Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Non-GRS type Euclidean and Hermitian LCD codes and Their Applications for EAQECCs

Zhonghao Liang, Dongmei Huang, Qunying Liao, Cuiling Fan, Zhengchun Zhou

Abstract

In recent years, the construction of non-GRS type linear codes has attracted considerable attention due to that they can effectively resist both the Sidelnikov-Shestakov attack and the Wieschebrink attack. Constructing linear complementary dual (LCD) codes and determining the hull of linear codes have long been important topics in coding theory, as they play the crucial role in constructing entanglement-assisted quantum error-correcting codes (EAQECCs), certain communication systems and cryptography. In this paper, by utilizing a class of non-GRS type linear codes, namely, generalized Roth-Lempel (in short, GRL) codes, we firstly construct several classes of Euclidean LCD codes, Hermitian LCD codes, and linear codes with small-dimensional hulls, generalized the main results given by Wu et al. in 2021. We also present an upper bound for the number of a class of Euclidean GRL codes with 1-dimensional hull, and then for several classes of Hermitian GRL codes, we firstly derive an upper bound for the dimension of the hull, and prove that the bound is attainable. Secondly, as an application, we obtain several families of EAQECCs. Thirdly, we prove that the GRL code is non-GRS for $k >\ell$. Finally, some corresponding examples for LCD MDS codes and LCD NMDS codes are presented.

Non-GRS type Euclidean and Hermitian LCD codes and Their Applications for EAQECCs

Abstract

In recent years, the construction of non-GRS type linear codes has attracted considerable attention due to that they can effectively resist both the Sidelnikov-Shestakov attack and the Wieschebrink attack. Constructing linear complementary dual (LCD) codes and determining the hull of linear codes have long been important topics in coding theory, as they play the crucial role in constructing entanglement-assisted quantum error-correcting codes (EAQECCs), certain communication systems and cryptography. In this paper, by utilizing a class of non-GRS type linear codes, namely, generalized Roth-Lempel (in short, GRL) codes, we firstly construct several classes of Euclidean LCD codes, Hermitian LCD codes, and linear codes with small-dimensional hulls, generalized the main results given by Wu et al. in 2021. We also present an upper bound for the number of a class of Euclidean GRL codes with 1-dimensional hull, and then for several classes of Hermitian GRL codes, we firstly derive an upper bound for the dimension of the hull, and prove that the bound is attainable. Secondly, as an application, we obtain several families of EAQECCs. Thirdly, we prove that the GRL code is non-GRS for . Finally, some corresponding examples for LCD MDS codes and LCD NMDS codes are presented.
Paper Structure (16 sections, 49 theorems, 137 equations, 1 figure)

This paper contains 16 sections, 49 theorems, 137 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Euclidean LCD GRL codes and their applications
  4. The first class of LCD GRL codes with the parameters $\left[k+\ell,k\right]$
  5. The second class of LCD GRL codes with the parameters $\left[k+1+\ell,k\right]$
  6. The third class of LCD GRL codes with the parameters $\left[2k+\ell,k\right]$
  7. Euclidean LCD GRL codes with the parameters $\left[3k+\ell,k\right]$
  8. Several classes of EAQECCs
  9. Hermitian LCD GRL codes and their applications
  10. Hermitian LCD GRL codes with the parameters $\left[k+\ell,k\right]$
  11. Hermitian LCD GRL codes with the parameters $\left[k+1+\ell,k\right]$
  12. Hermitian LCD GRL codes with the parameters $\left[2k+\ell,k\right]$
  13. Hermitian LCD GRL codes with the parameters $\left[(\delta+1)k+\ell,k\right]$
  14. Several classes of EAQECCs
  15. The non-RS property of the GRL code
  16. ...and 1 more sections

Key Result

Lemma 2.1

(nonGRS8, Definition 3) Let $\mathcal{C}_{1}$ and $\mathcal{C}_{2}$ be two linear codes of the same code length over $\mathbb{F}_{q}$, and let $\boldsymbol{M}$ be a generator matrix of $\mathcal{C}_{1}$. Then $\mathcal{C}_{1}$ and $\mathcal{C}_{2}$ are monomially equivalent if and only if there exis

Figures (1)

  • Figure 1: Classification of MDS codes

Theorems & Definitions (77)

  • Definition 2.1
  • Lemma 2.1
  • Remark 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Lemma 2.5
  • Lemma 2.6
  • Lemma 2.7
  • Lemma 2.8
  • ...and 67 more