DNA cyclic codes over rings

Nabil Bennenni; Kenza Guenda; Sihem Mesnager

doi:10.3934/amc.2017004

Article Contents

2017, Volume 11, Issue 1: 83-98. Doi: 10.3934/amc.2017004

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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
^* Corresponding author

Received: May 31, 2015

Revised: November 30, 2015

Published: May 2017

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Abstract

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:

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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	$u^5+u+1$

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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

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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

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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

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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

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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

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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

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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

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