{"paper":{"title":"The maximum-likelihood decoding threshold for graphic codes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO","math.IT"],"primary_cat":"cs.IT","authors_text":"Peter Nelson, Stefan H.M. van Zwam","submitted_at":"2015-04-20T20:47:17Z","abstract_excerpt":"For a class $\\mathcal{C}$ of binary linear codes, we write $\\theta_{\\mathcal{C}}\\colon (0,1) \\to [0,\\frac{1}{2}]$ for the maximum-likelihood decoding threshold function of $\\mathcal{C}$, the function whose value at $R \\in (0,1)$ is the largest bit-error rate $p$ that codes in $\\mathcal{C}$ can tolerate with a negligible probability of maximum-likelihood decoding error across a binary symmetric channel. We show that, if $\\mathcal{C}$ is the class of cycle codes of graphs, then $\\theta_{\\mathcal{C}}(R) \\le \\frac{(1-\\sqrt{R})^2}{2(1+R)}$ for each $R$, and show that equality holds only when $R$ is"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1504.05225","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}