Reversible DNA codes using skew polynomial rings

Abstract

In this study we determine the structure of reversible DNA codes obtained from skew cyclic codes. We show that the generators of such DNA codes enjoy some special properties. We study the structural properties of such family of codes and we also illustrate our results with examples.

Reversible DNA codes from skew cyclic codes over a ring of order 256

“…Proof. This can be proven similarly to Theorem 3 in [12].…”

“…The next two theorems show that palindromic and θ-palindromic polynomials come in pairs. They are analogous to Theorem 4 and Theorem 3 in [12] stated for skew polynomials over a field.…”

“…and the coefficient of x n−t is b n−t = t j=0 h 2k−1−j θ 2k−1−j (a 2s−1−(t−j) ). By using the fact that b 0 = b n = 0 and b i = 0 for all i = 1, 2, ..., n − 1, it can be shown that h i = h 2k−1−i for all i = 0, 1, .., k − 1 as in the proof of Theorem 4 in [12], by induction.…”

Reversible DNA codes over a family of non-chain rings $R_{k,s}$

“…Also, because of its natural error correcting structure, DNA has found many applications in coding theory. DNA single basis ( [2, 13, 27, 28]), double bases ( [5,14,22,30]), and multiple bases ( [6,16,23,24,25]) over various fields and recently rings have been studied by applying different constraints especially, Hamming distance constraint. In each case, the reversibility problem has been solved by different methods.…”

DNA codes from skew dihedral group ring

“…Therefore, numerous techniques have been explored and studied that lead to constructing good DNA codes. Some known methods for designing good DNA codes include the study of reversible self-dual codes over GF (4) [21], the study of cyclic and extended cyclic constructions [1,2], the study of linear constructions [17], and the study of skew polynomial rings [18]. Recently in [11,15,23,25,24], linear codes derived from group ring elements were considered to construct reversible DNA codes that satisfy some constraints, and many new lower bounds of DNA codes of various lengths are found.…”