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Acta Physica Sinica

Quantum Generalized Reed-Solomon codes

Zhuo Li, Lijuan Xing, 西安电子科技大学综合业务网国家重点实验室，西安 710071

Published Jan 1, 2008• 13 citations
Publisher

Abstract

We construct a family of quantum error-correcting codes with parameters ［［n，n-2k，k+1］］q which are defined in q-dimensional quantum systems，where q is an arbitrary prime power. These codes are optimal in the sense that the minimum distance is maximal. It is shown that codes exist for all n satisfying 2≤n≤q or q2-q+2≤n≤q2.

DOI
10.7498/aps.57.28

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We construct a family of quantum error-correcting codes with parameters ［［n，n-2k，k+1］］q which are defined in q-dimensional quantum systems，where q is an arbitrary prime power. These codes are optimal in the sense that the minimum distance is maximal. It is shown that codes exist for all n satisfying 2≤n≤q or q2-q+2≤n≤q2.
We present families of quantum error-correcting codes which are optimal in the sense that the minimum distance is maximal. These maximum distance separable (MDS) codes are defined over q-dimensional quantum systems, where q is an arbitrary prime power. It is shown that codes with parameters 〚n, n - 2d + 2, d〛q exist for all 3≤n≤q and 1≤d≤n/2+1. We also present quantum MDS codes with parameters 〚q2, q2-2d+2, d〛q for 1≤d≤q which additionally give rise to shortened codes 〚q2-s, q2-2d+2-s, d〛q for some s.
We construct a family of quantum maximum-distance-separable (MDS) codes from classical generalized Reed-Solomon codes and derive the necessary and sufficient condition under which these quantum codes exist. We also give code bounds and show how to construct them analytically. We find that existing quantum MDS codes can be unified under these codes in the sense that when a quantum MDS code exists, then a quantum code of this type with the same parameters also exists. Thus, as far as is known at present, they are the most important family of quantum MDS codes.
In this paper, we construct two classes of asymmetric quantum codes by using constacyclic codes. The first class is the asymmetric quantum codes with parameters [[q2 + 1, q2 + 1 - 2(t + k + 1), (2k + 2)/(2t + 2)]]q2 where q is an odd prime power, t, k are integers with [Formula: see text], which is a generalization of [J. Chen, J. Li and J. Lin, Int. J. Theor. Phys. 53 (2014) 72, Theorem 2] in the sense that we do not assume that q ≡1 ( mod 4). The second one is the asymmetric quantum codes with parameters [Formula: see text], where q ≥ 5 is an odd prime power, t, k are integers with 0 ≤ t ≤ k ≤ q - 1. The constructed asymmetric quantum codes are optimal and their parameters are not covered by the codes available in the literature.