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arXiv:1804.08213 · IEEE Transactions on Information Theory

Some New Constructions of Quantum MDS Codes

Weijun Fang, Fang‐Wei Fu

Published Sep 3, 2019• 58 citations
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Abstract

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 of$q$-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 than$\frac {q}{2}+1$. 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.

DOI
10.1109/tit.2019.2939114

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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.
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.
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.
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.
Quantum MDS Codes with length n≡0,1(n\equiv 0,1(n≡0,1(mod q±12)\,\frac{q\pm1}{2})2q±1​)
An important family of quantum codes is the quantum maximum-distance-separable (MDS) codes. In this paper, we construct some new classes of quantum MDS codes by generalized Reed-Solomon (GRS) codes and Hermitian construction. In addition, the length nnn of most of the quantum MDS codes we constructed satisfies n≡0,1(n\equiv 0,1(n≡0,1(mod q±12)\,\frac{q\pm1}{2})2q±1​), which is different from previously known code lengths. At the same time, the quantum MDS codes we construct have large minimum distances that are greater than q/2+1q/2+1q/2+1.
Quantum maximum-distance-separable (MDS) codes play an important role in the quantum codes. The previous quantum MDS codes had been constructed according to [Formula: see text] is odd or even. However, such classification omits to consider some special categories of quantum MDS codes. Because of this, we will discuss the other classifications of [Formula: see text]. In this paper, we construct some new [Formula: see text]-ary quantum MDS codes from generalized Reed–Solomon codes by using Hermitian construction, and prove that these quantum MDS codes have minimum distance greater than [Formula: see text], where [Formula: see text].