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

TL;DR

This paper investigates the structure and conditions for $\sigma$-self-orthogonality of constacyclic codes over a specific finite chain ring, providing classifications and extending results to codes of length $2 p^s$.

Contribution

It characterizes $\sigma$-self-orthogonal and $\sigma$-self-dual constacyclic codes over $_{p^m}+u_{p^m}$ of length $p^s$, including necessary and sufficient conditions.

Findings

Derived the structure of $\sigma$-dual codes.

Established criteria for $\sigma$-self-orthogonality.

Identified all $\sigma$-self-dual codes of length $p^s$.

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 $2 p^s$.