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On the Existence of Galois Self-Dual GRS and TGRS Codes

Shixin Zhu, Ruhao Wan

TL;DR

The paper develops a unified, extension-based method to study $e$-Galois self-duality of linear codes over $\mathbb{F}_q$, deriving a necessary-and-sufficient condition for the dual extended code $\underline{\mathcal{C}}[\bm{u}]$ to be $e$-Galois self-orthogonal and then leveraging it to obtain existence and nonexistence results. It proves nonexistence of $e$-Galois self-dual (extended) GRS codes when the length exceeds $\min\{p^e+1, p^{m-e}+1\}$, and shows many $e$-Galois self-dual (extended) TGRS codes do not exist under various parameter regimes. The authors also provide sufficient and necessary conditions for $(\ast)$-TGRS codes to be Hermitian self-dual and present several new classes of Hermitian self-dual $(+)$-TGRS and $(\ast)$-TGRS codes, including explicit constructions and non-GRS examples. Overall, the work advances systematic construction and nonexistence results for Galois self-dual codes and broadens the landscape of Hermitian self-dual non-GRS TGRS codes with potential applications in quantum error correction and secure communications.

Abstract

Let $q=p^m$ be a prime power and $e$ be an integer with $0\leq e\leq m-1$. $e$-Galois self-dual codes are generalizations of Euclidean $(e=0)$ and Hermitian ($e=\frac{m}{2}$ with even $m$) self-dual codes. In this paper, for a linear code $\C$ and a nonzero vector $\bm{u}\in \F_q^n$, we give a sufficient and necessary condition for the dual extended code $\underline{\C}[\bm{u}]$ of $\C$ to be $e$-Galois self-orthogonal. From this, a new systematic approach is proposed to prove the existence of $e$-Galois self-dual codes. By this method, we prove that $e$-Galois self-dual (extended) generalized Reed-Solomon (GRS) codes of length $n>\min\{p^e+1,p^{m-e}+1\}$ do not exist, where $1\leq e\leq m-1$. Moreover, based on the non-GRS properties of twisted GRS (TGRS) codes, we show that in many cases $e$-Galois self-dual (extended) TGRS codes do not exist. Furthermore, we present a sufficient and necessary condition for $(\ast)$-TGRS codes to be Hermitian self-dual, and then construct several new classes of Hermitian self-dual $(+)$-TGRS and $(\ast)$-TGRS codes.

On the Existence of Galois Self-Dual GRS and TGRS Codes

TL;DR

The paper develops a unified, extension-based method to study -Galois self-duality of linear codes over , deriving a necessary-and-sufficient condition for the dual extended code to be -Galois self-orthogonal and then leveraging it to obtain existence and nonexistence results. It proves nonexistence of -Galois self-dual (extended) GRS codes when the length exceeds , and shows many -Galois self-dual (extended) TGRS codes do not exist under various parameter regimes. The authors also provide sufficient and necessary conditions for -TGRS codes to be Hermitian self-dual and present several new classes of Hermitian self-dual -TGRS and -TGRS codes, including explicit constructions and non-GRS examples. Overall, the work advances systematic construction and nonexistence results for Galois self-dual codes and broadens the landscape of Hermitian self-dual non-GRS TGRS codes with potential applications in quantum error correction and secure communications.

Abstract

Let be a prime power and be an integer with . -Galois self-dual codes are generalizations of Euclidean and Hermitian ( with even ) self-dual codes. In this paper, for a linear code and a nonzero vector , we give a sufficient and necessary condition for the dual extended code of to be -Galois self-orthogonal. From this, a new systematic approach is proposed to prove the existence of -Galois self-dual codes. By this method, we prove that -Galois self-dual (extended) generalized Reed-Solomon (GRS) codes of length do not exist, where . Moreover, based on the non-GRS properties of twisted GRS (TGRS) codes, we show that in many cases -Galois self-dual (extended) TGRS codes do not exist. Furthermore, we present a sufficient and necessary condition for -TGRS codes to be Hermitian self-dual, and then construct several new classes of Hermitian self-dual -TGRS and -TGRS codes.
Paper Structure (14 sections, 33 theorems, 66 equations)

This paper contains 14 sections, 33 theorems, 66 equations.

Table of Contents

  1. Introduction
  2. Our motivation
  3. Our results
  4. Organization of this paper
  5. Galois self-orthogonal dual extended codes
  6. The existence of Galois self-dual GRS and TGRS codes
  7. Galois self-dual GRS and EGRS codes
  8. Galois self-dual TGRS and ETGRS codes
  9. Galois self-dual TGRS code with $h=0$ and $0\in A_{{\bm{\alpha}}}$
  10. Galois self-dual ETGRS code with $h=k-1$
  11. The construction of Hermitian self-dual $(+)$-TGRS and $(\ast)$-TGRS codes
  12. Hermitian self-dual $(+)$-TGRS codes
  13. Hermitian self-dual $(\ast)$-TGRS codes
  14. Conclusion

Key Result

Lemma 1

(RefJ (2018) Liu. Galois and RefJ (2020) Liu. Galois) Let ${\mathcal{C}}$ be an $[n,k]_q$ linear code with a generator matrix $G$. Then ${\mathcal{C}}$ is an $e$-Galois self-orthogonal code if and only if ${\mathcal{C}}$ is an $(m-e)$-Galois self-orthogonal code, if and only if $G\sigma^e(G^T)=\bm{0

Theorems & Definitions (60)

  • Lemma 1
  • Lemma 2
  • Definition 1
  • Lemma 3
  • Definition 2
  • Lemma 4
  • Theorem 1
  • proof
  • Corollary 1
  • proof
  • ...and 50 more