The self-duality and self-orthogonality of linear codes are very important properties. A linear code is called an MDS self-dual (self-orthogonal) code if it is both a code reached Singleton bound and a self-dual (self-orthogonal) code with respect to the Euclidean inner product. The parameters of MDS self-dual codes are completely determined by the code length. Recently, the new constructions of MDS self-dual (self-orthogonal) codes have been widely investigated, and several new classes of codes with different lengths have been found.In this paper, we utilize generalized Reed-Solomon (GRS) codes and extended GRS codes to construct MDS self-dual codes, MDS self-orthogonal codes and MDS almost self-dual codes over F q (q = r 2 , r is a power of an odd prime). The main idea of our constructions is to choose suitable evaluation points such that the corresponding (extended) GRS codes are Euclidean self-dual (self-orthogonal). The evaluation sets are consists of two subsets which satisfy some certain conditions and the length of these codes can be expressed as a linear combination of two factors of q − 1. Four families of MDS self-dual codes, two families of MDS selforthogonal codes and two families of MDS almost self-dual codes are obtained and they have new parameters.
<p style='text-indent:20px;'>Recently, the construction of new MDS Euclidean self-dual codes has been widely investigated. In this paper, for square <inline-formula><tex-math id="M1">\begin{document}$ q $\end{document}</tex-math></inline-formula>, we utilize generalized Reed-Solomon (GRS) codes and their extended codes to provide four generic families of <inline-formula><tex-math id="M2">\begin{document}$ q $\end{document}</tex-math></inline-formula>-ary MDS Euclidean self-dual codes of lengths in the form <inline-formula><tex-math id="M3">\begin{document}$ s\frac{q-1}{a}+t\frac{q-1}{b} $\end{document}</tex-math></inline-formula>, where <inline-formula><tex-math id="M4">\begin{document}$ s $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M5">\begin{document}$ t $\end{document}</tex-math></inline-formula> range in some interval and <inline-formula><tex-math id="M6">\begin{document}$ a, b \,|\, (q -1) $\end{document}</tex-math></inline-formula>. In particular, for large square <inline-formula><tex-math id="M7">\begin{document}$ q $\end{document}</tex-math></inline-formula>, our constructions take up a proportion of generally more than 34% in all the possible lengths of <inline-formula><tex-math id="M8">\begin{document}$ q $\end{document}</tex-math></inline-formula>-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.</p>
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