Table 1.
Identifying codons with the elements of the ring $R$.
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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
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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. |
| [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.
|
| [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. |
| [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. |
| [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.
|
| [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. |
| [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 |
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