Table 1.
Identifying codons with the elements of the ring $R$.
-
Previous Article
Cyclic codes over local Frobenius rings of order 16
- AMC Home
- This Issue
-
Next Article
Some designs and codes invariant under the Tits group
February
2017, 11(1): 83-98.
doi: 10.3934/amc.2017004
DNA cyclic codes over rings
| 1. | Faculty of Mathematics, University of Science and Technology, USTHB, Algeria |
| 2. | University of Paris Ⅷ and ⅩⅢ (Department of Mathematics), and Telecom ParisTech, Paris, France |
In this paper we construct new DNA cyclic codes over rings. Firstly, we introduce a new family of DNA cyclic codes over the ring $R=\mathbb{F}_2[u]/(u.6)$. A direct link between the elements of such a ring and the $64$ codons used in the amino acids of the living organisms is established. Using this correspondence we study the reverse-complement properties of our codes. We use the edit distance between the codewords which is an important combinatorial notion for the DNA strands. Next, we define the Lee weight, the Gray map over the ring $R$ as well as the binary image of the DNA cyclic codes allowing the transfer of studying DNA codes into studying binary codes. Secondly, we introduce another new family of DNA skew cyclic codes constructed over the ring $\tilde {R}=\mathbb{F}_2+v\mathbb{F}_2=\{0, 1, v, v+1\}, $ where $v^2=v$. The codes obtained are cyclic reverse-complement over the ring $\tilde {R}$. Further we find their binary images and construct some explicit examples of such codes.
Keywords:
DNA cyclic codes,
binary image of DNA codes,
DNA skew cyclic codes,
the reverse-complement skew cyclic codes.
Mathematics Subject Classification:
58F15, 58F17; Secondary: 53C35.
Citation:
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
![]()
References:
| [1] |
T. Abualrub, N. Aydin and P. Seneviratne,
On Θ-cyclic codes over $\mathbb{F}_2 + v\mathbb{F}_2$, Austral. J. Combin., 54 (2012), 115-126.
|
| [2] |
T. Abualrub, A. Ghrayeb and X. N. Zeng,
Construction of cyclic codes over $\mathbb F_4$ for DNA computing, J. Franklin Ins., 343 (2006), 488-457.
doi: 10.1016/j.jfranklin.2006.02.009. |
| [3] |
L. Adleman,
Molecular computation of the solutions to combinatorial problems, Science, 266 (1994), 1021-1024.
doi: 10.1126/science.7973651. |
| [4] |
C. Alf-Steinberger,
The genetic code and error transmission, Proc. Natl. Acad. Sci. USA, 64 (1969), 584-591.
doi: 10.1073/pnas.64.2.584. |
| [5] |
M. B. Bechet, Bias de codons et Régulation de la Traduction chez les Bactéries et le Phages, Ph. D thesis, Univ. Paris 7,2007. Google Scholar |
| [6] |
H. Q. Dinh and S. R. Lopez-Permouth,
Cyclic and negacyclic codes over finite chain rings, IEEE Trans. Inf. Theory, 50 (2004), 1728-1744.
doi: 10.1109/TIT.2004.831789. |
| [7] |
S. T. Dougherty, J. Lark Kim and H. Kulosman,
MDS code over finit principal ideal rings, Des. Codes Cryptogr., 50 (2009), 77-92.
doi: 10.1007/s10623-008-9215-5. |
| [8] |
K. Guenda and T. A. Gulliver,
Construction of cyclic codes over $\mathbb F_2+u\mathbb F_2$ for DNA computing, Appl. Algebra Eng. Commun. Comput, 24 (2013), 445-459.
doi: 10.1007/s00200-013-0188-x. |
| [9] |
K. Guenda and T. A. Gulliver, Repeated root constacyclic codes of length mps over $mp^s$ over $\mathbb F_p^r+u\mathbb F_p^r+\cdot\cdot\cdot+u^{e-1}\mathbb F_p^r$, J. Alg. App. , to appear.
doi: 10.1142/S0219498814500819. |
| [10] |
K. Guenda, T. A. Gulliver and P. Solé, On cyclic DNA codes, in Proc. IEEE Int. Symp. Inform. Theory, Istanbul, 2013,121-125.
doi: 10.1109/ISIT.2013.6620200. |
| [11] |
A. K. Konopka, Theory of the degenerate coding and information parameters of the protein coding genes, Biochimie, 67 (1985), 455-468. Google Scholar |
| [12] |
M. Mansuripur, P. K. Khulbe, S. M. Kuebler, J. W. Perry, M. S. Giridhar and N. Peyghambarian, Information Storage and Retrieval Using Macromolecules as Storage Media, Univ. Arizona Technical Report, 2003. Google Scholar |
| [13] |
J. L. Massey,
Reversible codes, Inf. Control, 7 (1964), 369-380.
doi: 10.1016/S0019-9958(64)90438-3. |
| [14] |
O. Milenkovic and N. Kashyap, On the design of codes for DNA computing, in IEEE Int. Symp. Inf. Theory (ISIT), 2006.
doi: 10.1007/11779360_9. |
| [15] |
G. H. Norton and A. Salagean,
On the structure of linear and cyclic codes over finite chain ring, AAECC, 10 (2000), 489-506.
doi: 10.1007/PL00012382. |
| [16] |
E. S. Ristad and P. N. Yianilos,
Learning string-edit distance, IEEE Trans. Anal. Mach. Intell, 20 (1998), 522-532.
doi: 10.1109/34.682181. |
| [17] |
V. Rykov, A. J. Macula, D. Torny and P. White, DNA sequences and quaternary cyclic codes, in IEEE Int. Symp. Inf. Theory (ISIT), 2001.
doi: 10.1109/ISIT.2001.936111. |
| [18] |
R. Sanchez, E. Morgado and R. Grau,
Gene algebra from a genetic code algebraic structure, J. Math. Biol., 51 (2005), 431-475.
doi: 10.1007/s00285-005-0332-8. |
| [19] |
I. Siap, T. Abualrub and A. Ghrayeb,
Cyclic DNA codes over ring $\mathbb{F}_2[u]/(u^2-1)$ based on the deletion distance, Franklin Institute, 36 (2009), 731-740.
doi: 10.1016/j.jfranklin.2009.07.002. |
| [20] |
show all references
References:
| [1] |
T. Abualrub, N. Aydin and P. Seneviratne,
On Θ-cyclic codes over $\mathbb{F}_2 + v\mathbb{F}_2$, Austral. J. Combin., 54 (2012), 115-126.
|
| [2] |
T. Abualrub, A. Ghrayeb and X. N. Zeng,
Construction of cyclic codes over $\mathbb F_4$ for DNA computing, J. Franklin Ins., 343 (2006), 488-457.
doi: 10.1016/j.jfranklin.2006.02.009. |
| [3] |
L. Adleman,
Molecular computation of the solutions to combinatorial problems, Science, 266 (1994), 1021-1024.
doi: 10.1126/science.7973651. |
| [4] |
C. Alf-Steinberger,
The genetic code and error transmission, Proc. Natl. Acad. Sci. USA, 64 (1969), 584-591.
doi: 10.1073/pnas.64.2.584. |
| [5] |
M. B. Bechet, Bias de codons et Régulation de la Traduction chez les Bactéries et le Phages, Ph. D thesis, Univ. Paris 7,2007. Google Scholar |
| [6] |
H. Q. Dinh and S. R. Lopez-Permouth,
Cyclic and negacyclic codes over finite chain rings, IEEE Trans. Inf. Theory, 50 (2004), 1728-1744.
doi: 10.1109/TIT.2004.831789. |
| [7] |
S. T. Dougherty, J. Lark Kim and H. Kulosman,
MDS code over finit principal ideal rings, Des. Codes Cryptogr., 50 (2009), 77-92.
doi: 10.1007/s10623-008-9215-5. |
| [8] |
K. Guenda and T. A. Gulliver,
Construction of cyclic codes over $\mathbb F_2+u\mathbb F_2$ for DNA computing, Appl. Algebra Eng. Commun. Comput, 24 (2013), 445-459.
doi: 10.1007/s00200-013-0188-x. |
| [9] |
K. Guenda and T. A. Gulliver, Repeated root constacyclic codes of length mps over $mp^s$ over $\mathbb F_p^r+u\mathbb F_p^r+\cdot\cdot\cdot+u^{e-1}\mathbb F_p^r$, J. Alg. App. , to appear.
doi: 10.1142/S0219498814500819. |
| [10] |
K. Guenda, T. A. Gulliver and P. Solé, On cyclic DNA codes, in Proc. IEEE Int. Symp. Inform. Theory, Istanbul, 2013,121-125.
doi: 10.1109/ISIT.2013.6620200. |
| [11] |
A. K. Konopka, Theory of the degenerate coding and information parameters of the protein coding genes, Biochimie, 67 (1985), 455-468. Google Scholar |
| [12] |
M. Mansuripur, P. K. Khulbe, S. M. Kuebler, J. W. Perry, M. S. Giridhar and N. Peyghambarian, Information Storage and Retrieval Using Macromolecules as Storage Media, Univ. Arizona Technical Report, 2003. Google Scholar |
| [13] |
J. L. Massey,
Reversible codes, Inf. Control, 7 (1964), 369-380.
doi: 10.1016/S0019-9958(64)90438-3. |
| [14] |
O. Milenkovic and N. Kashyap, On the design of codes for DNA computing, in IEEE Int. Symp. Inf. Theory (ISIT), 2006.
doi: 10.1007/11779360_9. |
| [15] |
G. H. Norton and A. Salagean,
On the structure of linear and cyclic codes over finite chain ring, AAECC, 10 (2000), 489-506.
doi: 10.1007/PL00012382. |
| [16] |
E. S. Ristad and P. N. Yianilos,
Learning string-edit distance, IEEE Trans. Anal. Mach. Intell, 20 (1998), 522-532.
doi: 10.1109/34.682181. |
| [17] |
V. Rykov, A. J. Macula, D. Torny and P. White, DNA sequences and quaternary cyclic codes, in IEEE Int. Symp. Inf. Theory (ISIT), 2001.
doi: 10.1109/ISIT.2001.936111. |
| [18] |
R. Sanchez, E. Morgado and R. Grau,
Gene algebra from a genetic code algebraic structure, J. Math. Biol., 51 (2005), 431-475.
doi: 10.1007/s00285-005-0332-8. |
| [19] |
I. Siap, T. Abualrub and A. Ghrayeb,
Cyclic DNA codes over ring $\mathbb{F}_2[u]/(u^2-1)$ based on the deletion distance, Franklin Institute, 36 (2009), 731-740.
doi: 10.1016/j.jfranklin.2009.07.002. |
| [20] |
| CCC | $u^5+u^4+u^3+u^2+u+1$ | GGG | 0 | ACT | $u^3+u^2+1$ | GTC | $u^4+u^2+u+1$ |
| GGA | $u^5+u^4+u^3+u^2+u$ | CCT | 1 | ACG | $u^3+u^2+u$ | ACA | $u^3+u^2+u+1$ |
| GGC | $u^5+u^4+u^3+u^2+1$ | CCG | $u$ | TTT | $u^4+u^2+1$ | GAC | $u^5+u^3+u^2+1$ |
| GGT | $u^5+u^4+u^3+u^2$ | CCA | $u+1$ | TTG | $u^4+u^2+u$ | AGG | $u^5+u^3+u+1$ |
| AGG | $u^5+u^4+u^3+u+1$ | TCC | $u^2$ | CTA | $u^4+u+1$ | GAT | $u^5+u^3+u^2$ |
| CGG | $u^5+u^4+u^2+u+1$ | GCC | $u^3$ | GTT | $u^4+u^3+1$ | GTA | $u^4+u^3+u+1$ |
| GAG | $u^5+u^3+u^2+u+1$ | CTC | $u^4$ | GTG | $u^4+u^3+u$ | ATT | $u^4+u^3+u^2+1$ |
| AGA | $u^5+u^4+u^3+u$ | TCT | $u^2+1$ | TCA | $u^2+u+1$ | ATA | $u^4+u^3+u^2+u$ |
| AGC | $u^5+u^4+u^3+1$ | TCG | $u^2+u$ | CAA | $u^5+u^2+u$ | ATC | $u^4+u^3+u^2$ |
| ATG | $u^4+u^3+u^2+u+1$ | TAC | $u^5$ | CAC | $u^5+u^2+u$ | TGA | $u^5+u^4+u$ |
| AGT | $u^5+u^4+u^3$ | TAT | $u^5+1$ | GCA | $u^3+u+1$ | AAT | $u^5+u^2+u+1$ |
| CGA | $u^5+u^4+u^2+u$ | GCT | $u^3+1$ | TTA | $u^4+u^3$ | AAA | $u^5+u^3+u$ |
| CGC | $u^5+u^4+u^2+1$ | GCG | $u^3+u$ | ACC | $u^3+u^2$ | TGC | $u^5+u^4+1$ |
| CGT | $u^5+u^4+u^2$ | TAA | $u^5+u$ | CAT | $u^5+u^2$ | AAC | $u^5+u^3+1$ |
| TGG | $u^5+u^4+u+1$ | CTG | $u^4+u$ | TGT | $u^5+u^4$ | TCC | $u^4+u^2$ |
| GAA | $u^5+u^3+u^2+u$ | CTT | $u^4+1$ | CAG | $u^5+u^3$ | TAG | $u^5+u+1$ |
| CCC | $u^5+u^4+u^3+u^2+u+1$ | GGG | 0 | ACT | $u^3+u^2+1$ | GTC | $u^4+u^2+u+1$ |
| GGA | $u^5+u^4+u^3+u^2+u$ | CCT | 1 | ACG | $u^3+u^2+u$ | ACA | $u^3+u^2+u+1$ |
| GGC | $u^5+u^4+u^3+u^2+1$ | CCG | $u$ | TTT | $u^4+u^2+1$ | GAC | $u^5+u^3+u^2+1$ |
| GGT | $u^5+u^4+u^3+u^2$ | CCA | $u+1$ | TTG | $u^4+u^2+u$ | AGG | $u^5+u^3+u+1$ |
| AGG | $u^5+u^4+u^3+u+1$ | TCC | $u^2$ | CTA | $u^4+u+1$ | GAT | $u^5+u^3+u^2$ |
| CGG | $u^5+u^4+u^2+u+1$ | GCC | $u^3$ | GTT | $u^4+u^3+1$ | GTA | $u^4+u^3+u+1$ |
| GAG | $u^5+u^3+u^2+u+1$ | CTC | $u^4$ | GTG | $u^4+u^3+u$ | ATT | $u^4+u^3+u^2+1$ |
| AGA | $u^5+u^4+u^3+u$ | TCT | $u^2+1$ | TCA | $u^2+u+1$ | ATA | $u^4+u^3+u^2+u$ |
| AGC | $u^5+u^4+u^3+1$ | TCG | $u^2+u$ | CAA | $u^5+u^2+u$ | ATC | $u^4+u^3+u^2$ |
| ATG | $u^4+u^3+u^2+u+1$ | TAC | $u^5$ | CAC | $u^5+u^2+u$ | TGA | $u^5+u^4+u$ |
| AGT | $u^5+u^4+u^3$ | TAT | $u^5+1$ | GCA | $u^3+u+1$ | AAT | $u^5+u^2+u+1$ |
| CGA | $u^5+u^4+u^2+u$ | GCT | $u^3+1$ | TTA | $u^4+u^3$ | AAA | $u^5+u^3+u$ |
| CGC | $u^5+u^4+u^2+1$ | GCG | $u^3+u$ | ACC | $u^3+u^2$ | TGC | $u^5+u^4+1$ |
| CGT | $u^5+u^4+u^2$ | TAA | $u^5+u$ | CAT | $u^5+u^2$ | AAC | $u^5+u^3+1$ |
| TGG | $u^5+u^4+u+1$ | CTG | $u^4+u$ | TGT | $u^5+u^4$ | TCC | $u^4+u^2$ |
| GAA | $u^5+u^3+u^2+u$ | CTT | $u^4+1$ | CAG | $u^5+u^3$ | TAG | $u^5+u+1$ |
Table 2.
DNA cyclic codes of length 7
| Code $\mathcal{C}$ | Size of $\mathcal{C}$ | $d_{H}$ |
| $\langle u^2f_0\rangle$ | 4096 | 2 |
| $\langle u^2f_1\rangle$ | 256 | 3 |
| $\langle u^2f_2\rangle$ | 256 | 3 |
| $\langle u^2f_1f_2\rangle$ | 4 | 7 |
| $\langle u^2f_0f_1 \rangle$ | 64 | 4 |
| $\langle u^2f_0f_2\rangle$ | 64 | 4 |
| $\langle u^4f_0f_1\rangle$ | 64 | 4 |
| Code $\mathcal{C}$ | Size of $\mathcal{C}$ | $d_{H}$ |
| $\langle u^2f_0\rangle$ | 4096 | 2 |
| $\langle u^2f_1\rangle$ | 256 | 3 |
| $\langle u^2f_2\rangle$ | 256 | 3 |
| $\langle u^2f_1f_2\rangle$ | 4 | 7 |
| $\langle u^2f_0f_1 \rangle$ | 64 | 4 |
| $\langle u^2f_0f_2\rangle$ | 64 | 4 |
| $\langle u^4f_0f_1\rangle$ | 64 | 4 |
Table 3.
A DNA Cyclic Code associate to $\mathcal{C}=\langle u^4f_0f_1\rangle$ given in (4)
| GGGGGGGGGGGGGGGGGGGGG | CCCCCCCCCCCCCCCCCCCCC |
| CTCGGGCTCCTCCTCGGGGGG | GAGCCCGAGGAGGAGCCCCCC |
| GGGCTCGGGCTCTGTTGTTGT | CCCGAGCCCGAGACAATAACA |
| TGTGGGCTCGGGCTCTGTTGT | ACACCCGAGCCCGAGACAACA |
| TGTTGTGGGCTCGGGCTCTGT | ACAACACCCGAGCCCGAGACA |
| TGTTGTTGTGGGCTCGGGCTC | ACAACAACACCCGAGCCCGAG |
| CTCTGTTGTTGTGGGCTCGGG | GAGACAACAACACCCGAGCCC |
| GGGCTCTGTTGTTGTGGGCTC | CCCGAGACAACAACACCCGAG |
| TATGGGTATTATTATGGGGGG | ATACCCATAATAATACCCCCC |
| GGGTATGGGTATTATTATGGG | CCCATACCCATAATAATACCC |
| GGGGGGTATGGGTATTATTAT | CCCCCCATACCCATAATAATA |
| TATGGGGGGTATGGGTATTAT | ATACCCCCCATACCCATAATA |
| TATTATGGGGGGTATGGGTAT | ATAATACCCCCCATACCCATA |
| TATTATTATGGGGGGTATGGG | ATAATAATACCCCCCATACCC |
| GGGTATTATTATGGGGGGTAT | CCCATAATAATACCCCCCATA |
| TGTGGGTGTTGTTGTGGGGGG | ACACCCACAACAACACCCCCC |
| GGGTGTGGGTGTTGTTGTGGG | CCCACACCCACAACAACACCC |
| GGGGGGTGTGGGTGTTGTTGT | CCCCCCACACCCACAACAACA |
| TGTGGGGGGTGTGGGTGTTGT | ACACCCCCCACACCCACAACA |
| TGTTGTGGGGGGTGTGGGTGT | ACAACACCCCCCACACCCACA |
| TGTTGTTGTGGGGGGTGTGGG | ACAACAACACCCCCCACACCC |
| GGGTGTTGTTGTGGGGGGTGT | CCCACAACAACACCCCCCACA |
| CTCGGGCTCTGTTGTTGTGGG | GAGCCCGAGACAACAACACCC |
| GGGCTCGGGCTCTGTTGTTGT | CCCGAGCCCGAGACAACAACA |
| TGTGGGCTCGGGCTCTGTTGT | ACACCCGAGCCCGAGACAACA |
| TGTTGTGGGCTCGGGCTCTGT | ACAACACCCGAGCCCGAGACA |
| TGTTGTTGTGGGCTCGGGCTC | ACAACAACACCCGAGCCCGAG |
| CTCTGTTGTTGTGGGCTCGGG | GAGACAACAACACCCGAGCCC |
| GGGCTCTGTTGTTGTGGGCTC | CCCGAGACAACAACACCCGAG |
| GGGGGGCTCGGGCTCCTCCTC | CCCCCCGAGCCCGAGGAGGAG |
| CTCGGGGGGCTCGGGCTCCTC | GAGCCCCCCGAGCCCGAGGAG |
| CTCCTCGGGGGGCTCGGGCTC | GAGGAGCCCCCCGAGCCCGAG |
| GGGGGGGGGGGGGGGGGGGGG | CCCCCCCCCCCCCCCCCCCCC |
| CTCGGGCTCCTCCTCGGGGGG | GAGCCCGAGGAGGAGCCCCCC |
| GGGCTCGGGCTCTGTTGTTGT | CCCGAGCCCGAGACAATAACA |
| TGTGGGCTCGGGCTCTGTTGT | ACACCCGAGCCCGAGACAACA |
| TGTTGTGGGCTCGGGCTCTGT | ACAACACCCGAGCCCGAGACA |
| TGTTGTTGTGGGCTCGGGCTC | ACAACAACACCCGAGCCCGAG |
| CTCTGTTGTTGTGGGCTCGGG | GAGACAACAACACCCGAGCCC |
| GGGCTCTGTTGTTGTGGGCTC | CCCGAGACAACAACACCCGAG |
| TATGGGTATTATTATGGGGGG | ATACCCATAATAATACCCCCC |
| GGGTATGGGTATTATTATGGG | CCCATACCCATAATAATACCC |
| GGGGGGTATGGGTATTATTAT | CCCCCCATACCCATAATAATA |
| TATGGGGGGTATGGGTATTAT | ATACCCCCCATACCCATAATA |
| TATTATGGGGGGTATGGGTAT | ATAATACCCCCCATACCCATA |
| TATTATTATGGGGGGTATGGG | ATAATAATACCCCCCATACCC |
| GGGTATTATTATGGGGGGTAT | CCCATAATAATACCCCCCATA |
| TGTGGGTGTTGTTGTGGGGGG | ACACCCACAACAACACCCCCC |
| GGGTGTGGGTGTTGTTGTGGG | CCCACACCCACAACAACACCC |
| GGGGGGTGTGGGTGTTGTTGT | CCCCCCACACCCACAACAACA |
| TGTGGGGGGTGTGGGTGTTGT | ACACCCCCCACACCCACAACA |
| TGTTGTGGGGGGTGTGGGTGT | ACAACACCCCCCACACCCACA |
| TGTTGTTGTGGGGGGTGTGGG | ACAACAACACCCCCCACACCC |
| GGGTGTTGTTGTGGGGGGTGT | CCCACAACAACACCCCCCACA |
| CTCGGGCTCTGTTGTTGTGGG | GAGCCCGAGACAACAACACCC |
| GGGCTCGGGCTCTGTTGTTGT | CCCGAGCCCGAGACAACAACA |
| TGTGGGCTCGGGCTCTGTTGT | ACACCCGAGCCCGAGACAACA |
| TGTTGTGGGCTCGGGCTCTGT | ACAACACCCGAGCCCGAGACA |
| TGTTGTTGTGGGCTCGGGCTC | ACAACAACACCCGAGCCCGAG |
| CTCTGTTGTTGTGGGCTCGGG | GAGACAACAACACCCGAGCCC |
| GGGCTCTGTTGTTGTGGGCTC | CCCGAGACAACAACACCCGAG |
| GGGGGGCTCGGGCTCCTCCTC | CCCCCCGAGCCCGAGGAGGAG |
| CTCGGGGGGCTCGGGCTCCTC | GAGCCCCCCGAGCCCGAGGAG |
| CTCCTCGGGGGGCTCGGGCTC | GAGGAGCCCCCCGAGCCCGAG |
Table 4.
A DNA Cyclic associate to $\mathcal{C}= < f_1f_2>$ given in (4)
| GGGGGGGGGGGGGGGGGGGGG | CCCCCCCCCCCCCCCCCCCCC |
| GGAGGAGGAGGAGGAGGAGGA | CCTCCTCCTCCTCCTCCTCCT |
| GGCGGCGGCGGCGGCGGCGGC | CCGCCGCCGCCGCCGCCGCCG |
| GGTGGTGGTGGTGGTGGTGGT | CCACCACCACCACCACCACCA |
| AGGAGGAGGAGGAGGAGGAGG | TCCTCCTCCTCCTCCTCCTCC |
| AGAAGAAGAAGAAGAAGAAGA | TCTTCTTCTTCTTCTTCTTCT |
| AGCAGCAGCAGCAGCAGCAGC | TCGTCGTCGTCGTCGTCGTCG |
| AGTAGTAGTAGTAGTAGTAGT | TCATCATCATCATCATCATCA |
| CGGCGGCGGCGGCGGCGGCGG | GCCGCCGCCGCCGCCGCCGCC |
| CGACGACGACGACGACGACGA | GCTGCTGCTGCTGCTGCTGCT |
| CGCCGCCGCCGCCGCCGCCGC | GCGGCGGCGGCGGCGGCGGCG |
| CGTCGTCGTCGTCGTCGTCGT | GCAGCAGCAGCAGCAGCAGCA |
| TGGTGGTGGTGGTGGTGGTGG | ACCACCACCACCACCACCACC |
| TGATGATGATGATGATGATGA | ACTACTACTACTACTACTACT |
| TGCTGCTGCTGCTGCTGCTGC | ACGACGACGACGACGACGACG |
| TGTTGTTGTTGTTGTTGTTGT | ACAACAACAACAACAACAACA |
| GAGGAGGAGGAGGAGGAGGAG | CTCCTCCTCCTCCTCCTCCTC |
| GAAGAAGAAGAAGAAGAAGAA | CTTCTTCTTCTTCTTCTTCTT |
| GACGACGACGACGACGACGAC | CTGCTGCTGCTGCTGCTGCTG |
| GATGATGATGATGATGATGAT | CTACTACTACTACTACTACTA |
| AGGAGGAGGAGGAGGAGGAGG | TCCTCCTCCTCCTCCTCCTCC |
| AAAAAAAAAAAAAAAAAAAAA | TTTTTTTTTTTTTTTTTTTTT |
| AACAACAACAACAACAACAAC | TTGTTGTTGTTGTTGTTGTTG |
| AATAATAATAATAATAATAAT | TTATTATTATTATTATTATTA |
| CAGCAGCAGCAGCAGCAGCAG | GTCGTCGTCGTCGTCGTCGTC |
| CAACAACAACAACAACAACAA | GTTGTTGTTGTTGTTGTTGTT |
| CACCACCACCACCACCACCAC | GTGGTGGTGGTGGTGGTGGTG |
| CATCATCATCATCATCATCAT | GTAGTAGTAGTAGTAGTAGTA |
| TAGTAGTAGTAGTAGTAGTAG | ATCATCATCATCATCATCATC |
| TAATAATAATAATAATAATAA | ATTATTATTATTATTATTATT |
| TACTACTACTACTACTACTAC | ATGATGATGATGATGATGATG |
| TATTATTATTATTATTATTAT | ATAATAATAATAATAATAATA |
| GGGGGGGGGGGGGGGGGGGGG | CCCCCCCCCCCCCCCCCCCCC |
| GGAGGAGGAGGAGGAGGAGGA | CCTCCTCCTCCTCCTCCTCCT |
| GGCGGCGGCGGCGGCGGCGGC | CCGCCGCCGCCGCCGCCGCCG |
| GGTGGTGGTGGTGGTGGTGGT | CCACCACCACCACCACCACCA |
| AGGAGGAGGAGGAGGAGGAGG | TCCTCCTCCTCCTCCTCCTCC |
| AGAAGAAGAAGAAGAAGAAGA | TCTTCTTCTTCTTCTTCTTCT |
| AGCAGCAGCAGCAGCAGCAGC | TCGTCGTCGTCGTCGTCGTCG |
| AGTAGTAGTAGTAGTAGTAGT | TCATCATCATCATCATCATCA |
| CGGCGGCGGCGGCGGCGGCGG | GCCGCCGCCGCCGCCGCCGCC |
| CGACGACGACGACGACGACGA | GCTGCTGCTGCTGCTGCTGCT |
| CGCCGCCGCCGCCGCCGCCGC | GCGGCGGCGGCGGCGGCGGCG |
| CGTCGTCGTCGTCGTCGTCGT | GCAGCAGCAGCAGCAGCAGCA |
| TGGTGGTGGTGGTGGTGGTGG | ACCACCACCACCACCACCACC |
| TGATGATGATGATGATGATGA | ACTACTACTACTACTACTACT |
| TGCTGCTGCTGCTGCTGCTGC | ACGACGACGACGACGACGACG |
| TGTTGTTGTTGTTGTTGTTGT | ACAACAACAACAACAACAACA |
| GAGGAGGAGGAGGAGGAGGAG | CTCCTCCTCCTCCTCCTCCTC |
| GAAGAAGAAGAAGAAGAAGAA | CTTCTTCTTCTTCTTCTTCTT |
| GACGACGACGACGACGACGAC | CTGCTGCTGCTGCTGCTGCTG |
| GATGATGATGATGATGATGAT | CTACTACTACTACTACTACTA |
| AGGAGGAGGAGGAGGAGGAGG | TCCTCCTCCTCCTCCTCCTCC |
| AAAAAAAAAAAAAAAAAAAAA | TTTTTTTTTTTTTTTTTTTTT |
| AACAACAACAACAACAACAAC | TTGTTGTTGTTGTTGTTGTTG |
| AATAATAATAATAATAATAAT | TTATTATTATTATTATTATTA |
| CAGCAGCAGCAGCAGCAGCAG | GTCGTCGTCGTCGTCGTCGTC |
| CAACAACAACAACAACAACAA | GTTGTTGTTGTTGTTGTTGTT |
| CACCACCACCACCACCACCAC | GTGGTGGTGGTGGTGGTGGTG |
| CATCATCATCATCATCATCAT | GTAGTAGTAGTAGTAGTAGTA |
| TAGTAGTAGTAGTAGTAGTAG | ATCATCATCATCATCATCATC |
| TAATAATAATAATAATAATAA | ATTATTATTATTATTATTATT |
| TACTACTACTACTACTACTAC | ATGATGATGATGATGATGATG |
| TATTATTATTATTATTATTAT | ATAATAATAATAATAATAATA |
Table 5.
DNA cyclic codes associate to $\mathcal{C}= \langle f_0, uf_1, u^2f_2, u^3f_3, u^4f_4, u^5f_5\rangle$
| The Code $\mathcal{C}$ | Size of the code $\mathcal{C}$ |
| $\langle u^3f_1, u^4f_2, u^5f_3\rangle$ | 1125899906842624 |
| $\langle u^5f_2\rangle$ | 512 |
| $\langle f_3, u^5f_2\rangle$ | 4611686018427387904 |
| $\langle u^4f_1, u^5f_3\rangle$ | 8589934592 |
| The Code $\mathcal{C}$ | Size of the code $\mathcal{C}$ |
| $\langle u^3f_1, u^4f_2, u^5f_3\rangle$ | 1125899906842624 |
| $\langle u^5f_2\rangle$ | 512 |
| $\langle f_3, u^5f_2\rangle$ | 4611686018427387904 |
| $\langle u^4f_1, u^5f_3\rangle$ | 8589934592 |
| GGG | 000000 | CCC | 111111 | TAT | 000001 | ATA | 111110 |
| GGA | 011111 | CCT | 100000 | TAC | 100001 | ATG | 011110 |
| GGC | 101111 | CCG | 010000 | TAA | 010001 | ATT | 101110 |
| GGT | 001111 | CCA | 110000 | TAG | 110001 | ATC | 001110 |
| AGG | 110111 | TCC | 001000 | CAT | 001001 | GTA | 110110 |
| AGA | 010111 | TCT | 101000 | CAC | 011001 | GTG | 100110 |
| AGC | 100111 | TCG | 011000 | CAA | 011001 | GTT | 100110 |
| AGT | 000111 | TCA | 111000 | CAG | 111001 | GTC | 000110 |
| CGG | 111011 | GCC | 000100 | AAT | 000101 | TTA | 111010 |
| CGA | 011011 | GCT | 100100 | AAC | 100101 | TTG | 011010 |
| CGC | 101011 | GCG | 010100 | AAA | 010101 | TTT | 101010 |
| CGT | 001011 | GCA | 110100 | AGG | 110101 | TCC | 001010 |
| TGG | 110011 | ACC | 001100 | GAT | 001101 | CTA | 110010 |
| TGA | 010011 | ACT | 101100 | GAC | 101101 | CTG | 010010 |
| TGC | 100011 | ACG | 011100 | GAA | 011101 | CTT | 100010 |
| TGT | 000011 | ACA | 111100 | GAG | 111101 | CTC | 000010 |
| GGG | 000000 | CCC | 111111 | TAT | 000001 | ATA | 111110 |
| GGA | 011111 | CCT | 100000 | TAC | 100001 | ATG | 011110 |
| GGC | 101111 | CCG | 010000 | TAA | 010001 | ATT | 101110 |
| GGT | 001111 | CCA | 110000 | TAG | 110001 | ATC | 001110 |
| AGG | 110111 | TCC | 001000 | CAT | 001001 | GTA | 110110 |
| AGA | 010111 | TCT | 101000 | CAC | 011001 | GTG | 100110 |
| AGC | 100111 | TCG | 011000 | CAA | 011001 | GTT | 100110 |
| AGT | 000111 | TCA | 111000 | CAG | 111001 | GTC | 000110 |
| CGG | 111011 | GCC | 000100 | AAT | 000101 | TTA | 111010 |
| CGA | 011011 | GCT | 100100 | AAC | 100101 | TTG | 011010 |
| CGC | 101011 | GCG | 010100 | AAA | 010101 | TTT | 101010 |
| CGT | 001011 | GCA | 110100 | AGG | 110101 | TCC | 001010 |
| TGG | 110011 | ACC | 001100 | GAT | 001101 | CTA | 110010 |
| TGA | 010011 | ACT | 101100 | GAC | 101101 | CTG | 010010 |
| TGC | 100011 | ACG | 011100 | GAA | 011101 | CTT | 100010 |
| TGT | 000011 | ACA | 111100 | GAG | 111101 | CTC | 000010 |
| The code $\mathcal{C}$ | Length of $\varphi(\mathcal{C})$ | $d_H(\varphi(\mathcal{C}))$ | Size of the Code $\varphi(\mathcal{C})$ |
| $\langle u^2f_0\rangle$ | 42 | 12 | 4096 |
| $\langle u^2f_1\rangle$ | 42 | 18 | 256 |
| $\langle u^2f_2\rangle$ | 42 | 18 | 256 |
| The code $\mathcal{C}$ | Length of $\varphi(\mathcal{C})$ | $d_H(\varphi(\mathcal{C}))$ | Size of the Code $\varphi(\mathcal{C})$ |
| $\langle u^2f_0\rangle$ | 42 | 12 | 4096 |
| $\langle u^2f_1\rangle$ | 42 | 18 | 256 |
| $\langle u^2f_2\rangle$ | 42 | 18 | 256 |
Table 8.
DNA skew cyclic code of length 10 and minimal distance 2
| GGGGGGGGGG | CCCCCCCCCC | CCCCCGGGGG | GGGGGCCCCC |
| GGGGCCCCCG | CCCCGGGGGC | CCCCGCCCCG | GGGGCGGGGC |
| GGGTTTTTGG | CCCAAAAACC | CCCATAAACG | GGGTATTTGC |
| GGGTAAAACG | CCCATTTTGC | CCGGGCCGGG | GGCCCGGCCC |
| GGCCCCCGGG | CCGGGGGCCC | CCGGCGGCCG | GGCCGCCGGC |
| GGCCGGGCCG | CCGGCCCGGC | CCGTATTACG | GGCATAATGC |
| GGCAAAATCG | CCGTTTTAGC | CCGTTAATTG | GGCAATTAAC |
| GGCAAAATGG | CCGTTTTACC | CATTAACGGG | GTAATTGCCC |
| GTAAAACGGG | CATTTTGCCC | CAGTATGCCG | GTCATACGGC |
| GTAACCATTG | CATTGGTAAC | CAAGCGTACG | GTTCGCATGC |
| GTACGGTACG | CATGCCATGC | CAATTACCGG | GTTAATGGCC |
| GTACCCATGG | CTAGGGTACC | CAAAATGGGG | GTTTTACCCC |
| GATTTTGGGG | CTAAAACCCC | CAAATACCCG | GTTTATGGGC |
| GTTTAACCCG | CAAATTGGGC | CAACGCAACG | GTTGCGTTGC |
| GTTGCCAACG | CAACGGTTGC | CAACCGTTGG | GTTGGCAACC |
| GTTGGGTTGG | CAACCCAACC | CCACGCAACG | GGTGCGTTGC |
| GGGGGGGGGG | CCCCCCCCCC | CCCCCGGGGG | GGGGGCCCCC |
| GGGGCCCCCG | CCCCGGGGGC | CCCCGCCCCG | GGGGCGGGGC |
| GGGTTTTTGG | CCCAAAAACC | CCCATAAACG | GGGTATTTGC |
| GGGTAAAACG | CCCATTTTGC | CCGGGCCGGG | GGCCCGGCCC |
| GGCCCCCGGG | CCGGGGGCCC | CCGGCGGCCG | GGCCGCCGGC |
| GGCCGGGCCG | CCGGCCCGGC | CCGTATTACG | GGCATAATGC |
| GGCAAAATCG | CCGTTTTAGC | CCGTTAATTG | GGCAATTAAC |
| GGCAAAATGG | CCGTTTTACC | CATTAACGGG | GTAATTGCCC |
| GTAAAACGGG | CATTTTGCCC | CAGTATGCCG | GTCATACGGC |
| GTAACCATTG | CATTGGTAAC | CAAGCGTACG | GTTCGCATGC |
| GTACGGTACG | CATGCCATGC | CAATTACCGG | GTTAATGGCC |
| GTACCCATGG | CTAGGGTACC | CAAAATGGGG | GTTTTACCCC |
| GATTTTGGGG | CTAAAACCCC | CAAATACCCG | GTTTATGGGC |
| GTTTAACCCG | CAAATTGGGC | CAACGCAACG | GTTGCGTTGC |
| GTTGCCAACG | CAACGGTTGC | CAACCGTTGG | GTTGGCAACC |
| GTTGGGTTGG | CAACCCAACC | CCACGCAACG | GGTGCGTTGC |
| [1] |
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 |
| [2] |
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 |
| [3] |
Jérôme Ducoat, Frédérique Oggier. On skew polynomial codes and lattices from quotients of cyclic division algebras. Advances in Mathematics of Communications, 2016, 10 (1) : 79-94. doi: 10.3934/amc.2016.10.79 |
| [4] |
Minjia Shi, Yaqi Lu. Cyclic DNA codes over $ \mathbb{F}_2[u,v]/\langle u^3, v^2-v, vu-uv\rangle$. Advances in Mathematics of Communications, 2019, 13 (1) : 157-164. doi: 10.3934/amc.2019009 |
| [5] |
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 |
| [6] |
Fatmanur Gursoy, Irfan Siap, Bahattin Yildiz. Construction of skew cyclic codes over $\mathbb F_q+v\mathbb F_q$. Advances in Mathematics of Communications, 2014, 8 (3) : 313-322. doi: 10.3934/amc.2014.8.313 |
| [7] |
Amit Sharma, Maheshanand Bhaintwal. A class of skew-cyclic codes over $\mathbb{Z}_4+u\mathbb{Z}_4$ with derivation. Advances in Mathematics of Communications, 2018, 12 (4) : 723-739. doi: 10.3934/amc.2018043 |
| [8] |
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 |
| [9] |
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 |
| [10] |
Alexis Eduardo Almendras Valdebenito, Andrea Luigi Tironi. On the dual codes of skew constacyclic codes. Advances in Mathematics of Communications, 2018, 12 (4) : 659-679. doi: 10.3934/amc.2018039 |
| [11] |
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 |
| [12] |
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 |
| [13] |
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 |
| [14] |
San Ling, Buket Özkaya. New bounds on the minimum distance of cyclic codes. Advances in Mathematics of Communications, 2019 doi: 10.3934/amc.2020038 |
| [15] |
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 |
| [16] |
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 |
| [17] |
Gerardo Vega, Jesús E. Cuén-Ramos. The weight distribution of families of reducible cyclic codes through the weight distribution of some irreducible cyclic codes. Advances in Mathematics of Communications, 2020, 14 (3) : 525-533. doi: 10.3934/amc.2020059 |
| [18] |
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, 2020 doi: 10.3934/amc.2020088 |
| [19] |
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, 2020 doi: 10.3934/amc.2020106 |
| [20] |
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 |
2019 Impact Factor: 0.734
Tools
Metrics
Other articles
by authors
[Back to Top]