May  2017, 11(2): 307-312. doi: 10.3934/amc.2017023

Reversible DNA codes over $F_{16}+uF_{16}+vF_{16}+uvF_{16}$

1. 

Department of Mathematics, Yildiz Technical University, Istanbul, Turkey

2. 

Department of Mathematics, Karamanoglu Mehmetbey University, Karaman, Turkey

3. 

JACODESMATH Institute, Istanbul, Turkey

Received  February 2016 Revised  March 2016 Published  May 2017

In this paper we study the structure of specific linear codes called DNA codes. The first attempts on studying such codes have been proposed over four element rings which are naturally matched with DNA four letters. Later, double (pair) DNA strings or more general $k$-DNA strings called $k$-mers have been matched with some special rings and codes over such rings with specific properties are studied. However, these matchings in general are not straightforward and because of the fact that the reverse of the codewords ($k$-mers) need to exist in the code, the matching problem is difficult and it is referred to as the reversibility problem. Here, $8$-mers (DNA 8-bases) are matched with the ring elements of $R_{16}=F_{16}+uF_{16}+vF_{16}+uvF_{16}.$ Furthermore, cyclic codes over the ring $R_{16}$ where the multiplication is taken to be noncommutative with respect to an automorphism $\theta$ are studied. The preference on the skewness is shown to be very useful and practical especially since this serves as a direct solution to the reversibility problem compared to the commutative approaches.

Citation: 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
References:
[1]

T. AbulraubA. Ghrayeb and X. N. Zeng, Construction of cyclic codes over $GF(4)$ for DNA computing, J. Franklin Inst., 343 (2006), 448-457.  doi: 10.1016/j.jfranklin.2006.02.009.

[2]

L. Adleman, Molecular computation of solutions to combinatorial problems, Science New Ser., 266 (1994), 1021-1024. 

[3]

A. BayramE. S. Oztas and I. Siap, Codes over $F_4 + v F_4$ and some DNA applications, Des.Codes Crypt., 80 (2016), 379-393.  doi: 10.1007/s10623-015-0100-8.

[4]

D. BoucherW. Geiselmann and F. Ulmer, Skew cyclic codes, Appl. Algebra Eng. Comm., 18 (2007), 379-389.  doi: 10.1007/s00200-007-0043-z.

[5]

D. BoucherP. Solé and F. Ulmer, Skew constacyclic codes over Galois rings, Adv. Math. Commun., 2 (2008), 273-292.  doi: 10.3934/amc.2008.2.273.

[6]

F. Gursoy, E. S. Oztas and I. Siap, Reversible DNA codes using skew polynomial rings Appl. Algebra Engrg. Comm. Comput. to appear.

[7]

F. GursoyI. Siap and B. Yildiz, Construction of skew cyclic codes over $F_q + vF_q$, Adv. Math. Commun., 44 (2014), 313-322.  doi: 10.3934/amc.2014.8.313.

[8]

N. Jacobson, Finite-Dimensional Division Algebras over Fields Springer, Berlin, 1996. doi: 10.1007/978-3-642-02429-0.

[9]

S. JitmanS. Ling and P. Udomkavanich, Skew constacyclic codes over finite chain rings, Adv. Math. Commun., 6 (2012), 29-63.  doi: 10.3934/amc.2012.6.39.

[10]

J. Lichtenberg, A. Yilmaz, J. Welch, K. Kurz, X. Liang, F. Drews, K. Ecker, S. Lee, M. Geisler, E. Grotewold and L. Welch, The word landscape of the non-coding segments of the Arabidopsis thaliana genome BMC Genomics 10 (2009), 463.

[11]

I. SiapT. AbualrubN. Aydin and P. Seneviratne, Skew cyclic codes of arbitrary length, Int. J. Inf. Coding Theory, 2 (2011), 10-20.  doi: 10.1504/IJICOT.2011.044674.

[12]

I. SiapT. Abulraub and A. Ghrayeb, Cyclic DNA codes over the ring $F_2[u]/(u^2-1)$ based on the deletion distance, J. Franklin Inst., 346 (2009), 731-740.  doi: 10.1016/j.jfranklin.2009.07.002.

[13]

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

[14]

T. YaoM. Shi and P. Solé, Skew cyclic codes over $F_q + uF_q + vF_q + uvF_q$, J. Algebra Comb. Discrete Appl., 2 (2015), 163-168.  doi: 10.13069/jacodesmath.90080.

[15]

B. Yildiz and I. Siap, Cyclic codes over $F_2[u]/(u^4-1)$ and applications to DNA codes, Comput. Math. Appl., 63 (2012), 1169-1176.  doi: 10.1016/j.camwa.2011.12.029.

show all references

References:
[1]

T. AbulraubA. Ghrayeb and X. N. Zeng, Construction of cyclic codes over $GF(4)$ for DNA computing, J. Franklin Inst., 343 (2006), 448-457.  doi: 10.1016/j.jfranklin.2006.02.009.

[2]

L. Adleman, Molecular computation of solutions to combinatorial problems, Science New Ser., 266 (1994), 1021-1024. 

[3]

A. BayramE. S. Oztas and I. Siap, Codes over $F_4 + v F_4$ and some DNA applications, Des.Codes Crypt., 80 (2016), 379-393.  doi: 10.1007/s10623-015-0100-8.

[4]

D. BoucherW. Geiselmann and F. Ulmer, Skew cyclic codes, Appl. Algebra Eng. Comm., 18 (2007), 379-389.  doi: 10.1007/s00200-007-0043-z.

[5]

D. BoucherP. Solé and F. Ulmer, Skew constacyclic codes over Galois rings, Adv. Math. Commun., 2 (2008), 273-292.  doi: 10.3934/amc.2008.2.273.

[6]

F. Gursoy, E. S. Oztas and I. Siap, Reversible DNA codes using skew polynomial rings Appl. Algebra Engrg. Comm. Comput. to appear.

[7]

F. GursoyI. Siap and B. Yildiz, Construction of skew cyclic codes over $F_q + vF_q$, Adv. Math. Commun., 44 (2014), 313-322.  doi: 10.3934/amc.2014.8.313.

[8]

N. Jacobson, Finite-Dimensional Division Algebras over Fields Springer, Berlin, 1996. doi: 10.1007/978-3-642-02429-0.

[9]

S. JitmanS. Ling and P. Udomkavanich, Skew constacyclic codes over finite chain rings, Adv. Math. Commun., 6 (2012), 29-63.  doi: 10.3934/amc.2012.6.39.

[10]

J. Lichtenberg, A. Yilmaz, J. Welch, K. Kurz, X. Liang, F. Drews, K. Ecker, S. Lee, M. Geisler, E. Grotewold and L. Welch, The word landscape of the non-coding segments of the Arabidopsis thaliana genome BMC Genomics 10 (2009), 463.

[11]

I. SiapT. AbualrubN. Aydin and P. Seneviratne, Skew cyclic codes of arbitrary length, Int. J. Inf. Coding Theory, 2 (2011), 10-20.  doi: 10.1504/IJICOT.2011.044674.

[12]

I. SiapT. Abulraub and A. Ghrayeb, Cyclic DNA codes over the ring $F_2[u]/(u^2-1)$ based on the deletion distance, J. Franklin Inst., 346 (2009), 731-740.  doi: 10.1016/j.jfranklin.2009.07.002.

[13]

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

[14]

T. YaoM. Shi and P. Solé, Skew cyclic codes over $F_q + uF_q + vF_q + uvF_q$, J. Algebra Comb. Discrete Appl., 2 (2015), 163-168.  doi: 10.13069/jacodesmath.90080.

[15]

B. Yildiz and I. Siap, Cyclic codes over $F_2[u]/(u^4-1)$ and applications to DNA codes, Comput. Math. Appl., 63 (2012), 1169-1176.  doi: 10.1016/j.camwa.2011.12.029.

Table 1.  The $\tau$ mapping between DNA pairs and $F_{16}$ [13]}
$F_{16}$(multiplicative) $F_{16}$(additive) Double DNA pair
0 0 AA
$\alpha^0$ 1 TT
$\alpha^1$ $\alpha$ AT
$\alpha^2$ $\alpha^2$ GC
$\alpha^3$ $\alpha^3$ AG
$\alpha^4$ $1+\alpha$ TA
$\alpha^5$ $ \alpha+\alpha^2$ CC
$\alpha^6$ $\alpha^2 +\alpha^3$ AC
$\alpha^7$ $1+\alpha +\alpha^3$ GT
$\alpha^8$ $1 +\alpha^2$ CG
$\alpha^9$ $\alpha +\alpha^3$ CA
$\alpha^{10}$ $1+\alpha +\alpha^2$ GG
$\alpha^{11}$ $\alpha +\alpha^2+\alpha^3$ CT
$\alpha^{12}$ $1+\alpha +\alpha^2+\alpha^3$ GA
$\alpha^{13}$ $1 +\alpha^2+\alpha^3$ TG
$\alpha^{14}$ $1+\alpha^3$ TC
$F_{16}$(multiplicative) $F_{16}$(additive) Double DNA pair
0 0 AA
$\alpha^0$ 1 TT
$\alpha^1$ $\alpha$ AT
$\alpha^2$ $\alpha^2$ GC
$\alpha^3$ $\alpha^3$ AG
$\alpha^4$ $1+\alpha$ TA
$\alpha^5$ $ \alpha+\alpha^2$ CC
$\alpha^6$ $\alpha^2 +\alpha^3$ AC
$\alpha^7$ $1+\alpha +\alpha^3$ GT
$\alpha^8$ $1 +\alpha^2$ CG
$\alpha^9$ $\alpha +\alpha^3$ CA
$\alpha^{10}$ $1+\alpha +\alpha^2$ GG
$\alpha^{11}$ $\alpha +\alpha^2+\alpha^3$ CT
$\alpha^{12}$ $1+\alpha +\alpha^2+\alpha^3$ GA
$\alpha^{13}$ $1 +\alpha^2+\alpha^3$ TG
$\alpha^{14}$ $1+\alpha^3$ TC
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