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International Conference on Natural Computation (ICNC)

Cyclotomic cosets and quantum BCH codes

Fangying Xiao, Hanwu Chen

Published Aug 1, 2010• 2 citations
Publisher

Abstract

In order to constructing quantum BCH codes from classical BCH codes using CSS constructing method it needs to determine whether the BCH codes contains its dual. Cyclotomic cosets play very important roles in classical error correction theory and quantum error correction theory. This letter presented several properties of cyclotomic cosets and introduced a method to determine whether a non-primitive BCH code over GF (q) contains its dual. The method presented in this paper with polynomial time complexity can also be applied to nonnarrow sense primitive BCH codes.

DOI
10.1109/icnc.2010.5584242

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In order to constructing quantum BCH codes from classical BCH codes using CSS constructing method it needs to determine whether the BCH codes contains its dual. Cyclotomic cosets play very important roles in classical error correction theory and quantum error correction theory. This letter presented several properties of cyclotomic cosets and introduced a method to determine whether a non-primitive BCH code over GF (q) contains its dual. The method presented in this paper with polynomial time complexity can also be applied to nonnarrow sense primitive BCH codes.
Cyclotomic cosets play very important roles in classical and quantum error correction theory.In order to constructing quantum BCH(Bose Chaudhuri Hocquenghem) codes with CSS constructing method from classical BCH codes,it needs to determine whether a BCH code contains its dual.It proposed several properties of cyclotomic cosets.And according to these properties,an algorithm with polynomial time complexity was presented to determine whether a non-primitive BCH code over finite field contains its dual code,which can also be applied to nonnarrow sense primitive BCH codes.
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.
Let q≥3q\geq 3q≥3 be a prime power. Maximal designed distances of primitive Hermitian dual containing q2q^{2}q2-ary BCH codes (narrow-sense or non-narrow-sense) are determined by a careful analysis of properties of cyclotomic cosets. Non-narrow-sense BCH codes which achieve these maximal designed distances are presented, and a sequence of nested non-narrow-sense BCH codes that contain these BCH codes with maximal designed distances are constructed and their parameters are computed. Consequently, new nonbinary quantum BCH codes are derived from these non-narrow-sense BCH codes. The nonbinary quantum BCH codes presented here have better parameters than those quantum BCH codes available in the literature.
In this paper, we use q -cyclotomic cosets modulo n = q − 1 and generator polynomial to describe repeated-root cyclic codes of length N = p α n . We determine the condition under which a repeated-root cyclic code contains its dual code. Then, for q = 2 a , we give a class of dual-containing repeated-root cyclic codes. Applying CSS construction and Steane construction to these underlying classical codes, many quantum codes with good parameters shall be constructed, which are new in the sense that they are not 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.
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.