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.
References
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.
L. Adleman, Molecular computation of the solutions to combinatorial problems, Science 266 (1994) 1021–1024.
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.
N. Bennenni, K. Guenda, S. Mesnager, DNA cyclic codes over rings, Advances in Mathematics of Communications 11(1) (2017) 83–98.
D. Boneh, C. Dunworth, R. Lipton, Breaking DES using molecular computer, Princeton CS Tech- Report, Number CS-TR-489-95 (1995).
Y. Cengellenmis, A. Dertli, On the cyclic DNA codes over the finite ring, Acta Universitatis Apulensis 58 (2019) 1–11.
A. Dertli, Y. Cengellenmis, On cyclic DNA codes over the rings Z4+wZ4 and Z4+wZ4+vZ4+wvZ4, Biomath 6(2) (2017) 1712167.
P. Gaborit, H. King, Linear constructions for DNA codes, Theor. Comput. Sci. 334(1âAS3) (2005) 99–113.
K. Guenda, T. A. Gulliver, Construction of cyclic codes over F2 +uF2 for DNA computing, AAECC 24 (2013) 445–459.
F. Gursoy, E. S. Oztas, I. Siap, Reversible DNA codes over F16 + uF16 + vF16 + uvF16, 11(2) 2017 307–312.
F. Gursoy, E. S. Oztas, B. Yildiz, Reversible DNA codes over a family of non-chain ring, arXiv:1711.02385.
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.
J. Liang, L. Wang, On cyclic DNA codes over F2 + uF2, J.Appl Math Comput. 52 (2016) 81–91.
D. Limbachiya, B. Rao, G. K. Manish, The Art of DNA Strings: Sixteen Years of DNA Coding Theory, arXiv:1607.00266.
Magma computer algebra system, online, http://magma.maths.usyd.edu.au/
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).
O. Milenkovic, N. Kashyap, On the design of codes for DNA computing, Lecture Notes in Computer Science 3969, Springer (2006) 100–119.
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
S. Pattanayak, A. K. Singh, Construction of cyclic DNA codes over the Ring Z4[u]= < u2
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.
I. Siap, T. Abualrub, A. Ghrayeb, Cyclic DNA codes over the ring F2[u]=(u2
B. Yildiz, I. Siap, Cyclic codes over F2[u]=(u4
S. Zhu, X. Chen, Cyclic DNA codes over F2+uF2+vF2+uvF2 and their applications, J. Appl.Math Comput. 55 (2017) 479–493.