American Institute of Mathematical Sciences

Advanced
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. Google Scholar

[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. Google Scholar

[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.Google Scholar

[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. Google Scholar

[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. Google Scholar

[6]

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

[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.Google Scholar

[8]

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

[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. Google Scholar

[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. Google Scholar

[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.Google Scholar

[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. Google Scholar

[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. Google Scholar

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. Google Scholar

[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. Google Scholar

[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.Google Scholar

[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. Google Scholar

[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. Google Scholar

[6]

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

[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.Google Scholar

[8]

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

[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. Google Scholar

[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. Google Scholar

[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.Google Scholar

[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. Google Scholar

[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. Google Scholar

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

Download as excel

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

Download as excel

$AAA$ $000000$ $CCC$ $010101$ $GGG$ $111111$ $TTT$ $101010$
$AAG$ $000011$ $CCT$ $010110$ $GGA$ $111100$ $TTC$ $101010$
$AAC$ $000001$ $CCA$ $010100$ $GGT$ $111110$ $TTG$ $101001$
$AAT$ $000010$ $CCG$ $010111$ $GGC$ $111101$ $TTA$ $101000$
$AGA$ $001100$ $CTC$ $011001$ $GAG$ $110011$ $TCT$ $100110$
$AGG$ $001111$ $CTT$ $011010$ $GAA$ $110000$ $TCC$ $101010$
$AGC$ $001101$ $CTA$ $011000$ $GAT$ $110010$ $TCG$ $100111$
$AGT$ $001110$ $CTG$ $011011$ $GAC$ $110001$ $TCA$ $100100$
$ACA$ $000100$ $CAC$ $010011$ $GTG$ $111011$ $TGT$ $101110$
$ACG$ $000111$ $CAT$ $010010$ $GTA$ $111000$ $TGC$ $101101$
$ACC$ $000101$ $CAA$ $010000$ $GTT$ $111010$ $TGG$ $101111$
$ACT$ $000110$ $CAG$ $010011$ $GTC$ $111001$ $TGA$ $101100$
$ATA$ $001000$ $CGA$ $011100$ $GCG$ $110111$ $TAT$ $100010$
$ATG$ $001011$ $CGT$ $011110$ $GCA$ $110100$ $TAC$ $100001$
$ATC$ $001001$ $CGC$ $011101$ $GCT$ $110110$ $TAG$ $100011$
$ATT$ $001010$ $CGG$ $011111$ $GCC$ $110101$ $TAA$ $100000$
Table 3.  The cyclic code $C$ over $R$ of length $4$
Codewords of $C$ $\phi(c)$
$(0,0,0,0)$ $AAAAAAAAAAAA$
$(1+v+u+uv+u^2+u^2v,1+v+u^2$
$+u^2v,1+v+u+uv,1+v)$		 $TATATATTAAAT$
$(u+uv+u^2+u^2v,u+uv,$
$u+uv+u^2+u^2v,u+uv)$		 $TAATTTTAATTT$
$(1+v,1+v+u+uv+u^2+u^2v,$
$1+v+u^2+u^2v,1+v+u+uv)$		 $AATTATATATTA$
$(u^2+u^2v,u^2+u^2v,u^2+u^2v,u^2+u^2v)$ $ATTATTATTATT$
$(1+v+u+uv,1+v,1+v+u+uv$
$+u^2+u^2v,1+v+u^2+u^2v)$		 $TTAAATTATATA$
$(u+uv,u+uv+u^2+u^2v,u+uv,$
$u+uv+u^2+u^2v)$		 $TTTTAATTTTAA$
$(1+v+u^2+u^2v,1+v+u+uv,1+v,$
$1+v+u+uv+u^2+u^2v)$		 $ATATTAAATTAT$
$(u^2+u^2v,0,u^2+u^2v,0)$ $ATTAAAATTAAA$
$(1+v+u+uv,1+v+u^2+u^2v,$
$1+v+u+uv+u^2+u^2v,1+v)$		 $TTAATATATAAT$
$(u+uv,u+uv,u+uv,u+uv)$ $TTTTTTTTTTTT$
$(1+v+u^2+u^2v,1+v+u+uv+$
$u^2+u^2v,1+v,1+v+u+uv)$		 $ATATATAATTTA$
$(0,u^2+u^2v,0,u^2+u^2v)$ $AAAATTAAAATT$
$(1+v+u+uv+u^2+u^2v,1+v,$
$1+v+u+uv,1+v+u^2+u^2v)$		 $TATAATTTAATA$
$(u+uv+u^2+u^2v,u+uv+u^2+u^2v,$
$u+uv+u^2+u^2v,u+uv+u^2+u^2v)$		 $TAATAATAATAA$
$(1+v,1+v+u+uv,1+v+u^2+u^2v,$
$1+v+u+uv+u^2+u^2v)$		 $AATTTAATATAT$
Table Options

Download as excel

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

Nabil Bennenni, Kenza Guenda, Sihem Mesnager. DNA cyclic codes over rings. Advances in Mathematics of Communications, 2017, 11 (1) : 83-98. doi: 10.3934/amc.2017004
[2]

Fatmanur Gursoy, Elif Segah Oztas, Irfan Siap. Reversible DNA codes over $F_{16}+uF_{16}+vF_{16}+uvF_{16}$. Advances in Mathematics of Communications, 2017, 11 (2) : 307-312. doi: 10.3934/amc.2017023
[3]

Rafael Arce-Nazario, Francis N. Castro, Jose Ortiz-Ubarri. On the covering radius of some binary cyclic codes. Advances in Mathematics of Communications, 2017, 11 (2) : 329-338. doi: 10.3934/amc.2017025
[4]

Steven T. Dougherty, Cristina Fernández-Córdoba. Codes over $\mathbb{Z}_{2^k}$, Gray map and self-dual codes. Advances in Mathematics of Communications, 2011, 5 (4) : 571-588. doi: 10.3934/amc.2011.5.571
[5]

Michael Braun. On lattices, binary codes, and network codes. Advances in Mathematics of Communications, 2011, 5 (2) : 225-232. doi: 10.3934/amc.2011.5.225
[6]

Cem Güneri, Ferruh Özbudak, Funda ÖzdemIr. On complementary dual additive cyclic codes. Advances in Mathematics of Communications, 2017, 11 (2) : 353-357. doi: 10.3934/amc.2017028
[7]

Heide Gluesing-Luerssen, Katherine Morrison, Carolyn Troha. Cyclic orbit codes and stabilizer subfields. Advances in Mathematics of Communications, 2015, 9 (2) : 177-197. doi: 10.3934/amc.2015.9.177
[8]

Joaquim Borges, Ivan Yu. Mogilnykh, Josep Rifà, Faina I. Solov'eva. Structural properties of binary propelinear codes. Advances in Mathematics of Communications, 2012, 6 (3) : 329-346. doi: 10.3934/amc.2012.6.329
[9]

Martianus Frederic Ezerman, San Ling, Patrick Solé, Olfa Yemen. From skew-cyclic codes to asymmetric quantum codes. Advances in Mathematics of Communications, 2011, 5 (1) : 41-57. doi: 10.3934/amc.2011.5.41
[10]

Yunwen Liu, Longjiang Qu, Chao Li. New constructions of systematic authentication codes from three classes of cyclic codes. Advances in Mathematics of Communications, 2018, 12 (1) : 1-16. doi: 10.3934/amc.2018001
[11]

Steven T. Dougherty, Esengül Saltürk, Steve Szabo. Codes over local rings of order 16 and binary codes. Advances in Mathematics of Communications, 2016, 10 (2) : 379-391. doi: 10.3934/amc.2016012
[12]

Heide Gluesing-Luerssen, Fai-Lung Tsang. A matrix ring description for cyclic convolutional codes. Advances in Mathematics of Communications, 2008, 2 (1) : 55-81. doi: 10.3934/amc.2008.2.55
[13]

Long Yu, Hongwei Liu. A class of $p$-ary cyclic codes and their weight enumerators. Advances in Mathematics of Communications, 2016, 10 (2) : 437-457. doi: 10.3934/amc.2016017
[14]

Umberto Martínez-Peñas. Rank equivalent and rank degenerate skew cyclic codes. Advances in Mathematics of Communications, 2017, 11 (2) : 267-282. doi: 10.3934/amc.2017018
[15]

Heide Gluesing-Luerssen, Uwe Helmke, José Ignacio Iglesias Curto. Algebraic decoding for doubly cyclic convolutional codes. Advances in Mathematics of Communications, 2010, 4 (1) : 83-99. doi: 10.3934/amc.2010.4.83
[16]

Stefka Bouyuklieva, Anton Malevich, Wolfgang Willems. On the performance of binary extremal self-dual codes. Advances in Mathematics of Communications, 2011, 5 (2) : 267-274. doi: 10.3934/amc.2011.5.267
[17]

Daniel Heinlein, Sascha Kurz. Binary subspace codes in small ambient spaces. Advances in Mathematics of Communications, 2018, 12 (4) : 817-839. doi: 10.3934/amc.2018048
[18]

Fernando Hernando, Tom Høholdt, Diego Ruano. List decoding of matrix-product codes from nested codes: An application to quasi-cyclic codes. Advances in Mathematics of Communications, 2012, 6 (3) : 259-272. doi: 10.3934/amc.2012.6.259
[19]

Yuncheng You. Dynamics of three-component reversible Gray-Scott model. Discrete & Continuous Dynamical Systems - B, 2010, 14 (4) : 1671-1688. doi: 10.3934/dcdsb.2010.14.1671
[20]

Jérôme Ducoat, Frédérique Oggier. On skew polynomial codes and lattices from quotients of cyclic division algebras. Advances in Mathematics of Communications, 2016, 10 (1) : 79-94. doi: 10.3934/amc.2016.10.79

2018 Impact Factor: 0.879

Tools

Metrics

  • PDF downloads (91)
  • HTML views (271)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences