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arXiv:2001.03445

Two new classes of quantum MDS codes from generalized Reed-Solomon codes

Weiwei Wang, Jiantao Li

Published Jan 6, 2020• 0 citations
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Abstract

In this paper, we mainly use classical Hermitian self-orthogonal generalized Reed-Solomon codes to construct two new classes of quantum MDS codes. Most of our quantum MDS codes have minimum distance larger than q/2+1. Compared with previously known constructions, these quantum MDS codes in our constructions have large minimum distance and their parameters are not covered by the previous codes.

DOI
10.48550/arxiv.2001.03445

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In this paper, we mainly use classical Hermitian self-orthogonal generalized Reed-Solomon codes to construct two new classes of quantum MDS codes. Most of our quantum MDS codes have minimum distance larger than q/2+1. Compared with previously known constructions, these quantum MDS codes in our constructions have large minimum distance and their parameters are not covered by the previous codes.
The construction of quantum maximum-distance-separable (MDS) codes has become one of the major goals in quantum coding theory. In this paper, we construct several classes of Hermitian self-orthogonal generalized Reed–Solomon codes. Based on these classical MDS codes, we obtain several new classes of quantum MDS codes with large minimum distance. It turns out that these constructed quantum MDS codes have fewer constraints on the selection of code length, and in some cases, have larger minimum distance than previous literature. Notably, some of the distance parameters of our codes are greater than (q/2)+1.
Quantum maximum-distance-separable (MDS for short) codes are an important class of quantum codes. In this paper, by using Hermitian self-orthogonal generalized Reed-Solomon (GRS for short) codes, we construct five new classes of qqq-ary quantum MDS codes with minimum distance larger than q/2+1q/2+1q/2+1. Furthermore, the parameters of our quantum MDS code cannot be obtained from the previous constructions.
In this paper, two classes of quantum MDS codes are constructed. The main tools are multiplicative structures on finite fields. Carefully choosing different cosets can make the corresponding generalized Reed-Solomon codes Hermitian self-orthogonal, which yields quantum MDS codes by utilizing Hermitian type CSS construction. In some cases, these codes have minimal distances larger than q/2q/2q/2. Also, the parameters of these codes have never been reported before.
It is an important task to construct quantum maximum-distance-separable (MDS) codes with good parameters. In the present paper, we provide six new classes ofqqq-ary quantum MDS codes by using generalized Reed–Solomon (GRS) codes and Hermitian construction. The minimum distances of our quantum MDS codes can be larger thanq2+1\frac {q}{2}+12q​+1. Three of these six classes of quantum MDS codes have longer lengths than the ones constructed in[1]and[2], hence some of their results can be easily derived from ours via the propagation rule. Moreover, some known quantum MDS codes of specific lengths can be seen as special cases of ours and the minimum distances of some known quantum MDS codes are also improved as well.
Abstract The construction of quantum Maximum Distance Separable (MDS) error-correcting codes is of great significance in practical applications, as it plays a vital role in enhancing the reliability and efficiency of quantum communication and quantum computing. As a result, it remains an active research topic in the field of quantum information science. In this paper, based on generalized Reed-Solomon (GRS) codes and Hermitian construction, we construct two new classes of quantum MDS codes. Our constructed quantum MDS codes are novel in the sense that their parameters differ from all previously known codes. Furthermore, most of our codes exhibit significantly larger minimum distance compared to the existing ones of the same length.