New self-dual codes from $ 2 \times 2 $ block circulant matrices, group rings and neighbours of neighbours

Abstract

<p style='text-indent:20px;'>In this paper, we construct new self-dual codes from a construction that involves a unique combination; <inline-formula><tex-math id="M1">\begin{document}$ 2 \times 2 $\end{document}</tex-math></inline-formula> block circulant matrices, group rings and a reverse circulant matrix. There are certain conditions, specified in this paper, where this new construction yields self-dual codes. The theory is supported by the construction of self-dual codes over the rings <inline-formula><tex-math id="M2">\begin{document}$ \mathbb{F}_2 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M3">\begin{document}$ \mathbb{F}_2+u \mathbb{F}_2 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M4">\begin{document}$ \mathbb{F}_4+u \mathbb{F}_4 $\end{document}</tex-math></inline-formula>. Using extensions and neighbours of codes, we construct <inline-formula><tex-math id="M5">\begin{document}$ 32 $\end{document}</tex-math></inline-formula> new self-dual codes of length <inline-formula><tex-math id="M6">\begin{document}$ 68 $\end{document}</tex-math></inline-formula>. We construct 48 new best known singly-even self-dual codes of length 96.</p>