Families of cyclic code over some finite rings

Cyclic codes are the most studied field among the codes because of their rich algebraic structure for coding. In this study, the family of cyclic codes over different rings are investigated. By using cyclic codes over these rings, both new codes and optimal codes are obtained. This thesis consists of six chapters and in the first chapter, some basic definitions and theorems related to algebra and coding theory are given. In the second chapter, Galois extensions of the ring Z4 + uZ4 +u^2Z4 are studied. Also, cyclic codes over this ring are investigated and the general form of the generator and a minimal spanning set of such codes are determined. Using these informations, new linear codes over Z4 are given in a table at the end of the chapter. In the third chapter, the algebraic structure of skew quasi cyclic codes over the ring Fq + vFq is investigated. The duals of skew quasi cyclic codes are discussed. Also from a different viewpoint, necessary and sufficient condition for skew cyclic codes over Fq + vFq is given to contain its dual. By using this information, quantum codes are constructed from skew cyclic codes. In the fourth chapter, the structure of linear codes over F3 + uF3 + vF3 + uvF3 is investigated and a new Gray map is given. Also, generator polynomials of cyclic codes over this ring are determined. Using these results, some parameters of quantum codes are found. In the fifth chapter, a new class of cyclic codes which is referred to as F2F2[u] -cyclic codes is discussed. Also a new Gray map is defined and some examples of optimal codes which are the binary Gray images of F2F2[u]-cyclic codes are presented in the form a table. In the last chapter, the conclusion and some recommendations are given.