{"paper":{"title":"Finite Commutative Rings with a MacWilliams Type Relation for the m-Spotty Hamming Weight Enumerators","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.IT","math.IT"],"primary_cat":"math.AC","authors_text":"Ashkan Nikseresht","submitted_at":"2016-05-10T07:02:24Z","abstract_excerpt":"Let $R$ be a finite commutative ring. We prove that a MacWilliams type relation between the m-spotty weight enumerators of a linear code over $R$ and its dual hold, if and only if, $R$ is a Frobenius (equivalently, Quasi-Frobenius) ring, if and only if, the number of maximal ideals and minimal ideals of $R$ are the same, if and only if, for every linear code $C$ over $R$, the dual of the dual $C$ is $C$ itself. Also as an intermediate step, we present a new and simpler proof for the commutative case of Wood's theorem which states that $R$ has a generating character if and only if $R$ is a Frob"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1605.02870","kind":"arxiv","version":3},"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"}