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A family of Hermitian dual-containing constacyclic codes and related quantum codes

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Abstract

In this paper, we study a family of constacyclic BCH codes over \({\mathbb {F}}_{q^2}\) of length \(n=\frac{q^{2m}-1}{q+1}\), where q is a prime power, and \(m\ge 2\) an even integer. The maximum designed distance of narrow-sense Hermitian dual-containing constacyclic BCH codes over \({\mathbb {F}}_{q^2}\) of length n is determined. Furthermore, the exact dimensions of these constacyclic BCH codes with given designed distance are obtained. As a consequence, we are able to derive the parameters of quantum codes as a function of their designed parameters of the associated constacyclic BCH codes. This improves a recent result by Yuan et al. (Des Codes Cryptogr 85(1): 179–190, 2017) for codes with the same lengths except three trivial cases (\(q=2, 3, 4\)). Moreover, some of our newly constructed quantum codes have better parameters compared with those constructed recently (Song et al. Quantum Inf Process 17(10): 1–24, 2018, Aly et al. IEEE Trans Inf Theory 53(3): 1183–1188, 2007, Li et al. Quantum Inf Process 18(5): 127, 2019, Wang et al. Quantum Inf Process 18(10): 1–40, 2019).

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Acknowledgements

We thank Markus Grassl, Xiaoshan Kai, Minjia Shi for their helpful suggestions. We are also very grateful to the associate editor Faisal Shah Khan and the anonymous referees, for their constructive comments and suggestions that improved the presentation of this paper. This work is supported by the National Natural Science Foundation of China under Grant No.61902429, No.11775306, the Shandong Provincial Natural Science Foundation of China under Grants No.ZR2019MF070, the Key Laboratory of Applied Mathematics of Fujian Province University (Putian University) under Grants No.SX201806, the Open Research Fund from Shandong provincial Key Laboratory of Computer Network under Grant No.SDKLCN-2018-02, and Fundamental Research Funds for the Central Universities No.17CX02030A.

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Authors and Affiliations

  1. The College of Science, China University of Petroleum (East China), Qingdao, 266580, Shandong, China

    Xubo Zhao, Xiaoping Li & Tongjiang Yan

  2. Key Laboratory of Applied Mathematics, Putian University, Putian, 351100, Fujian, China

    Xubo Zhao, Xiaoping Li & Tongjiang Yan

  3. Shandong Provincial Key Laboratory of Computer Networks, Jinan, 250014, Shandong, China

    Xubo Zhao, Xiaoping Li & Tongjiang Yan

  4. School of Mathematics and Statistics, Carleton University, 1125 Colonel By Drive, Ottawa, ON, K1S 5B6, Canada

    Qiang Wang

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  1. Xubo Zhao
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Zhao, X., Li, X., Wang, Q. et al. A family of Hermitian dual-containing constacyclic codes and related quantum codes. Quantum Inf Process 20, 186 (2021). https://doi.org/10.1007/s11128-021-03102-3

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