{"paper":{"title":"New nonbinary code bounds based on divisibility arguments","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Sven Polak","submitted_at":"2016-06-16T11:17:09Z","abstract_excerpt":"For $q,n,d \\in \\mathbb{N}$, let $A_q(n,d)$ be the maximum size of a code $C \\subseteq [q]^n$ with minimum distance at least $d$. We give a divisibility argument resulting in the new upper bounds $A_5(8,6) \\leq 65$, $A_4(11,8)\\leq 60$ and $A_3(16,11) \\leq 29$. These in turn imply the new upper bounds $A_5(9,6) \\leq 325$, $A_5(10,6) \\leq 1625$, $A_5(11,6) \\leq 8125$ and $A_4(12,8) \\leq 240$. Furthermore, we prove that for $\\mu,q \\in \\mathbb{N}$, there is a 1-1-correspondence between symmetric $(\\mu,q)$-nets (which are certain designs) and codes $C \\subseteq [q]^{\\mu q}$ of size $\\mu q^2$ with mi"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1606.05144","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"}