Constructions of Asymmetric Quantum Alternant Codes
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Asymmetric quantum error-correcting codes (AQCs) have been proposed to deal with the significant asymmetry in many quantum channels, which may have more flexbility than general quantum error-correcting codes (QECs). In this paper, we construct AQCs based on Alternant codes. Firstly, we propose a new subclass of Alternant codes and combine them with BCH codes to construct AQCs. Then we construct AQCs based on series of nested pairs of subclasses of Alternant codes such as nested Goppa codes. As an illustrative example, we get three [[55, 6, 19/4]], [[55, 10, 19/3]], [[55, 15, 19/2]] AQCs from the well known [55, 16, 19] binary Goppa code. At last, we get asymptotically good binary expansions of quantum GRS codes, which are quantum generalizations of Retter’s classical results.
KeywordsAlternant codes Asymmetric quantum error-correcting codes BCH codes Generalized Reed-Solomon codes Gilbert-Varshamov bound Goppa codes Quantum error-correcting codes
This work was supported by the National Natural Science Foundation of China under Grant No. 61170321, the Specialized Research Fund for the Doctoral Program of Higher Education under Grant No. 20110092110024 and the Graduate Research Innovation Plans of Colleges and Universities in Jiangsu Province 2013 under Grant No. CXZZ13_0105.
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