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

TL;DR

This paper constructs new classes of Euclidean and Hermitian LCD codes using non-GRS type linear codes, specifically GRL codes, and applies them to develop entanglement-assisted quantum error-correcting codes, advancing coding theory and quantum communication.

Contribution

It introduces new Euclidean and Hermitian LCD codes based on GRL codes, extending previous results, and provides bounds on hull dimensions, with applications to EAQECCs and examples of LCD MDS and NMDS codes.

Findings

Constructed Euclidean and Hermitian LCD codes from GRL codes.

Derived upper bounds for hull dimensions of certain codes.

Developed new families of EAQECCs using these codes.

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.