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Optimal constructions of quantum and synchronizable codes from repeated-root cyclic codes of length \(3p^s\)

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Abstract

In this paper, we use the CSS and Steane’s constructions to establish quantum error-correcting codes (briefly, QEC codes) from cyclic codes of length \(3p^s\) over \(\mathbb F_{p^m}\). We obtain several new classes of QEC codes in the sense that their parameters are different from all the previous constructions. Among them, we identify all quantum MDS (briefly, qMDS) codes, i.e., optimal quantum codes with respect to the quantum Singleton bound. In addition, we construct quantum synchronizable codes (briefly, QSCs) from cyclic codes of length \(3p^s\) over \(\mathbb F_{p^m}\). Furthermore, we give many new QSCs to enrich the variety of available QSCs. A lot of them are QSCs codes with shorter lengths and much larger minimum distances than known non-primitive narrow-sense BCH codes.

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Acknowledgements

H.Q. Dinh and W. Yamaka are grateful to the Centre of Excellence in Econometrics, Faculty of Economics, Chiang Mai University, for partial financial support.

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Authors and Affiliations

  1. Department of Mathematical Sciences, Kent State University, Kent, USA

    Hai Q. Dinh

  2. Institute of Fundamental and Applied Sciences, Duy Tan University, Ho Chi Minh City, 700000, Vietnam

    Bac T. Nguyen

  3. Faculty of Natural Sciences, Duy Tan University, Da Nang, 550000, Vietnam

    Bac T. Nguyen

  4. Centre of Excellence in Econometrics, Faculty of Economics, Chiang Mai University, Chiang Mai, 52000, Thailand

    M. Paravee

  5. Faculty of Education, Thu Dau Mot University, Binh Duong, Vietnam

    Hiep L. Thi

  6. Department of General Education, Industrial University of Vinh, Vinh City, Nghe An, Vietnam

    Thang M. Vo

  7. Mathematics Department, Ohio University, Athens, OH, 45701, Greece

    Thang M. Vo

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  1. Hai Q. Dinh
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Dinh, H.Q., Nguyen, B.T., Paravee, M. et al. Optimal constructions of quantum and synchronizable codes from repeated-root cyclic codes of length \(3p^s\). Quantum Inf Process 22, 257 (2023). https://doi.org/10.1007/s11128-023-03958-7

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