{"paper":{"title":"Lengths of divisible codes -- the missing cases","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.IT","math.IT"],"primary_cat":"math.CO","authors_text":"Sascha Kurz","submitted_at":"2023-11-03T14:38:05Z","abstract_excerpt":"A linear code $C$ over $\\mathbb{F}_q$ is called $\\Delta$-divisible if the Hamming weights $\\operatorname{wt}(c)$ of all codewords $c \\in C$ are divisible by $\\Delta$. The possible effective lengths of $q^r$-divisible codes have been completely characterized for each prime power $q$ and each non-negative integer $r$. The study of $\\Delta$ divisible codes was initiated by Harold Ward. If $c$ divides $\\Delta$ but is coprime to $q$, then each $\\Delta$-divisible code $C$ over $\\F_q$ is the $c$-fold repetition of a $\\Delta/c$-divisible code. Here we determine the possible effective lengths of $p^r$-"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2311.01947","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2311.01947/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"}