New Extremal Binary Self-Dual Codes of Length 72 from $M_6({\mathbb{F}}_2)G$ - Group Matrix Rings by a Hybrid Search Technique Based on a Neighbourhood-Virus Optimisation Algorithm

Abstract

In this paper, a new search technique based on the virus optimisation algorithm is proposed for calculating the neighbours of binary self-dual codes. The aim of this new technique is to calculate neighbours of self-dual codes without reducing the search field in the search process (this is a known in the literature approach due to the computational time constraint) but still obtaining results in a reasonable time (significantly faster when compared to the standard linear computational search). We employ this new search algorithm to the well-known neighbour method and its extension, the $k^{th}$-range neighbours and search for binary $[72,36,12]$ self-dual codes. In particular, we present six generator matrices of the form $[I_{36}|{\tau }_6(v)],$ where $I_{36}$ is the $36\times 36$ identity matrix, $v$ is an element in the group matrix ring $M_6({\mathbb{F}}_2)G$ and $G$ is a finite group of order 6, which we then employ to the proposed algorithm and search for binary $[72,36,12]$ self-dual codes directly over the finite field ${\mathbb{F}}_2$. We construct 1471 new Type I binary $[72,36,12]$ self-dual codes with the rare parameters $\gamma =11,13,14,15,17,19,20,21,22,23,25,26,28,29,30,31,32$ in their weight enumerators.

arXiv admin note: text overlap with arXiv:2103.07739, arXiv:2102.12863