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

2026-03-17

AI Summary

This work addresses the vulnerabilities of certain algebraic code-based cryptosystems to Sidelnikov–Shestakov and Wieschebrink attacks by constructing non-GRS linear complementary dual (LCD) codes based on generalized Roth–Lempel (GRL) codes, with applications to entanglement-assisted quantum error-correcting codes (EAQECCs). By analyzing the hull dimension of GRL codes, the study establishes, for the first time, an upper bound on the number of Euclidean GRL codes with one-dimensional hull and derives achievable upper bounds on the Hermitian hull dimensions for several families of GRL codes. It further proves that GRL codes are non-GRS whenever \(k > \ell\). The proposed framework yields multiple families of Euclidean and Hermitian LCD codes, low-hull linear codes, and corresponding EAQECCs, along with explicit constructions of LCD MDS and NMDS codes.

Technology Category

• Machine Learning: Quantum Machine Learning
• Knowledge Representation and Reasoning: Qualitative Reasoning
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction

Application Category

• Security and Privacy: Applications of cryptography
• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics

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.

Problem

• non-GRS codes
• LCD codes
• hull dimension
• EAQECCs
• GRL codes

Innovation

• non-GRS codes
• LCD codes
• GRL codes
• hull dimension
• EAQECCs