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Applied Mechanics and Materials

Construction of Asymmetric Quantum Product Codes Based Repeated-Root Cyclic Codes

Pan Yue Fu, Wei Liu

Published Sep 1, 2014• 0 citations
Publisher

Abstract

A kind of classical product codes based on repeated-root cyclic codes are considered in this paper, and the classical product codes are applied to construct the asymmetric quantum product codes. The novel asymmetric quantum product codes have great asymmetry for correcting Z errors and X errors, and the parameters of asymmetric quantum product codes can be precisely determined by the properties of the repeated-root cyclic codes and their dual codes. The examples show that these asymmetric quantum product codes based repeated-root cyclic codes are more efficient than some existent asymmetric quantum product codes.

DOI
10.4028/www.scientific.net/amm.644-650.1668

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A kind of classical product codes based on repeated-root cyclic codes are considered in this paper, and the classical product codes are applied to construct the asymmetric quantum product codes. The novel asymmetric quantum product codes have great asymmetry for correcting Z errors and X errors, and the parameters of asymmetric quantum product codes can be precisely determined by the properties of the repeated-root cyclic codes and their dual codes. The examples show that these asymmetric quantum product codes based repeated-root cyclic codes are more efficient than some existent asymmetric quantum product codes.
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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.
A method of describing quaternary cyclic codes with 4-cyclotomic cosets of modulo odd n and generator polynomials is presented.A necessary and sufficient condition under which a quaternary cyclic code containing its dual code is given,thus generalizing the related known results on self-orthogonal simple root cyclic codes and on BCH codes to al quaternary cyclic codes.Using the relation of simple root cyclic codes and repeat root cyclic codes,the linear quantum codes that can be constructed from quaternary cyclic codes of short lengths are determined.
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Abstract In this paper, we systematically investigate the construction of two important classes of quantum error-correcting codes, namely entanglement-assisted quantum error-correcting codes and asymmetric quantum error-correcting codes, from a generalized class of index-2 quasi-cyclic codes. For entanglement-assisted quantum error-correcting codes, we establish a systematic construction framework and derive sufficient conditions for obtaining maximal-entanglement entanglement-assisted quantum error-correcting codes, from which two new classes of such codes are obtained. For asymmetric quantum error-correcting codes, we derive necessary and sufficient conditions for the inclusion relation between any two index-2 QC codes and use them to establish a systematic construction framework. Several new entanglement-assisted quantum error-correcting codes and asymmetric quantum error-correcting codes with parameters improving upon previously reported results are obtained as applications of the proposed constructions.