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IEEE Transactions on Information Theory

New Quantum MDS Codes From Negacyclic Codes

Xiaoshan Kai, Shixin Zhu

Published Sep 24, 2012• 183 citations
Publisher

Abstract

Letqbe an odd prime power. Based on classical negacyclic codes, we construct two classes of quantum maximum-distance-separable (MDS) codes with parameters [[q2+1,q2-2d+3,d]]qwhereq≡ 1 (mod 4) and 2 ≤d≤q+1 is even, and [[(q2+1)/2,(q2+1)/2-2d+2,d]]qwhere 3 ≤d≤qis odd. Some of these quantum MDS codes are new in the sense that their parameters are different from all the previously known ones.

DOI
10.1109/tit.2012.2220519

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Letqbe an odd prime power. Based on classical negacyclic codes, we construct two classes of quantum maximum-distance-separable (MDS) codes with parameters [[q2+1,q2-2d+3,d]]qwhereq≡ 1 (mod 4) and 2 ≤d≤q+1 is even, and [[(q2+1)/2,(q2+1)/2-2d+2,d]]qwhere 3 ≤d≤qis odd. Some of these quantum MDS codes are new in the sense that their parameters are different from all the previously known ones.
In this paper, we construct two classes of new quantum maximum-distance-separable (MDS) codes with parameters [Formula: see text], where q is an odd prime power with q ≡ 3 (mod 4) and [Formula: see text]; [[8(q - 1), 8(q - 1) - 2d + 2, d]]q, where q is an odd prime power with the form q = 8t - 1 (t is an even positive integer) and [Formula: see text]. Comparing the parameters with all known quantum MDS codes, the quantum MDS codes exhibited here have minimum distances bigger than the ones available in the literature.
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.
Let p be an odd prime, and Fp(m)is the finite field of pmelements. In this paper, all maximum distance separable (briefly, MDS) cyclic and negacyclic codes of length 2psover Fp(m)are established. As an application, all quantum MDS (briefly, qMDS) codes are constructed from cyclic and negacyclic codes of length 2psover finite fields using the Calderbank Shor-Steane (briefly, CSS) and Hermitian constructions. These codes are new in the sense that their parameters are different from all the previous constructions. Furthermore, quantum synchronizable codes (briefly, QSCs) are obtained from cyclic codes of length 2psover Fp(m). To enrich the variety of available QSCs, many new QSCs are constructed to illustrate our results. Among them, there are QSCs codes with shorter lengths and much larger minimum distances than known primitive narrow-sense Bose-Chaudhuri-Hocquenghem (briefly, BCH) codes.
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.