Title: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.