Bounds and MacWilliams Identities for codes over Artinian rings

Abstract

This work develops new foundations for the theory of linear codes over local Artinian commutative rings. We use algebraic invariants such as the socle, type, length, and minimal number of generators to measure the size of codes. We prove a relation between the type of a code and the free rank of its dual over Frobenius rings, extending previous results for chain rings. We also provide new upper bounds for the Hamming distance in terms of the length and type of a code and and conditions under which the dual of an MDS code remains MDS for general Artinian rings. The latter result is obtained by reducing to Frobenius rings via Nagata idealizations, which, to the best of our knowledge, had not been used in coding theory before. We introduce a conceptually different version of the weight enumerator polynomial. This enumerator is meaningful even in the case of infinite rings and yields new applications in the finite setting. Using this polynomial, we prove a MacWilliams identity that holds over Frobenius rings.