{"paper":{"title":"Critical pairs for the Product Singleton Bound","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.IT"],"primary_cat":"cs.IT","authors_text":"Diego Mirandola, Gilles Z\\'emor","submitted_at":"2015-01-26T14:48:04Z","abstract_excerpt":"We characterize Product-MDS pairs of linear codes, i.e.\\ pairs of codes $C,D$ whose product under coordinatewise multiplication has maximum possible minimum distance as a function of the code length and the dimensions $\\dim C, \\dim D$. We prove in particular, for $C=D$, that if the square of the code $C$ has minimum distance at least $2$, and $(C,C)$ is a Product-MDS pair, then either $C$ is a generalized Reed-Solomon code, or $C$ is a direct sum of self-dual codes. In passing we establish coding-theory analogues of classical theorems of additive combinatorics."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1501.06419","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"}