$\sigma$-self-orthogonal constacyclic codes of length $p^s$ over ${\mathbb{F}}_{p^m}+u{\mathbb{F}}_{p^m}$

Abstract

In this paper, we study the $\sigma$-self-orthogonality of constacyclic codes of length $p^s$ over the finite commutative chain ring ${\mathbb{F}}_{p^m}+u{\mathbb{F}}_{p^m}$, where $u^2=0$ and $\sigma$ is a ring automorphism of ${\mathbb{F}}_{p^m}+u{\mathbb{F}}_{p^m}$. First, we obtain the structure of $\sigma$-dual code of a $\lambda$-constacyclic code of length $p^s$ over ${\mathbb{F}}_{p^m}+u{\mathbb{F}}_{p^m}$. Then, the necessary and sufficient conditions for a $\lambda$-constacyclic code to be $\sigma$-self-orthogonal are provided. In particular, we determine the $\sigma$-self-dual constacyclic codes of length $p^s$ over ${\mathbb{F}}_{p^m}+u{\mathbb{F}}_{p^m}$. Finally, we extend the results to constacyclic codes of length $2p^s$.