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Three classes of propagation rules for generalized Reed-Solomon codes and their applications to EAQECCs

Ruhao Wan, Shixin Zhu

TL;DR

This paper derives three new propagation rules for MDS entanglement-assisted quantum error correction codes (EAQECCs) constructed from GRS codes by extending the length, increasing the dimension, and extending the length and increasing the dimension at the same time.

Abstract

In this paper, we study the Hermitian hulls of generalized Reed-Solomon (GRS) codes over finite fields. For a given class of GRS codes, by extending the length, increasing the dimension, and extending the length and increasing the dimension at the same time, we obtain three classes of GRS codes with Hermitian hulls of arbitrary dimensions. Furthermore, based on some known $q^2$-ary Hermitian self-orthogonal GRS codes with dimension $q-1$, we obtain several classes of $q^2$-ary maximum distance separable (MDS) codes with Hermitian hulls of arbitrary dimensions. It is worth noting that the dimension of these MDS codes can be taken from $q$ to $\frac{n}{2}$, and the parameters of these MDS codes can be more flexible by propagation rules. As an application, we derive three new propagation rules for MDS entanglement-assisted quantum error correction codes (EAQECCs) constructed from GRS codes. Then, from the presently known GRS codes with Hermitian hulls, we can directly obtain many MDS EAQECCs with more flexible parameters. Finally, we present several new classes of (MDS) EAQECCs with flexible parameters, and the distance of these codes can be taken from $q+1$ to $\frac{n+2}{2}$.

Three classes of propagation rules for generalized Reed-Solomon codes and their applications to EAQECCs

TL;DR

This paper derives three new propagation rules for MDS entanglement-assisted quantum error correction codes (EAQECCs) constructed from GRS codes by extending the length, increasing the dimension, and extending the length and increasing the dimension at the same time.

Abstract

In this paper, we study the Hermitian hulls of generalized Reed-Solomon (GRS) codes over finite fields. For a given class of GRS codes, by extending the length, increasing the dimension, and extending the length and increasing the dimension at the same time, we obtain three classes of GRS codes with Hermitian hulls of arbitrary dimensions. Furthermore, based on some known -ary Hermitian self-orthogonal GRS codes with dimension , we obtain several classes of -ary maximum distance separable (MDS) codes with Hermitian hulls of arbitrary dimensions. It is worth noting that the dimension of these MDS codes can be taken from to , and the parameters of these MDS codes can be more flexible by propagation rules. As an application, we derive three new propagation rules for MDS entanglement-assisted quantum error correction codes (EAQECCs) constructed from GRS codes. Then, from the presently known GRS codes with Hermitian hulls, we can directly obtain many MDS EAQECCs with more flexible parameters. Finally, we present several new classes of (MDS) EAQECCs with flexible parameters, and the distance of these codes can be taken from to .
Paper Structure (14 sections, 32 theorems, 46 equations, 6 tables)

This paper contains 14 sections, 32 theorems, 46 equations, 6 tables.

Table of Contents

  1. Introduction
  2. Recent related works
  3. Our motivations and contributions
  4. Organization of this paper
  5. Preliminaries
  6. Three classes of GRS codes from known GRS codes
  7. Extend the length of the GRS code
  8. Increase the dimension of the GRS code
  9. Extend the length and increase the dimension at the same time of the GRS code
  10. MDS codes with Hermitian hulls of arbitrary dimensions
  11. Application to EAQECCs and comparison of results
  12. Application to EAQECCs
  13. Comparison of results
  14. Conclusion

Key Result

Lemma 1

(RefJ (2023) Liu Galois) Let $\bm{a}=(a_{1},a_{2},\dots,a_{n})\in {\mathbb{F}}_q^n$ with $a_1,a_2,\dots,a_n$ distinct elements and $\bm{v}=(v_1,v_2,\dots,v_n)\in ({\mathbb{F}}_{q}^*)^n$. Then for any $a\in {\mathbb{F}}_q^*$, $b\in {\mathbb{F}}_q$ and $\lambda\in {\mathbb{F}}_q^*$, we have and where $a\bm{a}+b\bm{1}=(aa_1+b,aa_2+b,\dots,aa_n+b)\in {\mathbb{F}}_q^n$ and $\lambda\bm{v}=(\lambda v_1

Theorems & Definitions (52)

  • Lemma 1
  • Remark 1
  • Lemma 2
  • Lemma 3
  • Lemma 4
  • Remark 2
  • Lemma 5
  • Corollary 1
  • proof
  • Proposition 1
  • ...and 42 more