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DNA cyclic codes over rings

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  • 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.

    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 $u^5+u+1$
     | Show Table
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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
     | Show Table
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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
     | Show Table
    DownLoad: CSV

    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
     | Show Table
    DownLoad: CSV

    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
     | Show Table
    DownLoad: CSV

    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
     | Show Table
    DownLoad: CSV

    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
     | Show Table
    DownLoad: CSV

    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
     | Show Table
    DownLoad: CSV
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