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Journal of Air Force Engineering University

Construction of S-chains From Self-dual Codes

Qiu Dang-qing

Published Jan 1, 2008• 0 citations

Abstract

It is a central problem in the research of coding to study the existence,structure and construction of the certain optimal property codes.In order to construct the quantum error-correcting codes,people have begun to study the self-orthogonal code that has the special self-dual distance.In this paper,two classes of sub-codes,which have optimal dual distance or intended optimal dual distance,of the binary non-decomposable self-dual codes of length n between 12 and 20,such as B12,D14,E16,F16,H18,I18,J20,K20,L20,M20 and S20,and the corresponding S-chain construction are studied.Based on the generator matrices of the self-dual codes,optimal sub-codes of dual distance 2,3 and 4 of these self-dual codes are constructed by using a combinational method.Then code chains of these optimal sub-codes and their dual codes are discussed.Consequently,S-chains are constructed from these sub-codes with optimal dual distance or intended optimal dual distance.Finally,some very good quantum error-correcting codes are constructed from the S-chains obtained.These quantum codes are the ones that the distance reaches the maximum when their n and k are given.

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It is a central problem in the research of coding to study the existence,structure and construction of the certain optimal property codes.In order to construct the quantum error-correcting codes,people have begun to study the self-orthogonal code that has the special self-dual distance.In this paper,two classes of sub-codes,which have optimal dual distance or intended optimal dual distance,of the binary non-decomposable self-dual codes of length n between 12 and 20,such as B12,D14,E16,F16,H18,I18,J20,K20,L20,M20 and S20,and the corresponding S-chain construction are studied.Based on the generator matrices of the self-dual codes,optimal sub-codes of dual distance 2,3 and 4 of these self-dual codes are constructed by using a combinational method.Then code chains of these optimal sub-codes and their dual codes are discussed.Consequently,S-chains are constructed from these sub-codes with optimal dual distance or intended optimal dual distance.Finally,some very good quantum error-correcting codes are constructed from the S-chains obtained.These quantum codes are the ones that the distance reaches the maximum when their n and k are given.

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 method of constructing quantum error-correcting code by the quaternary self-orthogonal sub-code chains and L-chains are investigated in this paper.Random searching method is used to find quaternary self-orthogonal codes and these self-orthogonal codes could form some self-orthogonal code chains.By using combinatorial method,quantum error-correcting codes are constructed in light of L-chains which are constructed by dual of these self-orthogonal codes.The code chains of these self-orthogonal sub-codes and the L-chains which are obtained from the dual of these self-orthogonal sub-codes of length n between 20 and 36 and n=40,45,50,55,60 and dual distance five and six are determined.Some quantum codes of distance five and six are constructed by the obtained L-chains,some quantum error-correcting codes are new.

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.

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.