Cyclic DNA codes over the ring $\mathbb{Z}_4+u\mathbb{Z}_4+u^2\mathbb{Z}_4$
DOI:
https://doi.org/10.13069/jacodesmath.1000959Keywords:
DNA codes, Skew codes, Reversible codesAbstract
In this work, we have investigated the one generator cyclic DNA codes with reverse and reverse complement constraints over the ring $R=\mathbb{Z}_4+u\mathbb{Z}_4+u^2\mathbb{Z}_4$ with $u^3=0$. Skew cyclic codes with reverse complement constraint are constructed over $R$. We have also determined a one-to-one correspondence between the elements of the ring $R$ and DNA codons satisfying the Watson-Crick complement. Finally, we have established some examples which satisfy the given constraints.
Received: 07 December 2020 | Accepted: 29 April 2021Downloads
References
T. Abualrub and R. Oehmke, On the generators of $\mathbb{Z}_4$ cyclic codes of length $2^e$, IEEE Trans. Inform. Theory 49(9) (2003) 2126–2133.
T. Abualrub and I. Siap, Cyclic codes over the rings $\mathbb{Z}_2 + u\mathbb{Z}_2$ and $\mathbb{Z}_2 + u\mathbb{Z}_2 + u^2\mathbb{Z}_2$, Des. Codes Cryptogr. 42(3) (2007) 273–287.
T. Abualrub and I. Siap, Reversible cyclic codes over $\mathbb{Z}_4$, Australas. J. Combin. 38 (2007) 195–205.
L. M. Adleman, Molecular computation of solutions to combinatorial problems, Science 266(5187) (1994) 1021–1024.
N. Bennenni, K. Guenda, and S. Mesnager, New DNA cyclic codes over rings, Adv. Math. Commun. 11(1) (2017) 83–98.
A. Bonnecaze and P. Udaya, Cyclic codes and self-dual codes over $\mathbb{F}_2 + u\mathbb{F}_2$, IEEE Trans. Inform. Theory 45(4) (1999) 1250–1255.
D. Boucher, W. Geiselmann, and F. Ulmer, Skew cyclic codes, Appl. Algebra Engrg. Comm. Comput. 18(4) (2007) 379–389.
D. Boucher and F. Ulmer, Coding with skew polynomial rings, J. Symbolic Comput. 44(12) (2009) 1644–1656.
Y. Cengellenmis, N. Aydin, and A. Dertli, Reversible DNA codes from skew cyclic codes over a ring of order 256, J. Algebra Comb. Discrete Struct. Appl. 8(1) (2021) 1–8.
H. Q. Dinh, S. Pattanayak, A. K. Singh, and S. Sriboonchitta, Construction of cyclic DNA codes over the ring $\mathbb{Z}_4[u]/(u^2-1)$ based in deletion distance, Theoret. Comput. Sci. 773 (2019) 27–42.
B. Feng, S. S. Bai, B. Y. Chen, and X. N. Zhou, The constructions of DNA codes from linear self-dual codes over $\mathbb{Z}_4$, in: Proceedings of the 2015 International Conference on Computer Information Systems and Industrial Applications (CISIA 2015), Atlantis Press, 2015, pp. 496–498.
K. Guenda, T. A. Gulliver, and P. Solé, On cyclic DNA codes, in: 2013 IEEE International Symposium on Information Theory (ISIT), IEEE, 2013, pp. 121–125.
A. R. Hammons, Jr., P. V. Kumar, A. R. Calderbank, N. J. A. Sloane, and P. Solé, The $\mathbb{Z}_4$-linearity of Kerdock, Preparata, Goethals, and related codes, IEEE Trans. Inform. Theory 40(2) (1994) 301–319.
J. Liang and L. Wang, On cyclic DNA codes over $\mathbb{F}_2 + u\mathbb{F}_2$, J. Appl. Math. Comput. 51(1-2) (2016) 81–91.
M. Özen, N. T. Özzaim, and N. Aydin, Cyclic codes over $\mathbb{Z}_4+u\mathbb{Z}_4+u^2\mathbb{Z}_4$, Turkish J. Math. 41(5) (2017) 1235–1247.
A. S. L. Rocha, L. C. B. Faria, J. H. Kleinschmidt, R. Palazzo, Jr., and M. C. Silva-Filho, DNA sequences generated by $\mathbb{Z}_4$-linear codes, in: 2010 IEEE International Symposium on Information Theory (ISIT), IEEE, 2010, pp. 1320–1324.
I. Siap, T. Abualrub, N. Aydin, and P. Seneviratne, Skew cyclic codes of arbitrary length, Int. J. Inf. Coding Theory 2(1) (2011) 10–20.
B. Yildiz and I. Siap, Cyclic codes over $\mathbb{F}_2[u]/(u^4-1)$ and applications to DNA codes, Comput. Math. Appl. 63(7) (2012) 1169–1176.
S. Zhu and X. Chen, 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(1-2) (2017) 479–493.
