The structure of codes over group rings

Due to the increasing importance of digital communications, the area of research in algebraic coding theory is broad and fast developing. In this sense, finding new construction methods, new parameters for codes or finding bounds on minimum distance or dimension of codes have a great importance. The notion of group rings was introduced by Arthur Cayley (see [1]) for the first time. Since group rings are rich in units and zero divisors, they are quite suitable algebraic structures to obtain new code parameters. Many code properties may more easily be expressed in terms of group ring properties (see [2], [3]). This work, in which we study the structure of codes over group rings, is organized as the following. The first section is devoted to the literature review, aim of the thesis and the hypothesis. Section 2 and Section 3 collects the notions and notations needed in the rest of the dissertation about group rings and algebraic coding theory. In Section 4 we study the structure of cyclic zero divisor codes over a family of group rings. In addition, we determine the number of elements of these codes and we introduced the dual codes. In Section 5 for the first time in the literature, we establish a relation between constacyclic codes and group algebras and study their algebraic structures. Further, we give a method for constructing constacyclic codes by using zero-divisors in group algebras. Some good parameters for constacyclic codes which are derived from the proposed construction are also listed. Then, we determine self dual and self orthogonal codes arising from constacyclic codes over group algebras. Also, based on these codes we obtained some good parameters for quantum error-correcting codes. Moreover, we determine self dual and self orthogonal codes arising from negacyclic codes over the group ring (F_q + vF_q)G. By taking gray image of these codes we obtained many parameters for quantum error-correcting codes over the finite field . In Section 6 we derive some necessary and sufficient conditions for LCD codes (linear codes with complementary duals) from group rings. The last section is reserved for the conclusions and future research directions.