New self-dual codes of length 68 from a $ 2 \times 2 $ block matrix construction and group rings
М. Ю. Бортош, Joe Gildea, Abidin Kaya, Adrian Korban, Alexander Tylyshchak · Advances in Mathematics of Communications · 2020
Many generator matrices for constructing extremal binary self-dual codes of different lengths have the form \begin{document}$ G = (I_n \ | \ A), $\end{document} where \begin{document}$ I_n $\end{document} is the \begin{document}$ n \times n $\end{document} identity matrix and \begin{document}$ A $\end{document} is the \begin{document}$ n \times n $\end{document} matrix fully determined by the first row. In this work, we define a generator matrix in which \begin{document}$ A $\end{document} is a block matrix, where the blocks come from group rings and also, \begin{document}$ A $\end{document} is not fully determined by the elements appearing in the first row. By applying our construction over \begin{document}$ \mathbb{F}_2+u\mathbb{F}_2 $\end{document} and by employing the extension method for codes, we were able to construct new extremal binary self-dual codes of length 68. Additionally, by employing a generalised neighbour method to the codes obtained, we were able to construct many new binary self-dual \begin{document}$ [68, 34, 12] $\end{document} -codes with the rare parameters \begin{document}$ \gamma = 7, 8 $\end{document} and \begin{document}$ 9 $\end{document} in \begin{document}$ W_{68, 2}. $\end{document} In particular, we find 92 new binary self-dual \begin{document}$ [68, 34, 12] $\end{document} -codes.