New type I binary $[72, 36, 12]$ self-dual codes from $M_6(\mathbb{F}_2)G$ - Group matrix rings by a hybrid search technique based on a neighbourhood-virus optimisation algorithm

Abstract

<p style='text-indent:20px;'>In this paper, a new search technique based on a 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 technique is known in the literature 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 <inline-formula><tex-math id="M1">\begin{document}$ k^{th} $\end{document}</tex-math></inline-formula>-range neighbours, and search for binary <inline-formula><tex-math id="M2">\begin{document}$ [72, 36, 12] $\end{document}</tex-math></inline-formula> self-dual codes. In particular, we present six generator matrices of the form <inline-formula><tex-math id="M3">\begin{document}$ [I_{36} \ | \ \tau_6(v)], $\end{document}</tex-math></inline-formula> where <inline-formula><tex-math id="M4">\begin{document}$ I_{36} $\end{document}</tex-math></inline-formula> is the <inline-formula><tex-math id="M5">\begin{document}$ 36 \times 36 $\end{document}</tex-math></inline-formula> identity matrix, <inline-formula><tex-math id="M6">\begin{document}$ v $\end{document}</tex-math></inline-formula> is an element in the group matrix ring <inline-formula><tex-math id="M7">\begin{document}$ M_6(\mathbb{F}_2)G $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M8">\begin{document}$ G $\end{document}</tex-math></inline-formula> is a finite group of order 6, to which we employ the proposed algorithm and search for binary <inline-formula><tex-math id="M9">\begin{document}$ [72, 36, 12] $\end{document}</tex-math></inline-formula> self-dual codes directly over the finite field <inline-formula><tex-math id="M10">\begin{document}$ \mathbb{F}_2 $\end{document}</tex-math></inline-formula>. We construct 1471 new Type I binary <inline-formula><tex-math id="M11">\begin{document}$ [72, 36, 12] $\end{document}</tex-math></inline-formula> self-dual codes with the rare parameters <inline-formula><tex-math id="M12">\begin{document}$ \gamma = 11, 13, 14, 15, 17, 19, 20, 21, 22, 23, 25, 26, 28, 29, 30, 31, 32 $\end{document}</tex-math></inline-formula> in their weight enumerators.</p>