In this thesis we use DNA structure as a model in order to construct a good error correcting codes. A DNA strand consist of four nucleotides with bases: Adenine (A), Guanine (G), Cytosine (C) and Thymine (T). It can be shown a DNA strand is oriented since, the strands have two tails that is called 5' end and 3' end. The strands are linked by their bases, i. e., every A is connected with a T, and every C with a G, and conversely. We denote the compliment of X by , i. e., = T, =A, = C and =G. In a complementary strand we reverse the direction of initial strand and replace the complements of bases. For example the complement of the strand 5'-TCGGGCTA-3' is 3'-TAGCCCGA-5'. A double strand of DNA occurs when a strand and its complement bind to each other, which is known as Hybridization. Recently the error correction capability of DNA has found interest in coding theory. Since many of chemical experiments about DNA structure are quite expensive, it's Worthwhile to study error correcting codes with similar properties as DNA. There are some constraints that are used in DNA codes, such as, The Hamming distance constraint, the reverse-complement constraint, the reverse constraint and the fixed GC-content constraint. In this thesis, DNA pairs are associated with a special 16 elements ring. Then, the cyclic DNA codes of odd lengths that follow some of the properties of DNA are investigated. Furthermore, the algebraic structure of these codes is studied. These codes are designed for use in DNA computing applications. Necessary and sufficient conditions are given under which cyclic codes have DNA properties. Finally, some cyclic DNA codes are introduced.