Weight Enumerators From Equivalence Relations and MacWilliams Identities

TL;DR

This paper explores how weight enumerators of codes over various algebraic structures relate to MacWilliams identities, defining new types of weight enumerators based on equivalence relations and identifying conditions for MacWilliams relations to hold.

Contribution

It introduces a generalized framework for weight enumerators based on equivalence relations and characterizes when MacWilliams identities are valid for these new enumerators.

Findings

Identifies conditions under which MacWilliams relations hold for equivalence relation-based weight enumerators.

Defines new weight enumerators for codes over finite fields, groups, and rings.

Studies specific cases of weight enumerators for particular equivalence relations.

Abstract

In this paper, we consider codes over finite fields, finite abelian groups, and finite Frobenius rings. For such codes, the complete weight enumerator and the Hamming weight enumerator serve as powerful tools. These two types of weight enumerators satisfy the MacWilliams relations. We define the weight enumerator of a code with respect to an equivalence relation and determine in which cases the MacWilliams relations hold for this weight enumerator. We also study some weight enumerators for specific equivalence relations.