-
Previous Article
New 2-designs over finite fields from derived and residual designs
- AMC Home
- This Issue
-
Next Article
The weight distribution of a class of p-ary cyclic codes and their applications
Cyclic DNA codes over $ \mathbb{F}_2[u,v]/\langle u^3, v^2-v, vu-uv\rangle$
School of Mathematical Sciences, Anhui University, Hefei 230601, China |
In this paper, we construct cyclic DNA codes over the ring $R = \mathbb{F}_2[u,v]/\langle u^3, v^2-v, vu-uv\rangle$. The correspondence between the elements of $R$ and the alphabet $\{A,T,G,C\}^{3}$ is obtained by a given Gray map. Moreover, some properties of binary images of the Condons under the Gray map are also discussed. Finally, two examples of cyclic DNA codes over $R$ are presented to illustrate the obtained results.
References:
[1] |
T. Abualrub, A. Ghrayeb and X. N. Zeng,
Construction of cyclic codes over $ \mathbb{F}_4$ for DNA computing, Appl. Algebra in Engrg. Comm. Comput., 24 (2006), 445-459.
|
[2] |
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 (2007), 273-287.
doi: 10.1007/s10623-006-9034-5. |
[3] |
A. D'Yachkov, A. Macula, T. Renz, P. Vilenkin and I. Ismagilov, New results on DNA codes, IEEE International Symposium on Information Theory, Adelaide, SA, Australia, 2005, 283-287. |
[4] |
H. Q. Dinh, A. K. Singh, S. Pattanayak and S. Sriboonchitta,
Cyclic DNA codes over the ring $ \mathbb{F}_2 + u\mathbb{F}_2 + v\mathbb{F}_2 + uv\mathbb{F}_2 + v^2\mathbb{F}_2 + uv^2\mathbb{F}_2$, Des. Codes Cryptogr., 86 (2018), 1451-1467.
doi: 10.1007/s10623-017-0405-x. |
[5] |
Q. Q. Feng and W. G. Zhou,
Cyclic code and self-dual code over $ \mathbb{F}_2+u\mathbb{F}_2+u^2\mathbb{F}_2$, Journal of Mathematical Research and Exposition, 29 (2009), 500-506.
|
[6] |
K. Guenda, T. A. Gulliver and P. Solé,
On cyclic DNA codes, IEEE International Symposium
on Information Theory, Istanbul, (2012), 121-125.
|
[7] |
H. Mostafanasab and A. Y. Darani, On cyclic DNA codes over $ \mathbb{F}_2 + u\mathbb{F}_2 + u^2\mathbb{F}_2$, arXiv: 1603.05894vl [cs.IT] 18 Mar 2016. |
[8] |
J. L. Massey,
Reversible codes, Information and Control, 7 (1964), 369-380.
doi: 10.1016/S0019-9958(64)90438-3. |
[9] |
E. S. Oztas and I. Siap,
Lifted polynomials over $ \mathbb{F}_{16}$ and their applications to DNA codes, Filomat, 27 (2013), 459-466.
doi: 10.2298/FIL1303459O. |
[10] |
J. F. Qian, L. N. Zhang and S. X. Zhu,
Constacyclic and cyclic codes over $ \mathbb{F}_2+u\mathbb{F}_2+u^2\mathbb{F}_2$, IEICE Transactions on Fundamentals of Electronics Communications and Computer Sciences, E89-A (2006), 1863-1865.
|
[11] |
V. Rykov, A. J. Macula, D. Torny and P. White, DNA sequence and quaternary cyclic codes, IEEE International Syposium on Information Theory, Washington, DC, USA, 2001,248. |
[12] |
I. Siap, T. Abualrub and A. Ghrayeb,
Cyclic DNA codes over the ring $ \mathbb{F}_2[u]/\langle u^2-1\rangle$ based on the deletion distance, J. Franklin Inst., 346 (2009), 731-740.
doi: 10.1016/j.jfranklin.2009.07.002. |
[13] |
S. X. Zhu and X. J. Chen,
Cyclic DNA codes over $ \mathbb{F}_2+u\mathbb{F}_2+v\mathbb{F}_2+uv\mathbb{F}_2$ and their applications, Journal of Applied Mathematics and Computing, 55 (2017), 479-493.
doi: 10.1007/s12190-016-1046-3. |
show all references
References:
[1] |
T. Abualrub, A. Ghrayeb and X. N. Zeng,
Construction of cyclic codes over $ \mathbb{F}_4$ for DNA computing, Appl. Algebra in Engrg. Comm. Comput., 24 (2006), 445-459.
|
[2] |
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 (2007), 273-287.
doi: 10.1007/s10623-006-9034-5. |
[3] |
A. D'Yachkov, A. Macula, T. Renz, P. Vilenkin and I. Ismagilov, New results on DNA codes, IEEE International Symposium on Information Theory, Adelaide, SA, Australia, 2005, 283-287. |
[4] |
H. Q. Dinh, A. K. Singh, S. Pattanayak and S. Sriboonchitta,
Cyclic DNA codes over the ring $ \mathbb{F}_2 + u\mathbb{F}_2 + v\mathbb{F}_2 + uv\mathbb{F}_2 + v^2\mathbb{F}_2 + uv^2\mathbb{F}_2$, Des. Codes Cryptogr., 86 (2018), 1451-1467.
doi: 10.1007/s10623-017-0405-x. |
[5] |
Q. Q. Feng and W. G. Zhou,
Cyclic code and self-dual code over $ \mathbb{F}_2+u\mathbb{F}_2+u^2\mathbb{F}_2$, Journal of Mathematical Research and Exposition, 29 (2009), 500-506.
|
[6] |
K. Guenda, T. A. Gulliver and P. Solé,
On cyclic DNA codes, IEEE International Symposium
on Information Theory, Istanbul, (2012), 121-125.
|
[7] |
H. Mostafanasab and A. Y. Darani, On cyclic DNA codes over $ \mathbb{F}_2 + u\mathbb{F}_2 + u^2\mathbb{F}_2$, arXiv: 1603.05894vl [cs.IT] 18 Mar 2016. |
[8] |
J. L. Massey,
Reversible codes, Information and Control, 7 (1964), 369-380.
doi: 10.1016/S0019-9958(64)90438-3. |
[9] |
E. S. Oztas and I. Siap,
Lifted polynomials over $ \mathbb{F}_{16}$ and their applications to DNA codes, Filomat, 27 (2013), 459-466.
doi: 10.2298/FIL1303459O. |
[10] |
J. F. Qian, L. N. Zhang and S. X. Zhu,
Constacyclic and cyclic codes over $ \mathbb{F}_2+u\mathbb{F}_2+u^2\mathbb{F}_2$, IEICE Transactions on Fundamentals of Electronics Communications and Computer Sciences, E89-A (2006), 1863-1865.
|
[11] |
V. Rykov, A. J. Macula, D. Torny and P. White, DNA sequence and quaternary cyclic codes, IEEE International Syposium on Information Theory, Washington, DC, USA, 2001,248. |
[12] |
I. Siap, T. Abualrub and A. Ghrayeb,
Cyclic DNA codes over the ring $ \mathbb{F}_2[u]/\langle u^2-1\rangle$ based on the deletion distance, J. Franklin Inst., 346 (2009), 731-740.
doi: 10.1016/j.jfranklin.2009.07.002. |
[13] |
S. X. Zhu and X. J. Chen,
Cyclic DNA codes over $ \mathbb{F}_2+u\mathbb{F}_2+v\mathbb{F}_2+uv\mathbb{F}_2$ and their applications, Journal of Applied Mathematics and Computing, 55 (2017), 479-493.
doi: 10.1007/s12190-016-1046-3. |
Elements over |
Gray images over |
Elements over |
Elements over |
Gray images over |
Elements over |
Codewords of |
|
Codewords of |
|
[1] |
Nabil Bennenni, Kenza Guenda, Sihem Mesnager. DNA cyclic codes over rings. Advances in Mathematics of Communications, 2017, 11 (1) : 83-98. doi: 10.3934/amc.2017004 |
[2] |
Fatmanur Gursoy, Elif Segah Oztas, Irfan Siap. Reversible DNA codes over $F_{16}+uF_{16}+vF_{16}+uvF_{16}$. Advances in Mathematics of Communications, 2017, 11 (2) : 307-312. doi: 10.3934/amc.2017023 |
[3] |
Yasemin Cengellenmis, Abdullah Dertli, Steven T. Dougherty, Adrian Korban, Serap Şahinkaya, Deniz Ustun. Reversible $ G $-codes over the ring $ {\mathcal{F}}_{j,k} $ with applications to DNA codes. Advances in Mathematics of Communications, 2021 doi: 10.3934/amc.2021056 |
[4] |
Rafael Arce-Nazario, Francis N. Castro, Jose Ortiz-Ubarri. On the covering radius of some binary cyclic codes. Advances in Mathematics of Communications, 2017, 11 (2) : 329-338. doi: 10.3934/amc.2017025 |
[5] |
Steven T. Dougherty, Cristina Fernández-Córdoba. Codes over $\mathbb{Z}_{2^k}$, Gray map and self-dual codes. Advances in Mathematics of Communications, 2011, 5 (4) : 571-588. doi: 10.3934/amc.2011.5.571 |
[6] |
Michael Braun. On lattices, binary codes, and network codes. Advances in Mathematics of Communications, 2011, 5 (2) : 225-232. doi: 10.3934/amc.2011.5.225 |
[7] |
Cem Güneri, Ferruh Özbudak, Funda ÖzdemIr. On complementary dual additive cyclic codes. Advances in Mathematics of Communications, 2017, 11 (2) : 353-357. doi: 10.3934/amc.2017028 |
[8] |
Heide Gluesing-Luerssen, Katherine Morrison, Carolyn Troha. Cyclic orbit codes and stabilizer subfields. Advances in Mathematics of Communications, 2015, 9 (2) : 177-197. doi: 10.3934/amc.2015.9.177 |
[9] |
Joaquim Borges, Ivan Yu. Mogilnykh, Josep Rifà, Faina I. Solov'eva. Structural properties of binary propelinear codes. Advances in Mathematics of Communications, 2012, 6 (3) : 329-346. doi: 10.3934/amc.2012.6.329 |
[10] |
Rong Wang, Xiaoni Du, Cuiling Fan. Infinite families of 2-designs from a class of non-binary Kasami cyclic codes. Advances in Mathematics of Communications, 2021, 15 (4) : 663-676. doi: 10.3934/amc.2020088 |
[11] |
Xiaoni Du, Rong Wang, Chunming Tang, Qi Wang. Infinite families of 2-designs from two classes of binary cyclic codes with three nonzeros. Advances in Mathematics of Communications, 2022, 16 (1) : 157-168. doi: 10.3934/amc.2020106 |
[12] |
Gustavo Terra Bastos, Reginaldo Palazzo Júnior, Marinês Guerreiro. Abelian non-cyclic orbit codes and multishot subspace codes. Advances in Mathematics of Communications, 2020, 14 (4) : 631-650. doi: 10.3934/amc.2020035 |
[13] |
Martianus Frederic Ezerman, San Ling, Patrick Solé, Olfa Yemen. From skew-cyclic codes to asymmetric quantum codes. Advances in Mathematics of Communications, 2011, 5 (1) : 41-57. doi: 10.3934/amc.2011.5.41 |
[14] |
Yunwen Liu, Longjiang Qu, Chao Li. New constructions of systematic authentication codes from three classes of cyclic codes. Advances in Mathematics of Communications, 2018, 12 (1) : 1-16. doi: 10.3934/amc.2018001 |
[15] |
Steven T. Dougherty, Esengül Saltürk, Steve Szabo. Codes over local rings of order 16 and binary codes. Advances in Mathematics of Communications, 2016, 10 (2) : 379-391. doi: 10.3934/amc.2016012 |
[16] |
Heide Gluesing-Luerssen, Fai-Lung Tsang. A matrix ring description for cyclic convolutional codes. Advances in Mathematics of Communications, 2008, 2 (1) : 55-81. doi: 10.3934/amc.2008.2.55 |
[17] |
Long Yu, Hongwei Liu. A class of $p$-ary cyclic codes and their weight enumerators. Advances in Mathematics of Communications, 2016, 10 (2) : 437-457. doi: 10.3934/amc.2016017 |
[18] |
Umberto Martínez-Peñas. Rank equivalent and rank degenerate skew cyclic codes. Advances in Mathematics of Communications, 2017, 11 (2) : 267-282. doi: 10.3934/amc.2017018 |
[19] |
Heide Gluesing-Luerssen, Uwe Helmke, José Ignacio Iglesias Curto. Algebraic decoding for doubly cyclic convolutional codes. Advances in Mathematics of Communications, 2010, 4 (1) : 83-99. doi: 10.3934/amc.2010.4.83 |
[20] |
San Ling, Buket Özkaya. New bounds on the minimum distance of cyclic codes. Advances in Mathematics of Communications, 2021, 15 (1) : 1-8. doi: 10.3934/amc.2020038 |
2020 Impact Factor: 0.935
Tools
Metrics
Other articles
by authors
[Back to Top]