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Quantum MDS and synchronizable codes from cyclic and negacyclic codes of length \(4p^s\) over \({\mathbb {F}}_{p^m}\)

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Abstract

In this paper, we establish all quantum error-correcting codes (briefly, QEC codes) from cyclic and negacyclic codes of length \(4p^s\) over \({\mathbb {F}}_{p^m}\) using CSS and Steane’s constructions. Some new QEC codes are given in the sense that their parameters are different from all the previous constructions. In addition, if we fix the length and the Hamming distance of a QEC code, some QEC codes are better than all QEC codes in the sense that their dimensions are larger than the dimensions of all the previous QEC codes. We also investigate quantum maximum distance separable (briefly, qMDS) codes from cyclic and negacyclic codes of length \(4p^s\) over finite fields using the CSS and Hermitian constructions. We provide all qMDS codes constructed from dual codes of cyclic and negacyclic codes of length \(4p^s\) over finite fields using the Hermitian construction. We also construct quantum synchronizable codes (briefly, QSCs) from cyclic codes of length \(4p^s\) over \({\mathbb {F}}_{p^m}\). To enrich the variety of available QSCs, many new QSCs are constructed to illustrate our results. Among them, there 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 S. Sriboonchitta are grateful to the Centre of Excellence in Econometrics, Chiang Mai University, for partial financial support. This research is partially supported by the Research Administration Centre, Chaing Mai University.

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Authors and Affiliations

  1. Division of Computational Mathematics and Engineering, Institute for Computational Science, Ton Duc Thang University, Ho Chi Minh City, Vietnam

    Hai Q. Dinh

  2. Faculty of Mathematics and Statistics, Ton Duc Thang University, Ho Chi Minh City, Vietnam

    Hai Q. Dinh

  3. Department of Basic Sciences, Thai Nguyen University of Economics and Business Administration, Thai Nguyen City, Thai Nguyen Province, 250000, Vietnam

    Ha T Le

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

    Bac T. Nguyen

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

    Bac T. Nguyen

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

    Roengchai Tansuchat

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  1. Hai Q. Dinh
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Dinh, H.Q., Le, H.T., Nguyen, B.T. et al. Quantum MDS and synchronizable codes from cyclic and negacyclic codes of length \(4p^s\) over \({\mathbb {F}}_{p^m}\). Quantum Inf Process 20, 373 (2021). https://doi.org/10.1007/s11128-021-03306-7

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