Periodicity of weight enumerators for codes generated by an integral matrix

TL;DR

This paper explores the periodic properties of weight enumerators in error-correcting codes generated by integral matrices, linking them to quasi-polynomials and matroid theory to facilitate calculations of code properties.

Contribution

It introduces a method to analyze weight enumerators as quasi-polynomials, reducing minimum weight computation to smaller residue rings and establishing a connection with Tutte quasi-polynomials.

Findings

Derived a transformation formula between Tutte quasi-polynomial and weight enumerator.

Computed the number of maximum weight codewords for specific matroid-related codes.

Linked the characteristic quasi-polynomial of hyperplane arrangements to codeword enumeration.

Abstract

In the theory of error-correcting codes, the minimum weight and the weight enumerator play a crucial role in evaluating the error-correcting capacity. In this paper, by viewing the weight enumerator as a quasi-polynomial, we reduce the calculation of the minimum weight to that of a code over a smaller integer residue ring. We also give a transformation formula between the Tutte quasi-polynomial and the weight enumerator. Furthermore, we compute the number of maximum weight codewords for the codes related to the matroids $N_k$ and $Z_k$. This is equivalent to computing the characteristic quasi-polynomial of the hyperplane arrangements related to $N_k$ and $Z_k$.