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A study of quantum codes obtained from cyclic codes over a non-chain ring

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Abstract

In this paper, the structure of cyclic codes of odd length n over \(\mathfrak {R}=\mathbb {Z}_{4} + v \mathbb {Z}_{4}\), \(v^2 = v\), is conferred. We define a Gray map \(\xi\), which is distance preserving from \(\mathfrak {R}^n\) (Gray distance) to \(\mathbb {Z}^{4n}_2\) (Hamming distance), and show that it is \(\mathbb {Z}_2\)-linear. We construct quantum codes over \(\mathbb {Z}_{2}\) by utilizing the Gray images \(\xi (C)\) of cyclic codes C over \(\mathfrak {R}\). As an application, we provide many new quantum MDS codes and quantum codes with good parameters comparing to known quantum codes.

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Acknowledgements

The authors would like to express their sincere approciation to the anonymous referees for their valuable comments and suggestions which helped highly improved the quality of this manuscript.

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Authors and Affiliations

  1. Department of Mathematical Sciences, Kent State University, 4314 Mahoning Avenue, 44483, Warren, Ohio, USA

    Hai Q. Dinh

  2. Department of Electrical Communication Engineering, Indian Institute of Science Bangalore, 560012, Bangalore, India

    Narendra Kumar

  3. Department of Mathematics & Computing, Indian Institute of Technology (ISM), 826004, Dhanbad, India

    Abhay Kumar Singh

Authors
  1. Hai Q. Dinh
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  2. Narendra Kumar
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  3. Abhay Kumar Singh
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Corresponding author

Correspondence to Abhay Kumar Singh.

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Dinh, H.Q., Kumar, N. & Singh, A.K. A study of quantum codes obtained from cyclic codes over a non-chain ring. Cryptogr. Commun. 14, 909–923 (2022). https://doi.org/10.1007/s12095-022-00567-6

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  • DOI: https://doi.org/10.1007/s12095-022-00567-6

Keywords

Mathematics Subject Classification (2010)