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

2026-05-13

AI Summary

This study investigates the structure of left ideals in skew polynomial quotient rings over the finite chain ring $\mathbb{F}_{p^m}[u]/\langle u^t \rangle$, induced by automorphisms, with a focus on classifying skew polycyclic codes modulo $x^{np^s} - \lambda$ where $\lambda_1 \neq 0$. By integrating finite chain ring theory, skew polynomial rings, and module-theoretic analysis, the work provides explicit generator forms for these left ideals, revealing their refined algebraic structure. It establishes necessary conditions for code distinctness absent in prior literature and computes, for the first time, $i$-th torsion codes for specific parameter choices. Notably, it completely characterizes all left ideals in the cases $n=1, t=3$ and $n=2, t=2$, correcting and extending existing classification frameworks, thereby laying a theoretical foundation for constructing novel error-correcting codes.

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 $θ$ of $\mathbb{F}_{p^m}$, we give the structure of the left ideals of the ring $\frac{R^t[x,Θ]}{\langle f(x) \rangle},$ where $f(x)$ is in the center of the skew polynomial ring $R^t[x,Θ]$ and $Θ$ is an automorphism of $R^t$ that extends $θ$ with $Θ(u)=u$. These left ideals are also referred to as skew polycyclic codes associated to $f(x).$ In particular, when the central element \( f(x)\) is \(x^{np^s}-λ\), where $λ=λ_0+uλ_1+\cdots +u^{t-1}λ_{t-1}$ with $λ_0\ne0,$ and \( n=1,2 \), we give a more refined form of the left ideals (which are also called skew constacyclic codes). Moreover, the case $λ_1 \neq 0$ 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.

Problem

• skew polycyclic codes
• finite chain rings
• left ideals
• skew constacyclic codes
• structural characterization

Innovation

• skew polycyclic codes
• finite chain rings
• skew polynomial rings
• constacyclic codes
• torsion codes