Skew Constacyclic Codes Of Length $np^s$ over $ \frac{\mathbb{F}_{p^m}[u]}{\langle u^k \rangle}

TL;DR

This paper introduces a unified algebraic framework for analyzing skew constacyclic codes over specific rings, classifying their structures and providing examples with optimal parameters.

Contribution

It develops a comprehensive approach to classify skew constacyclic codes over rings of the form _{p^m}[u]/l u^k, including explicit classifications and case analyses.

Findings

Classified all left ideals for certain skew constacyclic codes.

Established isomorphism between skew cyclic and skew constacyclic codes.

Provided examples of codes with optimal parameters.

Abstract

Let $\mathbb{F}_{p^m}$ be the field containing $p^m$ elements where $p$ is an odd prime and $m \in \mathbb{N}$. In this article, we propose a unified approach to the study of skew constacyclic codes of length $np^s$ over the ring $R_k = \mathbb{F}_{p^m}[u]/\langle u^k \rangle,$ where $n, s, k \in \mathbb{N}$ and $\gcd(n, p)=1$. Consider the skew polynomial ring $R_k[x;\Theta]$, where $\Theta$ is an automorphism of $R_k$ such that $xa = \Theta(a)x$ for all $a \in R_k$. Let $f(x)$ be a central irreducible divisor of $x^{np^s} - \lambda$ of degree $l$ and multiplicity $j$ in $R_k[x;\Theta]$, where $\lambda $ is an invertible element in $R_k$. In this article, we study skew constacyclic codes of length \(np^s\) over \(R_k\), which reduces to the study of skew polycyclic codes of length $jl$ associated with a polynomial \(f(x)^j\). Using the fact that skew polycyclic codes associated with a polynomial \(f(x)^j\) can be described by the left ideal structure of the quotient ring $R_k[x;\Theta]/\langle f(x)^{j}\rangle$, we investigate this class of codes for specific choices of $\Theta$. In particular, if $\lambda$ is an invertible element of $\mathbb{F}_{p^m}$, we classify all left ideals and establish an isomorphism between skew cyclic and skew constacyclic codes, under suitable conditions. Furthermore, we provide a comprehensive analysis of skew constacyclic codes of length $3p^s$ over $R_k$. Finally, we examine skew cyclic and skew negacyclic codes of length $6p^s$ over $R_k$ using the factorization of $x^{6p^s} - 1$ and $x^{6p^s} + 1$, respectively; with a complete case-by-case analysis. Examples demonstrating codes with optimal parameters are also included.