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Wuhan University Journal of Natural Sciences

Optimal Asymmetric Quantum Codes from the Euclidean Sums of Linear Codes

Peng Xu, Xiusheng Liu

Published Feb 1, 2024• 0 citations
Publisher

Abstract

In this paper, we first give the definition of the Euclidean sums of linear codes, and prove that the Euclidean sums of linear codes are Euclidean dual-containing. Then we construct two new classes of optimal asymmetric quantum error-correcting codes based on Euclidean sums of the Reed-Solomon codes, and two new classes of optimal asymmetric quantum error-correcting codes based on Euclidean sums of linear codes generated by Vandermonde matrices over finite fields. Moreover, these optimal asymmetric quantum error-correcting codes constructed in this paper are different from the ones in the literature.

DOI
10.1051/wujns/2024291045

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Matrix-product codes over finite fields are an important class of long linear codes by combining several commensurate shorter linear codes with a defining matrix over finite fields. The construction of matrix-product codes with certain self-orthogonality over finite fields is an effective way to obtain good qqq-ary quantum codes of large length. This article has two purposes: the first is to summarize some results of this topic obtained by the author of this article and his cooperators in [10-12]; the second is to add some new results on quasi-orthogonal matrices (resp. quasi-unitary matrices), Euclidean dual-containing (resp. Hermitian dual-containing) matrix-product codes and qqq-ary quantum codes derived from these matrix-product codes.
In this paper, we discuss two problems on asymmetric quantum error-correcting codes (AQECCs). The first one is on the construction of a [[12, 1, 5/3]]2asymmetric quantum code, we show an impure [[12, 1, 5/3 ]]2exists. The second one is on the construction of AQECCs from binary cyclic codes, we construct many families of new asymmetric quantum codes with dz> δmax+1 from binary primitive cyclic codes of length n = 2m-1, where δmax= 2⌈m/2⌉-1 is the maximal designed distance of dual containing narrow sense BCH code of length n = 2m-1. A number of known codes are special cases of the codes given here. Some of these AQECCs have parameters better than the ones available in the literature.
In 2019, Carletet al. introduced the concept of σ duals of linear codes involving the σ inner product, which generalizes the Euclidean, Hermitian and ℓ-Galois cases. This paper focuses on constructing new and improved classical codes and quantum codes within the framework of the σ inner product. We derive some general properties of linear codes, including matrix-product (MP) codes, with respect to the σ inner product. We develop general methods and design effective routes involving certain optimization problems for constructing σ self-orthogonal (SO) and σ dual-containing (DC) MP codes. Our schemes efficiently generate numerous such codes with new or optimal parameters. We establish the σ construction of quantum stabilizer codes from classical codes. We propose a unified method for constructing two general classes of entanglement-assisted quantum error-correcting codes (EAQECCs) based on the σ hulls of general linear codes. This further yields six types of EAQECCs with flexible parameters based on propagation rules using MP codes under the Euclidean and Hermitian cases. Compared to the best-known ternary EAQECCs, we obtain 17 new ones and 13 of them have improved parameters. Finally, we present two infinite families of q-ary EAQECCs with lengths (q2− 1)(q+ 2) andq2(q+2), respectively. These families include many q-ary QECCs that are not only new according to Grassl’s online database but also surpass those listed in Edel’s online database.