{"paper":{"title":"An Algebraic-Combinatorial Proof Technique for the GM-MDS Conjecture","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.IT"],"primary_cat":"cs.IT","authors_text":"Alex Sprintson, Anoosheh Heidarzadeh","submitted_at":"2017-02-06T18:36:10Z","abstract_excerpt":"This paper considers the problem of designing maximum distance separable (MDS) codes over small fields with constraints on the support of their generator matrices. For any given $m\\times n$ binary matrix $M$, the GM-MDS conjecture, due to Dau et al., states that if $M$ satisfies the so-called MDS condition, then for any field $\\mathbb{F}$ of size $q\\geq n+m-1$, there exists an $[n,m]_q$ MDS code whose generator matrix $G$, with entries in $\\mathbb{F}$, fits $M$ (i.e., $M$ is the support matrix of $G$). Despite all the attempts by the coding theory community, this conjecture remains still open "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1702.01734","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"}