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.

翻译：本文研究有限域、有限阿贝尔群及有限弗罗贝尼乌斯环上的编码理论。对于此类编码，完全重量枚举器与汉明重量枚举器是重要的分析工具，二者均满足麦克威廉姆斯关系。我们定义了基于等价关系的编码重量枚举器，并确定了该枚举器满足麦克威廉姆斯关系的条件。同时，针对特定等价关系下的若干重量枚举器进行了深入探讨。