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

Kundan Suxena, Indibar Debnath, Om Prakash

TL;DR

This work extends skew generalized quasi-cyclic codes to the non-chain ring $S = F_q + vF_q$ with $v^2 = v$ under a Galois automorphism $\theta_t$, developing a comprehensive algebraic framework for SGQC codes over $S$. By representing codes as left $S[x;\theta_t]$-submodules of a product ring and relating them to constituent SGQC codes over $F_q$ via the map $\mu$, the authors establish duality, self-duality criteria, and a robust generator theory that includes $1$-generator and $\rho$-generator polynomials, parity-check polynomials, and dimension bounds. They prove the dual of an SGQC code is SGQC and that self-duality reduces to self-duality of the constituent codes, while providing idempotent generators under coprimality conditions and practical $\rho$-generator constructions, along with MAGMA-based examples and BCH-type distance bounds. The results equip code designers with explicit construction tools over $S$ and indicate potential for improved parameters and quantum-code applications.

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 $θ_t$ 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 $ρ$-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 $2$.

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

TL;DR

This work extends skew generalized quasi-cyclic codes to the non-chain ring with under a Galois automorphism , developing a comprehensive algebraic framework for SGQC codes over . By representing codes as left -submodules of a product ring and relating them to constituent SGQC codes over via the map , the authors establish duality, self-duality criteria, and a robust generator theory that includes -generator and -generator polynomials, parity-check polynomials, and dimension bounds. They prove the dual of an SGQC code is SGQC and that self-duality reduces to self-duality of the constituent codes, while providing idempotent generators under coprimality conditions and practical -generator constructions, along with MAGMA-based examples and BCH-type distance bounds. The results equip code designers with explicit construction tools over and indicate potential for improved parameters and quantum-code applications.

Abstract

For a prime , let be the finite field of order . This paper presents the study on skew generalized quasi-cyclic (SGQC) codes of length over the non-chain ring where and is the Galois automorphism. Here, first, we prove the dual of an SGQC code of length 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 -generator polynomial and the -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 -generator SGQC codes of index .
Paper Structure (10 sections, 29 theorems, 64 equations, 3 tables)

This paper contains 10 sections, 29 theorems, 64 equations, 3 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Algebraic Structure of SGQC Codes
  4. Duality of SGQC codes
  5. Generator set for SGQC codes
  6. 1-Generator polynomial of SGQC codes over $S$
  7. Idempotent generators of SGQC codes over $S$
  8. $\rho$-Generator Polynomial over $S$
  9. Conclusion
  10. Declarations

Key Result

Theorem 2.3

Ozen19Right Division Algorithm: Suppose $a(x)$ and $b(x)$ are two nonzero polynomials in $S[x ; \theta_t]$ such that the leading coefficient of $b(x)$ is a unit, then there exist unique polynomials $q(x)$ and $r(x)$ such that $a(x)=q(x)b(x)+r(x)$ where $\deg r(x) < \deg a(x)$ or $r(x)=0$.

Theorems & Definitions (59)

  • Definition 2.1
  • Definition 2.2
  • Theorem 2.3
  • Definition 2.4
  • Definition 2.5
  • Theorem 2.6
  • Corollary 2.7
  • Lemma 2.8
  • Definition 3.1
  • Lemma 3.2
  • ...and 49 more