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Entropy

Some New Quantum BCH Codes over Finite Fields

Lijuan Xing, Zhuo Li

Published Jun 3, 2021• 4 citations
Publisher

Abstract

Quantum error correcting codes (QECCs) play an important role in preventing quantum information decoherence. Good quantum stabilizer codes were constructed by classical error correcting codes. In this paper, Bose-Chaudhuri-Hocquenghem (BCH) codes over finite fields are used to construct quantum codes. First, we try to find such classical BCH codes, which contain their dual codes, by studying the suitable cyclotomic cosets. Then, we construct nonbinary quantum BCH codes with given parameter sets. Finally, a new family of quantum BCH codes can be realized by Steane's enlargement of nonbinary Calderbank-Shor-Steane (CSS) construction and Hermitian construction. We have proven that the cyclotomic cosets are good tools to study quantum BCH codes. The defining sets contain the highest numbers of consecutive integers. Compared with the results in the references, the new quantum BCH codes have better code parameters without restrictions and better lower bounds on minimum distances. What is more, the new quantum codes can be constructed over any finite fields, which enlarges the range of quantum BCH codes.

DOI
10.3390/e23060712

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Quantum error correcting codes (QECCs) play an important role in preventing quantum information decoherence. Good quantum stabilizer codes were constructed by classical error correcting codes. In this paper, Bose-Chaudhuri-Hocquenghem (BCH) codes over finite fields are used to construct quantum codes. First, we try to find such classical BCH codes, which contain their dual codes, by studying the suitable cyclotomic cosets. Then, we construct nonbinary quantum BCH codes with given parameter sets. Finally, a new family of quantum BCH codes can be realized by Steane's enlargement of nonbinary Calderbank-Shor-Steane (CSS) construction and Hermitian construction. We have proven that the cyclotomic cosets are good tools to study quantum BCH codes. The defining sets contain the highest numbers of consecutive integers. Compared with the results in the references, the new quantum BCH codes have better code parameters without restrictions and better lower bounds on minimum distances. What is more, the new quantum codes can be constructed over any finite fields, which enlarges the range of quantum BCH codes.
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.
Four quantum code constructions generating several new families of good nonbinary quantum nonprimitive nonnarrow-sense, Bose-Chaudhuri-Hocquenghem codes, are presented in this paper. The first two are based on Calderbank-Shor-Steane (CSS) construction derived from two nonprimitive Bose-Chaudhuri-Hocquenghem codes. The third one is based on Steane's enlargement of nonbinary CSS codes applied to suitable subfamilies of nonprimitive nonnarrow-sense Bose-Chaudhuri-Hocquenghem codes. The fourth construction is derived from suitable subfamilies of Hermitian dual-containing nonprimitive nonnarrow-sense Bose-Chaudhuri-Hocquenghem codes. These constructions generate new families of quantum codes whose parameters are better than the ones available in the literature.
Four quantum code constructions generating several new families of good nonbinary quantum nonprimitive nonnarrow-sense, Bose-Chaudhuri-Hocquenghem codes, are presented in this paper. The first two are based on Calderbank-Shor-Steane (CSS) construction derived from two nonprimitive Bose-Chaudhuri-Hocquenghem codes. The third one is based on Steane's enlargement of nonbinary CSS codes applied to suitable subfamilies of nonprimitive nonnarrow-sense Bose-Chaudhuri-Hocquenghem codes. The fourth construction is derived from suitable subfamilies of Hermitian dual-containing nonprimitive nonnarrow-sense Bose-Chaudhuri-Hocquenghem codes. These constructions generate new families of quantum codes whose parameters are better than the ones available in the literature.
Four quantum code constructions generating several new families of good nonbinary quantum nonprimitive non-narrow-sense Bose-Chaudhuri-Hocquenghem (BCH) codes are presented in this paper. The first two ones are based on Calderbank-Shor-Steane (CSS) construction derived from two distinct nonprimitive BCH codes, not necessarily self-orthogonal. The third one is based on nonbinary Steane's enlargement of CSS codes applied to suitable sub-families of nonprimitive non-narrow-sense BCH codes. The fourth construction is derived from suitable sub-families of Hermitian self-orthogonal nonprimitive non-narrow-sense BCH codes. These constructions generate new families of quantum BCH codes whose parameters are better than the ones available in the literature.
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.
This paper concentrates on construction of new nonbinary quantum error-correcting codes (QECCs) from three classes of narrow-sense imprimitive BCH codes over finite field [Formula: see text] ([Formula: see text] is an odd prime power). By a careful analysis on properties of cyclotomic cosets in defining set [Formula: see text] of these BCH codes, the improved maximal designed distance of these narrow-sense imprimitive Hermitian dual-containing BCH codes is determined to be much larger than the result given according to Aly et al. [S. A. Aly, A. Klappenecker and P. K. Sarvepalli, IEEE Trans. Inf. Theory 53, 1183 (2007)] for each different code length. Thus families of new nonbinary QECCs are constructed, and the newly obtained QECCs have larger distance than those in previous literature.