Self-orthogonal codes from orbit matrices of Seidel and Laplacian matrices of strongly regular graphs

Abstract

<p style='text-indent:20px;'>In this paper we introduce the notion of orbit matrices of integer matrices such as Seidel and Laplacian matrices of some strongly regular graphs with respect to their permutation automorphism groups. We further show that under certain conditions these orbit matrices yield self-orthogonal codes over finite fields <inline-formula><tex-math id="M1">\begin{document}$ \mathbb{F}_q $\end{document}</tex-math></inline-formula>, where <inline-formula><tex-math id="M2">\begin{document}$ q $\end{document}</tex-math></inline-formula> is a prime power and over finite rings <inline-formula><tex-math id="M3">\begin{document}$ \mathbb{Z}_m $\end{document}</tex-math></inline-formula>. As a case study, we construct codes from orbit matrices of Seidel, Laplacian and signless Laplacian matrices of strongly regular graphs. In particular, we construct self-orthogonal codes from orbit matrices of Seidel and Laplacian matrices of the Higman-Sims and McLaughlin graphs.