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Galois self-orthogonal MDS codes with large dimensions

Ruhao Wan, Shixin Zhu

TL;DR

This work advances the theory of Galois self-orthogonal MDS codes in the $q$-ary setting by developing a general framework that tests $e$-Galois self-orthogonality via the map $\Phi_{\bm{v}}({\mathcal C})$ and its extension. It delivers necessary and sufficient conditions for (extended) GRS and EGRS codes to be $e$-Galois self-orthogonal, and partitions the parameter space into three cases based on $\frac{m}{s}$ and parity, enabling the construction of new MDS codes with dimensions exceeding the conventional bound $\big\lfloor\frac{n+p^e-1}{p^e+1}\big\rfloor$. The paper provides explicit constructions for lengths $1\le n\le p^s+1$, $p^s+1\le n\le p^{2s}+1$, and $n=q+1$, and demonstrates that propagation rules yield MDS codes with Galois hulls of arbitrary dimension, with significant implications for quantum codes via EAQECCs. Overall, the results enrich the catalog of high-dimension Galois self-orthogonal MDS codes and offer practical avenues for quantum-information applications.

Abstract

Let $q=p^m$ be a prime power, $e$ be an integer with $0\leq e\leq m-1$ and $s=\gcd(e,m)$. In this paper, for a vector $v$ and a $q$-ary linear code $C$, we give some necessary and sufficient conditions for the equivalent code $vC$ of $C$ and the extended code of $vC$ to be $e$-Galois self-orthogonal. From this, we directly obtain some necessary and sufficient conditions for (extended) generalized Reed-Solomon (GRS and EGRS) codes to be $e$-Galois self-orthogonal. Furthermore, for all possible $e$ satisfying $0\leq e\leq m-1$, we classify them into three cases (1) $\frac{m}{s}$ odd and $p$ even; (2) $\frac{m}{s}$ odd and $p$ odd; (3) $\frac{m}{s}$ even, and construct several new classes of $e$-Galois self-orthogonal maximum distance separable (MDS) codes. It is worth noting that our $e$-Galois self-orthogonal MDS codes can have dimensions greater than $\lfloor \frac{n+p^e-1}{p^e+1}\rfloor$, which are not covered by previously known ones. Moreover, by propagation rules, we obtain some new MDS codes with Galois hulls of arbitrary dimensions. As an application, many quantum codes can be obtained from these MDS codes with Galois hulls.

Galois self-orthogonal MDS codes with large dimensions

TL;DR

This work advances the theory of Galois self-orthogonal MDS codes in the -ary setting by developing a general framework that tests -Galois self-orthogonality via the map and its extension. It delivers necessary and sufficient conditions for (extended) GRS and EGRS codes to be -Galois self-orthogonal, and partitions the parameter space into three cases based on and parity, enabling the construction of new MDS codes with dimensions exceeding the conventional bound . The paper provides explicit constructions for lengths , , and , and demonstrates that propagation rules yield MDS codes with Galois hulls of arbitrary dimension, with significant implications for quantum codes via EAQECCs. Overall, the results enrich the catalog of high-dimension Galois self-orthogonal MDS codes and offer practical avenues for quantum-information applications.

Abstract

Let be a prime power, be an integer with and . In this paper, for a vector and a -ary linear code , we give some necessary and sufficient conditions for the equivalent code of and the extended code of to be -Galois self-orthogonal. From this, we directly obtain some necessary and sufficient conditions for (extended) generalized Reed-Solomon (GRS and EGRS) codes to be -Galois self-orthogonal. Furthermore, for all possible satisfying , we classify them into three cases (1) odd and even; (2) odd and odd; (3) even, and construct several new classes of -Galois self-orthogonal maximum distance separable (MDS) codes. It is worth noting that our -Galois self-orthogonal MDS codes can have dimensions greater than , which are not covered by previously known ones. Moreover, by propagation rules, we obtain some new MDS codes with Galois hulls of arbitrary dimensions. As an application, many quantum codes can be obtained from these MDS codes with Galois hulls.

Paper Structure

This paper contains 22 sections, 30 theorems, 51 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Related works
  3. Our motivation
  4. Our results
  5. Organization of this paper
  6. Preliminaries
  7. Galois dual codes and Galois hulls
  8. GRS and EGRS codes
  9. Extended code
  10. Main methods
  11. Necessary and sufficient conditions for the code $\Phi_{\bm{v}}({\mathcal{C}})$ to be Galois self-orthogonal
  12. Necessary and sufficient conditions for the code $\widetilde{\Phi_{\bm{v}}({\mathcal{C}})}(G diag(\bm{v}),\infty)$ to be Galois self-orthogonal
  13. Necessary and sufficient conditions for (extended) GRS codes of length $n$ to be Galois self-orthogonal when $p^e\leq k\leq \lfloor \frac{n+p^e-1}{p^e+1}\rfloor$, where $2e\leq m$.
  14. Construction of Galois self-orthogonal MDS code
  15. $e$-Galois self-orthogonal MDS code of length $1\leq n\leq p^s+1$
  16. ...and 7 more sections

Key Result

Lemma 1

(RefJ (2023) Yang DM) Let ${\mathcal{C}}$ be an $[n,k,d]_q$ code. Then ${\mathcal{C}}$ is an $e$-Galois self-orthogonal code if and only if ${\mathcal{C}}$ is an $(m-e)$-Galois self-orthogonal code.

Theorems & Definitions (57)

  • Lemma 1
  • Remark 1
  • Definition 1
  • Lemma 2
  • Remark 2
  • Lemma 3
  • Corollary 1
  • Lemma 4
  • Corollary 2
  • Remark 3
  • ...and 47 more