On $ \sigma $-self-orthogonal constacyclic codes over $ \mathbb F_{p^m}+u\mathbb F_{p^m} $

Abstract

<p style='text-indent:20px;'>In this paper, we generalize the notion of self-orthogonal codes to <inline-formula><tex-math id="M1">\begin{document}$ \sigma $\end{document}</tex-math></inline-formula>-self-orthogonal codes over an arbitrary finite ring. Then, we study the <inline-formula><tex-math id="M2">\begin{document}$ \sigma $\end{document}</tex-math></inline-formula>-self-orthogonality of constacyclic codes of length <inline-formula><tex-math id="M3">\begin{document}$ p^s $\end{document}</tex-math></inline-formula> over the finite commutative chain ring <inline-formula><tex-math id="M4">\begin{document}$ \mathbb F_{p^m} + u \mathbb F_{p^m} $\end{document}</tex-math></inline-formula>, where <inline-formula><tex-math id="M5">\begin{document}$ p $\end{document}</tex-math></inline-formula> is a prime, <inline-formula><tex-math id="M6">\begin{document}$ u^2 = 0 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M7">\begin{document}$ \sigma $\end{document}</tex-math></inline-formula> is an arbitrary ring automorphism of <inline-formula><tex-math id="M8">\begin{document}$ \mathbb F_{p^m} + u \mathbb F_{p^m} $\end{document}</tex-math></inline-formula>. We characterize the structure of <inline-formula><tex-math id="M9">\begin{document}$ \sigma $\end{document}</tex-math></inline-formula>-dual code of a <inline-formula><tex-math id="M10">\begin{document}$ \lambda $\end{document}</tex-math></inline-formula>-constacyclic code of length <inline-formula><tex-math id="M11">\begin{document}$ p^s $\end{document}</tex-math></inline-formula> over <inline-formula><tex-math id="M12">\begin{document}$ \mathbb F_{p^m} + u \mathbb F_{p^m} $\end{document}</tex-math></inline-formula>. Further, the necessary and sufficient conditions for a <inline-formula><tex-math id="M13">\begin{document}$ \lambda $\end{document}</tex-math></inline-formula>-constacyclic code to be <inline-formula><tex-math id="M14">\begin{document}$ \sigma $\end{document}</tex-math></inline-formula>-self-orthogonal are provided. In particular, we determine all <inline-formula><tex-math id="M15">\begin{document}$ \sigma $\end{document}</tex-math></inline-formula>-self-dual constacyclic codes of length <inline-formula><tex-math id="M16">\begin{document}$ p^s $\end{document}</tex-math></inline-formula> over <inline-formula><tex-math id="M17">\begin{document}$ \mathbb F_{p^m} + u \mathbb F_{p^m} $\end{document}</tex-math></inline-formula>. In the end of this paper, when <inline-formula><tex-math id="M18">\begin{document}$ p $\end{document}</tex-math></inline-formula> is an odd prime, we extend the results to constacyclic codes of length <inline-formula><tex-math id="M19">\begin{document}$ 2 p^s $\end{document}</tex-math></inline-formula>.</p>

Quantum codes from constacyclic codes over $S_{k}$

“…Calderbank et al [1] gave a way to construct quantum error correcting codes from classical error correcting codes, constructing quantum error correcting codes is a systematic and effective mathematical method by using constacyclic codes. There are a lot of works about constacyclic codes over finite fields and finite rings [2][3][4][5][6][7][8][9][10] and many good quantum codes constructed by using cyclic codes over finite rings [11][12][13][14]. Currently, some authors have obtained quantum codes from constacyclic codes over finite non-chain ring.…”