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On the Codes over a Family of Rings and Their Applications to DNA Codes

Year 2023, , 164 - 177, 02.09.2023
https://doi.org/10.36753/mathenot.903589

Abstract

In this paper, the structures of the linear codes over a family of the rings $A_{t}=Z_{4}\left[ u_{1},\ldots ,u_{t}\right] \left/ \left\langle u_{i}^{2}-u_{i},u_{i}u_{j}-u_{j}u_{i}\right\rangle \right. $ are given, where $i,j=1,2,\ldots ,t$, $i\neq j$, $Z_{4}=\{0,1,2,3\}$. A map between the elements of the $A_{t}$ and the alphabet $\left\{ A,T,C,G\right\} ^{2^{t}}$ is constructed. The DNA codes are obtained with three different methods, by using the cyclic, skew cyclic codes over a family of the rings $A_{t}$ and $\theta _{i}$-set, where $\theta _{i}$ is a non trivial automorphism on $A_{i}$, for $i=1,2,\ldots ,t$.

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References

  • [1] Yildiz, B., Siap, I.: Cyclic codes over $F_{2}[u]/(u^{4}-1)$ and applications to DNA codes. Computers Mathematics with Applications. 63(7), 1169-1176 (2012).
  • [2] Cengellenmis Y., Dertli A.: On the reversibility problem for DNA 4-bases. Erzincan University Journal of Science and Technology. 13(3), 1383-1388 (2020).
  • [3] Zhu, S., Chen, X.: Cyclic DNA codes over $F_{2}+uF_{2}+vF_{2}+uvF_{2}$ and their applications. Journal of Applied Mathematics and Computing. 55(1), 479-493 (2017).
  • [4] Benbelkacem, N., Ezerman, M. F., Abualrub, T., Aydin, N., Batoul, A.: Skew cyclic codes over $F_{4}R$. Journal of Algebra and Its Applications. 21(4), 2250065 (2020).
  • [5] Bennenni, N., Guenda, K., Mesnager, S.: New DNA cyclic codes over rings. Preprint arxiv:1505.06263 (2015).
  • [6] Guenda, K., Gulliver, T. A., Solé, P. : On cyclic DNA codes. In: 2013 IEEE International Symposium on Information Theory. 121-125 (2013).
  • [7] Dertli, A., Cengellenmis, Y.: On the cyclic DNA and skew cyclic DNA codes over $Z_{4}+vZ_{4}$. Biomath. (Submitted Article).
  • [8] Gao, J.: Quantum codes from cyclic codes over $F_{q}+vF_{q}+v^{2}F_{q}+v^{3}F_{q}$. International Journal of Quantum Information. 13(08), 1550063 (2015).
  • [9] Abualrub, T., Siap, I.: Reversible cyclic codes over $Z_{4}$. Australasian Journal of Combinatorics. 38, 195-206 (2007).
Year 2023, , 164 - 177, 02.09.2023
https://doi.org/10.36753/mathenot.903589

Abstract

Project Number

-

References

  • [1] Yildiz, B., Siap, I.: Cyclic codes over $F_{2}[u]/(u^{4}-1)$ and applications to DNA codes. Computers Mathematics with Applications. 63(7), 1169-1176 (2012).
  • [2] Cengellenmis Y., Dertli A.: On the reversibility problem for DNA 4-bases. Erzincan University Journal of Science and Technology. 13(3), 1383-1388 (2020).
  • [3] Zhu, S., Chen, X.: Cyclic DNA codes over $F_{2}+uF_{2}+vF_{2}+uvF_{2}$ and their applications. Journal of Applied Mathematics and Computing. 55(1), 479-493 (2017).
  • [4] Benbelkacem, N., Ezerman, M. F., Abualrub, T., Aydin, N., Batoul, A.: Skew cyclic codes over $F_{4}R$. Journal of Algebra and Its Applications. 21(4), 2250065 (2020).
  • [5] Bennenni, N., Guenda, K., Mesnager, S.: New DNA cyclic codes over rings. Preprint arxiv:1505.06263 (2015).
  • [6] Guenda, K., Gulliver, T. A., Solé, P. : On cyclic DNA codes. In: 2013 IEEE International Symposium on Information Theory. 121-125 (2013).
  • [7] Dertli, A., Cengellenmis, Y.: On the cyclic DNA and skew cyclic DNA codes over $Z_{4}+vZ_{4}$. Biomath. (Submitted Article).
  • [8] Gao, J.: Quantum codes from cyclic codes over $F_{q}+vF_{q}+v^{2}F_{q}+v^{3}F_{q}$. International Journal of Quantum Information. 13(08), 1550063 (2015).
  • [9] Abualrub, T., Siap, I.: Reversible cyclic codes over $Z_{4}$. Australasian Journal of Combinatorics. 38, 195-206 (2007).
There are 9 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Articles
Authors

Abdullah Dertli

Yasemin Cengellenmis This is me 0000-0002-8133-9836

Project Number -
Publication Date September 2, 2023
Submission Date March 26, 2021
Acceptance Date April 11, 2022
Published in Issue Year 2023

Cite

APA Dertli, A., & Cengellenmis, Y. (2023). On the Codes over a Family of Rings and Their Applications to DNA Codes. Mathematical Sciences and Applications E-Notes, 11(3), 164-177. https://doi.org/10.36753/mathenot.903589
AMA Dertli A, Cengellenmis Y. On the Codes over a Family of Rings and Their Applications to DNA Codes. Math. Sci. Appl. E-Notes. September 2023;11(3):164-177. doi:10.36753/mathenot.903589
Chicago Dertli, Abdullah, and Yasemin Cengellenmis. “On the Codes over a Family of Rings and Their Applications to DNA Codes”. Mathematical Sciences and Applications E-Notes 11, no. 3 (September 2023): 164-77. https://doi.org/10.36753/mathenot.903589.
EndNote Dertli A, Cengellenmis Y (September 1, 2023) On the Codes over a Family of Rings and Their Applications to DNA Codes. Mathematical Sciences and Applications E-Notes 11 3 164–177.
IEEE A. Dertli and Y. Cengellenmis, “On the Codes over a Family of Rings and Their Applications to DNA Codes”, Math. Sci. Appl. E-Notes, vol. 11, no. 3, pp. 164–177, 2023, doi: 10.36753/mathenot.903589.
ISNAD Dertli, Abdullah - Cengellenmis, Yasemin. “On the Codes over a Family of Rings and Their Applications to DNA Codes”. Mathematical Sciences and Applications E-Notes 11/3 (September 2023), 164-177. https://doi.org/10.36753/mathenot.903589.
JAMA Dertli A, Cengellenmis Y. On the Codes over a Family of Rings and Their Applications to DNA Codes. Math. Sci. Appl. E-Notes. 2023;11:164–177.
MLA Dertli, Abdullah and Yasemin Cengellenmis. “On the Codes over a Family of Rings and Their Applications to DNA Codes”. Mathematical Sciences and Applications E-Notes, vol. 11, no. 3, 2023, pp. 164-77, doi:10.36753/mathenot.903589.
Vancouver Dertli A, Cengellenmis Y. On the Codes over a Family of Rings and Their Applications to DNA Codes. Math. Sci. Appl. E-Notes. 2023;11(3):164-77.

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