New self-dual codes of length 68 from a $ 2 \times 2 $ block matrix construction and group rings

Abstract

<p style='text-indent:20px;'>Many generator matrices for constructing extremal binary self-dual codes of different lengths have the form <inline-formula><tex-math id="M1">\begin{document}$ G = (I_n \ | \ A), $\end{document}</tex-math></inline-formula> where <inline-formula><tex-math id="M2">\begin{document}$ I_n $\end{document}</tex-math></inline-formula> is the <inline-formula><tex-math id="M3">\begin{document}$ n \times n $\end{document}</tex-math></inline-formula> identity matrix and <inline-formula><tex-math id="M4">\begin{document}$ A $\end{document}</tex-math></inline-formula> is the <inline-formula><tex-math id="M5">\begin{document}$ n \times n $\end{document}</tex-math></inline-formula> matrix fully determined by the first row. In this work, we define a generator matrix in which <inline-formula><tex-math id="M6">\begin{document}$ A $\end{document}</tex-math></inline-formula> is a block matrix, where the blocks come from group rings and also, <inline-formula><tex-math id="M7">\begin{document}$ A $\end{document}</tex-math></inline-formula> is not fully determined by the elements appearing in the first row. By applying our construction over <inline-formula><tex-math id="M8">\begin{document}$ \mathbb{F}_2+u\mathbb{F}_2 $\end{document}</tex-math></inline-formula> and by employing the extension method for codes, we were able to construct new extremal binary self-dual codes of length 68. Additionally, by employing a generalised neighbour method to the codes obtained, we were able to construct many new binary self-dual <inline-formula><tex-math id="M9">\begin{document}$ [68, 34, 12] $\end{document}</tex-math></inline-formula>-codes with the rare parameters <inline-formula><tex-math id="M10">\begin{document}$ \gamma = 7, 8 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M11">\begin{document}$ 9 $\end{document}</tex-math></inline-formula> in <inline-formula><tex-math id="M12">\begin{document}$ W_{68, 2}. $\end{document}</tex-math></inline-formula> In particular, we find 92 new binary self-dual <inline-formula><tex-math id="M13">\begin{document}$ [68, 34, 12] $\end{document}</tex-math></inline-formula>-codes.</p>