Abstract
In this paper, we develop the method for constructing DNA codes of odd length over the finite chain ring $R=\mathbb {F}_{4}[u]/\langle u^{3}\rangle $ , which plays an important role in genetics, bioengineering and DNA computing. By using a Gray map $\eta $ from $R$ to $\mathbb {F}_{4}$ , we give a one-to-one correspondence between 64 DNA codons of the alphabet $\{A, T, G,C\}^{3}$ and the 64 elements of the chain ring $R$ . We also establish a map $\Theta $ from $R^{n}$ to $\{A, T, G,C\}^{3n}$ . Then we provide a necessary and sufficient condition for cyclic codes of odd length over the chain ring $R$ to be reversible. Finally, when $C$ is a reversible cyclic code over $R$ , we derive a necessary and sufficient condition for $\Theta (C)$ to be a DNA code.
Highlights
Deoxyribonucleic acid (DNA) contains the genetic program for the biological development of life
The DNA strand is held by the important feature called complementary base pairing which connects the Watson-Crick complementary bases with each other denoted by Ac = T and Gc = C
Based on the above works, in this paper, we present a construction method DNA codes over the finite chain ring of 64 elements, namely, R = F4/ u3
Summary
Deoxyribonucleic acid (DNA) contains the genetic program for the biological development of life It contains the information on how the cell runs, reproduces, builds and repairs itself, and every other function necessary for the cell life. Zhu and Chen studied cyclic DNA codes over the non-chain ring F2[u, v]/ u2, v2 − v, uv − vu in [26]. Studied the structure of cyclic DNA codes of odd length over the finite ring F2 + uF2 + vF2 + uvF2 + v2F2 + uv2F2, u2 = 0, v3 = v. Based on the above works, in this paper, we present a construction method DNA codes over the finite chain ring of 64 elements, namely, R = F4/ u3.
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