IEEE Transactions on Information Theory

Binary Construction of Quantum Codes of Minimum Distance Three and Four

Ruihu Li, Xinran Li

Published Jun 1, 2004• 67 citations

Abstract

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.

DOI: 10.1109/tit.2004.828149

Related papers

Quantum Codes of Minimum Distance Three Constructed from Binary Codes of Odd Length

Ruihu Li, Gang Chen

We give elementary recursive constructions of a pair of nested binary self-orthogonal codes with dual distance two and three for all odd n ¿ 11. Consequently, good quantum codes of minimum distance three for such length n are obtained via Steane's construction. Such quantum codes were explicitly constructed before only for a sparse set of lengths.

International Conference on Future NetworksJan 1, 20101 citation

LINEAR QUANTUM CODES OF MINIMUM DISTANCE THREE

Ruihu Li, Xueliang Li, Zongben Xu

We give elementary recursive constructions of quaternary self-orthogonal codes with dual distance three for all n ≥ 5. Consequently, good linear quantum codes of minimum distance three for such length n are obtained. Almost all of these linear quantum codes are optimal or near optimal.

International Journal of Quantum InformationDec 1, 20065 citations

Combinational Construction of Self-orthogonal Codes and Its Application

Jian Liu

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.

Journal of Air Force Engineering UniversityJan 1, 20090 citations

Subcodes of binary self-dual codes of minimum distance six

Xuejun Zhao

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.

Computer Engineering and Applications JournalJan 1, 20080 citations

TERNARY QUANTUM CODES OF MINIMUM DISTANCE THREE

Jian Liu

Based on finite field and combinatorics theory, an elementary recursive construction is used to construct Hermitian self-orthogonal codes over 𝔽9, the Galois field with nine elements, with dual distance three for all n ≥ 4. Consequently, good ternary linear quantum codes of minimum distance three for such length n are obtained. These ternary linear quantum codes are optimal or near optimal according to the quantum Singleton bound and quantum Hamming bound.

International Journal of Quantum InformationOct 1, 20106 citations

Self-Orthogonal Codes Constructed from Posets and Their Applications in Quantum Communication

Yansheng Wu, Yoonjin Lee

It is an important issue to search for self-orthogonal codes for construction of quantum codes by CSS construction (Calderbank-Sho-Steane codes); in quantum error correction, CSS codes are a special type of stabilizer codes constructed from classical codes with some special properties, and the CSS construction of quantum codes is a well-known construction. First, we employ hierarchical posets with two levels for construction of binary linear codes. Second, we find some necessary and sufficient conditions for these linear codes constructed using posets to be self-orthogonal, and we use these self-orthogonal codes for obtaining binary quantum codes. Finally, we obtain four infinite families of binary quantum codes for which the minimum distances are three or four by CSS construction, which include binary quantum Hamming codes with length n≥7. We also find some (almost) “optimal” quantum codes according to the current database of Grassl. Furthermore, we explicitly determine the weight distributions of these linear codes constructed using posets, and we present two infinite families of some optimal binary linear codes with respect to the Griesmer bound and a class of binary Hamming codes.

MathematicsSep 3, 20208 citations