On cyclic DNA codes over the rings Z4 + wZ4 and Z4 + wZ4 + vZ4 + wvZ4

Abstract

Abstract-The structures of the cyclic DNA codes of odd length over the finite rings R = Z 4 + wZ 4 , w 2 = 2 and S = Z 4 + wZ 4 + vZ 4 + wvZ 4 , w 2 = 2, v 2 = v, wv = vw are studied. The links between the elements of the rings R, S and 16 and 256 codons are established, respectively. The cyclic codes of odd length over the finite ring R satisfy reverse complement constraint and the cyclic codes of odd length over the finite ring S satisfy reverse constraint and reverse complement constraint are studied. The binary images of the cyclic DNA codes over the finite rings R and S are determined. Moreover, a family of DNA skew cyclic codes over R is constructed, its property of being reverse complement is studied.

DNA Codes over the Ring $\mathbb{Z}_4 + w\mathbb{Z}_4$

“…|A m | = 6 and |B m | = 9 for each m ∈ R θ . From Inequality(15), Equation(16), Equation(18) and Inequality(20), we obtain,M(M − 1)d G(θ) ≤ n l=1 3M Therefore, M ≤ 2d G(θ) 2d G(θ) −3n .Note that M, n and d G(θ) are finite positive integers. Therefore, M ≤2d G(θ) 2d G(θ) −3n when 2d G(θ) > 3n.…”

“…Furthermore, MacWilliams identities were also established in this paper. Cyclic DNA codes of odd length over the rings Z 4 + wZ 4 , w 2 = 2 and Z 4 + uZ 4 , u 2 = 0 have been constructed in [15], [16].…”

Cyclic and Quasi-Cyclic DNA Codes

“…In [2] cyclic DNA codes have been studied over the ring R 2 using the Gray image. Here, we shall use the Gau map which defined in [3] from the ring R 2 to the set of DNA strings of length 2 ,where λ = 2, relating it to the Gau distance to give some examples of cyclic and quasi-cyclic DNA codes.…”

“…Here, we shall use the Gau map which defined in [3] from the ring R 2 to the set of DNA strings of length 2 ,where λ = 2, relating it to the Gau distance to give some examples of cyclic and quasi-cyclic DNA codes. The zero divisors Z and the units U of the ring R 2 have been given in [2]. Clearly U = Z + 1.…”

“…Cyclic DNA codes have been intensively studied for many element sets such as fields and rings. In particular in [2], cyclic DNA codes were investigated using the Gray map over the ring Z 4 + ωZ 4 where ω 2 = 2. It is still an interesting problem to study this type of code using a different map.…”