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

Minjia Shi; Yaqi Lu

doi:10.3934/amc.2019009

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2019, Volume 13, Issue 1: 157-164. Doi: 10.3934/amc.2019009

This issue Previous Article The weight distribution of a class of p-ary cyclic codes and their applications Next Article New 2-designs over finite fields from derived and residual designs

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

Minjia Shi^, and
Yaqi Lu

School of Mathematical Sciences, Anhui University, Hefei 230601, China

* Corresponding author: Minjia Shi
* Corresponding author: Minjia Shi

Received: May 2018

Published: December 2018

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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Abstract

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.

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

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Table 2. Binary Images of the Codons

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

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

Codewords of $C$	$\phi(c)$
$(0,0,0,0)$	$AAAAAAAAAAAA$
$(1+v+u+uv+u^2+u^2v,1+v+u^2$ $+u^2v,1+v+u+uv,1+v)$	$TATATATTAAAT$
$(u+uv+u^2+u^2v,u+uv,$ $u+uv+u^2+u^2v,u+uv)$	$TAATTTTAATTT$
$(1+v,1+v+u+uv+u^2+u^2v,$ $1+v+u^2+u^2v,1+v+u+uv)$	$AATTATATATTA$
$(u^2+u^2v,u^2+u^2v,u^2+u^2v,u^2+u^2v)$	$ATTATTATTATT$
$(1+v+u+uv,1+v,1+v+u+uv$ $+u^2+u^2v,1+v+u^2+u^2v)$	$TTAAATTATATA$
$(u+uv,u+uv+u^2+u^2v,u+uv,$ $u+uv+u^2+u^2v)$	$TTTTAATTTTAA$
$(1+v+u^2+u^2v,1+v+u+uv,1+v,$ $1+v+u+uv+u^2+u^2v)$	$ATATTAAATTAT$
$(u^2+u^2v,0,u^2+u^2v,0)$	$ATTAAAATTAAA$
$(1+v+u+uv,1+v+u^2+u^2v,$ $1+v+u+uv+u^2+u^2v,1+v)$	$TTAATATATAAT$
$(u+uv,u+uv,u+uv,u+uv)$	$TTTTTTTTTTTT$
$(1+v+u^2+u^2v,1+v+u+uv+$ $u^2+u^2v,1+v,1+v+u+uv)$	$ATATATAATTTA$
$(0,u^2+u^2v,0,u^2+u^2v)$	$AAAATTAAAATT$
$(1+v+u+uv+u^2+u^2v,1+v,$ $1+v+u+uv,1+v+u^2+u^2v)$	$TATAATTTAATA$
$(u+uv+u^2+u^2v,u+uv+u^2+u^2v,$ $u+uv+u^2+u^2v,u+uv+u^2+u^2v)$	$TAATAATAATAA$
$(1+v,1+v+u+uv,1+v+u^2+u^2v,$ $1+v+u+uv+u^2+u^2v)$	$AATTTAATATAT$

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