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Entanglement-assisted quantum MDS codes from generalized Reed–Solomon codes

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Abstract

Entanglement-assisted quantum error-correcting (EAQEC) codes are a generalization of standard stabilizer quantum codes that can be obtained from arbitrary classical linear codes based on the entanglement-assisted stabilizer formalism. In this paper, by using generalized Reed–Solomon (GRS) codes, we construct two classes of entanglement-assisted quantum error-correcting MDS (EAQEC MDS) codes with parameters

$$\begin{aligned} \left[ \left[ \frac{q^2-1}{2a},\frac{q^2-1}{2a}-2d+c+2,d;c\right] \right] _q, \end{aligned}$$

where q is an odd prime power of the form \(q=2am-1>3\) with \(m\ge 2\), \(1\le c\le 2a-1\) and \(c m+2\le d\le (a+\lceil \frac{c}{2}\rceil )m\), and

$$\begin{aligned} \left[ \left[ \frac{q^2-1}{2a+1},\frac{q^2-1}{2a+1}-2d+c+2,d;c\right] \right] _q, \end{aligned}$$

where q is a prime power of the form \(q=(2a+1)m-1\), \(1\le c\le 2a\) and \(c m+2\le d\le (a+1+\lfloor \frac{c}{2}\rfloor )m\). The EAQEC MDS codes constructed have much larger minimum distance than the known quantum MDS codes with the same length, and most of them are new in the sense that the parameters of EAQEC codes are different from all the previously known ones. In particular, some of our EAQEC MDS codes have much larger d than the known ones that are of the same length and consume the same number of ebits.

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Authors and Affiliations

  1. School of Mathematics, Hefei University of Technology, Hefei, 230009, Anhui, People’s Republic of China

    Lanqiang Li, Shixin Zhu, Li Liu & Xiaoshan Kai

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  1. Lanqiang Li
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  2. Shixin Zhu
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Correspondence to Shixin Zhu.

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This research is supported in part by the National Natural Science Foundation of China under Project 61772168, Project 61572168, Project 11871187.

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Li, L., Zhu, S., Liu, L. et al. Entanglement-assisted quantum MDS codes from generalized Reed–Solomon codes. Quantum Inf Process 18, 153 (2019). https://doi.org/10.1007/s11128-019-2269-7

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