Generic constructions of MDS Euclidean self-dual codes via GRS codes

Abstract

Recently, the construction of new MDS Euclidean self-dual codes has been widely investigated. In this paper, for square \begin{document}$ q $\end{document}, we utilize generalized Reed-Solomon (GRS) codes and their extended codes to provide four generic families of \begin{document}$ q $\end{document}-ary MDS Euclidean self-dual codes of lengths in the form \begin{document}$ s\frac{q-1}{a}+t\frac{q-1}{b} $\end{document}, where \begin{document}$ s $\end{document} and \begin{document}$ t $\end{document} range in some interval and \begin{document}$ a, b \,|\, (q -1) $\end{document}. In particular, for large square \begin{document}$ q $\end{document}, our constructions take up a proportion of generally more than 34% in all the possible lengths of \begin{document}$ q $\end{document}-ary MDS Euclidean self-dual codes, which is larger than the previous results. Moreover, two new families of MDS Euclidean self-orthogonal codes and two new families of MDS Euclidean almost self-dual codes are given similarly.