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arXiv:quant-ph/0604102 · IEEE Transactions on Information Theory

On Quantum and Classical BCH Codes

Salah A. Aly, Andreas Klappenecker, Pradeep Kiran Sarvepalli

Published Mar 1, 2007• 383 citations
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Abstract

Classical Bose–Chaudhuri–Hocquenghem (BCH) codes that contain their (Euclidean or Hermitian) dual codes can be used to construct quantum stabilizer codes; this correspondence studies the properties of such codes. It is shown that a BCH code of length$n$can contain its dual code only if its designed distance$\delta =O(\sqrt {n})$, and the converse is proved in the case of narrow-sense codes. Furthermore, the dimension of narrow-sense BCH codes with small design distance is completely determined, and – consequently – the bounds on their minimum distance are improved. These results make it possible to determine the parameters of quantum BCH codes in terms of their design parameters.

DOI
10.1109/tit.2006.890730

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Classical Bose–Chaudhuri–Hocquenghem (BCH) codes that contain their (Euclidean or Hermitian) dual codes can be used to construct quantum stabilizer codes; this correspondence studies the properties of such codes. It is shown that a BCH code of lengthnnncan contain its dual code only if its designed distanceδ=O(n)\delta =O(\sqrt {n})δ=O(n​), and the converse is proved in the case of narrow-sense codes. Furthermore, the dimension of narrow-sense BCH codes with small design distance is completely determined, and – consequently – the bounds on their minimum distance are improved. These results make it possible to determine the parameters of quantum BCH codes in terms of their design parameters.
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.
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.
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.
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.
An attractive feature of BCH codes is that one can infer valuable information from their design parameters (length, size of the finite field, and designed distance), such as bounds on the minimum distance and dimension of the code. In this paper, it is shown that one can also deduce from the design parameters whether or not a primitive, narrow-sense BCH contains its Euclidean or Hermitian dual code. This information is invaluable in the construction of quantum BCH codes. A new proof is provided for the dimension of BCH codes with small designed distance, and simple bounds on the minimum distance of such codes and their duals are derived as a consequence. These results allow us to derive the parameters of two families of primitive quantum BCH codes as a function of their design parameters