On DNA codes from a family of chain rings

In this work, we focus on reversible cyclic codes which correspond to reversible DNA codes or reversible-complement DNA codes over a family of finite chain rings, in an effort to extend what was done by Yildiz and Siap in \cite{YildizSiap}. The ring family that we have considered are of size $2^{2^k}$, $k=1,2, \cdots$ and we match each ring element with a DNA $2^{k-1}$-mer. We use the so-called $u^2$-adic digit system to solve the reversibility problem and we characterize cyclic codes that correspond to reversible-complement DNA-codes. We then conclude our study with some examples.

Keywords:

Cyclic codes Chain rings, Reversible codes, DNA codes,

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[1] N. Aboluion, D. H. Smith, S. Perkins, Linear and nonlinear constructions of DNA codes with Hamming distance d, constant GC–content and a reverse–complement constraint, Discrete Math. 312(5) (2012) 1062–1075.
[2] T. Abulraub, A. Ghrayeb, X. N. Zeng, Construction of cyclic codes over GF(4) for DNA computing, J. Frankl. Inst. 343(4–5) (2006) 448–457.
[3] L. Adleman, Molecular computation of solutions to combinatorial problems, Science 266(5187) (1994) 1021–1024.
[4] L. Adleman, P. W. K. Rothemund, S. Roweis, E. Winfree, On applying molecular computation to the Data Encryption Standard, J. Comput. Biol. 6(1) (1999) 53–63.
[5] R. Alfaro, S. Bennett, J. Harvey, C. Thornburg, On distances and self–dual codes over Fq[u]=(ut), Involv. J. Math. 2(2) (2009) 177–194.
[6] D. Boneh, C. Dunworth, R. Lipton, Breaking DES using molecular computer, Princeton CS Tech–Report, Number CS–TR-489–95, 1995.
[7] A. G. Frutos, Q. Liu, A. J. Thiel, A. M. W. Sanner, A. E. Condon, L. M. Smith, R. M. Corn, Demonstration of a word design strategy for DNA computing on surfaces, Nucleic Acids Res. 25(23) (1997) 4748–4757.
[8] P. Gaborit, O. D. King, Linear construction for DNA codes, Theoret. Comput. Sci. 334(1–3) (2005) 99–113.
[9] O. D. King, Bounds for DNA codes with constant GC–content, Electron. J. Comb. 10 (2003) 1–13.
[10] M. Li, H. J. Lee, A. E. Condon, R. M. Corn, DNA word design strategy for creating sets of non–interacting oligonucleotides for DNA microarrays, Langmuir 18(3) (2002) 805–812.
[11] M. Mansuripur, P. K. Khulbe, S. M. Kuebler, J. W. Perry, M. S. Giridhar, N. Peyghambarian, Information storage and retrieval using acromolecules as storage media, in Optical Data Storage, OSA Technical Digest Series (Optical Society of America) paper TuC2, 2003.
[12] A. Marathe, A. E. Condon, R. M. Corn, On combinatorial DNA word design, J. Comput. Biol. 8(3) (2001) 201–220.
[13] J. L. Massey, Reversible codes, Inform. and Control 7(3) (1964) 369–380.
[14] G. H. Norton, A. Salagean, On the structure of linear and cyclic codes over a finite chain ring, Appl. Algebr. Eng. Comm. 10 (2000) 489–506.
[15] E. S. Oztas, I. Siap, Lifted polynomials Over F16 and their applications to DNA codes, Filomat 27(3) (2013) 459–466.
[16] E. S. Oztas, I. Siap, B. Yildiz, Reversible codes and applications to DNA, Lect. Notes Comput. Sc. 8592 (2014) 124–128.
[17] E. S. Oztas, I. Siap, On a generalization of lifted polynomials over finite fields and their applications to DNA codes, Int. J. Comput. Math. 92(9) (2015) 1976–1988.
[18] I. Siap, T. Abulraub, A. Ghayreb, Similarity cyclic DNA codes over rings, International Conference on Bioinformatics and Biomedical Engineering, in Shanghai, iCBBE 2008, PRC, May 16–18th 2008.
[19] I. Siap, T. Abulraub, A. Ghrayeb, Cyclic DNA codes over the ring F2[u]=(u^2-1) based on the deletion distance, J. Frankl. Inst. 346(8) (2009) 731–740.
[20] B. Yildiz, I. Siap, Cyclic codes over F2[u]=(u^4-1) and applications to DNA codes, Comput. Math. Appl. 63(7) (2012) 1169–1176.