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

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 $n{p}^s$ over the ring $R_k={\mathbb{F}}_{p^m}[u]/⟨u^k⟩,$ 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 $x^{n{p}^s}-\lambda$ 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 $R_k[x;\Theta ]/⟨f(x{)}^j⟩$, 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.