Genetic algorithms for finding the weight enumerator of binary linear block codes

TL;DR

This paper introduces a genetic algorithm-based method to determine the weight enumerator of binary linear block codes, leveraging automorphism groups and the MacWilliams identity to improve accuracy and efficiency.

Contribution

It presents a novel genetic algorithm approach combined with group automorphisms and MacWilliams identity to find weight enumerators of complex codes.

Findings

Successfully identified all possible weight enumerators for ERQ codes of lengths 192 and 200.

Provided accurate approximations of the true weight enumerators for these codes.

Demonstrated the effectiveness of genetic algorithms in coding theory problems.

Abstract

In this paper we present a new method for finding the weight enumerator of binary linear block codes by using genetic algorithms. This method consists in finding the binary weight enumerator of the code and its dual and to create from the famous MacWilliams identity a linear system (S) of integer variables for which we add all known information obtained from the structure of the code. The knowledge of some subgroups of the automorphism group, under which the code remains invariant, permits to give powerful restrictions on the solutions of (S) and to approximate the weight enumerator. By applying this method and by using the stability of the Extended Quadratic Residue codes (ERQ) by the Projective Special Linear group PSL2, we find a list of all possible values of the weight enumerators for the two ERQ codes of lengths 192 and 200. We also made a good approximation of the true value for these two enumerators.