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Physical Review A

Quantum negacyclic codes

Xiaoshan Kai, Shixin Zhu, Yongsheng Tang

Published Jul 24, 2013• 46 citations
Publisher

Abstract

In this paper, we construct families of nonbinary quantum codes from classical negacyclic codes. The BCH bound for negacyclic codes is applied to determine the dimension and distance of these quantum codes. The parameters of these quantum codes are good compared with the ones available in the literature. We also construct a family of quantum maximum-distance-separable codes with parameters ${[[q+1,q\ensuremath{-}2d+3,d]]}_{q}$, where $q\ensuremath{\equiv}1\phantom{\rule{0.28em}{0ex}}\mathrm{mod}\phantom{\rule{0.28em}{0ex}}4$ and $2\ensuremath{\le}d\ensuremath{\le}(q+3)/2$.

DOI
10.1103/physreva.88.012326

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In this paper, we construct families of nonbinary quantum codes from classical negacyclic codes. The BCH bound for negacyclic codes is applied to determine the dimension and distance of these quantum codes. The parameters of these quantum codes are good compared with the ones available in the literature. We also construct a family of quantum maximum-distance-separable codes with parameters [[q+1,q\ensuremath−2d+3,d]]q{[[q+1,q\ensuremath{-}2d+3,d]]}_{q}[[q+1,q\ensuremath−2d+3,d]]q​, where q\ensuremath≡1mod4q\ensuremath{\equiv}1\phantom{\rule{0.28em}{0ex}}\mathrm{mod}\phantom{\rule{0.28em}{0ex}}4q\ensuremath≡1mod4 and 2\ensuremath≤d\ensuremath≤(q+3)/22\ensuremath{\le}d\ensuremath{\le}(q+3)/22\ensuremath≤d\ensuremath≤(q+3)/2.
Let [Formula: see text] be an odd prime power and [Formula: see text] be a positive integer. Maximum designed distance such that negacyclic BCH codes over [Formula: see text] of length [Formula: see text] are Hermitian dual-containing codes is given. The dimension of such Hermitian dual-containing negacyclic codes is completely determined by analyzing cyclotomic cosets. Quantum negacyclic BCH codes of length [Formula: see text] are obtained by using Hermitian construction. The constructed quantum negacyclic BCH codes produce new quantum codes with parameters better than those obtained from quantum BCH codes.
Three code constructions generating new families of good nonbinary quantum codes are presented in this paper. The first two ones are derived from Hermitian self-orthogonal non-narrow-sense Bose-Chaudhuri-Hocquenghem (BCH) codes. The third one is derived from qqq-ary ($q\ensuremath{\ne}2$ is a prime power) Steane's enlargement of Calderbank-Shor-Steane codes applied to Euclidean self-orthogonal non-narrow-sense BCH codes. The quantum nonbinary BCH codes presented here have parameters better than the ones available in the literature.
Classical Bose-Chaudhuri-Hocquenghem (BCH) codes over finite fields have been studied extensively. One can construct quantum stabilizer codes with good parameters using classical BCH codes. In this paper, our goal is to find such classical BCH codes. We study some properties of suitable cyclotomic cosets at first. These results make it possible to construct nonbinary quantum BCH codes with a given parameter set. Several new families of quantum BCH codes obtained are based on Steane's enlargement of nonbinary Calderbank-Shor-Steane codes and Hermitian construction, respectively. Meanwhile, we have shown that the cyclotomic cosets given in our schemes are optimal to design quantum BCH codes. The defining set contains most consecutive integers. Therefore, corresponding quantum stabilizer codes have better lower bound of minimum distance. Furthermore, it is convenient to compute the dimension of new quantum codes. Compared with the ones available in the literature, the quantum BCH codes in our schemes have good parameters. In particular, we extend known results to more general case.
In this paper, the properties of q2-ary cyclotomic cosets modulo 2n are investigated in detail, where n=[(q2m-1)/(m(q-1))],q ≡ ±1mod m or q2≡ -1mod m when m is even and q is an odd power prime when m is power of 2. Then, non-narrow-sense negacyclic Bose-Chaudhuri-Hocquenghem (BCH) codes of length n that contain their Hermitian duals are determined. By the Hermitian construction, series of quantum codes are obtained from such negacyclic BCH codes. Compared with quantum codes in the [6], [10], [12], our constructed ones have better parameters.