fba\u306b\u6cbf\u3063\u3066\u9802\u70b9\u9593\u3092\u4f1d\u64ad\u3059\u308b\u5909\u6570\u3067\u3059\u3002<\/p>\n\n\n\n<p>\u3000\u5909\u6570\u30ce\u30fc\u30c9 $v_j$\u304b\u3089\u30d1\u30ea\u30c6\u30a3\u30ce\u30fc\u30c9 $u_i$\u3078\u306e\u30e1\u30c3\u30bb\u30fc\u30b8\u3092 $m_{v_j\\to u_i}$\u3068\u66f8\u304f\u3053\u3068\u306b\u3057\u307e\u3059\u3002\u3053\u306e\u3068\u304d\u3001\u30e1\u30c3\u30bb\u30fc\u30b8\u306f\u6b21\u306e\u56f3\u306e\u3088\u3046\u306b\u30d5\u30a1\u30af\u30bf\u30fc\u30b0\u30e9\u30d5\u4e0a\u3092\u4f1d\u64ad\u3057\u3066\u3044\u304d\u307e\u3059\u3002<\/p>\n\n\n\n<figure class=\"wp-block-image size-large\"><img loading=\"lazy\" decoding=\"async\" width=\"1024\" height=\"546\" src=\"https:\/\/qard.is.tohoku.ac.jp\/T-Wave\/wp-content\/uploads\/2026\/03\/bp_message_passing-1024x546.png\" alt=\"\" class=\"wp-image-8769\" srcset=\"https:\/\/qard.is.tohoku.ac.jp\/T-Wave\/wp-content\/uploads\/2026\/03\/bp_message_passing-1024x546.png 1024w, https:\/\/qard.is.tohoku.ac.jp\/T-Wave\/wp-content\/uploads\/2026\/03\/bp_message_passing-300x160.png 300w, https:\/\/qard.is.tohoku.ac.jp\/T-Wave\/wp-content\/uploads\/2026\/03\/bp_message_passing-768x410.png 768w, https:\/\/qard.is.tohoku.ac.jp\/T-Wave\/wp-content\/uploads\/2026\/03\/bp_message_passing-1536x819.png 1536w, https:\/\/qard.is.tohoku.ac.jp\/T-Wave\/wp-content\/uploads\/2026\/03\/bp_message_passing-2048x1092.png 2048w\" sizes=\"auto, (max-width: 1024px) 100vw, 1024px\" \/><figcaption class=\"wp-element-caption\">\u56f36 \u30e1\u30c3\u30bb\u30fc\u30b8\u306e\u4f1d\u64ad<\/figcaption><\/figure>\n\n\n\n<p>\u3000\u30e1\u30c3\u30bb\u30fc\u30b8\u306b\u306f\u3001\u5909\u6570\u30ce\u30fc\u30c9\u304b\u3089\u30d1\u30ea\u30c6\u30a3\u30ce\u30fc\u30c9\u3078\u4f1d\u64ad\u3059\u308b $m_{v_j\\to u_i}$\u3068\u30d1\u30ea\u30c6\u30a3\u30ce\u30fc\u30c9\u304b\u3089\u5909\u6570\u30ce\u30fc\u30c9\u3078\u4f1d\u64ad\u3059\u308b $m_{u_i\\to v_j}$\u306e2\u7a2e\u985e\u304c\u3042\u308a\u307e\u3059\u3002LLR-MSA\u306e\u5404\u53cd\u5fa9\u3067\u306f\u3053\u308c\u30892\u7a2e\u985e\u306e\u30e1\u30c3\u30bb\u30fc\u30b8\u3092\u4f1d\u64ad\u3055\u305b\u307e\u3059\u3002\u3053\u308c\u30892\u7a2e\u985e\u306e\u30e1\u30c3\u30bb\u30fc\u30b8\u304c\u3069\u306e\u3088\u3046\u306b\u8a08\u7b97\u3055\u308c\u308b\u304b\u3092\u898b\u3066\u3044\u304d\u307e\u3059\u3002<\/p>\n\n\n\n<p>\u3000\u5909\u6570\u30ce\u30fc\u30c9\u304b\u3089\u30d1\u30ea\u30c6\u30a3\u30ce\u30fc\u30c9\u3078\u306e\u30e1\u30c3\u30bb\u30fc\u30b8 $m_{v_j\\to u_i}$\u306fLLR-MSA\u3092\u8d77\u52d5\u3059\u308b\u3068\u6700\u521d\u306b\u9001\u4fe1\u3055\u308c\u308b\u30e1\u30c3\u30bb\u30fc\u30b8\u3067\u3059\u3002\u3053\u306e\u3068\u304d\u306e\u30e1\u30c3\u30bb\u30fc\u30b8\u306f\u6b21\u306e\u3088\u3046\u306b\u8a08\u7b97\u3055\u308c\u307e\u3059\u3002<br>$$ \\begin{align}m_{v_j \\to u_i} = \\log\\left(\\frac{1-p_j}{p_j}\\right)=\\log\\left(\\frac{P(e_j=0)}{P(e_j=1)}\\right)\\end{align} $$<br>\u3053\u3053\u3067\u3001 $p_j$\u306f$j$\u756a\u76ee\u306e\u91cf\u5b50\u30d3\u30c3\u30c8\u306b $X$\u30a8\u30e9\u30fc\u304c\u8d77\u3053\u308b\u78ba\u7387\u3092\u8868\u3057\u3066\u3044\u307e\u3059\u3002\u3053\u306e\u5f0f\u306f$j$\u756a\u76ee\u306e\u91cf\u5b50\u30d3\u30c3\u30c8\u306e\u30a8\u30e9\u30fc\u306e\u4e8b\u524d\u5206\u5e03\u306b\u5bfe\u5fdc\u3057\u3066\u3044\u307e\u3059\u3002$m_{v_j\\to u_i}&gt;0$\u306a\u3089$e_j=0$\u3067\u3042\u308b\u78ba\u7387\u304c\u9ad8\u304f\u3001$m_{v_j\\to u_i} &lt; 0$\u306a\u3089$e_j=1$\u3067\u3042\u308b\u78ba\u7387\u304c\u9ad8\u3044\u3053\u3068\u3092\u610f\u5473\u3057\u307e\u3059\u3002\u3053\u306e\u3088\u3046\u306b\u78ba\u7387\u306e\u6bd4\u3092\u3068\u3063\u3066\u5bfe\u6570\u3092\u3068\u3063\u305f\u5024\u306e\u3053\u3068\u3092\u5bfe\u6570\u5c24\u5ea6\u6bd4\u3068\u547c\u3073\u307e\u3059\u3002\u5f0f(12)\u3067\u8a08\u7b97\u3055\u308c\u308b\u5024\u306f\u3001\u5909\u6570\u30ce\u30fc\u30c9\u306e\u4e8b\u524d\u5206\u5e03 $p_j$\u3092\u7528\u3044\u3066\u3044\u308b\u306e\u3067\u3001\u4e8b\u524d\u5206\u5e03\u306e\u5bfe\u6570\u5c24\u5ea6\u6bd4\u3068\u3044\u3046\u3053\u3068\u306b\u306a\u308a\u307e\u3059\u3002LLR-MSA\u306e\u76ee\u6a19\u306f\u3001\u30e1\u30c3\u30bb\u30fc\u30b8\u4f1d\u64ad\u306b\u3088\u3063\u3066\u3001\u4e8b\u5f8c\u5206\u5e03\u306e\u5bfe\u6570\u5c24\u5ea6\u6bd4\u3092\u8a08\u7b97\u3059\u308b\u3053\u3068\u3067\u3059\u3002<\/p>\n\n\n\n<p>\u3000\u7d9a\u3044\u3066\u3001\u30d1\u30ea\u30c6\u30a3\u30ce\u30fc\u30c9\u304b\u3089\u5909\u6570\u30ce\u30fc\u30c9\u3078\u306e\u30e1\u30c3\u30bb\u30fc\u30b8 $m_{u_i\\to v_j}$\u304c\u66f4\u65b0\u3055\u308c\u307e\u3059\u3002\u3053\u306e\u3068\u304d\u306b\u3001\u5909\u6570\u30ce\u30fc\u30c9\u304b\u3089\u53d7\u3051\u53d6\u3063\u305f\u30e1\u30c3\u30bb\u30fc\u30b8 $m_{v_j\\to u_i}$\u3092\u7528\u3044\u3066\u3044\u308b\u3053\u3068\u306b\u6ce8\u76ee\u3057\u3066\u304f\u3060\u3055\u3044\u3002<\/p>\n\n\n\n<div class=\"wp-block-math\"><math display=\"block\"><semantics><mrow><msub><mi>m<\/mi><mrow><msub><mi>u<\/mi><mi>i<\/mi><\/msub><mo>\u2192<\/mo><msub><mi>v<\/mi><mi>j<\/mi><\/msub><\/mrow><\/msub><mo>=<\/mo><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mo form=\"prefix\" stretchy=\"false\">\u2212<\/mo><mn>1<\/mn><msup><mo form=\"postfix\" stretchy=\"false\">)<\/mo><msub><mi>s<\/mi><mi>i<\/mi><\/msub><\/msup><mi>\u03b1<\/mi><mrow><mo fence=\"true\" form=\"prefix\">(<\/mo><mrow><munder><mo movablelimits=\"false\">\u220f<\/mo><mrow><msubsup><mi>v<\/mi><mi>j<\/mi><mo lspace=\"0em\" rspace=\"0em\" class=\"tml-prime\">\u2032<\/mo><\/msubsup><mo>\u2208<\/mo><mi>V<\/mi><mo form=\"prefix\" stretchy=\"false\">(<\/mo><msub><mi>u<\/mi><mi>i<\/mi><\/msub><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mo>\u2216<\/mo><mo form=\"prefix\" stretchy=\"false\">{<\/mo><msub><mi>v<\/mi><mi>j<\/mi><\/msub><mo form=\"postfix\" stretchy=\"false\" lspace=\"0em\" rspace=\"0em\">}<\/mo><\/mrow><\/munder><\/mrow><mrow><mtext><\/mtext><mi>sign<\/mi><\/mrow><mo form=\"prefix\" stretchy=\"false\">(<\/mo><msub><mi>m<\/mi><mrow><msubsup><mi>v<\/mi><mi>j<\/mi><mo lspace=\"0em\" rspace=\"0em\" class=\"tml-prime\">\u2032<\/mo><\/msubsup><mo>\u2192<\/mo><msub><mi>u<\/mi><mi>i<\/mi><\/msub><\/mrow><\/msub><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace width=\"0.1667em\"><\/mspace><munder><mi>min<\/mi><mrow><msubsup><mi>v<\/mi><mi>j<\/mi><mo lspace=\"0em\" rspace=\"0em\" class=\"tml-prime\">\u2032<\/mo><\/msubsup><mo>\u2208<\/mo><mi>V<\/mi><mo form=\"prefix\" stretchy=\"false\">(<\/mo><msub><mi>u<\/mi><mi>i<\/mi><\/msub><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mo>\u2216<\/mo><mo form=\"prefix\" stretchy=\"false\">{<\/mo><msub><mi>v<\/mi><mi>j<\/mi><\/msub><mo form=\"postfix\" stretchy=\"false\" lspace=\"0em\" rspace=\"0em\">}<\/mo><\/mrow><\/munder><mo>\u2061<\/mo><mo form=\"prefix\" stretchy=\"false\">{<\/mo><mi>|<\/mi><msub><mi>m<\/mi><mrow><msubsup><mi>v<\/mi><mi>j<\/mi><mo lspace=\"0em\" rspace=\"0em\" class=\"tml-prime\">\u2032<\/mo><\/msubsup><mo>\u2192<\/mo><msub><mi>u<\/mi><mi>i<\/mi><\/msub><\/mrow><\/msub><mi>|<\/mi><mo form=\"postfix\" stretchy=\"false\">}<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">m_{u_i\\to v_j} = (-1)^{s_i} \\alpha \\left( \\prod_{v&#8217;_j \\in V(u_i) \\setminus \\{v_j\\}} \\mathrm{sign}(m_{v&#8217;_j \\to u_i}) \\right) \\min_{v&#8217;_j \\in V(u_i) \\setminus \\{v_j\\}} \\{ |m_{v&#8217;_j \\to u_i}| \\}<\/annotation><\/semantics><\/math><\/div>\n\n\n\n<p>\u3084\u3084\u8907\u96d1\u306a\u5f0f\u306a\u306e\u3067\u3053\u306e\u5f0f\u306e\u89e3\u91c8\u306b\u3064\u3044\u3066\u306fAppendix A\u3092\u53c2\u7167\u3057\u3066\u304f\u3060\u3055\u3044\u3002\u3056\u3063\u304f\u308a\u3068\u3057\u305f\u610f\u5473\u306f\u3001\u5909\u6570\u30ce\u30fc\u30c9 $v_j$\u306b\u5411\u304b\u3063\u3066\u3001\u30d1\u30ea\u30c6\u30a3\u691c\u67fb\u6761\u4ef6\u3092\u307f\u305f\u3059\u305f\u3081\u306b\u3069\u3046\u3044\u3046\u5024\u3067\u3042\u3063\u3066\u307b\u3057\u3044\u306e\u304b\u3092\u8868\u3057\u3066\u3044\u307e\u3059\u3002 $m_{u_i\\to v_j}$\u304c\u6b63\u306a\u3089 $v_j=0$\u3067\u3042\u3063\u3066\u307b\u3057\u304f\u3001\u8ca0\u306a\u3089 $v_j=1$\u3067\u3042\u3063\u3066\u307b\u3057\u3044\u3068\u3044\u3046\u3053\u3068\u3092\u610f\u5473\u3057\u3066\u3044\u307e\u3059\u3002<\/p>\n\n\n\n<p>\u3000\u3055\u3089\u306b\u3001\u30d1\u30ea\u30c6\u30a3\u30ce\u30fc\u30c9\u304b\u3089\u30e1\u30c3\u30bb\u30fc\u30b8\u3092\u53d7\u3051\u53d6\u3063\u305f\u5909\u6570\u30ce\u30fc\u30c9\u306f\u6b21\u306e\u3088\u3046\u306b\u30e1\u30c3\u30bb\u30fc\u30b8\u3092\u8a08\u7b97\u3057\u307e\u3059\u3002<br>$$\\begin{align}m_{v_j\\to u_i}=\\log\\left(\\frac{1-p_j}{p_j}\\right)+\\sum_{u_i&#8217; \\in U(v_j)\\setminus u_i}m_{u_i&#8217;\\to v_j}\\end{align}$$<br><\/p>\n\n\n\n<p>\u3000\u3053\u3053\u307e\u3067\u304c1\u56de\u306e\u53cd\u5fa9\u3067\u8a08\u7b97\u3055\u308c\u308b\u30e1\u30c3\u30bb\u30fc\u30b8\u3067\u3059\u3002\u5404\u53cd\u5fa9\u306e\u6700\u5f8c\u306b\u3001\u66ab\u5b9a\u7684\u306a\u4e8b\u5f8c\u5bfe\u6570\u5c24\u5ea6\u6bd4\u3092\u8a08\u7b97\u3057\u307e\u3059\u3002<br>$$ m_{v_j} = \\log\\left(\\frac{1-p_j}{p_j}\\right) + \\sum_{u_i \\in U(v_j)} m_{u_i\\to v_j} $$<br>\u3053\u3053\u3067\u3001\u4e8b\u5f8c\u5bfe\u6570\u5c24\u5ea6\u6bd4\u306f\u6b21\u306e\u3088\u3046\u306b\u5b9a\u7fa9\u3055\u308c\u307e\u3059\u3002<br>$$ \\mathrm{LLR}_i = \\frac{p(v_i=0 | s)}{p(v_i=1 | s)} $$<br>\u3057\u305f\u304c\u3063\u3066\u3001\u6b63\u3067\u3042\u308c\u3070 $v_i=0$\u3067\u3042\u308b\u53ef\u80fd\u6027\u304c\u9ad8\u304f\u3001\u8ca0\u3067\u3042\u308c\u3070 $v_i=1$\u3067\u3042\u308b\u53ef\u80fd\u6027\u304c\u9ad8\u304f\u306a\u308a\u307e\u3059\u3002\u3053\u306e\u3053\u3068\u304b\u3089\u3001\u66ab\u5b9a\u7684\u306a\u30a8\u30e9\u30fc\u306e\u63a8\u5b9a\u5024\u3092\u6b21\u306e\u3088\u3046\u306b\u6c7a\u5b9a\u3057\u307e\u3059\u3002<br>$$ \\hat{v_i} = \\begin{cases}0 &amp; \\mathrm{if} \\; m_{v_j} &gt; 0 \\\\ 1 &amp; \\mathrm{otherwise}\\end{cases} $$<br><\/p>\n\n\n\n<p>\u3000\u3053\u306e\u3088\u3046\u306b\u3057\u3066\u6c7a\u5b9a\u3057\u305f\u30a8\u30e9\u30fc\u304c\u30d1\u30ea\u30c6\u30a3\u691c\u67fb\u6761\u4ef6\u3092\u6e80\u305f\u3057\u3066\u3044\u308c\u3070\u30a2\u30eb\u30b4\u30ea\u30ba\u30e0\u3092\u7d42\u4e86\u3057\u3066\u3001\u30a8\u30e9\u30fc\u306e\u63a8\u5b9a\u5024\u3092\u51fa\u529b\u3057\u307e\u3059\u3002\u30d1\u30ea\u30c6\u30a3\u691c\u67fb\u6761\u4ef6\u3092\u6e80\u305f\u3057\u3066\u3044\u306a\u3051\u308c\u3070\u3001\u6b21\u306e\u53cd\u5fa9\u306b\u9032\u3093\u3067\u518d\u3073\u30e1\u30c3\u30bb\u30fc\u30b8\u306e\u4f1d\u64ad\u3092\u884c\u3044\u307e\u3059\u3002<\/p>\n\n\n\n<p>\u3000\u3053\u306e\u3088\u3046\u306b\u3057\u3066\u3001\u30d1\u30ea\u30c6\u30a3\u691c\u67fb\u6761\u4ef6\u3092\u6e80\u305f\u3059\u304b\u3042\u3089\u304b\u3058\u3081\u6c7a\u3081\u305f\u6700\u5927\u53cd\u5fa9\u56de\u6570\u306b\u5230\u9054\u3059\u308b\u307e\u3067\u30e1\u30c3\u30bb\u30fc\u30b8\u306e\u4f1d\u64ad\u3092\u7e70\u308a\u8fd4\u3057\u3066\u3044\u304d\u307e\u3059\u3002\u6b63\u3057\u3044\u30a8\u30e9\u30fc\u3092\u51fa\u529b\u3059\u308b\u4fdd\u8a3c\u306f\u3042\u308a\u307e\u305b\u3093\u304c\u3001\u591a\u304f\u306e\u5834\u5408\u3067\u6b63\u3057\u3044\u30a8\u30e9\u30fc\u3092\u51fa\u529b\u3067\u304d\u308b\u3053\u3068\u304c\u6570\u5024\u7684\u306b\u691c\u8a3c\u3067\u304d\u307e\u3059<sup data-fn=\"d000d753-0c30-4c6b-8300-256e84027910\" class=\"fn\"><a href=\"#d000d753-0c30-4c6b-8300-256e84027910\" id=\"d000d753-0c30-4c6b-8300-256e84027910-link\">3<\/a><\/sup>\u3002<\/p>\n\n\n\n<p>\u3000\u4ee5\u4e0a\u306e\u51e6\u7406\u3092\u307e\u3068\u3081\u305f\u7591\u4f3c\u30b3\u30fc\u30c9\u3092\u6b21\u306b\u793a\u3057\u307e\u3059\u3002<br><\/p>\n\n\n\n<figure class=\"wp-block-image size-large\"><img loading=\"lazy\" decoding=\"async\" width=\"671\" height=\"1024\" src=\"https:\/\/qard.is.tohoku.ac.jp\/T-Wave\/wp-content\/uploads\/2026\/03\/28cd482b2ebd5104d75d1569a215ff87-671x1024.png\" alt=\"\" class=\"wp-image-8770\" srcset=\"https:\/\/qard.is.tohoku.ac.jp\/T-Wave\/wp-content\/uploads\/2026\/03\/28cd482b2ebd5104d75d1569a215ff87-671x1024.png 671w, https:\/\/qard.is.tohoku.ac.jp\/T-Wave\/wp-content\/uploads\/2026\/03\/28cd482b2ebd5104d75d1569a215ff87-197x300.png 197w, https:\/\/qard.is.tohoku.ac.jp\/T-Wave\/wp-content\/uploads\/2026\/03\/28cd482b2ebd5104d75d1569a215ff87.png 716w\" sizes=\"auto, (max-width: 671px) 100vw, 671px\" \/><\/figure>\n\n\n\n<p><br>\uff08<a href=\"https:\/\/arxiv.org\/abs\/2005.07016%E3%81%8B%E3%82%89%E5%BC%95%E7%94%A8%EF%BC%89\">https:\/\/arxiv.org\/abs\/2005.07016\u304b\u3089\u5f15\u7528\uff09<\/a><br><\/p>\n\n\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"BP%E3%81%A0%E3%81%91%E3%81%A7%E3%81%AF%E3%83%80%E3%83%A1%E3%81%AA%E3%81%AE%E3%81%8B%EF%BC%9F\"><\/span>BP\u3060\u3051\u3067\u306f\u30c0\u30e1\u306a\u306e\u304b\uff1f<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p>\u3000LLR-MSA\u3092\u7528\u3044\u308b\u3053\u3068\u3067\u30a8\u30e9\u30fc\u306e\u63a8\u5b9a\u5024\u304c\u5f97\u3089\u308c\u308b\u306e\u3067\u3001LLR-MSA\u3060\u3051\u3067\u30c7\u30b3\u30fc\u30c0\u3092\u69cb\u6210\u3067\u304d\u307e\u3059\u3002\u3067\u3059\u304c\u3001\u3053\u306e\u8a18\u4e8b\u3067\u7d39\u4ecb\u3057\u3066\u3044\u308b\u30c7\u30b3\u30fc\u30c0\u306b\u306fLLR-MSA\u4ee5\u5916\u306bOSD\u3068\u547c\u3070\u308c\u308b\u30a2\u30eb\u30b4\u30ea\u30ba\u30e0\u304c\u4ed8\u3044\u3066\u3044\u307e\u3059\u3002\u306a\u305cLLR-MSA\u3060\u3051\u3067\u306f\u4e0d\u5341\u5206\u306a\u306e\u3067\u3057\u3087\u3046\u304b\uff1f<\/p>\n\n\n\n<p>\u3000LLR-MSA\u3092\u7528\u3044\u308b\u3053\u3068\u3067\u4e8b\u524d\u5206\u5e03 $\\mathrm{Pr}(e)$ \u304b\u3089\u4e8b\u5f8c\u5206\u5e03 $\\mathrm{Pr}(e|s)$\u306e\u63a8\u5b9a\u5024\u304c\u5f97\u3089\u308c\u307e\u3059\u3002\u3053\u3046\u3057\u3066\u5f97\u3089\u308c\u305f\u4e8b\u5f8c\u5206\u5e03\u3092\u7528\u3044\u3066\u3001\u4e8b\u5f8c\u5206\u5e03\u3092\u6700\u5927\u306b\u3059\u308b\u3088\u3046\u306a\u30a8\u30e9\u30fc\u30d9\u30af\u30c8\u30eb $\\hat e$ \u3092\u30a8\u30e9\u30fc\u306e\u63a8\u5b9a\u5024\u3068\u3059\u308b\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u3002<br>$$\\hat e = \\arg\\max_{e\\in\\mathbb Z_2^n} \\mathrm{Pr}(e|s)$$<br>\u3053\u306e\u3088\u3046\u306a\u63a8\u5b9a\u306f\u30d9\u30a4\u30ba\u7d71\u8a08\u306a\u3069\u3067\u7528\u3044\u3089\u308c\u308b\u624b\u6cd5\u3067\u3001MAP\u63a8\u5b9a\u3068\u547c\u3070\u308c\u307e\u3059\u3002MAP\u63a8\u5b9a\u3067\u5f97\u3089\u308c\u305f\u30a8\u30e9\u30fc\u30d9\u30af\u30c8\u30eb\u3092$\\hat e$\u3068\u3057\u3066\u3001<br>$$H\\cdot\\hat e=s$$<br>\u304c\u6210\u308a\u7acb\u3064\u306a\u3089\u3001\u63a8\u5b9a\u3057\u305f\u30a8\u30e9\u30fc\u306f\u30b7\u30f3\u30c9\u30ed\u30fc\u30e0\u5024\u306b\u5bfe\u5fdc\u3057\u305f\u9069\u5207\u306a\u7d50\u679c\u3067\u3042\u308b\u4e8b\u304c\u308f\u304b\u308a\u307e\u3059\u3002<\/p>\n\n\n\n<p>\u3000\u305f\u3060\u3001MAP\u63a8\u5b9a\u306e\u7d50\u679c\u5f97\u3089\u308c\u308b\u30a8\u30e9\u30fc\u30d9\u30af\u30c8\u30eb\u306f$H\\cdot \\hat e=s$\u304c\u6210\u308a\u7acb\u305f\u306a\u3044\u3053\u3068\u304c\u3042\u308a\u3001\u3053\u306e\u3068\u304d\u306f\u63a8\u5b9a\u306b\u5931\u6557\u3057\u305f\u3053\u3068\u306b\u306a\u308a\u307e\u3059\u3002\u5931\u6557\u3057\u305f\u5834\u5408\u306f\u3001\u30e1\u30c3\u30bb\u30fc\u30b8$m_{v_j\\to u_i}$\u3092<br><\/p>\n\n\n\n<div class=\"wp-block-math\"><math display=\"block\"><semantics><mrow><msubsup><mi>m<\/mi><mrow><msub><mi>v<\/mi><mi>j<\/mi><\/msub><mo>\u2192<\/mo><msub><mi>u<\/mi><mi>i<\/mi><\/msub><\/mrow><mo lspace=\"0em\" rspace=\"0em\" class=\"tml-prime\">\u2032<\/mo><\/msubsup><mo>=<\/mo><msub><mi>m<\/mi><mrow><msub><mi>v<\/mi><mi>j<\/mi><\/msub><mo>\u2192<\/mo><msub><mi>u<\/mi><mi>i<\/mi><\/msub><\/mrow><\/msub><mo>+<\/mo><mrow><munder><mo movablelimits=\"false\">\u2211<\/mo><mrow><msubsup><mi>u<\/mi><mi>i<\/mi><mo lspace=\"0em\" rspace=\"0em\" class=\"tml-prime\">\u2032<\/mo><\/msubsup><mo>\u2208<\/mo><mi>U<\/mi><mo form=\"prefix\" stretchy=\"false\">(<\/mo><msub><mi>v<\/mi><mi>j<\/mi><\/msub><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mo>\u2216<\/mo><mo form=\"prefix\" stretchy=\"false\">{<\/mo><msub><mi>u<\/mi><mi>i<\/mi><\/msub><mo form=\"postfix\" stretchy=\"false\" lspace=\"0em\" rspace=\"0em\">}<\/mo><\/mrow><\/munder><\/mrow><msub><mi>m<\/mi><mrow><msubsup><mi>u<\/mi><mi>i<\/mi><mo lspace=\"0em\" rspace=\"0em\" class=\"tml-prime\">\u2032<\/mo><\/msubsup><mo>\u2192<\/mo><msub><mi>v<\/mi><mi>j<\/mi><\/msub><\/mrow><\/msub><\/mrow><annotation encoding=\"application\/x-tex\">\nm&#8217;_{v_j \\to u_i} = m_{v_j \\to u_i} + \\sum_{u&#8217;_i \\in U(v_j) \\setminus \\{u_i\\}} m_{u&#8217;_i \\to v_j}\n<\/annotation><\/semantics><\/math><\/div>\n\n\n\n<p><br>\u3068\u66f4\u65b0\u3057\u3066\u3001\u3053\u3053\u307e\u3067\u306e\u624b\u9806\u3092\u7e70\u308a\u8fd4\u3057\u307e\u3059\u3002\u3053\u308c\u306f\u3001\u30d9\u30a4\u30ba\u7d71\u8a08\u3067\u306f\u9010\u6b21\u63a8\u5b9a\u3068\u547c\u3070\u308c\u308b\u624b\u6cd5\u3067\u3001\u5f97\u3089\u308c\u305f\u4e8b\u5f8c\u5206\u5e03\u3092\u4e8b\u524d\u5206\u5e03\u3068\u3059\u308b\u3053\u3068\u3067MAP\u63a8\u5b9a\u3092\u7e70\u308a\u8fd4\u3057\u307e\u3059\u3002\u3053\u308c\u306b\u3088\u3063\u3066\u3001\u6700\u521d\u306e\u4e8b\u524d\u5206\u5e03 $\\mathrm{Pr}(e)$ \u306e\u5f71\u97ff\u304c\u5f31\u307e\u308a\u3001\u5206\u5e03\u306e\u66f4\u65b0\u306b\u4f7f\u308f\u308c\u308b\u30b7\u30f3\u30c9\u30ed\u30fc\u30e0\u5024\u306e\u60c5\u5831\u306e\u5f71\u97ff\u304c\u5f37\u307e\u308a\u307e\u3059\u3002<\/p>\n\n\n\n<p>\u3000\u30a8\u30e9\u30fc\u306e\u63a8\u5b9a\u306b\u6210\u529f\u3057\u305f\u3089$\\hat e$\u3092\u51fa\u529b\u3057\u3066\u7d42\u4e86\u3057\u3001\u5931\u6557\u3092\u7d9a\u3051\u3066\u4e00\u5b9a\u306e\u7e70\u308a\u8fd4\u3057\u56de\u6570\u306b\u9054\u3057\u305f\u3089\u63a8\u5b9a\u306b\u5931\u6557\u3057\u305f\u3053\u3068\u3092\u5831\u544a\u3057\u3066\u30a2\u30eb\u30b4\u30ea\u30ba\u30e0\u3092\u7d42\u4e86\u3057\u307e\u3059\u3002\u4e8b\u524d\u306b\u7e70\u308a\u8fd4\u3057\u56de\u6570\u306e\u4e0a\u9650\u3092\u6c7a\u3081\u306a\u3044\u3068\u30a2\u30eb\u30b4\u30ea\u30ba\u30e0\u304c\u505c\u6b62\u3057\u306a\u3044\u53ef\u80fd\u6027\u304c\u3042\u308b\u306e\u3067\u6ce8\u610f\u304c\u5fc5\u8981\u3067\u3059\u3002<\/p>\n\n\n\n<p>\u3000\u3053\u306e\u3088\u3046\u306b\u3057\u3066\u3001\u30b7\u30f3\u30c9\u30ed\u30fc\u30e0\u5024$s$\u304b\u3089\u63a8\u5b9a\u30a8\u30e9\u30fc$\\hat e$\u3092\u6c42\u3081\u308b\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u3002\u3057\u304b\u3057\u3001\u91cf\u5b50\u8aa4\u308a\u8a02\u6b63\u7b26\u53f7\u306e\u6b21\u306e\u3088\u3046\u306a\u6027\u8cea\u304b\u3089\u3001\u5341\u5206\u306a\u7cbe\u5ea6\u3092\u5f97\u3089\u308c\u306a\u3044\u3068\u3044\u3046\u554f\u984c\u304c\u3042\u308a\u307e\u3059\u3002<br><\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>\u78ba\u7387\u4f1d\u64ad\u6cd5\u3092\u7528\u3044\u3066$\\mathrm{Pr}(e_j=1|s)$\u3092\u6c42\u3081\u308b\u3068\u304d\u3001\u30d5\u30a1\u30af\u30bf\u30fc\u30b0\u30e9\u30d5\u306b\u30b5\u30a4\u30af\u30eb\u304c\u3042\u308b\u3068\u7cbe\u5ea6\u304c\u4e0b\u304c\u308b\u3053\u3068\u304c\u77e5\u3089\u308c\u3066\u3044\u308b\u3002\u91cf\u5b50\u8aa4\u308a\u8a02\u6b63\u7b26\u53f7\u306e\u30d5\u30a1\u30af\u30bf\u30fc\u30b0\u30e9\u30d5\u306b\u306f\u30b5\u30a4\u30af\u30eb\u304c\u5b58\u5728\u3059\u308b\u305f\u3081\u3001\u91cf\u5b50\u8aa4\u308a\u8a02\u6b63\u7b26\u53f7\u306b\u5bfe\u3057\u3066\u306f\u7cbe\u5ea6\u304c\u4f4e\u3044<\/li>\n\n\n\n<li>\u91cf\u5b50\u8aa4\u308a\u8a02\u6b63\u7b26\u53f7\u306b\u306f\u3001\u7e2e\u9000\u3068\u3044\u3046\u6027\u8cea\u304c\u3042\u308a\u3001\u30b7\u30f3\u30c9\u30ed\u30fc\u30e0\u5024\u306b\u5bfe\u5fdc\u3059\u308b\u30a8\u30e9\u30fc\u304c\u8907\u6570\u5b58\u5728\u3059\u308b\u3053\u3068\u304c\u3042\u308b\u3002\u7e2e\u9000\u306fBP\u306e\u30e1\u30c3\u30bb\u30fc\u30b8\u4f1d\u64ad\u306b\u5bfe\u79f0\u6027\u3092\u751f\u3058\u3055\u305b\u3001\u5bfe\u79f0\u6027\u306e\u5f71\u97ff\u3067\u63a8\u5b9a\u3067\u304d\u308b\u30a8\u30e9\u30fc\u306b\u4e0d\u81ea\u7136\u306a\u5236\u7d04\u304c\u4ed8\u3044\u3066\u3057\u307e\u3046\u3002<br><\/li>\n<\/ul>\n\n\n\n<p>\u3000\u3064\u307e\u308a\u3001\u3053\u3053\u307e\u3067\u7d39\u4ecb\u3057\u305f\u78ba\u7387\u4f1d\u64ad\u6cd5\u306b\u3088\u308b\u30c7\u30b3\u30fc\u30c0\u3092\u305d\u306e\u307e\u307e\u4f7f\u3063\u305f\u3060\u3051\u3067\u306f\u5341\u5206\u306e\u6027\u80fd\u304c\u5f97\u3089\u308c\u306a\u3044\u3068\u3044\u3046\u3053\u3068\u3067\u3059\u3002\u305d\u3053\u3067\u3001\u8aa4\u308a\u8a02\u6b63\u306e\u7cbe\u5ea6\u3092\u5411\u4e0a\u3055\u305b\u308b\u305f\u3081\u306b\u69d8\u3005\u306a\u624b\u6cd5\u304c\u7814\u7a76\u3055\u308c\u3066\u3044\u307e\u3059\u3002\u6b21\u56de\u306e\u8a18\u4e8b\u3067\u306f\u3001\u305d\u306e\u4e2d\u3067\u3082\u300c\u9806\u5e8f\u7d71\u8a08\u91cf\u30c7\u30b3\u30fc\u30c7\u30a3\u30f3\u30b0 (ordered statistics decoding, OSD)\u300d\u3068\u547c\u3070\u308c\u308b\u624b\u6cd5\u3092\u7528\u3044\u305f\u30c7\u30b3\u30fc\u30c0\u3092\u7d39\u4ecb\u3057\u307e\u3059\u3002<br><\/p>\n\n\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"%E3%81%BE%E3%81%A8%E3%82%81\"><\/span>\u307e\u3068\u3081<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p>\u3000\u6700\u5f8c\u307e\u3067\u8aad\u3093\u3067\u3044\u305f\u3060\u304d\u3042\u308a\u304c\u3068\u3046\u3054\u3056\u3044\u307e\u3059\u3002\u4eca\u56de\u306e\u8a18\u4e8b\u3067\u306f\u3001BP\u3092\u4f7f\u3063\u305f\u30c7\u30b3\u30fc\u30c0\u3067\u3042\u308bLLR-MSA\u306b\u3064\u3044\u3066\u89e3\u8aac\u3057\u307e\u3057\u305f\u3002LLR-MSA\u306e\u524d\u63d0\u3068\u306a\u308b\u30d5\u30a1\u30af\u30bf\u30fc\u30b0\u30e9\u30d5\u3092\u5c0e\u5165\u3057\u3066\u3001LLR-MSA\u306e\u624b\u7d9a\u304d\u3092\u7d39\u4ecb\u3057\u307e\u3057\u305f\u3002\u6700\u5f8c\u306b\u3001LLR-MSA\u3060\u3051\u3067\u306f\u30c7\u30b3\u30fc\u30c0\u306e\u7cbe\u5ea6\u304c\u4e0d\u5341\u5206\u3067\u3042\u308b\u3053\u3068\u3092\u8aac\u660e\u3057\u307e\u3057\u305f\u3002<\/p>\n\n\n\n<p>\u3000\u6b21\u56de\u306e\u8a18\u4e8b\u3067\u306f\u3001LLR-MSA\u306e\u7cbe\u5ea6\u3092\u88dc\u3046\u9806\u5e8f\u7d71\u8a08\u91cf\u30c7\u30b3\u30fc\u30c7\u30a3\u30f3\u30b0 (ordered statistics decoding, OSD)\u3068\u547c\u3070\u308c\u308b\u624b\u6cd5\u3092\u7d39\u4ecb\u3057\u307e\u3059\u3002OSD\u306f\u3001LLR-MSA\u306e\u76f4\u5f8c\u306b\u5b9f\u884c\u3059\u308b\u3053\u3068\u3067\u30c7\u30b3\u30fc\u30c0\u306e\u7cbe\u5ea6\u3092\u5411\u4e0a\u3055\u305b\u308b\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u3002\u305c\u3072\u6b21\u56de\u306e\u8a18\u4e8b\u3082\u3054\u89a7\u304f\u3060\u3055\u3044\u3002<br><\/p>\n\n\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Appendix_A_Min-Sum%E3%81%AE%E6%9B%B4%E6%96%B0%E5%BC%8F%E3%81%AE%E8%A7%A3%E9%87%88\"><\/span>Appendix A: Min-Sum\u306e\u66f4\u65b0\u5f0f\u306e\u89e3\u91c8<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p>\u3053\u306e\u4ed8\u9332\u3067\u306f\u6b21\u306e\u66f4\u65b0\u5f0f\u306e\u610f\u5473\u3092\u8a73\u3057\u304f\u898b\u3066\u3044\u304d\u307e\u3059<br><\/p>\n\n\n\n<div class=\"wp-block-math\"><math display=\"block\"><semantics><mrow><msub><mi>m<\/mi><mrow><msub><mi>u<\/mi><mi>i<\/mi><\/msub><mo>\u2192<\/mo><msub><mi>v<\/mi><mi>j<\/mi><\/msub><\/mrow><\/msub><mo>=<\/mo><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mo form=\"prefix\" stretchy=\"false\">\u2212<\/mo><mn>1<\/mn><msup><mo form=\"postfix\" stretchy=\"false\">)<\/mo><msub><mi>s<\/mi><mi>i<\/mi><\/msub><\/msup><mi>\u03b1<\/mi><mrow><mo fence=\"true\" form=\"prefix\">(<\/mo><mrow><munder><mo movablelimits=\"false\">\u220f<\/mo><mrow><msubsup><mi>v<\/mi><mi>j<\/mi><mo lspace=\"0em\" rspace=\"0em\" class=\"tml-prime\">\u2032<\/mo><\/msubsup><mo>\u2208<\/mo><mi>V<\/mi><mo form=\"prefix\" stretchy=\"false\">(<\/mo><msub><mi>u<\/mi><mi>i<\/mi><\/msub><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mo>\u2216<\/mo><mo form=\"prefix\" stretchy=\"false\">{<\/mo><msub><mi>v<\/mi><mi>j<\/mi><\/msub><mo form=\"postfix\" stretchy=\"false\" lspace=\"0em\" rspace=\"0em\">}<\/mo><\/mrow><\/munder><\/mrow><mrow><mtext><\/mtext><mi>sign<\/mi><\/mrow><mo form=\"prefix\" stretchy=\"false\">(<\/mo><msub><mi>m<\/mi><mrow><msubsup><mi>v<\/mi><mi>j<\/mi><mo lspace=\"0em\" rspace=\"0em\" class=\"tml-prime\">\u2032<\/mo><\/msubsup><mo>\u2192<\/mo><msub><mi>u<\/mi><mi>i<\/mi><\/msub><\/mrow><\/msub><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace width=\"0.1667em\"><\/mspace><munder><mi>min<\/mi><mrow><msubsup><mi>v<\/mi><mi>j<\/mi><mo lspace=\"0em\" rspace=\"0em\" class=\"tml-prime\">\u2032<\/mo><\/msubsup><mo>\u2208<\/mo><mi>V<\/mi><mo form=\"prefix\" stretchy=\"false\">(<\/mo><msub><mi>u<\/mi><mi>i<\/mi><\/msub><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mo>\u2216<\/mo><mo form=\"prefix\" stretchy=\"false\">{<\/mo><msub><mi>v<\/mi><mi>j<\/mi><\/msub><mo form=\"postfix\" stretchy=\"false\" lspace=\"0em\" rspace=\"0em\">}<\/mo><\/mrow><\/munder><mo>\u2061<\/mo><mo form=\"prefix\" stretchy=\"false\">{<\/mo><mi>|<\/mi><msub><mi>m<\/mi><mrow><msubsup><mi>v<\/mi><mi>j<\/mi><mo lspace=\"0em\" rspace=\"0em\" class=\"tml-prime\">\u2032<\/mo><\/msubsup><mo>\u2192<\/mo><msub><mi>u<\/mi><mi>i<\/mi><\/msub><\/mrow><\/msub><mi>|<\/mi><mo form=\"postfix\" stretchy=\"false\">}<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">\nm_{u_i \\to v_j} = (-1)^{s_i} \\alpha \\left( \\prod_{v&#8217;_j \\in V(u_i) \\setminus \\{v_j\\}} \\mathrm{sign}(m_{v&#8217;_j \\to u_i}) \\right) \\min_{v&#8217;_j \\in V(u_i) \\setminus \\{v_j\\}} \\{ |m_{v&#8217;_j \\to u_i}| \\}<\/annotation><\/semantics><\/math><\/div>\n\n\n\n<p><br>\u3053\u3053\u3067\u3001$V(u_i)$\u306f\u30d1\u30ea\u30c6\u30a3\u30ce\u30fc\u30c9$u_i$\u306b\u96a3\u63a5\u3059\u308b\u5909\u6570\u30ce\u30fc\u30c9\u5168\u4f53\u306e\u96c6\u5408\u3001$\\mathrm{sign}(-)$\u306f\u5f15\u6570\u306e\u7b26\u53f7\u3092\u53d6\u308b\u95a2\u6570\u3067\u3001$s_i$\u306f\u30b7\u30f3\u30c9\u30ed\u30fc\u30e0\u5024\u306e$i$\u30d3\u30c3\u30c8\u76ee\u3067\u3059\u3002\u3053\u306e\u5f0f\u3092\u6b21\u306e4\u3064\u306e\u30d1\u30fc\u30c4\u306b\u5206\u3051\u3066\u3001\u8d64\u6587\u5b57\u3067\u66f8\u304b\u308c\u305f\u756a\u53f7\u306e\u9806\u306b\u8aac\u660e\u3057\u3066\u3044\u304d\u307e\u3059\u3002<br><\/p>\n\n\n\n<figure class=\"wp-block-image size-large\"><img loading=\"lazy\" decoding=\"async\" width=\"1024\" height=\"546\" src=\"https:\/\/qard.is.tohoku.ac.jp\/T-Wave\/wp-content\/uploads\/2026\/03\/bp_message_explanation-1024x546.png\" alt=\"\" class=\"wp-image-8772\" srcset=\"https:\/\/qard.is.tohoku.ac.jp\/T-Wave\/wp-content\/uploads\/2026\/03\/bp_message_explanation-1024x546.png 1024w, https:\/\/qard.is.tohoku.ac.jp\/T-Wave\/wp-content\/uploads\/2026\/03\/bp_message_explanation-300x160.png 300w, https:\/\/qard.is.tohoku.ac.jp\/T-Wave\/wp-content\/uploads\/2026\/03\/bp_message_explanation-768x410.png 768w, https:\/\/qard.is.tohoku.ac.jp\/T-Wave\/wp-content\/uploads\/2026\/03\/bp_message_explanation-1536x819.png 1536w, https:\/\/qard.is.tohoku.ac.jp\/T-Wave\/wp-content\/uploads\/2026\/03\/bp_message_explanation-2048x1092.png 2048w\" sizes=\"auto, (max-width: 1024px) 100vw, 1024px\" \/><\/figure>\n\n\n\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"%E2%91%A0_prod_mathrmsignm_vto_u\"><\/span>\u2460 $\\prod \\mathrm{sign}(m_{v&#8217;\\to u})$<span class=\"ez-toc-section-end\"><\/span><\/h3>\n\n\n\n<p>\u3000\u307e\u305a\u306f\u3001$\\prod \\mathrm{sign}(m_{v&#8217;\\to u})$\u306e\u90e8\u5206\u306b\u3064\u3044\u3066\u3067\u3059\u3002\u3053\u306e\u90e8\u5206\u306f\u3001\u30d1\u30ea\u30c6\u30a3\u30ce\u30fc\u30c9$u_i$\u306b\u9001\u3089\u308c\u305f$v_j$\u3092\u9664\u304f\u30e1\u30c3\u30bb\u30fc\u30b8\u306e\u7b26\u53f7\u3092\u53d6\u308a\u3001\u305d\u308c\u3089\u3092\u304b\u3051\u5408\u308f\u305b\u3066\u3044\u307e\u3059\u3002$m_{v_j\\to u_i}$\u306e\u7b26\u53f7\u306f<br><\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>$+1$\u306a\u3089$j$\u756a\u76ee\u306e\u91cf\u5b50\u30d3\u30c3\u30c8\u306b\u30a8\u30e9\u30fc\u304c\u8d77\u304d\u3066\u3044\u306a\u3044\u78ba\u7387\u306e\u307b\u3046\u304c\u9ad8\u3044\uff08$e_j=0$\u3067\u3042\u308b\u78ba\u7387\u304c\u9ad8\u3044\uff09<\/li>\n\n\n\n<li>$-1$\u306a\u3089$j$\u756a\u76ee\u306e\u91cf\u5b50\u30d3\u30c3\u30c8\u306b\u30a8\u30e9\u30fc\u304c\u8d77\u304d\u3066\u3044\u308b\u78ba\u7387\u306e\u307b\u3046\u304c\u9ad8\u3044\uff08$e_j=1$\u3067\u3042\u308b\u78ba\u7387\u304c\u9ad8\u3044\uff09<\/li>\n<\/ul>\n\n\n\n<p>\u3068\u3044\u3046\u3053\u3068\u3092\u610f\u5473\u3057\u3066\u3044\u307e\u3059\u3002\u3053\u306e\u3053\u3068\u304b\u3089$e_j$\u306e\u5024\u3092\u66ab\u5b9a\u7684\u306b<br>$$ \\begin{align} e_j=\\begin{cases}0&amp;\\mathrm{sign}(m_{v_j\\to u_i})=+1 \\\\ 1&amp;\\mathrm{sign}(m_{v_j\\to u_i})=-1\\end{cases}\\end{align} $$<br>\u3068\u6c7a\u3081\u3066\u3057\u307e\u3044\u307e\u3059\u3002\u3053\u306e\u3088\u3046\u306b\u3057\u305f\u3068\u304d\u3001$v_j$\u3092\u9664\u304f\u5909\u6570\u30ce\u30fc\u30c9\u306b\u3064\u3044\u3066\u30d1\u30ea\u30c6\u30a3\u3092\u53d6\u3063\u305f\u7d50\u679c\u304c<\/p>\n\n\n\n<div class=\"wp-block-math\"><math display=\"block\"><semantics><mrow><mrow><munder><mo movablelimits=\"false\">\u220f<\/mo><mrow><msubsup><mi>v<\/mi><mi>j<\/mi><mo lspace=\"0em\" rspace=\"0em\" class=\"tml-prime\">\u2032<\/mo><\/msubsup><mo>\u2208<\/mo><mi>V<\/mi><mo form=\"prefix\" stretchy=\"false\">(<\/mo><msub><mi>u<\/mi><mi>i<\/mi><\/msub><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mo>\u2216<\/mo><mo form=\"prefix\" stretchy=\"false\">{<\/mo><msub><mi>v<\/mi><mi>j<\/mi><\/msub><mo form=\"postfix\" stretchy=\"false\" lspace=\"0em\" rspace=\"0em\">}<\/mo><\/mrow><\/munder><\/mrow><mrow><mtext><\/mtext><mi>sign<\/mi><\/mrow><mo form=\"prefix\" stretchy=\"false\">(<\/mo><msub><mi>m<\/mi><mrow><msubsup><mi>v<\/mi><mi>j<\/mi><mo lspace=\"0em\" rspace=\"0em\" class=\"tml-prime\">\u2032<\/mo><\/msubsup><mo>\u2192<\/mo><msub><mi>u<\/mi><mi>i<\/mi><\/msub><\/mrow><\/msub><mo form=\"postfix\" stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">\n\\prod_{v&#8217;_j \\in V(u_i) \\setminus \\{v_j\\}} \\mathrm{sign}(m_{v&#8217;_j \\to u_i})\n<\/annotation><\/semantics><\/math><\/div>\n\n\n\n<p><br>\u306b\u5bfe\u5fdc\u3057\u307e\u3059\u3002<br><\/p>\n\n\n\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"%E2%91%A1_-1s_i\"><\/span>\u2461 $(-1)^{s_i}$<span class=\"ez-toc-section-end\"><\/span><\/h3>\n\n\n\n<p>\u3000\u7b26\u53f7\u3092\u5236\u5fa1\u3057\u3066\u3044\u308b\u30d1\u30fc\u30c4\u304c\u3082\u3046\u4e00\u3064\u3042\u308a\u307e\u3059\u3002$(-1)^{s_i}$\u306e\u90e8\u5206\u3067\u3059\u3002\u3053\u306e\u90e8\u5206\u306b\u306f\u3001\u30b7\u30f3\u30c9\u30ed\u30fc\u30e0\u5024$s_i$\u304c\u542b\u307e\u308c\u3066\u3044\u307e\u3059\u3002\u30b7\u30f3\u30c9\u30ed\u30fc\u30e0\u306e\u7d50\u679c\u306b\u5fdc\u3058\u3066\u3001<br><\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>$s_i = 0$\u306a\u3089\u7b26\u53f7\u3092\u305d\u306e\u307e\u307e\u306b<\/li>\n\n\n\n<li>$s_i=1$\u306a\u3089\u7b26\u53f7\u3092\u53cd\u8ee2\u3059\u308b<br><\/li>\n<\/ul>\n\n\n\n<p>\u3068\u3044\u3046\u6319\u52d5\u3092\u3057\u307e\u3059\u3002\u3053\u308c\u306b\u3088\u3063\u3066\u3001$\\prod \\mathrm{sign}(m_{v&#8217;\\to u})$\u3067\u8a08\u7b97\u3055\u308c\u305f$v_j$\u4ee5\u5916\u306e\u30d1\u30ea\u30c6\u30a3\u306b\u5bfe\u3057\u3066\u3001\u305d\u308c\u3092\u53cd\u8ee2\u3059\u308b\u304b\u3069\u3046\u304b\u3092\u6c7a\u5b9a\u3067\u304d\u307e\u3059\u3002\u305d\u3082\u305d\u3082\u3001\u30b7\u30f3\u30c9\u30ed\u30fc\u30e0\u5024\u306e\u610f\u5473\u306f<br><\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>$s_i=0$\u306a\u3089\u3001\u30d1\u30ea\u30c6\u30a3\u30ce\u30fc\u30c9$u_i$\u306b\u96a3\u63a5\u3059\u308b\u5909\u6570\u30ce\u30fc\u30c9\u306b\u751f\u3058\u305f\u30a8\u30e9\u30fc\u306e\u6570\u304c\u5076\u6570\u500b<\/li>\n\n\n\n<li>$s_i=1$\u306a\u3089\u5947\u6570\u500b<br><\/li>\n<\/ul>\n\n\n\n<p>\u3067\u3059\u3002\u3064\u307e\u308a\u3001<br>$$ \\begin{align}(-1)^{s_i}=\\prod_{v_j \\in V(u_i)}\\mathrm{sign}(m_{v_j\\to u_i})\\end{align} $$<br>\u3068\u306a\u308b\u3079\u304d\u3067\u3059\u3002\u5f93\u3063\u3066\u3001<br><\/p>\n\n\n\n<div class=\"wp-block-math\"><math display=\"block\"><semantics><mrow><mrow><mtext><\/mtext><mi>sign<\/mi><\/mrow><mo form=\"prefix\" stretchy=\"false\">(<\/mo><msub><mi>m<\/mi><mrow><msub><mi>v<\/mi><mi>j<\/mi><\/msub><mo>\u2192<\/mo><msub><mi>u<\/mi><mi>i<\/mi><\/msub><\/mrow><\/msub><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mo>=<\/mo><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mo form=\"prefix\" stretchy=\"false\">\u2212<\/mo><mn>1<\/mn><msup><mo form=\"postfix\" stretchy=\"false\">)<\/mo><msub><mi>s<\/mi><mi>i<\/mi><\/msub><\/msup><mrow><munder><mo movablelimits=\"false\">\u220f<\/mo><mrow><msubsup><mi>v<\/mi><mi>j<\/mi><mo lspace=\"0em\" rspace=\"0em\" class=\"tml-prime\">\u2032<\/mo><\/msubsup><mo>\u2208<\/mo><mi>V<\/mi><mo form=\"prefix\" stretchy=\"false\">(<\/mo><msub><mi>u<\/mi><mi>i<\/mi><\/msub><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mo>\u2216<\/mo><mo form=\"prefix\" stretchy=\"false\">{<\/mo><msub><mi>v<\/mi><mi>j<\/mi><\/msub><mo form=\"postfix\" stretchy=\"false\" lspace=\"0em\" rspace=\"0em\">}<\/mo><\/mrow><\/munder><\/mrow><mrow><mtext><\/mtext><mi>sign<\/mi><\/mrow><mo form=\"prefix\" stretchy=\"false\">(<\/mo><msub><mi>m<\/mi><mrow><msubsup><mi>v<\/mi><mi>j<\/mi><mo lspace=\"0em\" rspace=\"0em\" class=\"tml-prime\">\u2032<\/mo><\/msubsup><mo>\u2192<\/mo><msub><mi>u<\/mi><mi>i<\/mi><\/msub><\/mrow><\/msub><mo form=\"postfix\" stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">\n\\mathrm{sign}(m_{v_j \\to u_i}) = (-1)^{s_i} \\prod_{v&#8217;_j \\in V(u_i) \\setminus \\{v_j\\}} \\mathrm{sign}(m_{v&#8217;_j \\to u_i})\n<\/annotation><\/semantics><\/math><\/div>\n\n\n\n<p><br>\u304c\u6210\u308a\u7acb\u3061\u307e\u3059\u3002<\/p>\n\n\n\n<p>\u3000\u7b26\u53f7\u306b\u3064\u3044\u3066\u306f\u4ee5\u4e0a2\u3064\u306e\u30d1\u30fc\u30c4\u304b\u3089\u6c7a\u5b9a\u3067\u304d\u307e\u3057\u305f\u3002\u6b8b\u308b2\u3064\u306e\u30d1\u30fc\u30c4\u306f$m_{u_i\\to v_j}$\u306e\u5927\u304d\u3055\u3092\u5236\u5fa1\u3059\u308b\u90e8\u5206\u306b\u3042\u305f\u308a\u307e\u3059\u3002<br><\/p>\n\n\n\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"%E2%91%A2_alpha\"><\/span>\u2462 $\\alpha$<span class=\"ez-toc-section-end\"><\/span><\/h3>\n\n\n\n<p>\u3000\u307e\u305a\u306f\u3001\u30b9\u30b1\u30fc\u30ea\u30f3\u30b0\u306e\u305f\u3081\u306e\u56e0\u5b50\u3068\u3057\u3066\u3042\u308b$\\alpha$\u306b\u3064\u3044\u3066\u7c21\u5358\u306b\u8aac\u660e\u3057\u307e\u3059\u3002\u3053\u306e\u90e8\u5206\u306f\u3001Min-Sum\u30a2\u30eb\u30b4\u30ea\u30ba\u30e0\u306b\u76f4\u63a5\u95a2\u4fc2\u3057\u3066\u3044\u308b\u3082\u306e\u3067\u306f\u306a\u304f\u3001\u6027\u80fd\u5411\u4e0a\u306e\u305f\u3081\u306b\u8abf\u6574\u3055\u308c\u308b\u56e0\u5b50\u3067\u3059\u3002\u3053\u3053\u3067\u306f\u3001\u53cd\u5fa9\u56de\u6570 $i$\u306b\u5fdc\u3058\u3066<br>$$ \\begin{align}\\alpha = 1-2^{-i}\\end{align} $$<br>\u3068\u5b9a\u3081\u3066\u3044\u307e\u3059\u3002<br><\/p>\n\n\n\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"%E2%91%A3_minm_vto_u\"><\/span>\u2463 $\\min|m_{v\\to u}|$<span class=\"ez-toc-section-end\"><\/span><\/h3>\n\n\n\n<p>\u3000\u7d9a\u3044\u3066\u306f\u3001$\\min|m_{v\\to u}|$\u306e\u90e8\u5206\u306b\u3064\u3044\u3066\u3067\u3059\u3002\u3053\u306e\u90e8\u5206\u306f\u76f4\u63a5\u610f\u5473\u3092\u89e3\u91c8\u3059\u308b\u3053\u3068\u306f\u56f0\u96e3\u3067\u3059\u3002Min-Sum\u30a2\u30eb\u30b4\u30ea\u30ba\u30e0\u306b\u306f\u5143\u30cd\u30bf\u3068\u306a\u308bSum-Product\u30a2\u30eb\u30b4\u30ea\u30ba\u30e0\u3068\u547c\u3070\u308c\u308b\u30a2\u30eb\u30b4\u30ea\u30ba\u30e0\u304c\u3042\u308a\u3001\u305d\u306e\u8fd1\u4f3c\u304cMin-Sum\u30a2\u30eb\u30b4\u30ea\u30ba\u30e0\u306b\u306a\u3063\u3066\u3044\u307e\u3059\u3002\u3067\u306f\u3001\u5143\u30cd\u30bf\u306eSum-Product\u30a2\u30eb\u30b4\u30ea\u30ba\u30e0\u3067\u306f\u30e1\u30c3\u30bb\u30fc\u30b8\u304c\u3069\u3046\u306a\u3063\u3066\u3044\u308b\u304b\u3068\u8a00\u3046\u3068\u3001<br><\/p>\n\n\n\n<div class=\"wp-block-math\"><math display=\"block\"><semantics><mrow><msub><mi>m<\/mi><mrow><msub><mi>u<\/mi><mi>i<\/mi><\/msub><mo>\u2192<\/mo><msub><mi>v<\/mi><mi>j<\/mi><\/msub><\/mrow><\/msub><mo>=<\/mo><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mo form=\"prefix\" stretchy=\"false\">\u2212<\/mo><mn>1<\/mn><msup><mo form=\"postfix\" stretchy=\"false\">)<\/mo><msub><mi>s<\/mi><mi>i<\/mi><\/msub><\/msup><mrow><mo fence=\"true\" form=\"prefix\">(<\/mo><mrow><munder><mo movablelimits=\"false\">\u220f<\/mo><mrow><msubsup><mi>v<\/mi><mi>j<\/mi><mo lspace=\"0em\" rspace=\"0em\" class=\"tml-prime\">\u2032<\/mo><\/msubsup><mo>\u2208<\/mo><mi>V<\/mi><mo form=\"prefix\" stretchy=\"false\">(<\/mo><msub><mi>u<\/mi><mi>i<\/mi><\/msub><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mo>\u2216<\/mo><mo form=\"prefix\" stretchy=\"false\">{<\/mo><msub><mi>v<\/mi><mi>j<\/mi><\/msub><mo form=\"postfix\" stretchy=\"false\" lspace=\"0em\" rspace=\"0em\">}<\/mo><\/mrow><\/munder><\/mrow><mrow><mtext><\/mtext><mi>sign<\/mi><\/mrow><mo form=\"prefix\" stretchy=\"false\">(<\/mo><msub><mi>m<\/mi><mrow><msubsup><mi>v<\/mi><mi>j<\/mi><mo lspace=\"0em\" rspace=\"0em\" class=\"tml-prime\">\u2032<\/mo><\/msubsup><mo>\u2192<\/mo><msub><mi>u<\/mi><mi>i<\/mi><\/msub><\/mrow><\/msub><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mo>\u00d7<\/mo><mn>2<\/mn><mspace width=\"0.1667em\"><\/mspace><msup><mi>tanh<\/mi><mrow><mo lspace=\"0em\" rspace=\"0em\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup><mo>\u2061<\/mo><mrow><mo fence=\"true\" form=\"prefix\">(<\/mo><mrow><munder><mo movablelimits=\"false\">\u220f<\/mo><mrow><msubsup><mi>v<\/mi><mi>j<\/mi><mo lspace=\"0em\" rspace=\"0em\" class=\"tml-prime\">\u2032<\/mo><\/msubsup><mo>\u2208<\/mo><mi>V<\/mi><mo form=\"prefix\" stretchy=\"false\">(<\/mo><msub><mi>u<\/mi><mi>i<\/mi><\/msub><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mo>\u2216<\/mo><mo form=\"prefix\" stretchy=\"false\">{<\/mo><msub><mi>v<\/mi><mi>j<\/mi><\/msub><mo form=\"postfix\" stretchy=\"false\" lspace=\"0em\" rspace=\"0em\">}<\/mo><\/mrow><\/munder><\/mrow><mrow><mi>tanh<\/mi><mo>\u2061<\/mo><\/mrow><mrow><mo fence=\"true\" form=\"prefix\">(<\/mo><mfrac><mrow><mi>|<\/mi><msub><mi>m<\/mi><mrow><msubsup><mi>v<\/mi><mi>j<\/mi><mo lspace=\"0em\" rspace=\"0em\" class=\"tml-prime\">\u2032<\/mo><\/msubsup><mo>\u2192<\/mo><msub><mi>u<\/mi><mi>i<\/mi><\/msub><\/mrow><\/msub><mi>|<\/mi><\/mrow><mn>2<\/mn><\/mfrac><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mrow><annotation encoding=\"application\/x-tex\">m_{u_i \\to v_j} = (-1)^{s_i} \\left( \\prod_{v&#8217;_j \\in V(u_i) \\setminus \\{v_j\\}} \\mathrm{sign}(m_{v&#8217;_j \\to u_i}) \\right) \\times 2 \\tanh^{-1} \\left( \\prod_{v&#8217;_j \\in V(u_i) \\setminus \\{v_j\\}} \\tanh \\left( \\frac{|m_{v&#8217;_j \\to u_i}|}{2} \\right) \\right)<\/annotation><\/semantics><\/math><\/div>\n\n\n\n<p><br>\u3068\u306a\u3063\u3066\u3044\u307e\u3059\u3002Min-Sum\u3068\u6700\u5927\u306e\u9055\u3044\u306f\u3001$\\min$\u306e\u90e8\u5206\u304c$\\tanh$\u3092\u542b\u3080\u5225\u306e\u95a2\u6570\u306b\u7f6e\u304d\u63db\u308f\u3063\u3066\u3044\u308b\u3053\u3068\u3067\u3059\u3002\u306a\u305ctanh\u95a2\u6570\u304c\u4f7f\u308f\u308c\u3066\u3044\u308b\u304b\u3068\u3044\u3046\u3068\u3001\u5bfe\u6570\u5c24\u5ea6\u6bd4\u95a2\u6570$L(x)=\\log(P(x=0)\/P(x=1))$\u306b\u3064\u3044\u3066<br>$$ \\begin{align}\\tanh(L(x)\/2) = (e^{L(x)} &#8211; 1) \/ (e^{L(x)} + 1) = \\frac{P(x=0) &#8211; P(x=1)}{P(x=0) + P(x=1)}=P(x=0)-P(x=1)\\end{align} $$<br>\u304c\u6210\u308a\u7acb\u3061\u3001$x=0$\u3067\u3042\u308b\u78ba\u7387\u3068$x=1$\u3067\u3042\u308b\u78ba\u7387\u306e\u5dee\u306b\u5bfe\u5fdc\u3057\u3066\u3044\u308b\u304b\u3089\u3067\u3059\u3002\u3053\u306e\u5dee\u304c\u5927\u304d\u3051\u308c\u3070\u5927\u304d\u3044\u307b\u3069\u3001\u305d\u306e\u78ba\u7387\u306e\u4fe1\u983c\u6027\u306f\u9ad8\u307e\u308a\u3001\u9006\u306b\u5dee\u304c\u5c0f\u3055\u3044\u3068\u4fe1\u983c\u6027\u304c\u4f4e\u304f\u306a\u308a\u307e\u3059\u3002\u6975\u7aef\u306a\u4f8b\u3068\u3057\u3066\u3001$P(x=0)=P(x=1)=0.5$\u3067\u3042\u308c\u3070$\\tanh(L(x)\/2)=0$\u3068\u306a\u308a\u307e\u3059\u3002\u3053\u308c\u306f\u3001$x=0$\u306a\u306e\u304b$x=1$\u306a\u306e\u304b\u5168\u304f\u308f\u304b\u3089\u306a\u3044\u3068\u3044\u3046\u72b6\u6cc1\u3092\u610f\u5473\u3057\u3066\u3044\u307e\u3059\u3002<\/p>\n\n\n\n<p>\u3000\u3053\u306e\u3088\u3046\u306b$\\tanh$\u3092\u7528\u3044\u3066\u5b9a\u7fa9\u3055\u308c\u308bSum-Product\u30a2\u30eb\u30b4\u30ea\u30ba\u30e0\u306f\u7406\u8ad6\u7684\u306b\u306f\u512a\u308c\u3066\u3044\u307e\u3059\u304c\u3001\u5b9f\u7528\u7684\u306b\u306f$\\tanh$\u306e\u8a08\u7b97\u306b\u6642\u9593\u304c\u304b\u304b\u308a\u52b9\u7387\u7684\u3067\u306f\u3042\u308a\u307e\u305b\u3093\u3002\u305d\u3053\u3067\u3001$\\tanh$\u306e\u90e8\u5206\u3092$\\min$\u3092\u4f7f\u3063\u305f\u95a2\u6570\u306b\u7f6e\u304d\u63db\u3048\u305f\u3082\u306e\u304cMin-Sum\u30a2\u30eb\u30b4\u30ea\u30ba\u30e0\u3067\u3059\u3002$\\min$\u3092\u4f7f\u3063\u305f\u95a2\u6570\u3067\u8fd1\u4f3c\u3067\u304d\u308b\u3053\u3068\u3092\u78ba\u8a8d\u3059\u308b\u305f\u3081\u306b\u30012\u5909\u6570\u306e\u5358\u7d14\u306a\u5834\u5408\u3092\u8003\u3048\u307e\u3059\u3002Sum-Product\u306e\u30e1\u30c3\u30bb\u30fc\u30b8\u8a08\u7b97\u306b\u4f7f\u308f\u308c\u308b\u95a2\u6570\u306e2\u5909\u6570\u7248\u306f<br>$$ \\begin{align}z = 2\\tanh^{-1}\\left(\\tanh\\left(\\frac {|x|}2\\right)\\tanh\\left(\\frac {|y|}2\\right)\\right)\\end{align} $$<br>\u3067\u3042\u308a\u3001\u305d\u308c\u3092\u6b21\u306e\u5f0f\u3067\u8fd1\u4f3c\u3057\u307e\u3059\u3002<br>$$ \\begin{align}z = \\min\\{|x|,|y|\\}\\end{align} $$<br>\u304b\u306a\u308a\u5358\u7d14\u306a\u5f0f\u306b\u306a\u308a\u307e\u3059\u304c\u3001\u3053\u308c\u30892\u3064\u306e\u95a2\u6570\u306f\u5916\u5f62\u304c\u3088\u304f\u4f3c\u3066\u3044\u307e\u3059\u3002\u305d\u306e\u3053\u3068\u3092\u898b\u308b\u305f\u3081\u306b2\u3064\u306e\u95a2\u6570\u3092\u6bd4\u8f03\u3057\u305f\u30b0\u30e9\u30d5\u3092\u6b21\u306b\u793a\u3057\u307e\u3059\u3002\u305f\u3060\u3057\u3001 $x = y$\u3068\u306a\u308b2\u6b21\u5143\u5e73\u9762\u3067\u5207\u65ad\u3057\u305f\u30b0\u30e9\u30d5\u3067\u3042\u308b\u3053\u3068\u306b\u6ce8\u610f\u3057\u3066\u304f\u3060\u3055\u3044<br><\/p>\n\n\n\n<figure class=\"wp-block-image size-large\"><img loading=\"lazy\" decoding=\"async\" width=\"1024\" height=\"768\" src=\"https:\/\/qard.is.tohoku.ac.jp\/T-Wave\/wp-content\/uploads\/2026\/03\/tanh_vs_min-1024x768.png\" alt=\"\" class=\"wp-image-8773\" srcset=\"https:\/\/qard.is.tohoku.ac.jp\/T-Wave\/wp-content\/uploads\/2026\/03\/tanh_vs_min-1024x768.png 1024w, https:\/\/qard.is.tohoku.ac.jp\/T-Wave\/wp-content\/uploads\/2026\/03\/tanh_vs_min-300x225.png 300w, https:\/\/qard.is.tohoku.ac.jp\/T-Wave\/wp-content\/uploads\/2026\/03\/tanh_vs_min-768x576.png 768w, https:\/\/qard.is.tohoku.ac.jp\/T-Wave\/wp-content\/uploads\/2026\/03\/tanh_vs_min.png 1229w\" sizes=\"auto, (max-width: 1024px) 100vw, 1024px\" \/><figcaption class=\"wp-element-caption\">\u56f37 $\\tanh$\u3068$\\min$\u306e\u6bd4\u8f03<\/figcaption><\/figure>\n\n\n\n<p>\u9752\u3044\u7dda\u304c $\\tanh$\u3092\u7528\u3044\u305f\u771f\u306e\u5024\u3067\u3001\u30aa\u30ec\u30f3\u30b8\u306e\u7dda\u304c $\\min$\u3092\u7528\u3044\u305f\u8fd1\u4f3c\u3092\u8868\u3057\u3066\u3044\u307e\u3059\u3002\u5024\u306f\u7570\u306a\u308a\u307e\u3059\u304c\u3001\u30b0\u30e9\u30d5\u306e\u6982\u5f62\u304c\u3088\u304f\u4f3c\u3066\u3044\u308b\u4e8b\u304c\u308f\u304b\u308a\u307e\u3059\u3002<br>\u3053\u306e\u3053\u3068\u304b\u3089\u3001\u30e1\u30c3\u30bb\u30fc\u30b8\u8a08\u7b97\u306e\u305f\u3081\u306e\u95a2\u6570<br><\/p>\n\n\n\n<div class=\"wp-block-math\"><math display=\"block\"><semantics><mrow><mn>2<\/mn><mspace width=\"0.1667em\"><\/mspace><msup><mi>tanh<\/mi><mrow><mo lspace=\"0em\" rspace=\"0em\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup><mo>\u2061<\/mo><mrow><mo fence=\"true\" form=\"prefix\">(<\/mo><mrow><munder><mo movablelimits=\"false\">\u220f<\/mo><mrow><msubsup><mi>v<\/mi><mi>j<\/mi><mo lspace=\"0em\" rspace=\"0em\" class=\"tml-prime\">\u2032<\/mo><\/msubsup><mo>\u2208<\/mo><mi>V<\/mi><mo form=\"prefix\" stretchy=\"false\">(<\/mo><msub><mi>u<\/mi><mi>i<\/mi><\/msub><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mo>\u2216<\/mo><mo form=\"prefix\" stretchy=\"false\">{<\/mo><msub><mi>v<\/mi><mi>j<\/mi><\/msub><mo form=\"postfix\" stretchy=\"false\" lspace=\"0em\" rspace=\"0em\">}<\/mo><\/mrow><\/munder><\/mrow><mrow><mi>tanh<\/mi><mo>\u2061<\/mo><\/mrow><mrow><mo fence=\"true\" form=\"prefix\">(<\/mo><mfrac><mrow><mi>|<\/mi><msub><mi>m<\/mi><mrow><msubsup><mi>v<\/mi><mi>j<\/mi><mo lspace=\"0em\" rspace=\"0em\" class=\"tml-prime\">\u2032<\/mo><\/msubsup><mo>\u2192<\/mo><msub><mi>u<\/mi><mi>i<\/mi><\/msub><\/mrow><\/msub><mi>|<\/mi><\/mrow><mn>2<\/mn><\/mfrac><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mrow><annotation encoding=\"application\/x-tex\">\n2 \\tanh^{-1} \\left( \\prod_{v&#8217;_j \\in V(u_i) \\setminus \\{v_j\\}} \\tanh \\left( \\frac{|m_{v&#8217;_j \\to u_i}|}{2} \\right) \\right)\n<\/annotation><\/semantics><\/math><\/div>\n\n\n\n<p><br>\u3092\u8a08\u7b97\u304c\u7c21\u5358\u306a\u95a2\u6570<br><\/p>\n\n\n\n<div class=\"wp-block-math\"><math display=\"block\"><semantics><mrow><munder><mi>min<\/mi><mrow><msup><mi>v<\/mi><mo lspace=\"0em\" rspace=\"0em\" class=\"tml-prime\">\u2032<\/mo><\/msup><mi>j<\/mi><mo>\u2208<\/mo><mi>V<\/mi><mo form=\"prefix\" stretchy=\"false\">(<\/mo><msub><mi>u<\/mi><mi>i<\/mi><\/msub><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mo>\u2216<\/mo><msub><mi>v<\/mi><mi>j<\/mi><\/msub><\/mrow><\/munder><mo>\u2061<\/mo><mspace width=\"0.1667em\"><\/mspace><mrow><mi>|<\/mi><mi>m<\/mi><mrow><msubsup><mi>v<\/mi><mi>j<\/mi><mo lspace=\"0em\" rspace=\"0em\" class=\"tml-prime\">\u2032<\/mo><\/msubsup><mo>\u2192<\/mo><msub><mi>u<\/mi><mi>i<\/mi><\/msub><\/mrow><mi>|<\/mi><\/mrow><\/mrow><annotation encoding=\"application\/x-tex\">\\min_{v&#8217;j \\in V(u_i) \\setminus {v_j}} { |m{v&#8217;_j \\to u_i}| }\n<\/annotation><\/semantics><\/math><\/div>\n\n\n\n<p>\u3067\u8fd1\u4f3c\u3057\u305f\u3082\u306e\u304cMin-Sum\u30a2\u30eb\u30b4\u30ea\u30ba\u30e0\u306b\u306a\u308a\u307e\u3059\u3002<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n<ol class=\"wp-block-footnotes\"><li id=\"3b90fa1b-4bc1-4064-af22-e0d656deb491\">\u3053\u306e\u3088\u3046\u306a\u7b26\u53f7\u306f $X$ \u30a8\u30e9\u30fc\u3060\u3051\u3092\u691c\u51fa\u3067\u304d\u308b\u306e\u3067\u53e4\u5178\u8aa4\u308a\u8a02\u6b63\u7b26\u53f7\u3068\u540c\u4e00\u8996\u3067\u304d\u307e\u3059 <a href=\"#3b90fa1b-4bc1-4064-af22-e0d656deb491-link\" aria-label=\"\u811a\u6ce8\u53c2\u71671\u306b\u30b8\u30e3\u30f3\u30d7\">\u21a9\ufe0e<\/a><\/li><li id=\"dcd6f7e9-e6f8-4fad-9191-b0535b7e2807\">\u4e00\u822c\u306b\u306f\u3001\u5909\u6570\u30ce\u30fc\u30c9\u306f\u30a8\u30e9\u30fc\u306e\u767a\u751f\u7b87\u6240\u3068\u5bfe\u5fdc\u3057\u3066\u3044\u307e\u3059\u3002\u3057\u305f\u304c\u3063\u3066\u3001\u91cf\u5b50\u30d3\u30c3\u30c8\u4ee5\u5916\u306b\u3082\u30a8\u30e9\u30fc\u304c\u8d77\u3053\u308b\u5834\u5408\u3067\u3082\u8868\u73fe\u3067\u304d\u307e\u3059\u3002\u3053\u306e\u5834\u5408\u306f\u3001\u30d1\u30ea\u30c6\u30a3\u30c1\u30a7\u30c3\u30af\u884c\u5217\u3060\u3051\u3067\u306a\u304f\u3001\u64cd\u4f5c\u5168\u4f53\u306e\u30a8\u30e9\u30fc\u3068\u30b7\u30f3\u30c9\u30ed\u30fc\u30e0\u5024\u306e\u95a2\u4fc2\u3092\u8a18\u8ff0\u3059\u308b\u5fc5\u8981\u304c\u3042\u308a\u307e\u3059\u3002\u3053\u306e\u3088\u3046\u306a\u30e2\u30c7\u30eb\u3068\u3057\u3066detector error model\u3068\u547c\u3070\u308c\u308b\u3082\u306e\u304c\u77e5\u3089\u308c\u3066\u3044\u3066\u3001\u3088\u308a\u73fe\u5b9f\u7684\u306a\u30ce\u30a4\u30ba\u30e2\u30c7\u30eb\u3092\u8a18\u8ff0\u3067\u304d\u307e\u3059\u3002 <a href=\"#dcd6f7e9-e6f8-4fad-9191-b0535b7e2807-link\" aria-label=\"\u811a\u6ce8\u53c2\u71672\u306b\u30b8\u30e3\u30f3\u30d7\">\u21a9\ufe0e<\/a><\/li><li id=\"d000d753-0c30-4c6b-8300-256e84027910\">\u4e00\u822c\u306e\u30d5\u30a1\u30af\u30bf\u30fc\u30b0\u30e9\u30d5\u306b\u3064\u3044\u3066\u3001\u3053\u306e\u7e70\u308a\u8fd4\u3057\u3067\u6b63\u78ba\u306a\u4e8b\u5f8c\u5206\u5e03\u306b\u53ce\u675f\u3059\u308b\u3068\u306f\u9650\u3089\u306a\u3044\u3053\u3068\u306b\u6ce8\u610f\u304c\u5fc5\u8981\u3067\u3059\u3002\u3080\u3057\u308d\u3001\u6b63\u78ba\u306a\u4e8b\u5f8c\u5206\u5e03\u306b\u53ce\u675f\u3059\u308b\u3053\u3068\u304c\u4fdd\u8a3c\u3055\u308c\u3066\u3044\u308b\u306e\u306f\u30d5\u30a1\u30af\u30bf\u30fc\u30b0\u30e9\u30d5\u304c\u6728\u69cb\u9020\uff08\u9589\u8def\u304c\u306a\u3044\u30b0\u30e9\u30d5\uff09\u306e\u3068\u304d\u306e\u307f\u3067\u3059\u3002\u3064\u307e\u308a\u3001\u9589\u8def\u304c\u542b\u307e\u308c\u308b\u30d5\u30a1\u30af\u30bf\u30fc\u30b0\u30e9\u30d5\u306b\u3064\u3044\u3066\u306f\u4e0d\u6b63\u78ba\u306a\u30a2\u30eb\u30b4\u30ea\u30ba\u30e0\u306b\u306a\u308a\u307e\u3059\u3002\u3057\u304b\u3057\u3001\u9589\u8def\u304c\u542b\u307e\u308c\u308b\u30d5\u30a1\u30af\u30bf\u30fc\u30b0\u30e9\u30d5\u306b\u3064\u3044\u3066\u3082\u3042\u308b\u7a0b\u5ea6\u306e\u7cbe\u5ea6\u3067\u4e8b\u5f8c\u5206\u5e03\u3092\u6c42\u3081\u3089\u308c\u308b\u3053\u3068\u304c\u77e5\u3089\u308c\u3066\u3044\u3066\u3001loopy belief propagation\u3068\u547c\u3070\u308c\u3066\u3044\u307e\u3059\u3002loopy belief propagation\u306b\u3064\u3044\u3066\u306f\u53ce\u675f\u6027\u306b\u3064\u3044\u3066\u89e3\u6790\u3059\u308b\u3053\u3068\u304c\u96e3\u3057\u304f\u3001\u53ce\u675f\u306e\u6b63\u78ba\u306a\u6761\u4ef6\u306f\u308f\u304b\u3063\u3066\u3044\u306a\u3044\u3088\u3046\u3067\u3059\u3002 <a href=\"#d000d753-0c30-4c6b-8300-256e84027910-link\" aria-label=\"\u811a\u6ce8\u53c2\u71673\u306b\u30b8\u30e3\u30f3\u30d7\">\u21a9\ufe0e<\/a><\/li><\/ol>\n\n\n<p><\/p>\n\n\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"%E6%9C%AC%E8%A8%98%E4%BA%8B%E3%81%AE%E6%8B%85%E5%BD%93%E8%80%85\"><\/span>\u672c\u8a18\u4e8b\u306e\u62c5\u5f53\u8005<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n<div class=\"sabox-plus-item\"><div class=\"saboxplugin-wrap\" itemtype=\"http:\/\/schema.org\/Person\" itemscope itemprop=\"author\"><div class=\"saboxplugin-tab\"><div class=\"saboxplugin-gravatar\"><img alt='Shun Hasegawa' src='https:\/\/secure.gravatar.com\/avatar\/86624558a04468337363ffec6f104bc50416c2dd36dd047dbd1fa04c75d58ed2?s=100&#038;d=mm&#038;r=g' srcset='https:\/\/secure.gravatar.com\/avatar\/86624558a04468337363ffec6f104bc50416c2dd36dd047dbd1fa04c75d58ed2?s=200&#038;d=mm&#038;r=g 2x' class='avatar avatar-100 photo' height='100' width='100' itemprop=\"image\"\/><\/div><div class=\"saboxplugin-authorname\"><a href=\"https:\/\/qard.is.tohoku.ac.jp\/T-Wave\/author\/shun-hasegawa\/\" class=\"vcard author\" rel=\"author\"><span class=\"fn\">Shun Hasegawa<\/span><\/a><\/div><div class=\"saboxplugin-desc\"><div itemprop=\"description\"><p>\u6771\u4eac\u79d1\u5b66\u5927\u5b66 \u60c5\u5831\u7406\u5de5\u5b66\u9662 \u6570\u7406\u30fb\u8a08\u7b97\u79d1\u5b66\u7cfb \u5b66\u58eb\u8ab2\u7a0b<\/p>\n<\/div><\/div><div class=\"clearfix\"><\/div><\/div><\/div><\/div>\n\n\n<p><\/p>\n","protected":false},"excerpt":{"rendered":"<p>qLDPC\u7b26\u53f7\u306e\u30c7\u30b3\u30fc\u30c0\u3067\u3042\u308bBP+OSD\u306b\u3064\u3044\u3066\u89e3\u8aac\u3057\u307e\u3059\u3002#2\u3067\u306f\u3001\u30c7\u30b3\u30fc\u30c0\u306e\u4e2d\u5fc3\u3067\u3042\u308bBP\u30c7\u30b3\u30fc\u30c0\u306b\u3064\u3044\u3066\u89e3\u8aac\u3057\u307e\u3059\u3002<\/p>\n","protected":false},"author":25,"featured_media":8768,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":"[{\"content\":\"\u3053\u306e\u3088\u3046\u306a\u7b26\u53f7\u306f $X$ \u30a8\u30e9\u30fc\u3060\u3051\u3092\u691c\u51fa\u3067\u304d\u308b\u306e\u3067\u53e4\u5178\u8aa4\u308a\u8a02\u6b63\u7b26\u53f7\u3068\u540c\u4e00\u8996\u3067\u304d\u307e\u3059\",\"id\":\"3b90fa1b-4bc1-4064-af22-e0d656deb491\"},{\"content\":\"\u4e00\u822c\u306b\u306f\u3001\u5909\u6570\u30ce\u30fc\u30c9\u306f\u30a8\u30e9\u30fc\u306e\u767a\u751f\u7b87\u6240\u3068\u5bfe\u5fdc\u3057\u3066\u3044\u307e\u3059\u3002\u3057\u305f\u304c\u3063\u3066\u3001\u91cf\u5b50\u30d3\u30c3\u30c8\u4ee5\u5916\u306b\u3082\u30a8\u30e9\u30fc\u304c\u8d77\u3053\u308b\u5834\u5408\u3067\u3082\u8868\u73fe\u3067\u304d\u307e\u3059\u3002\u3053\u306e\u5834\u5408\u306f\u3001\u30d1\u30ea\u30c6\u30a3\u30c1\u30a7\u30c3\u30af\u884c\u5217\u3060\u3051\u3067\u306a\u304f\u3001\u64cd\u4f5c\u5168\u4f53\u306e\u30a8\u30e9\u30fc\u3068\u30b7\u30f3\u30c9\u30ed\u30fc\u30e0\u5024\u306e\u95a2\u4fc2\u3092\u8a18\u8ff0\u3059\u308b\u5fc5\u8981\u304c\u3042\u308a\u307e\u3059\u3002\u3053\u306e\u3088\u3046\u306a\u30e2\u30c7\u30eb\u3068\u3057\u3066detector error model\u3068\u547c\u3070\u308c\u308b\u3082\u306e\u304c\u77e5\u3089\u308c\u3066\u3044\u3066\u3001\u3088\u308a\u73fe\u5b9f\u7684\u306a\u30ce\u30a4\u30ba\u30e2\u30c7\u30eb\u3092\u8a18\u8ff0\u3067\u304d\u307e\u3059\u3002\",\"id\":\"dcd6f7e9-e6f8-4fad-9191-b0535b7e2807\"},{\"content\":\"\u4e00\u822c\u306e\u30d5\u30a1\u30af\u30bf\u30fc\u30b0\u30e9\u30d5\u306b\u3064\u3044\u3066\u3001\u3053\u306e\u7e70\u308a\u8fd4\u3057\u3067\u6b63\u78ba\u306a\u4e8b\u5f8c\u5206\u5e03\u306b\u53ce\u675f\u3059\u308b\u3068\u306f\u9650\u3089\u306a\u3044\u3053\u3068\u306b\u6ce8\u610f\u304c\u5fc5\u8981\u3067\u3059\u3002\u3080\u3057\u308d\u3001\u6b63\u78ba\u306a\u4e8b\u5f8c\u5206\u5e03\u306b\u53ce\u675f\u3059\u308b\u3053\u3068\u304c\u4fdd\u8a3c\u3055\u308c\u3066\u3044\u308b\u306e\u306f\u30d5\u30a1\u30af\u30bf\u30fc\u30b0\u30e9\u30d5\u304c\u6728\u69cb\u9020\uff08\u9589\u8def\u304c\u306a\u3044\u30b0\u30e9\u30d5\uff09\u306e\u3068\u304d\u306e\u307f\u3067\u3059\u3002\u3064\u307e\u308a\u3001\u9589\u8def\u304c\u542b\u307e\u308c\u308b\u30d5\u30a1\u30af\u30bf\u30fc\u30b0\u30e9\u30d5\u306b\u3064\u3044\u3066\u306f\u4e0d\u6b63\u78ba\u306a\u30a2\u30eb\u30b4\u30ea\u30ba\u30e0\u306b\u306a\u308a\u307e\u3059\u3002\u3057\u304b\u3057\u3001\u9589\u8def\u304c\u542b\u307e\u308c\u308b\u30d5\u30a1\u30af\u30bf\u30fc\u30b0\u30e9\u30d5\u306b\u3064\u3044\u3066\u3082\u3042\u308b\u7a0b\u5ea6\u306e\u7cbe\u5ea6\u3067\u4e8b\u5f8c\u5206\u5e03\u3092\u6c42\u3081\u3089\u308c\u308b\u3053\u3068\u304c\u77e5\u3089\u308c\u3066\u3044\u3066\u3001loopy 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