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 online: May 2017

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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.

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

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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

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