{"id":8813,"date":"2026-04-30T18:30:46","date_gmt":"2026-04-30T09:30:46","guid":{"rendered":"https:\/\/qard.is.tohoku.ac.jp\/T-Wave\/?p=8813"},"modified":"2026-04-30T18:30:46","modified_gmt":"2026-04-30T09:30:46","slug":"bposd-3-osd%e3%81%a8bp%e3%81%ae%e9%80%a3%e6%90%ba","status":"publish","type":"post","link":"https:\/\/qard.is.tohoku.ac.jp\/T-Wave\/2026\/04\/30\/bposd-3-osd%e3%81%a8bp%e3%81%ae%e9%80%a3%e6%90%ba\/","title":{"rendered":"BP+OSD #3 OSD\u3068BP\u306e\u9023\u643a"},"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-3-osd%e3%81%a8bp%e3%81%ae%e9%80%a3%e6%90%ba\/#%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-3-osd%e3%81%a8bp%e3%81%ae%e9%80%a3%e6%90%ba\/#%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-3-osd%e3%81%a8bp%e3%81%ae%e9%80%a3%e6%90%ba\/#OSD%E3%81%AE%E3%83%A2%E3%83%81%E3%83%99%E3%83%BC%E3%82%B7%E3%83%A7%E3%83%B3\" >OSD\u306e\u30e2\u30c1\u30d9\u30fc\u30b7\u30e7\u30f3<\/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-3-osd%e3%81%a8bp%e3%81%ae%e9%80%a3%e6%90%ba\/#%E6%83%85%E5%A0%B1%E9%9B%86%E5%90%88\" >\u60c5\u5831\u96c6\u5408<\/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-3-osd%e3%81%a8bp%e3%81%ae%e9%80%a3%e6%90%ba\/#OSD%E3%81%AE%E6%89%8B%E9%A0%86\" >OSD\u306e\u624b\u9806<\/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-3-osd%e3%81%a8bp%e3%81%ae%e9%80%a3%e6%90%ba\/#OSD%E3%81%AE%E3%83%90%E3%83%AA%E3%82%A8%E3%83%BC%E3%82%B7%E3%83%A7%E3%83%B3\" >OSD\u306e\u30d0\u30ea\u30a8\u30fc\u30b7\u30e7\u30f3<\/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-3-osd%e3%81%a8bp%e3%81%ae%e9%80%a3%e6%90%ba\/#%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%B8%E3%81%AE%E6%8B%A1%E5%BC%B5\" >\u91cf\u5b50\u8aa4\u308a\u8a02\u6b63\u7b26\u53f7\u3078\u306e\u62e1\u5f35<\/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-3-osd%e3%81%a8bp%e3%81%ae%e9%80%a3%e6%90%ba\/#BP%E3%81%A8OSD%E3%81%AE%E7%B5%B1%E5%90%88%EF%BC%9ABPOSD%E3%83%87%E3%82%B3%E3%83%BC%E3%83%80\" >BP\u3068OSD\u306e\u7d71\u5408\uff1aBP+OSD\u30c7\u30b3\u30fc\u30c0<\/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-3-osd%e3%81%a8bp%e3%81%ae%e9%80%a3%e6%90%ba\/#%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-3-osd%e3%81%a8bp%e3%81%ae%e9%80%a3%e6%90%ba\/#%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\u306e\u8a18\u4e8b\u3067\u306f\u3001LLR-MSA\u3068\u547c\u3070\u308c\u308bBP\u3092\u4f7f\u3063\u305f\u30c7\u30b3\u30fc\u30c0\u306b\u3064\u3044\u3066\u7d39\u4ecb\u3057\u307e\u3057\u305f\u3002LLR-MSA\u306f\u30b7\u30f3\u30c9\u30ed\u30fc\u30e0\u5024\u304b\u3089\u30a8\u30e9\u30fc\u306e\u8d77\u304d\u305f\u78ba\u7387\u3092\u52b9\u7387\u3088\u304f\u6c42\u3081\u308b\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u3002\u4e00\u65b9\u3067\u3001\u91cf\u5b50\u8aa4\u308a\u8a02\u6b63\u7b26\u53f7\u306e\u7e2e\u9000\u3068\u547c\u3070\u308c\u308b\u6027\u8cea\u3084\u30d5\u30a1\u30af\u30bf\u30fc\u30b0\u30e9\u30d5\u306e\u9589\u8def\u306e\u5f71\u97ff\u3067\u30a8\u30e9\u30fc\u78ba\u7387\u304c\u6b63\u3057\u304f\u63a8\u5b9a\u3067\u304d\u306a\u3044\u3068\u3044\u3046\u554f\u984c\u304c\u3042\u308a\u307e\u3057\u305f\u3002<\/p>\n\n\n\n<p>\u4eca\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\uff08Ordered Statistics Decoding, \u4ee5\u4e0b\u300cOSD\u300d\uff09\u3068\u547c\u3070\u308c\u308b\u30a2\u30eb\u30b4\u30ea\u30ba\u30e0\u3092\u7d39\u4ecb\u3057\u307e\u3059\u3002OSD\u3092BP\u306e\u76f4\u5f8c\u306b\u5b9f\u884c\u3059\u308b\u3053\u3068\u3067\u3001\u30c7\u30b3\u30fc\u30c0\u306e\u7cbe\u5ea6\u3092\u5411\u4e0a\u3055\u305b\u308b\u4e8b\u304c\u3067\u304d\u307e\u3059\u3002\u306f\u3058\u3081\u306b\u53e4\u5178\u8aa4\u308a\u8a02\u6b63\u7b26\u53f7\u306eOSD\u306b\u3064\u3044\u3066\u3001\u305d\u306e\u30e2\u30c1\u30d9\u30fc\u30b7\u30e7\u30f3\u3068\u5177\u4f53\u7684\u306a\u4ed5\u7d44\u307f\u3064\u3044\u3066\u89e3\u8aac\u3057\u307e\u3059\u3002\u7d9a\u3044\u3066\u3001\u53e4\u5178OSD\u3092\u91cf\u5b50\u8aa4\u308a\u8a02\u6b63\u7b26\u53f7\u306b\u5bfe\u5fdc\u3059\u308b\u5f62\u306b\u62e1\u5f35\u3057\u307e\u3059\u3002\u6700\u5f8c\u306b\u3001BP\u3068OSD\u3092\u7d71\u5408\u3057\u3066\u3053\u306e\u8a18\u4e8b\u306e\u4e3b\u5f79\u3067\u3042\u308bBP+OSD\u30c7\u30b3\u30fc\u30c0\u306e\u5168\u4f53\u50cf\u3092\u89e3\u8aac\u3057\u307e\u3059\u3002<\/p>\n\n\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"OSD%E3%81%AE%E3%83%A2%E3%83%81%E3%83%99%E3%83%BC%E3%82%B7%E3%83%A7%E3%83%B3\"><\/span>OSD\u306e\u30e2\u30c1\u30d9\u30fc\u30b7\u30e7\u30f3<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p>\u3053\u3053\u307e\u3067\u3067\u7d39\u4ecb\u3057\u305f\u78ba\u7387\u4f1d\u64ad\u6cd5\u3092\u7528\u3044\u305f\u30a2\u30eb\u30b4\u30ea\u30ba\u30e0\u3092\u7528\u3044\u3066\u3001\u30b7\u30f3\u30c9\u30ed\u30fc\u30e0\u5024$s$\u304b\u3089 $\\mathrm{Pr}(e_j=1|s)$\u3092\u6c42\u3081\u308b\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u3002\u3053\u306e\u4e8b\u5f8c\u78ba\u7387\u3092 $p_j$\u3068\u3057\u3066\u3001<\/p>\n\n\n\n<p>$$ \\begin{align}\\hat e_j=\\begin{cases}0&amp;p_j \\le 0.5 \\\\ 1&amp;p_j&gt;0.5\\end{cases}\\end{align} $$<\/p>\n\n\n\n<p>\u306e\u3088\u3046\u306b\u3057\u3066\u63a8\u5b9a\u30a8\u30e9\u30fc$\\hat e$\u3092\u5f97\u308b\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u3002MAP\u63a8\u5b9a\u3068\u3057\u3066\u306f\u3042\u308b\u7a0b\u5ea6\u306e\u7cbe\u5ea6\u304c\u3042\u308b\u3068\u671f\u5f85\u3067\u304d\u307e\u3059\u304c\u3001\u5b9f\u969b\u306b<\/p>\n\n\n\n<p>$$ \\begin{align}H\\hat e=s\\end{align} $$<\/p>\n\n\n\n<p>\u3092\u6e80\u305f\u3055\u306a\u3044\u3068\u30a8\u30e9\u30fc\u3068\u3057\u3066\u306e\u6761\u4ef6\u3092\u6e80\u305f\u3057\u307e\u305b\u3093\u3002<\/p>\n\n\n\n<p>\u4eee\u306b\u63a8\u5b9a\u30a8\u30e9\u30fc$\\hat e$\u304c\u3053\u306e\u6761\u4ef6\u3092\u6e80\u305f\u3055\u306a\u304b\u3063\u305f\u3068\u3057\u305f\u3089\u3069\u3046\u3057\u305f\u3089\u3044\u3044\u3067\u3057\u3087\u3046\u304b\uff1f\u30c7\u30b3\u30fc\u30c0\u306e\u8a02\u6b63\u6027\u80fd\u3092\u9ad8\u3081\u308b\u305f\u3081\u306b\u306f\u3001\u3053\u306e\u63a8\u5b9a\u30a8\u30e9\u30fc$\\hat e$\u306b\u50c5\u304b\u306a\u4fee\u6b63\u3092\u52a0\u3048\u3066\u30b7\u30f3\u30c9\u30ed\u30fc\u30e0\u6761\u4ef6\u3092\u6e80\u305f\u3059\u3088\u3046\u306b\u3059\u308b\u3053\u3068\u304c\u8003\u3048\u3089\u308c\u307e\u3059\u3002\u63a8\u5b9a\u30a8\u30e9\u30fc\u306f\u6761\u4ef6\u3092\u6e80\u305f\u3057\u3066\u3044\u306a\u304f\u3066\u3082\u3001\u771f\u306e\u30a8\u30e9\u30fc\u306b\u8fd1\u3044\u3053\u3068\u304c\u671f\u5f85\u3067\u304d\u307e\u3059\u3002\u305d\u3053\u3067\u3001\u4fee\u6b63\u306e\u57fa\u672c\u6226\u7565\u3092\u300c<strong>$\\hat e$\u306e\u6210\u5206\u306e\u4e2d\u3067\u4fe1\u983c\u6027\u306e\u9ad8\u3044\u6210\u5206\u3092\u6b8b\u3057\u3066\u3001\u305d\u308c\u4ee5\u5916\u3092\u6761\u4ef6\u3092\u6e80\u305f\u3059\u3088\u3046\u306b\u7f6e\u304d\u63db\u3048\u308b<\/strong>\u300d\u3068\u3057\u307e\u3059\u3002\u3053\u308c\u304c<strong>\u9806\u5e8f\u7d71\u8a08\u91cf\u30c7\u30b3\u30fc\u30c7\u30a3\u30f3\u30b0<\/strong>(ordered statistics decoding, OSD)\u306e\u57fa\u672c\u7684\u306a\u30a2\u30a4\u30c7\u30a2\u306b\u306a\u308a\u307e\u3059\u3002\u3053\u306e\u30a2\u30eb\u30b4\u30ea\u30ba\u30e0\u306f\u63a2\u7d22\u7a7a\u9593\u306e\u7279\u6b8a\u306a\u69cb\u9020\u306e\u304a\u304b\u3052\u3067\u8caa\u6b32\u6cd5\u3092\u9069\u7528\u3059\u308b\u3053\u3068\u304c\u3067\u304d\u3001\u3042\u308b\u7a0b\u5ea6\u52b9\u7387\u304c\u826f\u304f\u7cbe\u5ea6\u3082\u3044\u3044\u30a2\u30eb\u30b4\u30ea\u30ba\u30e0\u306b\u306a\u3063\u3066\u3044\u307e\u3059\u3002<\/p>\n\n\n\n<p>\u4ee5\u4e0b\u3067\u306f\u53e4\u5178\u8aa4\u308a\u8a02\u6b63\u7b26\u53f7\u306b\u3064\u3044\u3066\u306e\u9806\u5e8f\u7d71\u8a08\u91cf\u30c7\u30b3\u30fc\u30c7\u30a3\u30f3\u30b0\u3092\u7d39\u4ecb\u3057\u305f\u5f8c\u306b\u3001\u305d\u308c\u3092\u91cf\u5b50\u8aa4\u308a\u8a02\u6b63\u7b26\u53f7\u306b\u9069\u7528\u3059\u308b\u65b9\u6cd5\u3092\u8aac\u660e\u3057\u307e\u3059\u3002\u3053\u3053\u304b\u3089\u5148\u306f\u9806\u5e8f\u7d71\u8a08\u91cf\u30c7\u30b3\u30fc\u30c7\u30a3\u30f3\u30b0\u306e\u3053\u3068\u3092OSD\u3068\u7565\u3057\u3066\u8868\u8a18\u3057\u307e\u3059\u3002<\/p>\n\n\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"%E6%83%85%E5%A0%B1%E9%9B%86%E5%90%88\"><\/span>\u60c5\u5831\u96c6\u5408<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p>\u307e\u305a\u306f\u3001OSD\u306e\u30a2\u30eb\u30b4\u30ea\u30ba\u30e0\u306e\u8aac\u660e\u306b\u5fc5\u8981\u3068\u306a\u308b\u60c5\u5831\u96c6\u5408\u3068\u547c\u3070\u308c\u308b\u6982\u5ff5\u3092\u5c0e\u5165\u3057\u307e\u3059\u3002<\/p>\n\n\n\n<p>\u6dfb\u5b57\u96c6\u5408$I\\subseteq [n]$\u304c\u3001\u30d1\u30e9\u30e1\u30fc\u30bf\u304c$[n,k,d]$\u3067\u3042\u308b\u53e4\u5178\u8aa4\u308a\u8a02\u6b63\u7b26\u53f7$\\mathcal C$\u306b\u5bfe\u3057\u3066\u3001<\/p>\n\n\n\n<p>$$ \\begin{align}\\mathcal C_I=\\{c_I\\mid c\\in \\mathcal C\\} = \\mathbb Z^k_2\\end{align} $$<\/p>\n\n\n\n<p>\u3092\u307f\u305f\u3059\u3068\u304d\u3001$I$\u306f\u60c5\u5831\u96c6\u5408\u3067\u3042\u308b\u3068\u3044\u3044\u307e\u3059\u3002\u3053\u3053\u3067\u3001$c_I$\u306f\u30d3\u30c3\u30c8\u5217$c$\u3092\u6dfb\u5b57\u96c6\u5408$I$\u3067\u9650\u5b9a\u3057\u305f\u3082\u306e\u3092\u8868\u3057\u307e\u3059\u3002\u3064\u307e\u308a\u3001$I=\\{i_1,&#8230;,i_k\\}$\u306a\u3089\u3001<\/p>\n\n\n\n<p>$$ \\begin{align}c_I=(c_{i_1},&#8230;,c_{i_k})\\end{align} $$<\/p>\n\n\n\n<p>\u3068\u3044\u3046\u3053\u3068\u306b\u306a\u308a\u307e\u3059\u3002\u540c\u69d8\u306b\u3057\u3066\u884c\u5217$H$\u306b\u3064\u3044\u3066\u3082\u3001$H=(h_1,&#8230;,h_n)$\u306b\u3064\u3044\u3066<\/p>\n\n\n\n<p>$$ \\begin{align}H_I=(h_{i_1},&#8230;,h_{i_k})\\end{align} $$<\/p>\n\n\n\n<p>\u3068\u3057\u307e\u3059\u3002<\/p>\n\n\n\n<p>\u60c5\u5831\u96c6\u5408\u306b\u306f\u30d1\u30ea\u30c6\u30a3\u30c1\u30a7\u30c3\u30af\u884c\u5217\u3092\u7528\u3044\u305f\u540c\u5024\u306a\u8a00\u3044\u63db\u3048\u304c\u77e5\u3089\u308c\u3066\u3044\u307e\u3059\u3002\u8aa4\u308a\u8a02\u6b63\u7b26\u53f7 $\\mathcal C$\u306b\u3064\u3044\u3066\u306e\u6dfb\u5b57\u96c6\u5408 $I$\u3068\u30d1\u30ea\u30c6\u30a3\u30c1\u30a7\u30c3\u30af\u884c\u5217 $H$\u306b\u3064\u3044\u3066\u3001<\/p>\n\n\n\n<p>$$ \\begin{align}I\u304c\u60c5\u5831\u96c6\u5408\u3067\u3042\u308b \\Leftrightarrow \\mathrm{rank}(H_{I^c})=\\mathrm{rank}(H)\\end{align} $$<\/p>\n\n\n\n<p>\u304c\u6210\u308a\u7acb\u3061\u307e\u3059\u3002\u3053\u3053\u3067\u3001 $I^c$\u306f $I$\u306e\u88dc\u96c6\u5408\u3092\u8868\u3057\u307e\u3059\u3002\u3064\u307e\u308a\u3001 $H$\u306e\u30e9\u30f3\u30af\u304c\u5909\u308f\u3089\u306a\u3044\u3088\u3046\u306b $I^c$\u3092\u9078\u3079\u3070 $I$\u304c\u60c5\u5831\u96c6\u5408\u306b\u306a\u308a\u307e\u3059\u3002\u3053\u306e\u3053\u3068\u304b\u3089\u3001\u30d1\u30ea\u30c6\u30a3\u30c1\u30a7\u30c3\u30af\u884c\u5217 $H$\u306e\u7dda\u5f62\u72ec\u7acb\u306a\u5217\u3092\u9078\u629e\u3059\u308b\u3053\u3068\u3067 $\\mathcal C$\u306e\u60c5\u5831\u96c6\u5408\u306e\u88dc\u96c6\u5408\u304c\u5f97\u3089\u308c\u307e\u3059\u3002\u3053\u306e\u3088\u3046\u306b\u3001\u7dda\u5f62\u72ec\u7acb\u306a\u5217\u306e\u9078\u629e\u3067\u60c5\u5831\u96c6\u5408\u3092\u69cb\u6210\u3067\u304d\u308b\u3068\u3044\u3046\u4e8b\u5b9f\u306f\u3001\u30a2\u30eb\u30b4\u30ea\u30ba\u30e0\u306e\u5185\u90e8\u3067\u60c5\u5831\u96c6\u5408\u3092\u69cb\u6210\u3059\u308b\u3068\u304d\u306b\u4fbf\u5229\u3067\u3059\u3002<\/p>\n\n\n\n<p>\u307e\u305f\u3001\u4fbf\u5229\u306a\u8a18\u6cd5\u3068\u3057\u3066\u6b21\u306e\u8a18\u6cd5\u3092\u5c0e\u5165\u3057\u307e\u3059\u3002<\/p>\n\n\n\n<p>$$ \\begin{align}\\mathcal E^s_I(u)=v \\;:\\Leftrightarrow v_I=u,Hv=s\\end{align} $$<\/p>\n\n\n\n<p>\u3053\u306e\u8a18\u6cd5\u306e\u610f\u5473\u306f\u3001\u30d3\u30c3\u30c8\u5217$u$\u3092\u542b\u307f\u306a\u304c\u3089\u30b7\u30f3\u30c9\u30ed\u30fc\u30e0\u6761\u4ef6\u3092\u6e80\u305f\u3059\u3088\u3046\u306a\u30d3\u30c3\u30c8\u5217$v$\u3092\u8868\u3057\u3066\u3044\u307e\u3059\u3002\u3053\u308c\u306b\u3088\u3063\u3066\u3001$\\hat e$\u306e$I$\u3067\u6307\u5b9a\u3055\u308c\u308b\u6210\u5206\u4ee5\u5916\u3092\u7121\u8996\u3057\u3066\u3001\u30b7\u30f3\u30c9\u30ed\u30fc\u30e0\u6761\u4ef6\u3092\u6e80\u305f\u3059\u3088\u3046\u306b\u4fee\u6b63\u3057\u305f\u30d3\u30c3\u30c8\u5217\u3092\u8868\u3057\u3084\u3059\u304f\u306a\u308a\u307e\u3059\u3002<\/p>\n\n\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"OSD%E3%81%AE%E6%89%8B%E9%A0%86\"><\/span>OSD\u306e\u624b\u9806<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p>OSD\u3067\u307e\u305a\u6700\u521d\u306b\u3084\u308a\u305f\u3044\u3053\u3068\u306f\u3001$\\hat e$\u306e\u4e2d\u304b\u3089\u4fe1\u983c\u6027\u306e\u9ad8\u3044\u6210\u5206\u3092\u9078\u3076\u3053\u3068\u3067\u3059\u3002\u4fe1\u983c\u6027\u306e\u9ad8\u3044\u6210\u5206\u3092\u6b8b\u3057\u3066\u3001\u305d\u306e\u4ed6\u306e\u6210\u5206\u3092\u7f6e\u304d\u63db\u3048\u308b\u3068\u3044\u3046\u51e6\u7406\u3092\u884c\u3044\u307e\u3059\u3002\u305d\u306e\u305f\u3081\u306b\u307e\u305a\u306f\u3001$\\hat e$\u306e\u6210\u5206\u306e\u4fe1\u983c\u6027\u3092\u8003\u3048\u307e\u3059\u3002\u771f\u306e\u30a8\u30e9\u30fc\u3092$e$\u3068\u3057\u3066\u3001\u7b2c $i$\u6210\u5206\u306e\u4fe1\u983c\u6027\u3092<\/p>\n\n\n\n<p>$$ \\begin{align}\\rho_i:=\\mathrm{Pr}(e_i=\\hat {e_i} \\mid s)=\\max\\{p_i,1-p_i\\}\\end{align} $$<\/p>\n\n\n\n<p>\u3067\u8a55\u4fa1\u3059\u308b\u3053\u3068\u306b\u3057\u307e\u3059\u3002\u3053\u3053\u3067\u3001 $p_i = \\mathrm{Pr}(e_i = 1\\mid s)$\u3067\u3042\u308a\u3001\u30b7\u30f3\u30c9\u30ed\u30fc\u30e0\u5024\u304c\u5f97\u3089\u308c\u305f\u3046\u3048\u3067 $i$\u756a\u76ee\u306e\u91cf\u5b50\u30d3\u30c3\u30c8\u306b\u30a8\u30e9\u30fc\u304c\u8d77\u304d\u3066\u3044\u308b\u78ba\u7387\u3067\u3059\u3002\u6ce8\u610f\u3059\u3079\u304d\u3053\u3068\u3068\u3057\u3066\u3001\u5f0f\u306e\u9014\u4e2d\u3067\u771f\u306e\u30a8\u30e9\u30fc $e$ \u304c\u767b\u5834\u3057\u3066\u3044\u307e\u3059\u304c\u3053\u308c\u306f\u77e5\u308a\u5f97\u306a\u3044\u60c5\u5831\u3067\u3059\u3002\u305d\u308c\u3067\u3082\u3001\u6700\u53f3\u8fba\u3067\u306f $p_i$ \u3057\u304b\u4f7f\u308f\u308c\u3066\u3044\u306a\u3044\u305f\u3081\u8a08\u7b97\u3059\u308b\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u3002<\/p>\n\n\n\n<p>$\\hat e$\u306e\u6210\u5206\u3067\u305d\u306e\u307e\u307e\u6b8b\u3059\u3082\u306e\u306f $\\hat e$\u306e\u60c5\u5831\u3092\u6301\u3063\u3066\u3044\u3066\u307b\u3057\u3044\u306e\u3067\u3001\u3053\u3053\u3067\u306f\u7b26\u53f7 $\\mathcal C$\u306e\u60c5\u5831\u96c6\u5408\u306b\u306a\u3063\u3066\u3044\u308b\u3068\u8003\u3048\u307e\u3059\u3002\u3053\u306e\u8003\u3048\u306b\u57fa\u3065\u3044\u3066\u6700\u3082\u4fe1\u983c\u6027\u306e\u9ad8\u3044\u60c5\u5831\u96c6\u5408\u3060\u3051\u3092\u6b8b\u3059\u3088\u3046\u306b\u3057\u307e\u3059\u3002\u5404\u6210\u5206\u306e\u4fe1\u983c\u6027\u304c\u72ec\u7acb\u3067\u3042\u308b\u3068\u4eee\u5b9a\u3057\u305f\u3068\u304d\u3001\u60c5\u5831\u96c6\u5408$I$\u306e\u4fe1\u983c\u6027\u306f<\/p>\n\n\n\n<p>$$ \\begin{align}\\rho(I):=\\mathrm{Pr}(\\mathcal E^s_I(\\hat e_I)=e)=\\prod_{i \\in I}\\rho_i\\end{align} $$<\/p>\n\n\n\n<p>\u3068\u66f8\u304f\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u3002\u4fe1\u983c\u6027\u306e\u9ad8\u3044\u6210\u5206\u3092\u9078\u3076\u3068\u3044\u3046\u554f\u984c\u306f$\\rho(I)$\u3092\u6700\u5927\u306b\u3059\u308b\u60c5\u5831\u96c6\u5408$I$\u3092\u63a2\u3059\u3068\u3044\u3046\u554f\u984c\u306b\u7f6e\u304d\u63db\u3048\u3089\u308c\u307e\u3057\u305f\u3002<\/p>\n\n\n\n<p>$\\rho(I)$\u306e\u6700\u5927\u5316\u306f\u4e00\u898b\u3059\u308b\u3068\u8907\u96d1\u3067\u8a08\u7b97\u306e\u52b9\u7387\u304c\u60aa\u305d\u3046\u3067\u3059\u304c\u3001\u5b9f\u306f\u60c5\u5831\u96c6\u5408\u5168\u4f53\u306e\u96c6\u5408$\\mathcal B$\u304c\u30de\u30c8\u30ed\u30a4\u30c9\u3068\u3044\u3046\u69cb\u9020\u3092\u6301\u3063\u3066\u3044\u308b\u305f\u3081\u3001\u8caa\u6b32\u6cd5\u3092\u7528\u3044\u3066\u52b9\u7387\u7684\u306b\u8a08\u7b97\u3067\u304d\u308b\u3053\u3068\u304c\u77e5\u3089\u308c\u3066\u3044\u307e\u3059\u3002<\/p>\n\n\n\n<p>\u307e\u305a\u306f\u7a4d\u306e\u6700\u5927\u5316\u306b\u306a\u3063\u3066\u3044\u308b\u3068\u3053\u308d\u3092\u3001\u5bfe\u6570\u3092\u53d6\u3063\u3066\u548c\u306e\u6700\u5927\u5316\u306b\u7f6e\u304d\u63db\u3048\u3066<\/p>\n\n\n\n<p>$$ \\begin{align}\\max_{I\\in \\mathcal B}\\sum_{i\\in I}\\ln \\rho_i\\end{align} $$<\/p>\n\n\n\n<p>\u3068\u3044\u3046\u6700\u9069\u5316\u554f\u984c\u306b\u66f8\u304d\u63db\u3048\u307e\u3059\u3002\u3053\u308c\u306f\u30de\u30c8\u30ed\u30a4\u30c9\u4e0a\u306e\u6700\u5927\u72ec\u7acb\u96c6\u5408\u554f\u984c\u3068\u547c\u3070\u308c\u308b\u554f\u984c\u3067\u3001\u8caa\u6b32\u6cd5\u3092\u7528\u3044\u3066\u52b9\u7387\u7684\u306b\u89e3\u304f\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u3002\uff08\u30a2\u30eb\u30b4\u30ea\u30ba\u30e0\u306e\u8a73\u7d30\u306f<a href=\"https:\/\/speakerdeck.com\/seekworser\/jing-ji-puroguramingutomatoroido\">\u3053\u3061\u3089\u306e\u30b9\u30e9\u30a4\u30c9<\/a>\u306a\u3069\u3092\u53c2\u7167\u3057\u3066\u304f\u3060\u3055\u3044\uff09<\/p>\n\n\n\n<p>\u3053\u306e\u554f\u984c\u3092<\/p>\n\n\n\n<p>\u3053\u306e\u3088\u3046\u306b\u3057\u3066\u5f97\u3089\u308c\u305f\u4fe1\u983c\u6027\u306e\u9ad8\u3044\u60c5\u5831\u96c6\u5408$I$\u306b\u3064\u3044\u3066\u3001$\\hat e$\u306e$I$\u4ee5\u5916\u306e\u8981\u7d20\u3092\u7f6e\u304d\u63db\u3048\u308b\u3053\u3068\u3067\u30b7\u30f3\u30c9\u30ed\u30fc\u30e0\u6761\u4ef6\u3092\u6e80\u305f\u3059\u3088\u3046\u306b\u3059\u308b\u3053\u3068\u3092\u76ee\u6307\u3057\u307e\u3059\u3002$I^c$\u3067$I$\u306e\u88dc\u96c6\u5408\u3092\u8868\u3059\u3053\u3068\u306b\u3057\u3066\u3001<\/p>\n\n\n\n<p>$$ \\begin{align}H(\\hat e_I+x_{I^c})=H\\hat e_I+Hx_{I^c}=H_I\\hat e_I+H_{I^c}x_{I^c}=s\\end{align} $$<\/p>\n\n\n\n<p>\u3092\u307f\u305f\u3059\u3088\u3046\u306a\u30d3\u30c3\u30c8\u5217$x$\u3092\u6c42\u3081\u3066\u3001<\/p>\n\n\n\n<p>$$ \\begin{align}\\hat e&#8217;=\\hat e_I+x_{I^c}\\end{align} $$<\/p>\n\n\n\n<p>\u3068\u3059\u308b\u3053\u3068\u3067\u7f6e\u304d\u63db\u3048\u3092\u5b9f\u73fe\u3057\u307e\u3059\u3002\u3053\u306e\u3088\u3046\u306a$x$\u306f<\/p>\n\n\n\n<p>$$ \\begin{align}H_{I^c}x_{I^c}=s+H_I\\hat e_I\\end{align} $$<\/p>\n\n\n\n<p>\u3092\u307f\u305f\u3057\u3001\u3053\u306e\u65b9\u7a0b\u5f0f\u3092\u89e3\u304f\u3053\u3068\u306b\u3088\u3063\u3066\u5f97\u3089\u308c\u307e\u3059\u3002\u4ee5\u4e0a\u3067\u30b7\u30f3\u30c9\u30ed\u30fc\u30e0\u6761\u4ef6\u3092\u307f\u305f\u3059\u63a8\u5b9a\u30a8\u30e9\u30fc$\\hat e&#8217;$\u304c\u5f97\u3089\u308c\u307e\u3059\u3002<\/p>\n\n\n\n<p>\u3053\u3053\u307e\u3067\u7d39\u4ecb\u3057\u305f\u624b\u9806\u306b\u5f93\u3046OSD\u306f**order-0 OSD (OSD-0)**\u3068\u547c\u3070\u308c\u3066\u3044\u307e\u3059\u3002\u5177\u4f53\u7684\u306a\u30a2\u30eb\u30b4\u30ea\u30ba\u30e0\u306f\u6b21\u306e\u3088\u3046\u306b\u306a\u308a\u307e\u3059\u3002\u6ce8\u610f\u3059\u3079\u304d\u3053\u3068\u3068\u3057\u3066\u3001\u5165\u529b\u306e\u30d1\u30ea\u30c6\u30a3\u30c1\u30a7\u30c3\u30af\u884c\u5217 $H$\u306f\u4fe1\u983c\u5ea6\u304c $\\rho_1 \\le \\cdots \\le \\rho_n$ \u3068\u306a\u308b\u3088\u3046\u306b\u5217 $(h_1,&#8230;,h_n)$\u304c\u4e26\u3073\u66ff\u3048\u3089\u308c\u3066\u3044\u307e\u3059\u3002\u305d\u306e\u305f\u3081\u5b9f\u88c5\u3059\u308b\u969b\u306b\u306f\u3001\u4e8b\u524d\u306b\u30bd\u30fc\u30c6\u30a3\u30f3\u30b0\u304c\u5fc5\u8981\u306b\u306a\u308a\u307e\u3059\u3002<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Input: binary parity-check matrix $H$, syndrome vector $s \\in \\mathbb F_2^m$, vector of hard decisions $\\hat e \\in \\mathbb F^n_2$<\/li>\n\n\n\n<li>Output: error vector $e \\in \\mathbb F_2^n$ such that $He=s$<\/li>\n<\/ul>\n\n\n\n<ol class=\"wp-block-list\">\n<li>$J \\leftarrow \\emptyset$<\/li>\n\n\n\n<li>for $i \\leftarrow 1$ to $n$ do\n<ol class=\"wp-block-list\">\n<li>if $\\mathrm{rk}H_J = \\mathrm{rk}H$ then\n<ol class=\"wp-block-list\">\n<li>break;<\/li>\n<\/ol>\n<\/li>\n\n\n\n<li>$J&#8217; \\leftarrow J \\cup \\{i\\}$<\/li>\n\n\n\n<li>if $\\mathrm{rk} H_{J&#8217;} &gt; \\mathrm{rk}H_J$ then\n<ol class=\"wp-block-list\">\n<li>$J \\leftarrow J&#8217;$<\/li>\n<\/ol>\n<\/li>\n<\/ol>\n<\/li>\n\n\n\n<li>end<\/li>\n\n\n\n<li>$s&#8217; \\leftarrow s_J$<\/li>\n\n\n\n<li>$x \\leftarrow$ solution of $H_Jx = s&#8217;$<\/li>\n\n\n\n<li>return $e = \\mathcal R^x_J(\\hat e)$<\/li>\n<\/ol>\n\n\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"OSD%E3%81%AE%E3%83%90%E3%83%AA%E3%82%A8%E3%83%BC%E3%82%B7%E3%83%A7%E3%83%B3\"><\/span>OSD\u306e\u30d0\u30ea\u30a8\u30fc\u30b7\u30e7\u30f3<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p>\u5b9f\u306f\u3001OSD-0\u3092\u62e1\u5f35\u3059\u308b\u3053\u3068\u3067\u8aa4\u308a\u8a02\u6b63\u306e\u7cbe\u5ea6\u3092\u3055\u3089\u306b\u5411\u4e0a\u3055\u305b\u308b\u4e8b\u304c\u3067\u304d\u307e\u3059\u3002\u3069\u306e\u3088\u3046\u306b\u62e1\u5f35\u3059\u308b\u304b\u3068\u3044\u3046\u3068\u3001 $\\hat e$\u306e\u6210\u5206\u306e\u3046\u3061\u3001\u7f6e\u304d\u63db\u3048\u308b\u6210\u5206\u306e\u52a0\u5897\u3092\u5897\u3084\u3057\u307e\u3059\u3002\u3053\u306e\u3088\u3046\u306a\u62e1\u5f35\u7248\u306eOSD\u3092<strong>order-w OSD (OSD-w)<\/strong>\u3068\u547c\u3073\u307e\u3059\u3002<\/p>\n\n\n\n<p>OSD-0\u3067\u306f\u3001\u4fe1\u983c\u6027\u306e\u4f4e\u3044\u6210\u5206\u3092 $J$\u3067\u6307\u5b9a\u3057\u3066\u7f6e\u304d\u63db\u3048\u307e\u3057\u305f\u3002OSD-w\u3067\u306f $J$\u306b\u52a0\u3048\u3066\u3001\u4fe1\u983c\u6027\u306e\u4f4e\u3044\u6210\u5206\u3092\u3055\u3089\u306b $w$\u500b\u6307\u5b9a\u3057\u3066\u7f6e\u304d\u63db\u3048\u308b\u3068\u3044\u3046\u3053\u3068\u3092\u884c\u3044\u307e\u3059\u3002<\/p>\n\n\n\n<figure class=\"wp-block-image size-large\"><img loading=\"lazy\" decoding=\"async\" width=\"1024\" height=\"887\" src=\"https:\/\/qard.is.tohoku.ac.jp\/T-Wave\/wp-content\/uploads\/2026\/04\/fae0c82cc690d77ada605fbed0466e2f-1024x887.png\" alt=\"\" class=\"wp-image-8935\" srcset=\"https:\/\/qard.is.tohoku.ac.jp\/T-Wave\/wp-content\/uploads\/2026\/04\/fae0c82cc690d77ada605fbed0466e2f-1024x887.png 1024w, https:\/\/qard.is.tohoku.ac.jp\/T-Wave\/wp-content\/uploads\/2026\/04\/fae0c82cc690d77ada605fbed0466e2f-300x260.png 300w, https:\/\/qard.is.tohoku.ac.jp\/T-Wave\/wp-content\/uploads\/2026\/04\/fae0c82cc690d77ada605fbed0466e2f-768x665.png 768w, https:\/\/qard.is.tohoku.ac.jp\/T-Wave\/wp-content\/uploads\/2026\/04\/fae0c82cc690d77ada605fbed0466e2f.png 1208w\" sizes=\"auto, (max-width: 1024px) 100vw, 1024px\" \/><\/figure>\n\n\n\n<p>\uff08\u5f15\u7528\u5143\uff1a<a href=\"https:\/\/arxiv.org\/abs\/1904.02703%EF%BC%89\">https:\/\/arxiv.org\/abs\/1904.02703\uff09<\/a><\/p>\n\n\n\n<p>\u305d\u306e\u4ed6\u306e\u30d0\u30fc\u30b8\u30e7\u30f3\u3068\u3057\u3066\u3001OSD-0\u306e\u6539\u826f\u7248\u30a2\u30eb\u30b4\u30ea\u30ba\u30e0\u304c\u3042\u308a\u307e\u3059\u3002\u3053\u306e\u30a2\u30eb\u30b4\u30ea\u30ba\u30e0\u3067\u306f\u3001\u60c5\u5831\u96c6\u5408\u306e\u88dc\u96c6\u5408\u3092\u5b8c\u5168\u306b\u6c42\u3081\u308b\u3068\u3044\u3046\u3053\u3068\u306f\u305b\u305a\u3001 $x$\u3092\u6c42\u3081\u308b\u305f\u3081\u306e\u65b9\u7a0b\u5f0f $H_Jx = s&#8217;$\u306b\u89e3\u304c\u5b58\u5728\u3059\u308b\u3088\u3046\u306b\u306a\u308b\u307e\u3067 $J$\u3092\u62e1\u5927\u3055\u305b\u307e\u3059\u3002\u3053\u306e\u3088\u3046\u306b\u3057\u3066\u5f97\u3089\u308c\u308b\u6dfb\u5b57\u96c6\u5408 $J$\u306f\u3001\u4fe1\u983c\u6027\u306e\u9ad8\u3044\u60c5\u5831\u96c6\u5408 $I$\u306b\u3064\u3044\u3066 $I \\subseteq J^c$\u3092\u307f\u305f\u3059\u306e\u3067\u3001 $\\hat e_J$\u3092\u7f6e\u304d\u63db\u3048\u3066\u3082\u60c5\u5831\u96c6\u5408\u306b\u5f71\u97ff\u3057\u307e\u305b\u3093\u3002<\/p>\n\n\n\n<p>\u6539\u826f\u7248\u306e\u30a2\u30eb\u30b4\u30ea\u30ba\u30e0\u306f\u6b21\u306e\u3088\u3046\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=\"992\" src=\"https:\/\/qard.is.tohoku.ac.jp\/T-Wave\/wp-content\/uploads\/2026\/04\/6e40d9006e88ba368d19bcea9afda3df-1024x992.png\" alt=\"\" class=\"wp-image-8936\" srcset=\"https:\/\/qard.is.tohoku.ac.jp\/T-Wave\/wp-content\/uploads\/2026\/04\/6e40d9006e88ba368d19bcea9afda3df-1024x992.png 1024w, https:\/\/qard.is.tohoku.ac.jp\/T-Wave\/wp-content\/uploads\/2026\/04\/6e40d9006e88ba368d19bcea9afda3df-300x291.png 300w, https:\/\/qard.is.tohoku.ac.jp\/T-Wave\/wp-content\/uploads\/2026\/04\/6e40d9006e88ba368d19bcea9afda3df-768x744.png 768w, https:\/\/qard.is.tohoku.ac.jp\/T-Wave\/wp-content\/uploads\/2026\/04\/6e40d9006e88ba368d19bcea9afda3df.png 1216w\" sizes=\"auto, (max-width: 1024px) 100vw, 1024px\" \/><\/figure>\n\n\n\n<p>\uff08\u5f15\u7528\u5143\uff1a<a href=\"https:\/\/arxiv.org\/abs\/1904.02703%EF%BC%89\">https:\/\/arxiv.org\/abs\/1904.02703\uff09<\/a><\/p>\n\n\n\n<p>\u3053\u3053\u3067\u3001\u65b0\u3057\u3044\u8a18\u6cd5\u3068\u3057\u3066 $\\mathcal R^x_J$\u3068\u3044\u3046\u95a2\u6570\u304c\u767b\u5834\u3057\u3066\u3044\u307e\u3059\u3002\u3053\u306e\u95a2\u6570\u306e\u610f\u5473\u306f\u3001\u30d9\u30af\u30c8\u30eb\u3092\u53d7\u3051\u53d6\u3063\u3066\u3001\u305d\u306e\u4e2d\u306e $J$\u3067\u6307\u5b9a\u3055\u308c\u308b\u6210\u5206\u3092\u30d9\u30af\u30c8\u30eb $x$\u3067\u7f6e\u304d\u63db\u3048\u308b\u3068\u3044\u3046\u3082\u306e\u3067\u3059\u3002<\/p>\n\n\n\n<p>\u3053\u306e\u30a2\u30eb\u30b4\u30ea\u30ba\u30e0\u306e\u5404\u30b9\u30c6\u30c3\u30d7\u3067\u884c\u3063\u3066\u3044\u308b\u51e6\u7406\u306f\u6b21\u306e\u3088\u3046\u306b\u89e3\u91c8\u3067\u304d\u307e\u3059<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>1\u884c\u76ee\uff1a $J$\u306e\u521d\u671f\u5316\u3002 $J$\u306f\u3053\u306e\u5f8c\u3001\u60c5\u5831\u96c6\u5408\u306e\u88dc\u96c6\u5408\u3068\u3057\u3066\u4f7f\u308f\u308c\u308b\u96c6\u5408\u3067\u3059<\/li>\n\n\n\n<li>2\u884c\u76ee\uff1a $s&#8217;$\u306e\u521d\u671f\u5316\u3002 $s&#8217;$\u306f\u6700\u7d42\u7684\u306b $s+ H_{J^c}\\hat e_{J^c}$\u306b\u3059\u308b\u3053\u3068\u304c\u76ee\u6a19\u3067\u3059<\/li>\n\n\n\n<li>3\u884c\u76ee\u304b\u308912\u884c\u76ee\uff1a\u60c5\u5831\u96c6\u5408\u306e\u88dc\u96c6\u5408 $J$\u3092\u6c42\u3081\u3066\u3044\u307e\u3059\n<ul class=\"wp-block-list\">\n<li>$J$\u306f $H_J$\u304c\u7dda\u5f62\u72ec\u7acb\u306b\u306a\u308b\u3088\u3046\u306b\u9078\u3073\u307e\u3059\u3002\u305d\u306e\u7406\u7531\u306f\u3001 $\\mathrm{rk}H_J=\\mathrm{rk}H$\u306a\u3089\u3070 $J$\u306f\u60c5\u5831\u96c6\u5408\u306e\u88dc\u96c6\u5408\u306b\u306a\u308b\u304b\u3089\u3067\u3059<\/li>\n\n\n\n<li>4\u884c\u76ee\u304b\u30896\u884c\u76ee\uff1a\u62e1\u5927\u4fc2\u6570\u884c\u5217 $(H_J|s&#8217;)$\u306e\u30e9\u30f3\u30af\u304c $H_J$\u306e\u30e9\u30f3\u30af\u3068\u4e00\u81f4\u3057\u305f\u3089\u30eb\u30fc\u30d7\u3092\u629c\u3051\u307e\u3059\u3002\n<ul class=\"wp-block-list\">\n<li>\u3053\u306e\u6761\u4ef6\u3067\u30eb\u30fc\u30d7\u3092\u629c\u3051\u308b\u7406\u7531\u306f$\\mathrm{rk}(H_J|s&#8217;) = \\mathrm{rk}H_J$\u306e\u3068\u304d $H_Jx = s&#8217;$\u304c\u89e3\u3092\u6301\u3064\u304b\u3089\u3067\u3059<\/li>\n<\/ul>\n<\/li>\n\n\n\n<li>7\u884c\u76ee\uff1a\u6dfb\u5b57 $i$\u3092\u6dfb\u5b57\u96c6\u5408 $J$ \u306b\u8ffd\u52a0\u3057\u305f\u3082\u306e\u3092\u65b0\u305f\u306a\u96c6\u5408 $J&#8217;$\u306b\u4ee3\u5165<\/li>\n\n\n\n<li>8\u884c\u76ee\u304b\u308911\u884c\u76ee\uff1a$H_{J&#8217;}$\u306e\u30e9\u30f3\u30af\u304c $H_J$\u306e\u30e9\u30f3\u30af\u3088\u308a\u3082\u5927\u304d\u3044\u3068\u304d\u3001 $J$\u3092 $J&#8217;$\u3067\u7f6e\u304d\u63db\u3048\u3066\u3001 $s&#8217;$\u306b $\\hat e_i h_i$\u3092\u8db3\u3057\u307e\u3059\u3002\u3053\u3053\u3067\u3001 $\\hat e_i$\u306f \u30b9\u30ab\u30e9\u30fc\u3001 $h_i$\u306f $H$\u306e $i$\u756a\u76ee\u306e\u5217\u30d9\u30af\u30c8\u30eb\u3067\u3059\n<ul class=\"wp-block-list\">\n<li>\u306a\u305c $\\mathrm{rk}H_{J&#8217;}$\u3068 $\\mathrm{rk}H_J$\u3092\u6bd4\u8f03\u3057\u3066\u3044\u308b\u304b\u3068\u3044\u3046\u3068\u3001 $\\mathrm{rk}H_{J&#8217;} &gt; \\mathrm{rk}H_J$\u306a\u3089\u3070 $H_{J&#8217;}$\u306e\u5217\u306f\u7dda\u5f62\u72ec\u7acb\u3067\u3042\u308b\u3053\u3068\u304c\u4fdd\u8a3c\u3067\u304d\u308b\u304b\u3089\u3067\u3059\uff08 $J$\u306f\u7a7a\u96c6\u5408\u3067\u521d\u671f\u5316\u3055\u308c\u3066\u3044\u3066\u3001 $H_J$\u306e\u5217\u304c\u7dda\u5f62\u72ec\u7acb\u3067 $\\mathrm{rk}H_{J&#8217;} &gt; \\mathrm{rk}H_J$\u306a\u3089\u3070 $H_{J&#8217;}$\u306e\u5217\u304c\u7dda\u5f62\u72ec\u7acb\u3067\u3042\u308b\u3053\u3068\u304c\u5e30\u7d0d\u7684\u306b\u8a3c\u660e\u3067\u304d\u307e\u3059\uff09<\/li>\n\n\n\n<li>$s&#8217;$\u306b $\\hat e_i h_i$\u3092\u8db3\u3057\u3066\u3044\u308b\u7406\u7531\u306f\u3001 $H\\hat e=\\sum_{k=1}^n \\hat e_k h_k$\u3067\u3042\u308b\u3053\u3068\u304b\u3089\u3001 $J$\u306b\u8ffd\u52a0\u3057\u305f $i$\u306e\u9805\u3092\u6253\u3061\u6d88\u3057\u305f\u3044\u304b\u3089\u3067\u3059\u3002\u3053\u3046\u3059\u308b\u3053\u3068\u3067\u3001\u6700\u7d42\u7684\u306b\u5f97\u3089\u308c\u308b $s&#8217;$\u304c $s + H_{J^c}\\hat e_{J^c}$\u306b\u306a\u308a\u307e\u3059<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/li>\n\n\n\n<li>13\u884c\u76ee\uff1a $H_J x = s&#8217;$\u3092\u89e3\u304f\u3053\u3068\u3067 $x$\u3092\u6c42\u3081\u3066\u3044\u307e\u3059<\/li>\n\n\n\n<li>14\u884c\u76ee\uff1a $\\hat e$\u306e\u3046\u3061\u3001\u4fe1\u983c\u6027\u306e\u4f4e\u3044 $\\hat e_J$\u306e\u90e8\u5206\u3092 $x$\u306b\u7f6e\u304d\u63db\u3048\u308b\u3053\u3068\u3067\u3001 $e$\u304c\u30b7\u30f3\u30c9\u30ed\u30fc\u30e0\u6761\u4ef6 $He=s$\u3092\u307f\u305f\u3059\u3088\u3046\u306b\u3057\u307e\u3059<\/li>\n<\/ul>\n\n\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"%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%B8%E3%81%AE%E6%8B%A1%E5%BC%B5\"><\/span>\u91cf\u5b50\u8aa4\u308a\u8a02\u6b63\u7b26\u53f7\u3078\u306e\u62e1\u5f35<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p>\u3053\u3053\u307e\u3067\u7d39\u4ecb\u3057\u305fOSD\u306f\u53e4\u5178\u8aa4\u308a\u8a02\u6b63\u7b26\u53f7\u306b\u5bfe\u3059\u308b\u30a2\u30eb\u30b4\u30ea\u30ba\u30e0\u3067\u3057\u305f\u3002\u672c\u984c\u306f\u91cf\u5b50\u8aa4\u308a\u8a02\u6b63\u7b26\u53f7\u306b\u5bfe\u3059\u308b\u30a2\u30eb\u30b4\u30ea\u30ba\u30e0\u306a\u306e\u3067\u3001\u3053\u3053\u307e\u3067\u3067\u7d39\u4ecb\u3057\u305f\u3082\u306e\u3092\u91cf\u5b50\u8aa4\u308a\u8a02\u6b63\u7b26\u53f7\u3078\u62e1\u5f35\u3057\u307e\u3059\u3002<\/p>\n\n\n\n<p>\u62e1\u5f35\u3068\u8a00\u3063\u3066\u3082\u3001\u30a2\u30a4\u30c7\u30a2\u306f\u7c21\u5358\u3067\u3001\u91cf\u5b50\u8aa4\u308a\u8a02\u6b63\u7b26\u53f7\u306e\u8a00\u8449\u3092BS\u8868\u73fe\u306e\u8981\u9818\u3067\u30d3\u30c3\u30c8\u5217\u306b\u5909\u63db\u3057\u3066\u3001\u305d\u308c\u3092OSD\u306e\u5165\u529b\u3068\u3057\u307e\u3059\u3002\u3055\u3089\u306b\u3001OSD\u306e\u51fa\u529b\u3067\u3042\u308b\u63a8\u5b9a\u30a8\u30e9\u30fc\u306e\u30d3\u30c3\u30c8\u5217\u3092\u91cf\u5b50\u306e\u30a8\u30e9\u30fc\u306b\u5909\u63db\u3059\u308b\u3053\u3068\u3067\u3001\u5168\u4f53\u3068\u3057\u3066\u91cf\u5b50\u8aa4\u308a\u8a02\u6b63\u306e\u305f\u3081\u306e\u30a2\u30eb\u30b4\u30ea\u30ba\u30e0\u3068\u3057\u3066\u52d5\u4f5c\u3059\u308b\u3088\u3046\u306b\u3057\u307e\u3059\u3002\u307e\u3068\u3081\u308b\u3068\u3001\u51e6\u7406\u306e\u6d41\u308c\u306f\u6b21\u306e\u56f3\u306e\u3088\u3046\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\/04\/osd_flow-1024x546.png\" alt=\"\" class=\"wp-image-8937\" srcset=\"https:\/\/qard.is.tohoku.ac.jp\/T-Wave\/wp-content\/uploads\/2026\/04\/osd_flow-1024x546.png 1024w, https:\/\/qard.is.tohoku.ac.jp\/T-Wave\/wp-content\/uploads\/2026\/04\/osd_flow-300x160.png 300w, https:\/\/qard.is.tohoku.ac.jp\/T-Wave\/wp-content\/uploads\/2026\/04\/osd_flow-768x410.png 768w, https:\/\/qard.is.tohoku.ac.jp\/T-Wave\/wp-content\/uploads\/2026\/04\/osd_flow-1536x819.png 1536w, https:\/\/qard.is.tohoku.ac.jp\/T-Wave\/wp-content\/uploads\/2026\/04\/osd_flow-2048x1092.png 2048w\" sizes=\"auto, (max-width: 1024px) 100vw, 1024px\" \/><figcaption class=\"wp-element-caption\">\u56f31 OSD\u3092\u542b\u3080\u8aa4\u308a\u8a02\u6b63\u306e\u6d41\u308c<\/figcaption><\/figure>\n\n\n\n<p>\u5165\u529b\u3068\u3057\u3066\u53d7\u3051\u53d6\u3063\u305f\u30d1\u30ea\u30c6\u30a3\u30c1\u30a7\u30c3\u30af\u884c\u5217 $H$\u3068\u63a8\u5b9a\u30a8\u30e9\u30fc $\\hat e$\u306b\u3064\u3044\u3066\u3001\u53e4\u5178\u30d3\u30c3\u30c8\u5217\u3078\u306e\u5909\u63db $\\mathbb b$\u3092\u5b9a\u7fa9\u3057\u307e\u3059\u3002\u57fa\u672c\u7684\u306a\u30a2\u30a4\u30c7\u30a2\u306fBS\u8868\u73fe\u3068\u540c\u69d8\u3067\u3059\u304c\u3001\u6210\u5206\u306e\u4e26\u3073\u65b9\u304c\u3059\u3053\u3057\u9055\u3046\u306e\u3067\u6ce8\u610f\u3057\u3066\u304f\u3060\u3055\u3044\u3002<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>$n$\u91cf\u5b50\u30d3\u30c3\u30c8\u30d1\u30a6\u30ea\u6f14\u7b97\u5b50 $\\hat e = \\hat e_1 \\otimes \\cdots \\otimes \\hat e_n \\in P_n$ \u306b\u5bfe\u3057\u3066\u3001 $\\hat e_i = i^{l_i}X^{a_i}Z^{b_i}\\quad (a_i,b_i \\in \\mathbb Z_2, \\;l\\in\\mathbb Z_4)$\u3067\u5b9a\u307e\u308b $\\{a_i,b_i\\}$\u3092\u7528\u3044\u3066\u5909\u63db $b$\u3092\u6b21\u306e\u3088\u3046\u306b\u5b9a\u3081\u307e\u3059<\/li>\n<\/ul>\n\n\n\n<p>$$ \\begin{align}b(\\hat e) = (a_1,b_1,&#8230;,a_n,b_n) \\in \\mathbb F_2^{2n}\\end{align} $$<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>\u30d1\u30ea\u30c6\u30a3\u30c1\u30a7\u30c3\u30af\u884c\u5217 $H = (H_{i,j})\\in \\mathbb F^{m\\times 2n}$\u306b\u5bfe\u3057\u3066\u3001\u5404\u884c\u3092\u6b21\u306e\u3088\u3046\u306b\u4e26\u3079\u66ff\u3048\u305f\u3082\u306e\u3092 $b(H)$\u3068\u3057\u307e\u3059<\/li>\n<\/ul>\n\n\n\n<p>$$ \\begin{align}(H_{i,1}\\;H_{i,2}\\;\\cdots \\; H_{i,2n-1}\\;H_{i,2n}) \\mapsto (H_{i,1}\\;H_{i,n+1}\\;\\cdots \\; H_{i,n}\\;H_{i,2n})\\end{align} $$<\/p>\n\n\n\n<p>\u3053\u306e\u3088\u3046\u306a\u5909\u63db\u3092\u7528\u3044\u308b\u3053\u3068\u3067\u3001\u30d1\u30ea\u30c6\u30a3\u30c1\u30a7\u30c3\u30af\u884c\u5217 $H$\u3068 \u30a8\u30e9\u30fc $\\hat e \\in P_n$\u3068\u3001\u305d\u306e\u30b7\u30f3\u30c9\u30ed\u30fc\u30e0\u5024 $s$\u306e\u9593\u306b<\/p>\n\n\n\n<p>$$ \\begin{align}s =b(H)b(\\hat e)\\end{align} $$<\/p>\n\n\n\n<p>\u3068\u3044\u3046\u95a2\u4fc2\u304c\u6210\u308a\u7acb\u3064\u3053\u3068\u304c\u793a\u305b\u307e\u3059\u3002<\/p>\n\n\n\n<p>\u3053\u306e\u5909\u63db $b$\u3092\u7528\u3044\u3066\u3001\u91cf\u5b50\u7248\u306eOSD\u3092\u69cb\u6210\u3057\u307e\u3059\u3002\u53e4\u5178\u7248\u306eOSD-w\u3092 $\\mathrm{OSD}_w(|\\cdot|_{\\mathsf{Sp}}, H,e)$ \u3068\u66f8\u304f\u3053\u3068\u306b\u3057\u3066\u3001\u91cf\u5b50\u7248\u306eOSD-w\u3067\u3042\u308b $\\mathrm{qOSD}_w$ \u3092<\/p>\n\n\n\n<p>$$ \\begin{align}\\mathrm{qOSD}_w=b^{-1}(\\mathrm{OSD}_w(|\\cdot|_{\\mathsf {Sp}},b(H), b(e)))\\end{align} $$<\/p>\n\n\n\n<p>\u3067\u5b9a\u7fa9\u3057\u307e\u3059\u3002\u3053\u3053\u3067\u3001 $\\mathrm{OSD}_w$\u306b\u91cd\u307f\u95a2\u6570\u3068\u3057\u3066\u6e21\u3055\u308c\u3066\u3044\u308b $|\\cdot |_{\\mathsf{Sp}}$\u306f<strong>\u30b7\u30f3\u30d7\u30ec\u30af\u30c6\u30a3\u30c3\u30af\u91cd\u307f\u95a2\u6570<\/strong>\u3068\u547c\u3070\u308c\u3001<\/p>\n\n\n\n<p>$$ \\begin{align}|x|_{\\mathsf{Sp}}=\\sum^n_{i=1}x_{2i-1}\\lor x_{2i} \\end{align} $$<\/p>\n\n\n\n<p>\u3067\u5b9a\u7fa9\u3055\u308c\u307e\u3059\u3002\u91cd\u307f\u95a2\u6570\u306e\u9078\u629e\u306f\u30a8\u30e9\u30fc\u306e\u7279\u6027\u306b\u5fdc\u3058\u3066\u884c\u3044\u307e\u3059\u3002\u3053\u3053\u3067\u306f\u3001\u30a8\u30e9\u30fc $X,Z,Y$\u304c\u5f53\u78ba\u7387\u3067\u5404\u91cf\u5b50\u30d3\u30c3\u30c8\u306b\u72ec\u7acb\u306b\u304b\u304b\u308b\u30a8\u30e9\u30fc\u3092\u4eee\u5b9a\u3057\u3066\u30b7\u30f3\u30d7\u30ec\u30af\u30c6\u30a3\u30c3\u30af\u91cd\u307f\u95a2\u6570\u304c\u9078\u629e\u3055\u308c\u3066\u3044\u307e\u3059\u3002\u3088\u308a\u4e00\u822c\u306b\u306f\u3001\u6700\u3082\u78ba\u7387\u304c\u9ad8\u3044\u30a8\u30e9\u30fc\u304c\u6700\u3082\u5c0f\u3055\u3044\u91cd\u307f\u3092\u6301\u3064\u3088\u3046\u306a\u91cd\u307f\u95a2\u6570\u3092\u30a8\u30e9\u30fc\u306e\u7279\u6027\u306b\u5fdc\u3058\u3066\u9078\u629e\u3057\u307e\u3059\u3002<\/p>\n\n\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"BP%E3%81%A8OSD%E3%81%AE%E7%B5%B1%E5%90%88%EF%BC%9ABPOSD%E3%83%87%E3%82%B3%E3%83%BC%E3%83%80\"><\/span>BP\u3068OSD\u306e\u7d71\u5408\uff1aBP+OSD\u30c7\u30b3\u30fc\u30c0<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p>\u3064\u3044\u306b\u3053\u306e\u8a18\u4e8b\u306e\u4e3b\u5f79\u3067\u3042\u308bBP+OSD\u30c7\u30b3\u30fc\u30c0\u3092\u7d39\u4ecb\u3057\u307e\u3059\u3002\u524d\u56de\u306e\u8a18\u4e8b\u3067\u7d39\u4ecb\u3057\u305fBP\u30c7\u30b3\u30fc\u30c0\u3067\u3042\u308bLLR-MSA\u3068\u3001\u4eca\u56de\u306e\u8a18\u4e8b\u3067\u7d39\u4ecb\u3057\u305fOSD\u3092\u7d71\u5408\u3057\u307e\u3059\u3002<\/p>\n\n\n\n<p>\u524d\u56de\u306e\u8a18\u4e8b\u3092\u632f\u308a\u8fd4\u308b\u3068\u3001LLR-MSA\u306f\u30b7\u30f3\u30c9\u30ed\u30fc\u30e0\u5024\u3092\u53d7\u3051\u53d6\u3063\u3066\u3001\u5404\u91cf\u5b50\u30d3\u30c3\u30c8\u306b\u30a8\u30e9\u30fc\u304c\u8d77\u304d\u3066\u3044\u308b\u78ba\u7387\u3092\u51fa\u529b\u3059\u308b\u30a2\u30eb\u30b4\u30ea\u30ba\u30e0\u3067\u3057\u305f\u3002\u3053\u306eLLR-MSA\u3092 $X$\u30a8\u30e9\u30fc\u3068 $Z$\u30a8\u30e9\u30fc\u306e\u305d\u308c\u305e\u308c\u306b\u3064\u3044\u3066\u5b9f\u884c\u3057\u307e\u3059\u3002\u3059\u308b\u3068\u3001 $i$\u756a\u76ee\u306e\u91cf\u5b50\u30d3\u30c3\u30c8\u306b $X$\u30a8\u30e9\u30fc\u304c\u8d77\u304d\u3066\u3044\u308b\u78ba\u7387 $p_{i,X}$\u3068 $Z$\u30a8\u30e9\u30fc\u304c\u8d77\u304d\u3066\u3044\u308b\u78ba\u7387 $p_{i,Z}$\u304c\u5f97\u3089\u308c\u307e\u3059\u3002\u30a8\u30e9\u30fc\u306e\u8d77\u304d\u65b9\u306f4\u7a2e\u985e\u3067\u3001\u305d\u308c\u305e\u308c\u6b21\u306e\u3088\u3046\u306b\u306a\u308a\u307e\u3059<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>$X$\u30a8\u30e9\u30fc\u3082 $Z$\u30a8\u30e9\u30fc\u3082\u8d77\u304d\u306a\u3044\n<ul class=\"wp-block-list\">\n<li>$I$\u30a8\u30e9\u30fc\uff08\u30a8\u30e9\u30fc\u304c\u8d77\u3053\u3089\u306a\u3044\uff09<\/li>\n\n\n\n<li>\u78ba\u7387\u306f $(1-p_{i,X})(1-p_{i,Z})$<\/li>\n<\/ul>\n<\/li>\n\n\n\n<li>$X$\u30a8\u30e9\u30fc\u304c\u8d77\u304d\u3066 $Z$\u30a8\u30e9\u30fc\u306f\u8d77\u304d\u306a\u3044\n<ul class=\"wp-block-list\">\n<li>$X$\u30a8\u30e9\u30fc<\/li>\n\n\n\n<li>\u78ba\u7387\u306f$p_{i,X}(1-p_{i,Z})$<\/li>\n<\/ul>\n<\/li>\n\n\n\n<li>$X$\u30a8\u30e9\u30fc\u3082 $Z$\u30a8\u30e9\u30fc\u3082\u8d77\u304d\u308b\n<ul class=\"wp-block-list\">\n<li>\u5b9f\u8cea\u7684\u306b $Y$\u30a8\u30e9\u30fc<\/li>\n\n\n\n<li>\u78ba\u7387\u306f $p_{i,X}p_{i,Z}$<\/li>\n<\/ul>\n<\/li>\n\n\n\n<li>$X$\u30a8\u30e9\u30fc\u306f\u8d77\u304d\u305a $Z$\u30a8\u30e9\u30fc\u306f\u8d77\u304d\u308b\n<ul class=\"wp-block-list\">\n<li>$Z$\u30a8\u30e9\u30fc<\/li>\n\n\n\n<li>\u78ba\u7387\u306f $(1-p_{i,X})p_{i,Z}$<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n\n\n\n<p>\u3053\u306e4\u7a2e\u985e\u306e\u30a8\u30e9\u30fc\u306e\u78ba\u7387\u3092\u65b0\u305f\u306b $p_{i,I},p_{i,X},p_{i,Y},p_{i,Z}$\u3068\u3057\u307e\u3059\u3002\u3053\u306e\u5024\u3092\u7528\u3044\u3066\u3001 $i$\u756a\u76ee\u306e\u91cf\u5b50\u30d3\u30c3\u30c8\u306e\u63a8\u5b9a\u30a8\u30e9\u30fc $\\hat e_i$\u3092\u6b21\u306e\u3088\u3046\u306b\u6c7a\u5b9a\u3057\u307e\u3059\u3002<\/p>\n\n\n\n<p>$$ \\hat{e}_i = \\mathop{\\arg\\max}\\limits_{E\\in \\{I,X,Y,Z\\}} p_{i,E} $$<\/p>\n\n\n\n<p>\u3053\u3046\u3057\u3066\u5f97\u3089\u308c\u305f\u3001\u30a8\u30e9\u30fc\u78ba\u7387 $p_{i,I},p_{i,X},p_{i,Y},p_{i,Z}$\u3068\u63a8\u5b9a\u30a8\u30e9\u30fc $\\hat e_i$\u3092\u4f7f\u3063\u3066OSD\u306e\u5165\u529b\u30c7\u30fc\u30bf\u3092\u4f5c\u308a\u307e\u3059\u3002OSD\u306e\u5165\u529b\u3068\u3057\u3066\u5fc5\u8981\u3060\u3063\u305f\u306e\u306f\u30b7\u30f3\u30c9\u30ed\u30fc\u30e0\u5024 $s=(s_1,&#8230;,s_m)$\u3001\u63a8\u5b9a\u30a8\u30e9\u30fc $\\hat e=(\\hat e_1,&#8230;,\\hat e_n)$\u3001\u305d\u3057\u3066\u5217\u304c\u4fe1\u983c\u5ea6\u9806\u306b\u4e26\u3093\u3060\u30d1\u30ea\u30c6\u30a3\u30c1\u30a7\u30c3\u30af\u884c\u5217 $H$\u3067\u3059\u3002\u6700\u521d\u306e2\u3064\u306f\u3059\u3067\u306b\u5f97\u3089\u308c\u3066\u3044\u308b\u306e\u3067\u3001\u4f5c\u308b\u5fc5\u8981\u304c\u3042\u308b\u306e\u306f\u4e26\u3073\u66ff\u3048\u3089\u308c\u305f $H$\u3060\u3051\u3067\u3059\u3002<\/p>\n\n\n\n<p>\u30d1\u30ea\u30c6\u30a3\u30c1\u30a7\u30c3\u30af\u884c\u5217 $H$\u306e\u5217\u3092\u4fe1\u983c\u5ea6\u9806\u306b\u4e26\u3073\u66ff\u3048\u308b\u305f\u3081\u306b\u306f\u4fe1\u983c\u5ea6\u3092\u6c42\u3081\u308b\u5fc5\u8981\u304c\u3042\u308a\u307e\u3059\u304c\u3001\u4fe1\u983c\u5ea6\u306f\u30a8\u30e9\u30fc\u78ba\u7387 $p_{i,I},p_{i,X},p_{i,Y},p_{i,Z}$\u304b\u3089\u6c42\u3081\u308b\u4e8b\u304c\u3067\u304d\u307e\u3059\u3002 $i$\u756a\u76ee\u306e\u91cf\u5b50\u30d3\u30c3\u30c8\u306e\u4fe1\u983c\u5ea6 $\\rho_i$\u306f\u3001 $p_i=p_{i,X}+p_{i,Y}+p_{i,Z}$\u3092\u7528\u3044\u3066\u3001<\/p>\n\n\n\n<p>$$ \\rho_i = \\max\\{p_i, 1-p_i\\} $$<\/p>\n\n\n\n<p>\u3067\u6c42\u3081\u308b\u4e8b\u304c\u3067\u304d\u307e\u3059\u3002\u3053\u3046\u3057\u3066\u5f97\u3089\u308c\u305f\u4fe1\u983c\u5ea6 $\\rho_i$\u3092\u7528\u3044\u3066\u3001\u30d1\u30ea\u30c6\u30a3\u30c1\u30a7\u30c3\u30af\u884c\u5217 $H=(h_1,&#8230;,h_n)$\u306e\u5217\u3092\u4fe1\u983c\u5ea6\u9806\u306b\u4e26\u3073\u66ff\u3048\u307e\u3059\u3002\u4e26\u3073\u66ff\u3048\u306f\u9ad8\u901f\u306a\u30bd\u30fc\u30c6\u30a3\u30f3\u30b0\u30a2\u30eb\u30b4\u30ea\u30ba\u30e0\u3067\u884c\u3048\u3070 $O(n \\log n)$\u3067\u5b9f\u884c\u3067\u304d\u307e\u3059\u3002\u4ee5\u4e0a\u3067\u3001OSD\u306e\u5165\u529b\u3092\u3059\u3079\u3066\u63c3\u3048\u308b\u4e8b\u304c\u3067\u304d\u307e\u3057\u305f\u3002<\/p>\n\n\n\n<p>\u3053\u3053\u307e\u3067\u306e\u624b\u9806\u3092\u307e\u3068\u3081\u308b\u3068\u6b21\u306e\u3088\u3046\u306b\u306a\u308a\u307e\u3059\u3002<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"998\" height=\"854\" src=\"https:\/\/qard.is.tohoku.ac.jp\/T-Wave\/wp-content\/uploads\/2026\/04\/75bc779b9059227a9c675be14aa40f32.png\" alt=\"\" class=\"wp-image-8938\" srcset=\"https:\/\/qard.is.tohoku.ac.jp\/T-Wave\/wp-content\/uploads\/2026\/04\/75bc779b9059227a9c675be14aa40f32.png 998w, https:\/\/qard.is.tohoku.ac.jp\/T-Wave\/wp-content\/uploads\/2026\/04\/75bc779b9059227a9c675be14aa40f32-300x257.png 300w, https:\/\/qard.is.tohoku.ac.jp\/T-Wave\/wp-content\/uploads\/2026\/04\/75bc779b9059227a9c675be14aa40f32-768x657.png 768w\" sizes=\"auto, (max-width: 998px) 100vw, 998px\" \/><\/figure>\n\n\n\n<p>\uff08\u5f15\u7528\u5143\uff1a<a href=\"https:\/\/arxiv.org\/abs\/1904.02703%EF%BC%89\">https:\/\/arxiv.org\/abs\/1904.02703\uff09<\/a><\/p>\n\n\n\n<p>\u7c21\u5358\u306b\u8a08\u7b97\u91cf\u306b\u3064\u3044\u3066\u8003\u5bdf\u3057\u307e\u3059\u3002BP\u90e8\u5206\u306e\u8a08\u7b97\u91cf\u306f\u3001\u7e70\u308a\u8fd4\u3057\u56de\u6570\u306e\u4e0a\u9650\u3092\u6c7a\u3081\u308b\u3053\u3068\u3067\u7dda\u5f62\u6642\u9593\u3067\u52d5\u4f5c\u3057\u307e\u3059\u3002\u4e00\u65b9\u306eOSD\u306e\u90e8\u5206\u3067\u306f\u3001\u30a2\u30eb\u30b4\u30ea\u30ba\u30e0\u306e\u5185\u90e8\u3067\u30e9\u30f3\u30af\u3092\u8a08\u7b97\u3057\u3066\u3044\u3066\u3001\u305d\u306e\u969b\u306b\u30ac\u30a6\u30b9\u306e\u6d88\u53bb\u6cd5\u3092\u4f7f\u3046\u305f\u3081 $O(n^3)$\u304b\u304b\u308a\u307e\u3059\u3002\u7d50\u679c\u7684\u306b\u3001BP+OSD\u5168\u4f53\u3067\u8a08\u7b97\u91cf\u304c $O(n^3)$\u306b\u306a\u308a\u307e\u3059\u3002\u8a08\u7b97\u91cf\u304c $O(n^3)$\u3067\u3042\u308b\u3053\u3068\u306f\u3001\u591a\u9805\u5f0f\u6642\u9593\u3067\u3042\u308b\u3068\u3044\u3046\u70b9\u3067\u306f\u52b9\u7387\u7684\u3067\u3059\u304c\u3001\u5b9f\u7528\u4e0a\u306f\u5927\u304d\u3059\u304e\u308b\u3068\u8003\u3048\u3089\u308c\u3066\u3044\u307e\u3059\u3002\u30c7\u30b3\u30fc\u30c0\u306e\u8a08\u7b97\u91cf\u306f\u91cf\u5b50\u30b3\u30f3\u30d4\u30e5\u30fc\u30bf\u306e\u6027\u80fd\u306b\u76f4\u7d50\u3059\u308b\u91cd\u8981\u306a\u8981\u7d20\u306a\u306e\u3067\u3001OSD\u306e\u4ee3\u308f\u308a\u306b\u3088\u308a\u8a08\u7b97\u91cf\u306e\u5c11\u306a\u3044\u51e6\u7406\u304c\u591a\u304f\u63d0\u6848\u3055\u308c\u3066\u3044\u3066\u3001\u73fe\u5728\u3082\u7814\u7a76\u304c\u7d9a\u3044\u3066\u3044\u307e\u3059\u3002<\/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>\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\u3001OSD\u306b\u3064\u3044\u3066\u305d\u306e\u4ed5\u7d44\u307f\u3092\u8aac\u660e\u3057\u3066\u3001\u6700\u5f8c\u306b\u3053\u306e\u8a18\u4e8b\u306e\u4e3b\u5f79\u3067\u3042\u308bBP+OSD\u30c7\u30b3\u30fc\u30c0\u3092\u898b\u307e\u3057\u305f\u3002OSD\u306f\u60c5\u5831\u96c6\u5408\u304c\u6301\u3064\u30de\u30c8\u30ed\u30a4\u30c9\u3068\u3044\u3046\u69cb\u9020\u3092\u3046\u307e\u304f\u4f7f\u3046\u3053\u3068\u3067\u6b63\u3057\u3044\u63a8\u5b9a\u30a8\u30e9\u30fc\u3092\u52b9\u7387\u7684\u306b\u5f97\u308b\u30a2\u30eb\u30b4\u30ea\u30ba\u30e0\u3067\u3057\u305f\u3002OSD\u3092\u4f7f\u3046\u3053\u3068\u3067\u3001\u6761\u4ef6\u3092\u6e80\u305f\u3055\u306a\u3044\u63a8\u5b9a\u30a8\u30e9\u30fc\u3092\u6761\u4ef6\u3092\u307f\u305f\u3059\u3088\u3046\u306b\u5909\u63db\u3059\u308b\u4e8b\u304c\u3067\u304d\u3001\u3053\u308c\u306b\u3088\u3063\u3066BP\u306e\u30c7\u30b3\u30fc\u30c9\u7cbe\u5ea6\u3092\u88dc\u3046\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u3002\u4e00\u65b9\u3067\u3001\u5b9f\u7528\u4e0a\u306f\u8a08\u7b97\u91cf\u304c\u91cd\u3044\u306e\u3067\u3001\u3088\u308a\u52b9\u7387\u7684\u306a\u30a2\u30eb\u30b4\u30ea\u30ba\u30e0\u304c\u7814\u7a76\u3055\u308c\u3066\u3044\u307e\u3059\u3002<\/p>\n\n\n\n<p>3\u672c\u306e\u8a18\u4e8b\u3092\u901a\u3057\u3066\u3001\u91cf\u5b50\u8aa4\u308a\u8a02\u6b63\u7b26\u53f7\u306e\u57fa\u790e\u3001\u3000LLR-MSA\u3001OSD\u306b\u3064\u3044\u3066\u7d39\u4ecb\u3057\u3001BP+OSD\u30c7\u30b3\u30fc\u30c0\u306e\u4ed5\u7d44\u307f\u3092\u898b\u3066\u304d\u307e\u3057\u305f\u3002\u306a\u308b\u3079\u304f\u81ea\u5df1\u5b8c\u7d50\u3059\u308b\u3088\u3046\u306b\u66f8\u3044\u305f\u7d50\u679c\u3001\u3068\u3066\u3082\u9577\u3044\u8a18\u4e8b\u306b\u306a\u3063\u3066\u3057\u307e\u3044\u307e\u3057\u305f\u304c\u3001\u6539\u3081\u3066\u6700\u5f8c\u307e\u3067\u8aad\u3093\u3067\u3044\u305f\u3060\u304d\u3042\u308a\u304c\u3068\u3046\u3054\u3056\u3044\u307e\u3059\u3002\u91cf\u5b50\u8aa4\u308a\u8a02\u6b63\u7b26\u53f7\u3084\u305d\u306e\u30c7\u30b3\u30fc\u30c0\u306b\u3064\u3044\u3066\u8208\u5473\u3092\u6301\u3064\u304d\u3063\u304b\u3051\u306b\u306a\u308c\u308c\u3070\u5e78\u3044\u3067\u3059\u3002<\/p>\n\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 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