Weight Enumerators From Equivalence Relations and MacWilliams Identities

2025-10-31

AI Summary

This work addresses coding problems over finite fields, finite abelian groups, and finite Frobenius rings by systematically constructing generalized weight enumerators based on arbitrary equivalence relations. Methodologically, it integrates algebraic coding theory, group representation theory, and combinatorics to characterize the necessary and sufficient conditions under which such enumerators satisfy MacWilliams identities—establishing, for the first time, an equivalence-relation-driven framework for weight counting. By moving beyond classical reliance on Hamming or Lee metrics, the approach develops a unified duality analysis paradigm applicable across diverse algebraic structures. The theory’s completeness and consistency are rigorously verified under several canonical equivalence relations, including coordinate permutations and group-action orbits. The results substantially extend the scope of MacWilliams theory, providing novel analytical and constructive tools for non-standard code spaces.

Technology Category

Machine Learning: Probabilistic Circuits and Graphical ModelsGame Theory and Economic Paradigms: EquilibriumConstraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction

Application Category

Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphsWeb Mining and Content Analysis: Models for Web evolutionSecurity and Privacy: Applications of cryptography

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.

Problem

• Defining weight enumerators based on equivalence relations
• Determining when MacWilliams relations hold
• Studying specific enumerators for particular equivalence relations

Innovation

• Defines weight enumerators via equivalence relations
• Determines MacWilliams relations applicability conditions
• Studies enumerators for specific equivalence cases