Ring F <sub> 2 </ Sub> + Uf <sub> 2 </ Sub> + The U <sup> 2 </ Sup> F <sub> 2 </ Sub> + The U <sup> 3 </ Sup> The F <sub> 2 </ Sub> Cyclic Code

Cyclic codes play an important role in studying the coding theory. In this thesis, some properties of cyclic codes over the finite ring R are investigated. The main results are as follows:Firstly, a finite ring R with 16 elements is constructed by using some basic knowledge of algebra and it is proved that this ring is a commutative ring with unit element. A Gray mapping from ring R[x] to ring F2[x] is defined by extending the concept of linear codes over the number field F2 to the ring R. On this basis, all of the ideals in a residue class ring are obtained and the structure of each ideal among these ideals is investigated. In addition, the specific form of code words of cyclic codes is given over the ring R by using modulus of congruence. Cyclic codes with odd length are constructed by making use of the Chinese remainder theorem and the uniqueness of the polynomial decomposition. As well as, the number of code words of cyclic codes is given over the ring R. A necessary condition and a necessary and sufficient condition for linear codes becoming cyclic codes are given over the ring R. Meanwhile, we establish a generating polynomial of cyclic codes. Finally, applying the reciprocal polynomials we discuss the dual codes over the finite ring R. The numbers of the code words of dual codes are obtained and a necessary and sufficient condition for the existence of self-dual codes is proved over the ring R.