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 $R_k[x;Θ]$, where $Θ$ is an automorphism of $R_k$ such that $xa = Θ(a)x$ for all $a \in R_k$. Let $f(x)$ be a central irreducible divisor of $x^{np^s} - λ$ of degree $l$ and multiplicity $j$ in $R_k[x;Θ]$, where $λ$ 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 $R_k[x;Θ]/\langle f(x)^{j}\rangle$, we investigate this class of codes for specific choices of $Θ$. In particular, if $λ$ 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.

翻译：设$\mathbb{F}_{p^m}$为包含$p^m$个元素的域，其中$p$为奇素数且$m \in \mathbb{N}$。本文提出统一方法研究环$R_k = \mathbb{F}_{p^m}[u]/\langle u^k \rangle$上长度为$np^s$的斜常量循环码，其中$n, s, k \in \mathbb{N}$且$\gcd(n, p)=1$。考虑斜多项式环$R_k[x;Θ]$，其中$Θ$是$R_k$的自同构，满足对所有$a \in R_k$有$xa = Θ(a)x$。设$f(x)$为$x^{np^s} - λ$在$R_k[x;Θ]$中的次数为$l$、重数为$j$的中心不可约因子，其中$λ$为$R_k$中可逆元。本文研究$R_k$上长度为$np^s$的斜常量循环码，将问题简化为与多项式$f(x)^j$相关联的长度为$jl$的斜多循环码的研究。利用与多项式$f(x)^j$相关联的斜多循环码可由商环$R_k[x;Θ]/\langle f(x)^{j}\rangle$的左理想结构描述这一事实，我们对特定$Θ$选择下的这类码进行探究。特别地，当$λ$为$\mathbb{F}_{p^m}$中可逆元时，我们在适当条件下对所有左理想进行分类，并建立斜循环码与斜常量循环码之间的同构关系。此外，我们对$R_k$上长度为$3p^s$的斜常量循环码进行全面分析。最后，我们分别利用$x^{6p^s} - 1$和$x^{6p^s} + 1$的因式分解，通过完整的案例分析研究$R_k$上长度为$6p^s$的斜循环码与斜负循环码。文中还包含具有最优参数的码的示例。