American Institute of Mathematical Sciences

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February  2019, 13(1): 157-164. doi: 10.3934/amc.2019009

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

Received  May 2018 Published  December 2018

Fund Project: This paper is supported by National Natural Science Foundation of China (61672036), Excellent Youth Foundation of Natural Science Foundation of Anhui Province (1808085J20).

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.

Keywords: Cyclic DNA codes, Gray map, reversible codes, reversible-complement codes, binary images.
Mathematics Subject Classification: Primary: 94B05; Secondary: 94B15.
Citation: Minjia Shi, Yaqi Lu. Cyclic DNA codes over $ \mathbb{F}_2[u,v]/\langle u^3, v^2-v, vu-uv\rangle$. Advances in Mathematics of Communications, 2019, 13 (1) : 157-164. doi: 10.3934/amc.2019009
References:
[1]

T. Abualrub, A. Ghrayeb and X. N. Zeng, Construction of cyclic codes over $ \mathbb{F}_4$ for DNA computing, Appl. Algebra in Engrg. Comm. Comput., 24 (2006), 445-459. 

[2]

T. Abualrub and I. Siap, Cyclic codes over the rings $ \mathbb{Z}_2+u\mathbb{Z}_2$ and $ \mathbb{Z}_2+u\mathbb{Z}_2+u^2\mathbb{Z}_2$, Des. Codes Cryptogr., 42 (2007), 273-287.  doi: 10.1007/s10623-006-9034-5.

[3]

A. D'Yachkov, A. Macula, T. Renz, P. Vilenkin and I. Ismagilov, New results on DNA codes, IEEE International Symposium on Information Theory, Adelaide, SA, Australia, 2005, 283-287.

[4]

H. Q. Dinh, A. K. Singh, S. Pattanayak and S. Sriboonchitta, Cyclic DNA codes over the ring $ \mathbb{F}_2 + u\mathbb{F}_2 + v\mathbb{F}_2 + uv\mathbb{F}_2 + v^2\mathbb{F}_2 + uv^2\mathbb{F}_2$, Des. Codes Cryptogr., 86 (2018), 1451-1467.  doi: 10.1007/s10623-017-0405-x.

[5]

Q. Q. Feng and W. G. Zhou, Cyclic code and self-dual code over $ \mathbb{F}_2+u\mathbb{F}_2+u^2\mathbb{F}_2$, Journal of Mathematical Research and Exposition, 29 (2009), 500-506. 

[6]

K. Guenda, T. A. Gulliver and P. Solé, On cyclic DNA codes, IEEE International Symposium on Information Theory, Istanbul, (2012), 121-125. 

[7]

H. Mostafanasab and A. Y. Darani, On cyclic DNA codes over $ \mathbb{F}_2 + u\mathbb{F}_2 + u^2\mathbb{F}_2$, arXiv: 1603.05894vl [cs.IT] 18 Mar 2016.

[8]

J. L. Massey, Reversible codes, Information and Control, 7 (1964), 369-380.  doi: 10.1016/S0019-9958(64)90438-3.

[9]

E. S. Oztas and I. Siap, Lifted polynomials over $ \mathbb{F}_{16}$ and their applications to DNA codes, Filomat, 27 (2013), 459-466.  doi: 10.2298/FIL1303459O.

[10]

J. F. Qian, L. N. Zhang and S. X. Zhu, Constacyclic and cyclic codes over $ \mathbb{F}_2+u\mathbb{F}_2+u^2\mathbb{F}_2$, IEICE Transactions on Fundamentals of Electronics Communications and Computer Sciences, E89-A (2006), 1863-1865. 

[11]

V. Rykov, A. J. Macula, D. Torny and P. White, DNA sequence and quaternary cyclic codes, IEEE International Syposium on Information Theory, Washington, DC, USA, 2001,248.

[12]

I. Siap, T. Abualrub and A. Ghrayeb, Cyclic DNA codes over the ring $ \mathbb{F}_2[u]/\langle u^2-1\rangle$ based on the deletion distance, J. Franklin Inst., 346 (2009), 731-740.  doi: 10.1016/j.jfranklin.2009.07.002.

[13]

S. X. Zhu and X. J. Chen, Cyclic DNA codes over $ \mathbb{F}_2+u\mathbb{F}_2+v\mathbb{F}_2+uv\mathbb{F}_2$ and their applications, Journal of Applied Mathematics and Computing, 55 (2017), 479-493.  doi: 10.1007/s12190-016-1046-3.

show all references

References:
[1]

T. Abualrub, A. Ghrayeb and X. N. Zeng, Construction of cyclic codes over $ \mathbb{F}_4$ for DNA computing, Appl. Algebra in Engrg. Comm. Comput., 24 (2006), 445-459. 

[2]

T. Abualrub and I. Siap, Cyclic codes over the rings $ \mathbb{Z}_2+u\mathbb{Z}_2$ and $ \mathbb{Z}_2+u\mathbb{Z}_2+u^2\mathbb{Z}_2$, Des. Codes Cryptogr., 42 (2007), 273-287.  doi: 10.1007/s10623-006-9034-5.

[3]

A. D'Yachkov, A. Macula, T. Renz, P. Vilenkin and I. Ismagilov, New results on DNA codes, IEEE International Symposium on Information Theory, Adelaide, SA, Australia, 2005, 283-287.

[4]

H. Q. Dinh, A. K. Singh, S. Pattanayak and S. Sriboonchitta, Cyclic DNA codes over the ring $ \mathbb{F}_2 + u\mathbb{F}_2 + v\mathbb{F}_2 + uv\mathbb{F}_2 + v^2\mathbb{F}_2 + uv^2\mathbb{F}_2$, Des. Codes Cryptogr., 86 (2018), 1451-1467.  doi: 10.1007/s10623-017-0405-x.

[5]

Q. Q. Feng and W. G. Zhou, Cyclic code and self-dual code over $ \mathbb{F}_2+u\mathbb{F}_2+u^2\mathbb{F}_2$, Journal of Mathematical Research and Exposition, 29 (2009), 500-506. 

[6]

K. Guenda, T. A. Gulliver and P. Solé, On cyclic DNA codes, IEEE International Symposium on Information Theory, Istanbul, (2012), 121-125. 

[7]

H. Mostafanasab and A. Y. Darani, On cyclic DNA codes over $ \mathbb{F}_2 + u\mathbb{F}_2 + u^2\mathbb{F}_2$, arXiv: 1603.05894vl [cs.IT] 18 Mar 2016.

[8]

J. L. Massey, Reversible codes, Information and Control, 7 (1964), 369-380.  doi: 10.1016/S0019-9958(64)90438-3.

[9]

E. S. Oztas and I. Siap, Lifted polynomials over $ \mathbb{F}_{16}$ and their applications to DNA codes, Filomat, 27 (2013), 459-466.  doi: 10.2298/FIL1303459O.

[10]

J. F. Qian, L. N. Zhang and S. X. Zhu, Constacyclic and cyclic codes over $ \mathbb{F}_2+u\mathbb{F}_2+u^2\mathbb{F}_2$, IEICE Transactions on Fundamentals of Electronics Communications and Computer Sciences, E89-A (2006), 1863-1865. 

[11]

V. Rykov, A. J. Macula, D. Torny and P. White, DNA sequence and quaternary cyclic codes, IEEE International Syposium on Information Theory, Washington, DC, USA, 2001,248.

[12]

I. Siap, T. Abualrub and A. Ghrayeb, Cyclic DNA codes over the ring $ \mathbb{F}_2[u]/\langle u^2-1\rangle$ based on the deletion distance, J. Franklin Inst., 346 (2009), 731-740.  doi: 10.1016/j.jfranklin.2009.07.002.

[13]

S. X. Zhu and X. J. Chen, Cyclic DNA codes over $ \mathbb{F}_2+u\mathbb{F}_2+v\mathbb{F}_2+uv\mathbb{F}_2$ and their applications, Journal of Applied Mathematics and Computing, 55 (2017), 479-493.  doi: 10.1007/s12190-016-1046-3.

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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Elements over $R$ Gray images over $R_1^{'}$ Elements over $S_{D_{4}}^{3}$
$0$ $(0,0,0)$ $AAA$
$1$ $(0,0,1)$ $AAG$
$v$ $(0,0,v)$ $AAC$
$u$ $(1,1,1)$ $GGG$
$uv$ $(v,v,v)$ $CCC$
$u^{2}$ $(0,1,1)$ $AGG$
$u^{2}v$ $(0,v,v)$ $ACC$
$1+v$ $(0,0,1+v)$ $AAT$
$1+u$ $(1,1,0)$ $GGA$
$1+uv$ $(v,v,1+v)$ $CCT$
$1+u^{2}$ $(0,1,0)$ $AGA$
$1+u^{2}v$ $(0,v,1+v)$ $ACT$
$v+u$ $(1,1,1+v)$ $GGT$
$v+uv$ $(v,v,0)$ $CCA$
$v+u^{2}$ $(0,1,1+v)$ $AGT$
$v+u^{2}v$ $(0,v,0)$ $ACA$
$u+uv$ $(1+v,1+v,1+v)$ $TTT$
$u+u^{2}$ $(1,0,0)$ $GAA$
$u+u^{2}v$ $(1,1+v,1+v)$ $GTT$
$uv+u^{2}$ $(v,1+v,1+v)$ $CTT$
$uv+u^{2}v$ $(v,0,0)$ $CAA$
$u^{2}+u^{2}v$ $(0,1+v,1+v)$ $ATT$
$1+v+u$ $(1,1,v)$ $GGC$
$1+v+uv$ $(v,v,1)$ $CCG$
$1+v+u^{2}$ $(0,1,v)$ $AGC$
$1+v+u^{2}v$ $(0,v,1)$ $ACG$
$v+u+uv$ $(1+v,1+v,1)$ $TTG$
$v+u+u^{2}$ $(1,0,v)$ $GAC$
$v+u+u^{2}v$ $(1,1+v,1)$ $GTG$
$u+uv+u^{2}$ $(1+v,v,0)$ $TCA$
$v+uv+u^{2}v$ $(1+v,1,1)$ $TGG$
$uv+u^{2}+u^{2}v$ $(v,1,1)$ $CGG$
$1+u+uv$ $(1+v,1+v,1)$ $TTG$
$1+u+u^{2}$ $(1,0,1)$ $GAG$
$1+u+u^{2}v$ $(1,1+v,v)$ $GTC$
$v+uv+u^{2}$ $(v,1+v,1)$ $CTG$
$v+uv+u^{2}v$ $(v,0,v)$ $CAC$
$u+u^{2}+u^{2}v$ $(1,v,v)$ $GCC$
$1+uv+u^{2}$ $(v,1+v,v)$ $CTC$
$1+uv+u^{2}v$ $(v,0,1)$ $CAG$
$v+u^{2}+u^{2}v$ $(0,1+v,1)$ $ATG$
$1+u^{2}+u^{2}v$ $(0,1+v,v)$ $ATC$
$u+uv+u^{2}+u^{2}v$ $(1+v,0,0)$ $TAA$
$v+uv+u^{2}+u^{2}v$ $(v,1,1+v)$ $CGT$
$v+u+u^{2}+u^{2}v$ $(1,v,0)$ $GCA$
$v+u+uv+u^{2}v$ $(1+v,1,1+v)$ $TGT$
$v+u+uv+u^{2}$ $(1+v,v,0)$ $TCA$
$1+v+u+uv$ $(1+v,1+v,0)$ $TTA$
$1+uv+u^{2}+u^{2}v$ $(v,1,0)$ $CGA$
$1+u+u^{2}+u^{2}v$ $(1,v,1+v)$ $GCT$
$1+u+uv+u^{2}v$ $(1+v,1,0)$ $TGA$
$1+u+uv+u^{2}$ $(1+v,v,1+v)$ $TCT$
$1+v+u^{2}+u^{2}v$ $(0,1+v,0)$ $ATA$
$1+v+uv+u^{2}v$ $(v,0,1+v)$ $CAT$
$1+v+uv+u^{2}$ $(v,1+v,0)$ $CTA$
$1+v+u+u^{2}v$ $(1,1+v,0)$ $GTA$
$1+v+u+u^{2}$ $(1,0,1+v)$ $GAT$
$v+u+uv+u^{2}+u^{2}v$ $(1+v,0,v)$ $TAC$
$1+u+uv+u^{2}+u^{2}v$ $(1+v,0,1)$ $TAG$
$1+v+uv+u^{2}+u^{2}v$ $(v,1,v)$ $CGC$
$1+v+u+u^{2}+u^{2}v$ $(1,v,1)$ $GCG$
$1+v+u+uv+u^{2}v$ $(1+v,1,v)$ $TGC$
$1+v+u+uv+u^{2}$ $(1+v,v,1)$ $TCG$
$1+v+u+uv+u^{2}+u^{2}v$ $(1+v,0,1+v)$ $TAT$
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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$AAA$ $000000$ $CCC$ $010101$ $GGG$ $111111$ $TTT$ $101010$
$AAG$ $000011$ $CCT$ $010110$ $GGA$ $111100$ $TTC$ $101010$
$AAC$ $000001$ $CCA$ $010100$ $GGT$ $111110$ $TTG$ $101001$
$AAT$ $000010$ $CCG$ $010111$ $GGC$ $111101$ $TTA$ $101000$
$AGA$ $001100$ $CTC$ $011001$ $GAG$ $110011$ $TCT$ $100110$
$AGG$ $001111$ $CTT$ $011010$ $GAA$ $110000$ $TCC$ $101010$
$AGC$ $001101$ $CTA$ $011000$ $GAT$ $110010$ $TCG$ $100111$
$AGT$ $001110$ $CTG$ $011011$ $GAC$ $110001$ $TCA$ $100100$
$ACA$ $000100$ $CAC$ $010011$ $GTG$ $111011$ $TGT$ $101110$
$ACG$ $000111$ $CAT$ $010010$ $GTA$ $111000$ $TGC$ $101101$
$ACC$ $000101$ $CAA$ $010000$ $GTT$ $111010$ $TGG$ $101111$
$ACT$ $000110$ $CAG$ $010011$ $GTC$ $111001$ $TGA$ $101100$
$ATA$ $001000$ $CGA$ $011100$ $GCG$ $110111$ $TAT$ $100010$
$ATG$ $001011$ $CGT$ $011110$ $GCA$ $110100$ $TAC$ $100001$
$ATC$ $001001$ $CGC$ $011101$ $GCT$ $110110$ $TAG$ $100011$
$ATT$ $001010$ $CGG$ $011111$ $GCC$ $110101$ $TAA$ $100000$
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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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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