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Computer Engineering and Applications Journal

Subcode chains of self-orthogonal codes with even length over GF(4)

LI Rui-hu

Published Jan 1, 2010• 0 citations
OpenAlex

Abstract

In this paper,self-orthogonal codes tree of given self-orthogonal codes over GF(4) is firstly constructed in order to study decreasing dimension method of generator matrix.Then subcode chains and reduced L-chains of self-dual codes whose code length n satisfies 20≤n≤30 are built up by using greedy strategy and breadth-first-search algorithm.There are 45 good quantum-error-correcting codes constructed from the obtained L-chains.

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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.

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In this paper,self-orthogonal codes tree of given self-orthogonal codes over GF(4) is firstly constructed in order to study decreasing dimension method of generator matrix.Then subcode chains and reduced L-chains of self-dual codes whose code length n satisfies 20≤n≤30 are built up by using greedy strategy and breadth-first-search algorithm.There are 45 good quantum-error-correcting codes constructed from the obtained L-chains.
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.
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.
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.
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.
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.