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AIMS Mathematics

New entanglement-assisted quantum codes constructed from Hermitian LCD codes

Yuezhen Ren, Ruihu Li, Guanmin Guo

Published Jan 1, 2023• 1 citation
Publisher

Abstract

Hermitian linear complementary dual (LCD) codes are a class of linear codes that intersect with their Hermitian dual trivially. Each Hermitian LCD code can give an entanglement-assisted quantum error-correcting code (EAQECC) with maximal entanglement. Methods of constructing Hermitian LCD codes from known codes were developed, and seven new Hermitian LCD codes with parameters $ [119,4,88]_{4} $, $ [123,4,91]_{4} $, $ [124,4,92]_{4} $, $ [136,4,101]_{4} $, $ [140,4,104]_{4} $, $ [188,4,140]_{4} $ and $ [212,4,158]_{4} $ were constructed. Seven families of Hermitian LCD codes and their related EAQECCs were derived from these codes. These new EAQECCs have better parameters than those known in the literature.

DOI
10.3934/math.20231578

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Hermitian linear complementary dual (LCD) codes are a class of linear codes that intersect with their Hermitian dual trivially. Each Hermitian LCD code can give an entanglement-assisted quantum error-correcting code (EAQECC) with maximal entanglement. Methods of constructing Hermitian LCD codes from known codes were developed, and seven new Hermitian LCD codes with parameters [119,4,88]4[119,4,88]_{4}[119,4,88]4​, [123,4,91]4[123,4,91]_{4}[123,4,91]4​, [124,4,92]4[124,4,92]_{4}[124,4,92]4​, [136,4,101]4[136,4,101]_{4}[136,4,101]4​, [140,4,104]4[140,4,104]_{4}[140,4,104]4​, [188,4,140]4[188,4,140]_{4}[188,4,140]4​ and [212,4,158]4[212,4,158]_{4}[212,4,158]4​ were constructed. Seven families of Hermitian LCD codes and their related EAQECCs were derived from these codes. These new EAQECCs have better parameters than those known in the literature.
The classical linear codes that construct quantum error correction codes need to meet certain specific conditions. The proposal of entanglement-assisted quantum error correcting codes eliminates the restriction of these conditions, so all classical linear codes can be constructed into entanglement-assisted quantum error correcting codes. The Hadamard matrix is a special type of orthogonal matrix. Hadamard codes, constructed based on the standard form of Hadamard matrices, are a type of linear complementary dual (LCD) codes. By extending the generator and parity-check matrices of Hadamard codes, two types of classical codes whose parameters are [(l+1)(n−1), n−1] and [3n−3, n−1] maintain the LCD property. Utilizing the relationship between their LCD property and the dimensions of the Euclidean hull, this paper constructs two types of EAQECCs based on Hadamard matrices. These EAQECCs can achieve maximal entanglement, resulting in higher code rates and net rates compared to other EAQECCs. Their parameters are [[3n−3, 2n−2, d≥1; n−1]] and [[2n−2, n−1, d≥1; n−1]], respectively.
We introduce some methods for constructing quaternary Hermitian self-orthogonal (HSO) codes, and construct quaternary [n, 5] HSO for 342≤n≤492. Furthermore, we present methods of constructing Hermitian linear complementary dual (HLCD) codes from known HSO codes, and obtain many HLCD codes with good parameters. As an application, 31 classes of entanglement-assisted quantum error correction codes (EAQECCs) with maximal entanglement can be obtained from these HLCD codes. These new EAQECCs have better parameters than those in the literature.
Abstract It is known that entanglement-assisted quantum error-correcting codes (EAQECCs), a type of quantum error correction codes, can be easily constructed using linear codes that satisfy the property called linear complementary duals (LCD). For quasi-cyclic (QC) codes, which are a class of linear codes, we have already published the methods for constructing the codes with properties such as self-orthogonality, self-duality and reversibility according to the prime-factor decomposition of −1+xm-1+x^m−1+xm - 1 + x m , which reduce the amount of calculation by assembling several small generator polynomial matrices into a large generator polynomial matrix. In this paper, we propose a method to construct LCD–QC codes according to the prime-factor decomposition. The main idea of this method is to decompose the generator polynomial matrix of the QC code into several small generator polynomial matrices corresponding to the prime factors and perform LCD determination, which leads to a reduction in the amount of calculation. As an application of our construction method, we create EAQECCs from the constructed LCD–QC codes, compare those minimum weights with the maximum values of the minimum weights of existing EAQECCs and find 15 EAQECCs with larger minimum weights than the existing ones.
Entanglement-assisted quantum error-correcting codes (EAQECCs) make use of preexisting entanglement between the sender and receiver to boost the rate of transmission. It is possible to construct an EAQECC from any classical linear code, unlike standard quantum error-correcting codes, which can only be constructed from dual-containing codes. However, the parameter of ebits ccc is usually calculated by computer search. In this paper, we construct four classes of MDS entanglement-assisted quantum error-correcting codes (MDS EAQECCs) based on kkk-Galois LCD MDS codes for some certain code lengths, where the parameter of ebits ccc can be easily generated algebraically and not by computational search. Moreover, the constructed four classes of EAQECCs are also maximal-entanglement EAQECCs.
Linear complementary-dual (LCD for short) codes are linear codes that intersect with their duals trivially. There has been renewed interest in LCD codes due to an important application in cryptography. On the other hand, it is an important task to construct maximum-distance-separable (MDS) codes with good parameters in both coding theory and practice. The main purpose of this paper is to construct several classes of MDS codes with complimentary duals (i.e., LCD MDS codes). Then we give the new results on EAQEC MDS codes with maximal entanglement.
The entanglement-assisted (EA) formalism is a generalization of the standard stabilizer formalism, and it can transform arbitrary quaternary classical linear codes into entanglement-assisted quantum error correcting codes (EAQECCs) by using of shared entanglement between the sender and the receiver. Using a decomposition of the 126-cap in PG(5,4), we firstly constructed LCD n-cap with 6The entanglement-assisted (EA) formalism is a generalization of the standard stabilizer formalism, and it can transform arbitrary quaternary classical linear codes into entanglement-assisted quantum error correcting codes (EAQECCs) by using of shared entanglement between the sender and the receiver. Using a decomposition of the 126-cap in PG(5,4), we firstly constructed LCD n-cap with 6≤n≤120. From these LCD caps, we then derived the related maximal entanglement [[n, n-6,4;6]] EAQECCs. Finally, using LCD LCD subcaps of 126-cap obtained, we constructed maximal entanglement EAQECCs with parameters [[n, n-;k,4;k]] for 6≤k≤11.