{"paper":{"title":"Counterexamples to Charpin's Conjecture on BCH codes","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.IT"],"primary_cat":"cs.IT","authors_text":"Maosheng Xiong, Run Zheng, Yaoran Yang, Yutong Zhang","submitted_at":"2026-07-30T18:01:50Z","abstract_excerpt":"Determining the exact minimum distance of BCH codes is a longstanding and challenging problem. In this paper, we construct an infinite family of primitive narrow-sense BCH codes whose minimum distance strictly exceeds their Bose distance. Let $q$ be a prime power, let $m$ be an integer with $m \\geq 10$ and $m \\neq 12$, and set $u = \\lfloor m/4 \\rfloor$ and $t = \\lfloor (m-1)/3 \\rfloor$. For each integer $s$ with $u \\leq s < t$, we define$$\\delta = q^m - q^{m-1} - q^{m-1-u} - q^s - 1.$$We prove that the primitive narrow-sense BCH code with designed distance $\\delta$ has Bose distance $\\delta$ a"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.28741","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2607.28741/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}