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International Journal of Quantum Information

On non-binary quantum repeated-root cyclic codes

Liqi Wang, Shixin Zhu

Published Apr 1, 2014• 5 citations
Publisher

Abstract

In this paper, based on the Steane's enlargement construction, three classes of non-binary quantum codes are constructed from classical repeated-root cyclic codes of length 2ps over 𝔽q with odd characteristic p. The exact minimum distances of these quantum codes are determined. This construction yields a quantum MDS code with parameters [[2p, 2p - 4, 3]]p and two good quantum codes with parameters [[2p, 2p - 7, 4]]p and [[2p, 2p - 10, 5]]p.

DOI
10.1142/s0219749914500105

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Many good quantum error-correcting codes were constructed from cyclic codes. However, it is a difficult problem to determine the true minimum distance of quantum cyclic codes for large length n. In this work, we construct nonbinary quantum cyclic codes and asymmetric quantum cyclic codes that are derived from repeated-root cyclic codes for an arbitrary length ps, and determine the true minimum distance of all those codes. Some proposed quantum cyclic codes are optimal. Additionally, some proposed asymmetric quantum cyclic codes have better parameters than the ones available in the literature.
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In this paper, we use q -cyclotomic cosets modulo n = q − 1 and generator polynomial to describe repeated-root cyclic codes of length N = p α n . We determine the condition under which a repeated-root cyclic code contains its dual code. Then, for q = 2 a , we give a class of dual-containing repeated-root cyclic codes. Applying CSS construction and Steane construction to these underlying classical codes, many quantum codes with good parameters shall be constructed, which are new in the sense that they are not available in the literature.