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IEEE Transactions on Information Theory

Generalization of Steane's Enlargement Construction of Quantum Codes and Applications

San Ling, Jinquan Luo, Chaoping Xing

Published Jul 21, 2010• 63 citations
Publisher

Abstract

We generalize Steane’s enlargement construction of binary quantum codes to q-ary quantum codes. We then apply this result to BCH codes and the study of asymptotic bounds, and obtain improvements to the quantum BCH codes constructed by Aly and Klappenecker and the quantum asymptotic bounds from algebraic geometry codes obtained by Feng, Ling and Xing.

DOI
10.1109/tit.2010.2050828

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We generalize Steane’s enlargement construction of binary quantum codes to q-ary quantum codes. We then apply this result to BCH codes and the study of asymptotic bounds, and obtain improvements to the quantum BCH codes constructed by Aly and Klappenecker and the quantum asymptotic bounds from algebraic geometry codes obtained by Feng, Ling and Xing.
Three code constructions generating new families of good nonbinary quantum codes are presented in this paper. The first two ones are derived from Hermitian self-orthogonal non-narrow-sense Bose-Chaudhuri-Hocquenghem (BCH) codes. The third one is derived from qqq-ary ($q\ensuremath{\ne}2$ is a prime power) Steane's enlargement of Calderbank-Shor-Steane codes applied to Euclidean self-orthogonal non-narrow-sense BCH codes. The quantum nonbinary BCH codes presented here have parameters better than the ones available in the literature.
The stabilizer method for constructing a class of asymmetric quantum codes (AQC), called additive AQC, has been established by Aly et.al. In this paper, we present a new characterization of AQC, which generalizes a result of the symmetric case known previously. As an application of the characterization, we establish a relationship of AQC with classical error-correcting codes and show a few examples of good AQC with specific parameters. By using this relationship, we obtain an asymptotic bound on AQCs from algebraic geometry codes.
By a careful analysis on cyclotomic cosets, the maximal designed distance δnew of narrow-sense imprimitive Euclidean dual containing q-ary BCH code of length [Formula: see text] is determined, where q is a prime power and l is odd. Our maximal designed distance δnew of dual containing narrow-sense BCH codes of length n improves upon the lower bound δmax for maximal designed distances of dual containing narrow-sense BCH codes given by Aly et al. [IEEE Trans. Inf. Theory53 (2007) 1183]. A series of non-narrow-sense dual containing BCH codes of length n, including the ones whose designed distances can achieve or exceed δnew, are given, and their dimensions are computed. Then new quantum BCH codes are constructed from these non-narrow-sense imprimitive BCH codes via Steane construction, and these new quantum codes are better than previous results in the literature.