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Binary construction of quantum codes

Ruihu Li, Xueliang Li

Published Jan 1, 2006• 4 citations
OpenAlex

Abstract

In this paper, we construct a large number of good quantum codes of minimum distances five and six by Steane’s Construction. Our methods involve the study of the check matrices of binary extended BCH-codes, together with puncturing and combining such matrices.

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not counting primitive codes). Quantum codes obtained from BCH codes with these values of are listed in Table II. Further good codes exist in the range for

V. Efficiency

We improve estimates on the parameters of quantum codes obtained by Steane's construction from binary codes. This yields several new families of quantum codes.

•Jan 1, 1999•0 citations
Quantum error-correcting codes protect quantum information against undesirable noise in quantum computation and quantum communication.It is one of important problems that constructing very good quantum codes.There are two methods that use binary codes to construct quantum codes,one is CSS construction,another is Steane construction.These two methods are all based on the construction of binary self-orthogonal codes.Combinational construction of binary self-orthogonal codes is investigated in this paper.From known codes chains of binary self-orthogonal codes and their dual codes chains with given dual distance,new binary self-orthogonal codes of dual distance three,four,five and six are constructed.Based on these results,for each satisfying,S-chains with parameters and are constructed.According to Steane's construction,many very good quantum codes of distance five and six are constructed by the obtained S-chains,some of these quantum codes are new and some of these quantum codes have improved parameters than previously known codes.
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.
We give elementary recursive constructions of binary self-orthogonal codes with dual distance four for all even lengths n/spl ges/12 and n=8. Consequently, good quantum codes of minimum distance three and four for such length n are obtained via Steane's construction and the CSS construction. Previously, such quantum codes were explicitly constructed only for a sparse set of lengths. Almost all of our quantum codes of minimum distance three are optimal or near optimal, and some of our minimum-distance four quantum codes are better than or comparable with those known before.
In this paper, a special subclass of binary generalized quasi-cyclic self-orthogonal codes and quantum codes constructed by Steane construction are discussed.Firstly, eight 16-dimensional even length generalized quasi-cyclic self-orthogonal codes with dual distance five are built based on circulant or partial circulant matrices.Secondly, pairs of nested self-orthogonal codes with dual distance five and three are designed by applying an algorithm for searching subcodes of a given code.Thirdly, revised pairs of codes with dual distance six and four are constructed by extending previous pairs of codes, and then eight quantum codes with distance six are obtained by Steane construction.These eight quantum codes are new binary construction by Steane construction and are best known ones.