Bounds and Equivalence of Skew Polycyclic Codes over Finite Fields

2025-07-23

AI Summary

This work investigates distance bounds and equivalence classification of skew multi-cyclic codes over skew polynomial rings defined over finite fields. Specifically, it establishes— for the first time—the Roos-like lower bounds on minimum Hamming and rank distances of such codes. It then defines and characterizes equivalence relations between two classes of skew multi-cyclic codes under both metrics, fully describing the algebraic structure of their equivalence classes. Methodologically, the paper integrates skew polynomial ring theory, finite field automorphisms, and algebraic coding techniques, augmented by rigorous distance analysis tools. The theoretical results are validated via concrete examples, confirming the tightness of the derived bounds and the computability of the equivalence classification. This study fills a fundamental gap in the distance analysis and structural classification of skew multi-cyclic codes within the noncommutative setting, thereby providing a novel framework for constructing high-performance error-correcting codes.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Machine Learning: Multi-instance/Multi-view Learning
• Cognitive Modeling & Cognitive Systems: Neural Spike Coding

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for heterogeneous, signed, attributed, multi-relational, temporal, higher-order, and annotated Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Security and Privacy: Applications of cryptography

Abstract

We study skew polycyclic codes over a finite field $mathbb{F}_q$, associated with a skew polynomial $f(x) in mathbb{F}_q[x;σ]$, where $σ$ 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.

Problem

• Study skew polycyclic codes over finite fields
• Prove Roos-like bounds for Hamming and rank metrics
• Analyze equivalence relations between polycyclic codes

Innovation

• Proving Roos-like bounds for skew polycyclic codes
• Introducing equivalence relations for polycyclic codes
• Illustrating theory applications with concrete examples