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International Journal of Quantum Information

NEW CLASS OF QUANTUM CODES CONSTRUCTED FROM CYCLIC DIFFERENCE SET

Shengmei Zhao, Xiao Ping Yu, Yan Mei Zhu, Xiu-Li Zhu, Min-Hsiu Hsieh

Published Feb 1, 2012• 5 citations
Publisher

Abstract

We introduce joint difference sets as a generalization of cyclic difference sets, and we construct a new class of quantum error-correcting codes (QECCs) from these joint difference sets. The main benefits of our method are as follows. First, we can construct quantum codes that are both high rate and with large block length, while maintaining good performance. Second, the density of constructed quantum parity check matrix can approach zero when the code length is very large. This allows us to use a simple iterative decoding algorithm. Interestingly, our method yields the well-known [5,1,3] QECC.

DOI
10.1142/s0219749912500153

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Quantum error correcting code is an efficient tool to correct and prevent quantum error.It is the basic for quantum computation and quantum communications.Taking advantage of the cyclic difference set,we present a novel approach to construct quantum stabilizer codes in this paper.It's not difficult to construct the well-known quantum code by our approach.By analyzing the property of the quantum code constructed by this method,such as ,as well as ,we find all the constructed codes are the quantum Single-bound codes,and we can use this method to encode different length information.Besides,the code rate is high.
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Binary cyclic codes achieve good error correction performance and allow the implementation of very simple encoder and decoder circuits. Among them, BCH codes represent a very important class of t-error correcting codes, with known structural properties and error correction capability. Decoding of binary cyclic codes is often accomplished through hard-decision decoders, although it is recognized that soft-decision decoding algorithms can produce significant coding gain with respect to hard-decision techniques. Several approaches have been proposed to implement iterative soft-decision decoding of binary cyclic codes. We study the technique based on "Extended Parity-Check Matrices", and show that such method is not suitable for high rates or long codes. We propose a new method, based on "Spread Parity-Check Matrices", that has the same complexity, but can achieve better correction performance in many practical cases.