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Computer Engineering and Applications Journal

Construction of S-chains and quantum codes based on three self-dual codes

LI Rui-hu

Published Jan 1, 2009• 0 citations
OpenAlex

Abstract

By using of a random searching algorithm and classical group theory,the subcodes which dual distance between 3 and 7 of three double circulant self-dual codes D3,D4 and D5 are studied.The relation of these subcodes and the subcode chains of these self-orthogonal codes are determined,and the S-chains deduced from these self-orthogonal codes are also presented.Accord- ing to Steane’s construction and the S-chains obtained,some new quantum codes with good parameters are constructed.

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By using of a random searching algorithm and classical group theory,the subcodes which dual distance between 3 and 7 of three double circulant self-dual codes D3,D4 and D5 are studied.The relation of these subcodes and the subcode chains of these self-orthogonal codes are determined,and the S-chains deduced from these self-orthogonal codes are also presented.Accord- ing to Steane’s construction and the S-chains obtained,some new quantum codes with good parameters are constructed.
The subcodes of binary self-dual codes whose length n between 22 and 30 and minimum distance six are studied by using of a random search algorithm,and the subcodes of dual distance 3,4,5 and 6 of these self-dual codes are constructed.The code chains of these self-orthogonal subcodes and the S-chains obtained from the dual of these self-orthogonal subcodes are also determined.According to Steane’s construction,some very good quantum codes of distance five and six are constructed by the obtained S-chains,and some of these quantum codes have improved parameters than previously known codes.
Subcodes of some known maximal self-orthogonal codes and optimal self-dual codes of length n between 9 and 22 over F5 are studied. Using a random search algorithm, we construct subcodes with given dual distance 2, 3, 4, 5 and 6 from these known codes, and discuss code chains formed by obtained subcodes and their dual codes. Based on these obtained code chains, we construct many quantum codes of distance 3, 4, 5, 6 and 8 by using of Steane's construction, some of these quantum codes are present known best codes.
In this paper, we consider $ S $-chains of extremal self-dual and self-orthogonal codes and their applications in the construction of quantum codes. Then, by virtue of covering radius, we determine necessary conditions for linear codes to have subcodes with large dual distances and design a new $ S $-chain search method. As computational results, 18 $ S $-chains with large distances are obtained, and many good quantum codes can be derived from those $ S $-chains by Steane construction, some of which improve the previous results.
By using construction arithmetic,the subcodes of quaternary self-dual codes with code length 10≤n≤20 are studied and seve-ral self-orthogonal codes' matrixes with dual distance 3,4,5 and 6 are constructed.The series of the self-dual and self-orthogonal chains are constructed by these matrixes and the subcodes' L-chains with maximal dual distance are deduced.In the end,some quantum error correct codes are constructed by using these quaternary subcodes' L-chains and the construction theory of additive quantum error correct code.Some quantum error correct codes with good parameters are better than previously known.
The equivalence of maximal self-orthogonal codes obtained from binary self-dual codes by truncating are discussed.Many non-equivalent maximal self-orthogonal codes of length 21≤n≤29 and dual distance five are constructed by truncating self-dual codes of dual distance six.Subcodes of these maximal self-orthogonal codes are studied by using of a random search algorithm,and the subcodes of dual distance 3 and 5 of these maximal self-orthogonal codes are constructed.The code chains of these self-orthogonal subcodes and the S-chains obtained from the dual of these self-orthogonal subcodes are also determined.According to Steane’s construction,some very good quantum codes of distance five are constructed by the obtained S-chains.