Now showing items 1-3 of 3

• #### A novel genetic search scheme based on nature-inspired evolutionary algorithms for binary self-dual codes

&lt;p style='text-indent:20px;'&gt;In this paper, a genetic algorithm, one of the evolutionary algorithm optimization methods, is used for the first time for the problem of computing extremal binary self-dual codes. We present a comparison of the computational times between the genetic algorithm and a linear search for different size search spaces and show that the genetic algorithm is capable of computing binary self-dual codes significantly faster than the linear search. Moreover, by employing a known matrix construction together with the genetic algorithm, we are able to obtain new binary self-dual codes of lengths 68 and 72 in a significantly short time. In particular, we obtain 11 new binary self-dual codes of length 68 and 17 new binary self-dual codes of length 72.&lt;/p&gt;
• #### Binary self-dual codes of various lengths with new weight enumerators from a modified bordered construction and neighbours

&lt;p style='text-indent:20px;'&gt;In this work, we define a modification of a bordered construction for self-dual codes which utilises &lt;inline-formula&gt;&lt;tex-math id="M1"&gt;\begin{document}$\lambda$\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;-circulant matrices. We provide the necessary conditions for the construction to produce self-dual codes over finite commutative Frobenius rings of characteristic 2. Using the modified construction together with the neighbour construction, we construct many binary self-dual codes of lengths 54, 68, 82 and 94 with weight enumerators that have previously not been known to exist.&lt;/p&gt;
• #### 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

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