Skew Polycyclic Codes over $\frac{\mathbb{F}_{p^m}[u]}{\langle u^t \rangle}$

Abstract

Let $R^t$ denote the finite chain ring $\frac{\mathbb{F}_{p^m}[u]}{\langle u^t \rangle},$ where $p$ is a prime and $t$ is a positive integer. In this article, for a prime $p$ and an automorphism $\theta$ of $\mathbb{F}_{p^m}$, we give the structure of the left ideals of the ring $\frac{R^t[x,\Theta]}{\langle f(x) \rangle},$ where $f(x)$ is in the center of the skew polynomial ring $R^t[x,\Theta]$ and $\Theta$ is an automorphism of $R^t$ that extends $\theta$ with $\Theta(u)=u$. These left ideals are also referred to as skew polycyclic codes associated to $f(x).$ In particular, when the central element \( f(x)\) is \(x^{np^s}-\lambda \), where $\lambda=\lambda_0+u\lambda_1+\cdots +u^{t-1}\lambda_{t-1}$ with $\lambda_0\ne0,$ and \( n=1,2 \), we give a more refined form of the left ideals (which are also called skew constacyclic codes). Moreover, the case $\lambda_1 \neq 0$ is analyzed in detail, yielding a simpler form of generators that reveals a more refined structural characterization of the left ideals. As an application, for $n=1,t=3$ and $n=2,t=2$ we give a full description of the left ideals by including certain necessary conditions that were omitted in available literature, preventing the different classes of left ideals from being mutually disjoint and in certain cases, we also compute $i$-th torsion codes.