Finite Commutative Rings with a MacWilliams Type Relation for the m-Spotty Hamming Weight Enumerators

TL;DR

This paper establishes a precise condition under which a MacWilliams type relation for m-spotty Hamming weight enumerators holds over finite commutative rings, linking it to Frobenius ring properties.

Contribution

It proves that the MacWilliams relation holds if and only if the ring is Frobenius, providing a new proof of Wood's theorem in the commutative case.

Findings

MacWilliams relation holds iff the ring is Frobenius

Ring is Frobenius iff number of maximal and minimal ideals are equal

New simplified proof of Wood's theorem for commutative rings

Abstract

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 Frobenius ring.