Bounds and Equivalence of Skew Polycyclic Codes over Finite Fields

TL;DR

This paper investigates skew polycyclic codes over finite fields, establishing bounds for their error-correcting capabilities and exploring conditions for their equivalence, with practical examples illustrating the theoretical results.

Contribution

It introduces Roos-like bounds for skew polycyclic codes and characterizes Hamming and rank equivalences between different classes of these codes.

Findings

Established Roos-like bounds for skew polycyclic codes

Characterized Hamming and rank equivalences between code classes

Provided examples demonstrating theoretical applications

Abstract

We study skew polycyclic codes over a finite field $\mathbb{F}_q$, associated with a skew polynomial $f(x) \in \mathbb{F}_q[x;\sigma]$, where $\sigma$ is an automorphism of $\mathbb{F}_q$. We start by proving the Roos-like bound for both the Hamming and the rank metric for this class of codes. Next, we focus on the Hamming and rank equivalence between two classes of polycyclic codes by introducing an equivalence relation and describing its equivalence classes. Finally, we present examples that illustrate applications of the theory developed in this paper.