Research Article
Year 2023,
Volume: 11 Issue: 3, 164 - 177, 02.09.2023
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,
Volume: 11 Issue: 3, 164 - 177, 02.09.2023
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 | |
| Project Number | - |
| Publication Date | September 2, 2023 |
| Submission Date | March 26, 2021 |
| Acceptance Date | April 11, 2022 |
| Published in Issue | Year 2023 Volume: 11 Issue: 3 |
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