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Cyclic DNA codes over \(\mathbb {F}_2+u\mathbb {F}_2+v\mathbb {F}_2+uv\mathbb {F}_2\) and their applications

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Abstract

In this paper, we study the structure of cyclic DNA codes of arbitrary length over the ring \(R=\mathbb {F}_2+u\mathbb {F}_2+v\mathbb {F}_2+uv\mathbb {F}_2\), \(u^{2}=0, v^{2}=v, uv=vu\). By defining a Gray map, we establish a relation between R and \(R^{2}_{1}\), where \(R_{1}=\mathbb {F}_2+u\mathbb {F}_2\) is a ring with four elements. Cyclic codes of arbitrary length over R satisfying the reverse constraint and the reverse-complement constraint are studied in this paper. Furthermore, we introduce reversible codes which provide a rich source for DNA codes. The GC content constraint is also considered. We give some examples to support our study in the last.

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Acknowledgments

The authors would like to thank the referees for their helpful comments and a very meticulous reading of this manuscript.

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Authors and Affiliations

  1. Department of Mathematics, Hefei University of Technology, Hefei, 230009, Anhui, People’s Republic of China

    Shixin Zhu & Xiaojing Chen

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  1. Shixin Zhu
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  2. Xiaojing Chen
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Correspondence to Xiaojing Chen.

Additional information

This research is supported by the National Natural Science Foundation of China (No. 61370089), the Anhui Provincial Natural Science Foundation under Grant (No.1508085SQA198, 1508085MA13).

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Zhu, S., Chen, X. Cyclic DNA codes over \(\mathbb {F}_2+u\mathbb {F}_2+v\mathbb {F}_2+uv\mathbb {F}_2\) and their applications. J. Appl. Math. Comput. 55, 479–493 (2017). https://doi.org/10.1007/s12190-016-1046-3

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  • DOI: https://doi.org/10.1007/s12190-016-1046-3

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