Construction of cyclic DNA codes over the Ring ${\mathbb{Z}}_4[u]/⟨u^2-1⟩$ Based on the deletion distance

Abstract

In this paper, we develop the theory for constructing DNA cyclic codes of odd length over $R={\mathbb{Z}}_4[u]/⟨u^2-1⟩$ based on the deletion distance. Firstly, we relate DNA pairs with a special 16 elements of ring $R$. Cyclic codes of odd length over $R$ satisfy the reverse constraint and the reverse-complement constraint are discussed in this paper. We also study the $GC$-content of these codes and their deletion distance. The paper concludes with some examples of cyclic DNA codes with $GC$-content and their respective deletion distance.

15 pages, 3 tables. arXiv admin note: substantial text overlap with arXiv:1508.02015, arXiv:1511.03937