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International Journal of Quantum Information

TERNARY QUANTUM CODES OF MINIMUM DISTANCE THREE

Jian Liu

Published Oct 1, 2010• 6 citations
Publisher

Abstract

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.

DOI
10.1142/s0219749910006137

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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.
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.
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.
Self-orthogonal codes have nice applications in many areas including quantum codes, lattices and LCD codes. For a prime powerq, it is in general difficult to constructq-ary self-orthogonal codes. In the literature, there exists no simple method to judge whether a generalq-ary linear code is self-orthogonal or not. In this paper, we mainly present several families ofq-ary self-orthogonal codes and study their applications in quantum codes and LCD codes. Firstly, several families ofq-ary linear codes are constructed by some special defining sets. These codes are proved to be self-orthogonal. To this end, we determine the numbers of solutions of some systems of equations over finite fields. Secondly, three families ofq-ary quantum codes with unbounded length and minimum distance three are constructed from the self-orthogonal codes. These quantum codes are optimal according to the quantum Hamming bound. In particular, some of them have better parameters than known ones. Thirdly, several families ofq-ary LCD codes are constructed from the self-orthogonal codes. Many optimal or almost optimal binary and ternary LCD codes are produced by our constructions. Some binary and ternary LCD codes have better parameters than known ones.