Weight Enumerators From Equivalence Relations and MacWilliams Identities

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.