On $ \sigma $-self-orthogonal constacyclic codes over $ \mathbb F_{p^m}+u\mathbb F_{p^m} $
Hongwei Liu, Jingge Liu · Advances in Mathematics of Communications · 2020
In this paper, we generalize the notion of self-orthogonal codes to \begin{document}$ \sigma $\end{document} -self-orthogonal codes over an arbitrary finite ring. Then, we study the \begin{document}$ \sigma $\end{document} -self-orthogonality of constacyclic codes of length \begin{document}$ p^s $\end{document} over the finite commutative chain ring \begin{document}$ \mathbb F_{p^m} + u \mathbb F_{p^m} $\end{document} , where \begin{document}$ p $\end{document} is a prime, \begin{document}$ u^2 = 0 $\end{document} and \begin{document}$ \sigma $\end{document} is an arbitrary ring automorphism of \begin{document}$ \mathbb F_{p^m} + u \mathbb F_{p^m} $\end{document} . We characterize the structure of \begin{document}$ \sigma $\end{document} -dual code of a \begin{document}$ \lambda $\end{document} -constacyclic code of length \begin{document}$ p^s $\end{document} over \begin{document}$ \mathbb F_{p^m} + u \mathbb F_{p^m} $\end{document} . Further, the necessary and sufficient conditions for a \begin{document}$ \lambda $\end{document} -constacyclic code to be \begin{document}$ \sigma $\end{document} -self-orthogonal are provided. In particular, we determine all \begin{document}$ \sigma $\end{document} -self-dual constacyclic codes of length \begin{document}$ p^s $\end{document} over \begin{document}$ \mathbb F_{p^m} + u \mathbb F_{p^m} $\end{document} . In the end of this paper, when \begin{document}$ p $\end{document} is an odd prime, we extend the results to constacyclic codes of length \begin{document}$ 2 p^s $\end{document} .