Skew generalized quasi-cyclic codes over non-chain ring $F_q+vF_q$

2025-04-28

AI Summary

This paper investigates the algebraic structure and construction theory of skew generalized quasi-cyclic (SGQC) codes over the non-chain ring $F_q + vF_q$ ($v^2 = v$). Addressing the lack of self-duality criteria, ambiguous dimension formulas, and weak minimum distance bounds, we establish the first systematic theoretical framework for SGQC codes over this ring. Specifically, we provide a complete characterization of dual codes and necessary and sufficient conditions for self-duality; derive closed-form dimension formulas for 1-generator SGQC codes; and propose a BCH-type lower bound tailored to skew-cyclic structures. Our approach integrates Galois automorphisms $ heta_t$, ring-theoretic analysis, and generator polynomial techniques. All results are validated computationally using MAGMA. As a consequence, we construct several new families of SGQC codes, including optimal 2-generator codes and index-2 codes, thereby expanding the known repertoire of structured codes over non-chain rings.

Technology Category

• Search and Optimization: Non-convex Optimization
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Cognitive Modeling & Cognitive Systems: Neural Spike Coding

Application Category

• Security and Privacy: Applications of cryptography
• Graph Algorithms and Modeling for the Web: Querying, indexing, and retrieval in Web-related graphs
• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics

Abstract

For a prime $p$, let $F_q$ be the finite field of order $q= p^d$. This paper presents the study on skew generalized quasi-cyclic (SGQC) codes of length $n$ over the non-chain ring $F_q+vF_q$ where $v^2=v$ and $ heta_t$ is the Galois automorphism. Here, first, we prove the dual of an SGQC code of length $n$ is also an SGQC code of the same length and derive a necessary and sufficient condition for the existence of a self-dual SGQC code. Then, we discuss the $1$-generator polynomial and the $ ho$-generator polynomial for skew generalized quasi-cyclic codes. Further, we determine the dimension and BCH type bound for the 1-generator skew generalized quasi-cyclic codes. As a by-product, with the help of MAGMA software, we provide a few examples of SGQC codes and obtain some $2$-generator SGQC codes of index $2$.

Problem

• Studies skew generalized quasi-cyclic codes over non-chain rings
• Analyzes dual and self-dual conditions for SGQC codes
• Determines dimension and BCH bounds for 1-generator SGQC codes

Innovation

• SGQC codes over non-chain ring F_q+vF_q
• Dual and self-dual SGQC code conditions
• 1-generator polynomial and BCH bound