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

TL;DR

This paper characterizes the structure of skew polycyclic codes over a finite chain ring, providing explicit descriptions of left ideals for specific polynomial cases and correcting previous literature inaccuracies.

Contribution

It offers a detailed structural analysis of skew polycyclic codes over finite chain rings, including explicit generator forms and corrections to prior classifications.

Findings

Explicit structure of left ideals for specific polynomial cases

Refined generator forms for skew constacyclic codes

Correction of previous literature inaccuracies in classifying left ideals

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.