{"paper":{"title":"Asymptotic Improvement of the Gilbert-Varshamov Bound on the Size of Permutation Codes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.IT","math.IT"],"primary_cat":"math.CO","authors_text":"Alexander Vardy, Jacques Verstraete, Michael Tait","submitted_at":"2013-11-20T00:33:16Z","abstract_excerpt":"Given positive integers $n$ and $d$, let $M(n,d)$ denote the maximum size of a permutation code of length $n$ and minimum Hamming distance $d$. The Gilbert-Varshamov bound asserts that $M(n,d) \\geq n!/V(n,d-1)$ where $V(n,d)$ is the volume of a Hamming sphere of radius $d$ in $\\S_n$.\n  Recently, Gao, Yang, and Ge showed that this bound can be improved by a factor $\\Omega(\\log n)$, when $d$ is fixed and $n \\to \\infty$. Herein, we consider the situation where the ratio $d/n$ is fixed and improve the Gilbert-Varshamov bound by a factor that is \\emph{linear in $n$}. That is, we show that if $d/n <"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1311.4925","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":""},"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"}