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Journal of Air Force Engineering University

Quaternary Self-orthogonal Code Chains and Construction of Quantum Error-correcting Codes

Youqian Feng

Published Jan 1, 2008• 0 citations
OpenAlex

Abstract

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.

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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.
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.
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.
A heuristic algorithm is designed to searching for good higher dimensional self-orthogonal codes over GF(A) from low dimensional self-orthogonal codes. Many self-orthogonal codes of length 20 les n les 36 and dual distance 5 or 6 are obtained, and several have improved dual distance. Consequently, using these self-orthogonal codes and their dual codes, some linear quantum codes of minimum distance five or six for such length n are obtained, and three of these codes [[24, 8, 5]], [[25, 9, 5]], [[27, 7, 6]] are record-breaking. Quaternary code, self-orthogonal code, linear quantum error-correcting code.
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.
The optimal quaternary self-orthogonal code reaching the bound of Griesmer is studied.By the methods of combination and the random search algorithm,the self-orthogonal code chains of short code length n(10≤n≤19) are constructed in the filed of GF(4).One result of the optimal self-orthogonal code chains with code length n(10≤n≤19) is given in this paper,and the parameters of codes in which meet the Griesmer bound.The results are important references to farther researching the construction of the self-orthogonal code chain and quantum code.