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Cyclic DNA codes over $ \mathbb{F}_2[u,v]/\langle u^3, v^2-v, vu-uv\rangle$

  • * Corresponding author: Minjia Shi

    * Corresponding author: Minjia Shi 
This paper is supported by National Natural Science Foundation of China (61672036), Excellent Youth Foundation of Natural Science Foundation of Anhui Province (1808085J20).
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  • In this paper, we construct cyclic DNA codes over the ring $R = \mathbb{F}_2[u,v]/\langle u^3, v^2-v, vu-uv\rangle$ . The correspondence between the elements of $R$ and the alphabet $\{A,T,G,C\}^{3}$ is obtained by a given Gray map. Moreover, some properties of binary images of the Condons under the Gray map are also discussed. Finally, two examples of cyclic DNA codes over $R$ are presented to illustrate the obtained results.

    Mathematics Subject Classification: Primary: 94B05; Secondary: 94B15.


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  • Table 1.  $\xi$-table for DNA correspondence

    Elements over $R$ Gray images over $R_1^{'}$ Elements over $S_{D_{4}}^{3}$
    $0$ $(0,0,0)$ $AAA$
    $1$ $(0,0,1)$ $AAG$
    $v$ $(0,0,v)$ $AAC$
    $u$ $(1,1,1)$ $GGG$
    $uv$ $(v,v,v)$ $CCC$
    $u^{2}$ $(0,1,1)$ $AGG$
    $u^{2}v$ $(0,v,v)$ $ACC$
    $1+v$ $(0,0,1+v)$ $AAT$
    $1+u$ $(1,1,0)$ $GGA$
    $1+uv$ $(v,v,1+v)$ $CCT$
    $1+u^{2}$ $(0,1,0)$ $AGA$
    $1+u^{2}v$ $(0,v,1+v)$ $ACT$
    $v+u$ $(1,1,1+v)$ $GGT$
    $v+uv$ $(v,v,0)$ $CCA$
    $v+u^{2}$ $(0,1,1+v)$ $AGT$
    $v+u^{2}v$ $(0,v,0)$ $ACA$
    $u+uv$ $(1+v,1+v,1+v)$ $TTT$
    $u+u^{2}$ $(1,0,0)$ $GAA$
    $u+u^{2}v$ $(1,1+v,1+v)$ $GTT$
    $uv+u^{2}$ $(v,1+v,1+v)$ $CTT$
    $uv+u^{2}v$ $(v,0,0)$ $CAA$
    $u^{2}+u^{2}v$ $(0,1+v,1+v)$ $ATT$
    $1+v+u$ $(1,1,v)$ $GGC$
    $1+v+uv$ $(v,v,1)$ $CCG$
    $1+v+u^{2}$ $(0,1,v)$ $AGC$
    $1+v+u^{2}v$ $(0,v,1)$ $ACG$
    $v+u+uv$ $(1+v,1+v,1)$ $TTG$
    $v+u+u^{2}$ $(1,0,v)$ $GAC$
    $v+u+u^{2}v$ $(1,1+v,1)$ $GTG$
    $u+uv+u^{2}$ $(1+v,v,0)$ $TCA$
    $v+uv+u^{2}v$ $(1+v,1,1)$ $TGG$
    $uv+u^{2}+u^{2}v$ $(v,1,1)$ $CGG$
    $1+u+uv$ $(1+v,1+v,1)$ $TTG$
    $1+u+u^{2}$ $(1,0,1)$ $GAG$
    $1+u+u^{2}v$ $(1,1+v,v)$ $GTC$
    $v+uv+u^{2}$ $(v,1+v,1)$ $CTG$
    $v+uv+u^{2}v$ $(v,0,v)$ $CAC$
    $u+u^{2}+u^{2}v$ $(1,v,v)$ $GCC$
    $1+uv+u^{2}$ $(v,1+v,v)$ $CTC$
    $1+uv+u^{2}v$ $(v,0,1)$ $CAG$
    $v+u^{2}+u^{2}v$ $(0,1+v,1)$ $ATG$
    $1+u^{2}+u^{2}v$ $(0,1+v,v)$ $ATC$
    $u+uv+u^{2}+u^{2}v$ $(1+v,0,0)$ $TAA$
    $v+uv+u^{2}+u^{2}v$ $(v,1,1+v)$ $CGT$
    $v+u+u^{2}+u^{2}v$ $(1,v,0)$ $GCA$
    $v+u+uv+u^{2}v$ $(1+v,1,1+v)$ $TGT$
    $v+u+uv+u^{2}$ $(1+v,v,0)$ $TCA$
    $1+v+u+uv$ $(1+v,1+v,0)$ $TTA$
    $1+uv+u^{2}+u^{2}v$ $(v,1,0)$ $CGA$
    $1+u+u^{2}+u^{2}v$ $(1,v,1+v)$ $GCT$
    $1+u+uv+u^{2}v$ $(1+v,1,0)$ $TGA$
    $1+u+uv+u^{2}$ $(1+v,v,1+v)$ $TCT$
    $1+v+u^{2}+u^{2}v$ $(0,1+v,0)$ $ATA$
    $1+v+uv+u^{2}v$ $(v,0,1+v)$ $CAT$
    $1+v+uv+u^{2}$ $(v,1+v,0)$ $CTA$
    $1+v+u+u^{2}v$ $(1,1+v,0)$ $GTA$
    $1+v+u+u^{2}$ $(1,0,1+v)$ $GAT$
    $v+u+uv+u^{2}+u^{2}v$ $(1+v,0,v)$ $TAC$
    $1+u+uv+u^{2}+u^{2}v$ $(1+v,0,1)$ $TAG$
    $1+v+uv+u^{2}+u^{2}v$ $(v,1,v)$ $CGC$
    $1+v+u+u^{2}+u^{2}v$ $(1,v,1)$ $GCG$
    $1+v+u+uv+u^{2}v$ $(1+v,1,v)$ $TGC$
    $1+v+u+uv+u^{2}$ $(1+v,v,1)$ $TCG$
    $1+v+u+uv+u^{2}+u^{2}v$ $(1+v,0,1+v)$ $TAT$
     | Show Table
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    Table 2.  Binary Images of the Codons

    $AAA$ $000000$ $CCC$ $010101$ $GGG$ $111111$ $TTT$ $101010$
    $AAG$ $000011$ $CCT$ $010110$ $GGA$ $111100$ $TTC$ $101010$
    $AAC$ $000001$ $CCA$ $010100$ $GGT$ $111110$ $TTG$ $101001$
    $AAT$ $000010$ $CCG$ $010111$ $GGC$ $111101$ $TTA$ $101000$
    $AGA$ $001100$ $CTC$ $011001$ $GAG$ $110011$ $TCT$ $100110$
    $AGG$ $001111$ $CTT$ $011010$ $GAA$ $110000$ $TCC$ $101010$
    $AGC$ $001101$ $CTA$ $011000$ $GAT$ $110010$ $TCG$ $100111$
    $AGT$ $001110$ $CTG$ $011011$ $GAC$ $110001$ $TCA$ $100100$
    $ACA$ $000100$ $CAC$ $010011$ $GTG$ $111011$ $TGT$ $101110$
    $ACG$ $000111$ $CAT$ $010010$ $GTA$ $111000$ $TGC$ $101101$
    $ACC$ $000101$ $CAA$ $010000$ $GTT$ $111010$ $TGG$ $101111$
    $ACT$ $000110$ $CAG$ $010011$ $GTC$ $111001$ $TGA$ $101100$
    $ATA$ $001000$ $CGA$ $011100$ $GCG$ $110111$ $TAT$ $100010$
    $ATG$ $001011$ $CGT$ $011110$ $GCA$ $110100$ $TAC$ $100001$
    $ATC$ $001001$ $CGC$ $011101$ $GCT$ $110110$ $TAG$ $100011$
    $ATT$ $001010$ $CGG$ $011111$ $GCC$ $110101$ $TAA$ $100000$
     | Show Table
    DownLoad: CSV

    Table 3.  The cyclic code $C$ over $R$ of length $4$

    Codewords of $C$ $\phi(c)$
    $(0,0,0,0)$ $AAAAAAAAAAAA$
    $(u^2+u^2v,u^2+u^2v,u^2+u^2v,u^2+u^2v)$ $ATTATTATTATT$
    $(u^2+u^2v,0,u^2+u^2v,0)$ $ATTAAAATTAAA$
    $(u+uv,u+uv,u+uv,u+uv)$ $TTTTTTTTTTTT$
    $(0,u^2+u^2v,0,u^2+u^2v)$ $AAAATTAAAATT$
     | Show Table
    DownLoad: CSV
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