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

Fatmanur Gursoy; Elif Segah Oztas; Irfan Siap

doi:10.3934/amc.2017023

Article Contents

2017, Volume 11, Issue 2: 307-312. Doi: 10.3934/amc.2017023

This issue Previous Article Explicit formulas for monomial involutions over finite fields Next Article Arrays composed from the extended rational cycle

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: September 2017

Abstract / Introduction Full Text(HTML) Figure(0) / Table(1) Related Papers Cited by

Abstract

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.

Keywords:

Mathematics Subject Classification: Primary: 94B15, 94B05; Secondary: 92D20.

Citation:

Full Text(HTML)

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

| Show Table

DownLoad: CSV

Related Papers

Cited by

References

	T. Abulraub , A. 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.
	L. Adleman , Molecular computation of solutions to combinatorial problems, Science New Ser., 266 (1994) , 1021-1024.
	A. Bayram , E. 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.
	D. Boucher , W. Geiselmann and F. Ulmer , Skew cyclic codes, Appl. Algebra Eng. Comm., 18 (2007) , 379-389. doi: 10.1007/s00200-007-0043-z.
	D. Boucher , P. Solé and F. Ulmer , Skew constacyclic codes over Galois rings, Adv. Math. Commun., 2 (2008) , 273-292. doi: 10.3934/amc.2008.2.273.
	F. Gursoy, E. S. Oztas and I. Siap, Reversible DNA codes using skew polynomial rings Appl. Algebra Engrg. Comm. Comput. to appear.
	F. Gursoy , I. 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.
	N. Jacobson, Finite-Dimensional Division Algebras over Fields Springer, Berlin, 1996. doi: 10.1007/978-3-642-02429-0.
	S. Jitman , S. 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.
	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.
	I. Siap , T. Abualrub , N. 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.
	I. Siap , T. 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.
	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.
	T. Yao , M. 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.
	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.