Reversible DNA codes from skew cyclic codes over a ring of order 256

Abstract

We introduce skew cyclic codes over the finite ring $\R$, where $u^{2}=0,v^{2}=v,w^{2}=w,uv=vu,uw=wu,vw=wv$ and use them to construct reversible DNA codes. The 4-mers are matched with the elements of this ring. The reversibility problem for DNA 4-bases is solved and some examples are provided.

Keywords

• DNA codes
• Skew codes
• Reversibility

References

1. [1] T. Abualrub, A. Ghrayeb, X. N. Zeng, Construction of cyclic codes over GF(4) for DNA computing, J. Frankl. Inst. 343(4-5) (2006) 448–457.
2. [2] L. Adleman, Molecular computation of the solutions to combinatorial problems, Science 266 (1994) 1021–1024.
3. [3] L. Adleman, P. W. K. Rothemund, S. Roweis, E. Winfree, On applying molecular computation to the data encryption standard, J. Comp. Biology 6(1) (1999) 53–63.
4. [4] N. Bennenni, K. Guenda, S. Mesnager, DNA cyclic codes over rings, Advances in Mathematics of Communications 11(1) (2017) 83–98.
5. [5] D. Boneh, C. Dunworth, R. Lipton, Breaking DES using molecular computer, Princeton CS Tech- Report, Number CS-TR-489-95 (1995).
6. [6] Y. Cengellenmis, A. Dertli, On the cyclic DNA codes over the finite ring, Acta Universitatis Apulensis 58 (2019) 1–11.
7. [7] A. Dertli, Y. Cengellenmis, On cyclic DNA codes over the rings Z4+wZ4 and Z4+wZ4+vZ4+wvZ4, Biomath 6(2) (2017) 1712167.
8. [8] P. Gaborit, H. King, Linear constructions for DNA codes, Theor. Comput. Sci. 334(1âAS3) (2005) 99–113.

9. [9] K. Guenda, T. A. Gulliver, Construction of cyclic codes over F2 +uF2 for DNA computing, AAECC 24 (2013) 445–459.
10. [10] F. Gursoy, E. S. Oztas, I. Siap, Reversible DNA codes over F16 + uF16 + vF16 + uvF16, 11(2) 2017 307–312.
11. [11] F. Gursoy, E. S. Oztas, B. Yildiz, Reversible DNA codes over a family of non-chain ring, arXiv:1711.02385.
12. [12] F. Gursoy, E. S. Oztas, I. Siap, Reversible DNA codes using skew polynomial rings, Applicable Algebra in Engineering, Communication and Computing 28 (2017) 311–320.
13. [13] J. Liang, L. Wang, On cyclic DNA codes over F2 + uF2, J.Appl Math Comput. 52 (2016) 81–91.
14. [14] D. Limbachiya, B. Rao, G. K. Manish, The Art of DNA Strings: Sixteen Years of DNA Coding Theory, arXiv:1607.00266.
15. [15] Magma computer algebra system, online, http://magma.maths.usyd.edu.au/
16. [16] M. Mansuripur, P. K. Khulbe, S. M. Kuebler, J. W. Perry, M. S. Giridhar, N. Peyghambarian, Information storage and retrieval using macromolecules as storage media, in Optical Data Storage, OSA Technical Digest Series (Optical Society of America), paper TuC2 (2003).
17. [17] O. Milenkovic, N. Kashyap, On the design of codes for DNA computing, Lecture Notes in Computer Science 3969, Springer (2006) 100–119.
18. [18] E. S. Oztas, B. Yildiz and I. Siap, A novel approach for constructing reversible codes and applications to DNA codes over the ring F2[u]=(u2k 􀀀1), Finite Fields and Their Applications 46 (2017) 217–234.
19. [19] S. Pattanayak, A. K. Singh, Construction of cyclic DNA codes over the Ring Z4[u]= < u2 􀀀 1 > based on the deletion distance, arXiv:1603.04055.
20. [20] A. Sharma, B. Maheshanand, A class of skew-constacyclic codes over Z4+uZ4, International Journal of Information and Coding Theory 4(4) (2017) 289–303.
21. [21] I. Siap, T. Abualrub, A. Ghrayeb, Cyclic DNA codes over the ring F2[u]=(u2 􀀀 1) based on the deletion distance, J. Frankl. Inst. 346 (2009) 731–740.
22. [22] B. Yildiz, I. Siap, Cyclic codes over F2[u]=(u4 􀀀 1) and applications to DNA codes, Comput. Math. Appl. 63 (2012) 1169–1176.
23. [23] S. Zhu, X. Chen, Cyclic DNA codes over F2+uF2+vF2+uvF2 and their applications, J. Appl.Math Comput. 55 (2017) 479–493.