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Cyclic DNA codes over the ring \(\mathbb {F}_2+u\mathbb {F}_2+v\mathbb {F}_2+uv\mathbb {F}_2+v^2\mathbb {F}_2+uv^2\mathbb {F}_2\)

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Abstract

We study the structure of cyclic DNA codes of odd length over the finite commutative ring \(R=\mathbb {F}_2+u\mathbb {F}_2+v\mathbb {F}_2+uv\mathbb {F}_2 + v^2\mathbb {F}_2+uv^2\mathbb {F}_2,~u^2=0, v^3=v\), which plays an important role in genetics, bioengineering and DNA computing. A direct link between the elements of the ring R and 64 codons used in the amino acids of living organisms is established by introducing a Gray map from R to \(R_1=\mathbb {F}_2+u\mathbb {F}_2 ~(u^2=0)\). The reversible and the reversible-complement codes over R are investigated. We also discuss the binary image of the cyclic DNA codes over R. Among others, some examples of DNA codes obtained via Gray map are provided.

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Acknowledgements

The authors would like to sincerely thank the referees for a very meticulous reading of this manuscript, and for valuable suggestions which help to create an improved final version.

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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 Mathematical Sciences, Kent State University, 4314 Mahoning Avenue, Warren, OH, 44483, USA

    Hai Q. Dinh

  4. Department of Applied Mathematics, Indian Institute of Technology, Dhanbad, India

    Abhay Kumar Singh & Sukhamoy Pattanayak

  5. Faculty of Economics, Chiang Mai University, Chiang Mai, 52000, Thailand

    Songsak Sriboonchitta

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  1. Hai Q. Dinh
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  2. Abhay Kumar Singh
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Correspondence to Hai Q. Dinh.

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Communicated by J.-L. Kim.

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Dinh, H.Q., Singh, A.K., Pattanayak, S. et al. Cyclic DNA codes over the ring \(\mathbb {F}_2+u\mathbb {F}_2+v\mathbb {F}_2+uv\mathbb {F}_2+v^2\mathbb {F}_2+uv^2\mathbb {F}_2\) . Des. Codes Cryptogr. 86, 1451–1467 (2018). https://doi.org/10.1007/s10623-017-0405-x

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  • DOI: https://doi.org/10.1007/s10623-017-0405-x

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