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

* Corresponding author

Received  June 2015 Revised  December 2015 Published  February 2017

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.

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. AbualrubN. Aydin and P. Seneviratne, On Θ-cyclic codes over $\mathbb{F}_2 + v\mathbb{F}_2$, Austral. J. Combin., 54 (2012), 115-126.   Google Scholar

[2]

T. AbualrubA. 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.  Google Scholar

[3]

L. Adleman, Molecular computation of the solutions to combinatorial problems, Science, 266 (1994), 1021-1024.  doi: 10.1126/science.7973651.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

S. T. DoughertyJ. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

R. SanchezE. 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.  Google Scholar

[19]

I. SiapT. 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.  Google Scholar

[20]

http://www.codetables.de Google Scholar

show all references

References:
[1]

T. AbualrubN. Aydin and P. Seneviratne, On Θ-cyclic codes over $\mathbb{F}_2 + v\mathbb{F}_2$, Austral. J. Combin., 54 (2012), 115-126.   Google Scholar

[2]

T. AbualrubA. 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.  Google Scholar

[3]

L. Adleman, Molecular computation of the solutions to combinatorial problems, Science, 266 (1994), 1021-1024.  doi: 10.1126/science.7973651.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

S. T. DoughertyJ. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

R. SanchezE. 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.  Google Scholar

[19]

I. SiapT. 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.  Google Scholar

[20]

http://www.codetables.de Google Scholar

Table 1.  Identifying codons with the elements of the ring $R$.
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
Table 6.  Binary image of the codons given by Table 1
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
Table 7.  A binary image of DNA cyclic codes of length 7 given Table 2
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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