Genetic algorithms for finding the weight enumerator of binary linear block codes
This work addresses a specific challenge in coding theory for researchers, but it is incremental as it builds on existing methods like genetic algorithms and MacWilliams identity.
The authors tackled the problem of determining the weight enumerator of binary linear block codes by developing a genetic algorithm-based method that incorporates structural constraints and automorphism group information, successfully listing all possible values for Extended Quadratic Residue codes of lengths 192 and 200 and approximating their true values.
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.