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MDS Constacyclic Codes of Prime Power Lengths Over Finite Fields and Construction of Quantum MDS Codes

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Abstract

If we fix the code length n and dimension k, maximum distance separable (briefly, MDS) codes form an important class of codes because the class of MDS codes has the greatest error-correcting and detecting capabilities. In this paper, we establish all MDS constacyclic codes of length ps over \(\mathbb {F}_{p^{m}}\). We also give some examples of MDS constacyclic codes over finite fields. As an application, we construct all quantum MDS codes from repeated-root codes of prime power lengths over finite fields using the CSS and Hermitian constructions. We provide all quantum MDS codes constructed from dual codes of repeated-root codes of prime power lengths over finite fields using the Hermitian construction. They are new in the sense that their parameters are different from all the previous constructions. Moreover, some of them have larger Hamming distances than the well known quantum error-correcting codes in the literature.

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Acknowledgement

This work was supported by the Ministry of Education and Training of Vietnam under Grant No. B2019-TNA-02.T. H.Q. Dinh and R. Tansuchat 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. 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. Faculty of Engineering, Ain Shams University, Cairo, Egypt

    Ramy Taki ElDin

  4. Department of Basic Sciences, Thai Nguyen University of Economics and Business Administration, Thai Nguyen, Thai Nguyen Province, Vietnam

    Bac T. Nguyen

  5. 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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  4. Roengchai Tansuchat
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Correspondence to Bac T. Nguyen.

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Dinh, H.Q., ElDin, R.T., Nguyen, B.T. et al. MDS Constacyclic Codes of Prime Power Lengths Over Finite Fields and Construction of Quantum MDS Codes. Int J Theor Phys 59, 3043–3078 (2020). https://doi.org/10.1007/s10773-020-04524-y

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