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Dianzi xuebao

Dual-Containing Determination Method for Non-Primitive BCH Codes over Finite Field

Wenjie Liu

Published Jan 1, 2010• 0 citations
OpenAlex

Abstract

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.

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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.
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.
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.
The Bose–Chaudhuri–Hocquenghem (BCH) codes have been studied for more than 57 years and have found wide application in classical communication system and quantum information theory. In this paper, we study the construction of quantum codes from a family of [Formula: see text]-ary BCH codes with length [Formula: see text] (also called antiprimitive BCH codes in the literature), where [Formula: see text] is a power of 2 and [Formula: see text]. By a detailed analysis of some useful properties about [Formula: see text]-ary cyclotomic cosets modulo [Formula: see text], Hermitian dual-containing conditions for a family of non-narrow-sense antiprimitive BCH codes are presented, which are similar to those of [Formula: see text]-ary primitive BCH codes. Consequently, via Hermitian Construction, a family of new quantum codes can be derived from these dual-containing BCH codes. Some of these new antiprimitive quantum BCH codes are comparable with those derived from primitive BCH codes.
Dual-containing classical error-correcting codes can be used in the Calderbank-Shor-Steane (CSS) construction for quantum error-correcting codes (QECCs). Dual-containing Bose-Chaudhuri-Hocquenghem (BCH) codes have been extensively studied in this regard. Research on the more general alternant codes is far less complete, despite the perspective of finding superior codes among them. In this paper, conditions for an alternant code to contain its Euclidean dual code are derived. The necessity of having dual-containing generalized Reed-Solomon (GRS) codes is shown and the structure of such codes is studied. An approach for constructing dual-containing alternant codes from BCH codes is proposed. In addition, a method for finding GRS codes over a given finite field with dual-containing alternant subfield-subcodes is presented. This method encompasses dual-containing BCH codes as a special case.
Classical Bose-Chaudhuri-Hocquenghem (BCH) codes C that contain their dual codes can be used to construct quantum stabilizer codes this chapter studies the properties of such codes. It had been shown that a BCH code of length n which contains its dual code satisfies the bound on weight of any non-zero codeword in C and converse is also true. One impressive difficulty in quantum communication and computation is to protect informationcarrying quantum states against undesired interactions with the environment. To address this difficulty, many good quantum errorcorrecting codes have been derived as binary stabilizer codes. We were able to shed more light on the structure of dual containing BCH codes. These results make it possible to determine the parameters of quantum BCH codes in terms of weight of non-zero dual codeword.