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

      Korban, Adrian; Şahinkaya, Serap; Ustun, Deniz (American Institute of Mathematical Sciences (AIMS), 2022)
      <p style='text-indent:20px;'>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.</p>
    • Binary self-dual codes of various lengths with new weight enumerators from a modified bordered construction and neighbours

      Gildea, Joe; Korban, Adrian; Roberts, Adam M.; Tylyshchak, Alexander (American Institute of Mathematical Sciences (AIMS), 2022)
      <p style='text-indent:20px;'>In this work, we define a modification of a bordered construction for self-dual codes which utilises <inline-formula><tex-math id="M1">\begin{document}$ \lambda $\end{document}</tex-math></inline-formula>-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.</p>
    • 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

      Korban, Adrian; Sahinkaya, Serap; Ustun, Deniz (American Institute of Mathematical Sciences (AIMS), 2022)
      <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>