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On DNA codes from a family of chain rings

Year 2017, Volume: 4 Issue: 1, 93 - 102, 11.01.2017
https://doi.org/10.13069/jacodesmath.96056

Abstract

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.

References

  • [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.
Year 2017, Volume: 4 Issue: 1, 93 - 102, 11.01.2017
https://doi.org/10.13069/jacodesmath.96056

Abstract

References

  • [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.
There are 20 citations in total.

Details

Subjects Engineering
Journal Section Articles
Authors

Elif Segah Oztas

Bahattin Yildiz This is me

Irfan Siap

Publication Date January 11, 2017
Published in Issue Year 2017 Volume: 4 Issue: 1

Cite

APA Oztas, E. S., Yildiz, B., & Siap, I. (2017). On DNA codes from a family of chain rings. Journal of Algebra Combinatorics Discrete Structures and Applications, 4(1), 93-102. https://doi.org/10.13069/jacodesmath.96056
AMA Oztas ES, Yildiz B, Siap I. On DNA codes from a family of chain rings. Journal of Algebra Combinatorics Discrete Structures and Applications. January 2017;4(1):93-102. doi:10.13069/jacodesmath.96056
Chicago Oztas, Elif Segah, Bahattin Yildiz, and Irfan Siap. “On DNA Codes from a Family of Chain Rings”. Journal of Algebra Combinatorics Discrete Structures and Applications 4, no. 1 (January 2017): 93-102. https://doi.org/10.13069/jacodesmath.96056.
EndNote Oztas ES, Yildiz B, Siap I (January 1, 2017) On DNA codes from a family of chain rings. Journal of Algebra Combinatorics Discrete Structures and Applications 4 1 93–102.
IEEE E. S. Oztas, B. Yildiz, and I. Siap, “On DNA codes from a family of chain rings”, Journal of Algebra Combinatorics Discrete Structures and Applications, vol. 4, no. 1, pp. 93–102, 2017, doi: 10.13069/jacodesmath.96056.
ISNAD Oztas, Elif Segah et al. “On DNA Codes from a Family of Chain Rings”. Journal of Algebra Combinatorics Discrete Structures and Applications 4/1 (January 2017), 93-102. https://doi.org/10.13069/jacodesmath.96056.
JAMA Oztas ES, Yildiz B, Siap I. On DNA codes from a family of chain rings. Journal of Algebra Combinatorics Discrete Structures and Applications. 2017;4:93–102.
MLA Oztas, Elif Segah et al. “On DNA Codes from a Family of Chain Rings”. Journal of Algebra Combinatorics Discrete Structures and Applications, vol. 4, no. 1, 2017, pp. 93-102, doi:10.13069/jacodesmath.96056.
Vancouver Oztas ES, Yildiz B, Siap I. On DNA codes from a family of chain rings. Journal of Algebra Combinatorics Discrete Structures and Applications. 2017;4(1):93-102.