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Quasi-cyclic constructions of asymmetric quantum error-correcting codes

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Abstract

Asymmetric quantum error-correcting codes (AQECCs) are an extremely efficient coding scheme against the different occurring probabilities of qubit-flip and phase-shift errors in quantum asymmetric channel. It is generally known that good quantum codes play a decisive role in ensuring the reliability and the authenticity of quantum communication. In this paper, our main objective is to obtain good AQECCs from quasi-cyclic (QC) codes over small fields, which will effectively fill some gaps of the constructions of AQECCs. At first, a suitable family of r-generator QC codes is proposed and their parameters are determined. Moreover, we show that their dual codes are 1-generator QC codes. In particular, when r = 2 and 3, we embed these dual codes into 2-generator and 3-generator QC codes, and obtain two pairs of nested codes. Then two explicit constructions of AQECCs from QC codes are presented, respectively. As for computational results, many new binary and ternary AQECCs with good parameters are constructed, which all can not be deduced by the asymmetric quantum Gilbert-Varshamov (GV) bound in Matsumoto (Quantum Inf. Process. 16, 1–7, 2017).

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Acknowledgments

The authors would like to thank the anonymous referees for their invaluable suggestions and comments. Without these suggestions and comments, this paper would not have been in the current form. This work is supported by National Natural Science Foundation of China (Nos. 11801564, 11901579).

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  1. Department of Basic Sciences, Air Force Engineering University, Xi’an, 710051, People’s Republic of China

    Jingjie Lv, Ruihu Li & Yu Yao

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  1. Jingjie Lv
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  2. Ruihu Li
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Correspondence to Ruihu Li.

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Lv, J., Li, R. & Yao, Y. Quasi-cyclic constructions of asymmetric quantum error-correcting codes. Cryptogr. Commun. 13, 661–680 (2021). https://doi.org/10.1007/s12095-021-00489-9

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