Skew Cyclic Codes Over Finite Rings

As a kind of generalized codes of cyclic codes, skew cyclic codes are similar to cyclic codes in the way of algebraic structure. However, they are different naturally because of joined automorphism mapping. Since its first appearance, many foreign and domestic researchers have been paying wide attention on it. Gradually, it becomes a new representation of codes in encoding theory. Skew cyclic codes are constructed based on non-communicative skew polynomial rings. So far, the main content of skew cyclic codes is about the existence conditions of skew cyclic codes, the corresponding relation between skew cyclic codes and skew polynomial rings, and generating polynomials. In all, these conclusions provide more theoretical foundation and guide practical applications for constructing many codes over finite rings and fields. In addition, the existence of the self-dual code of a skew cyclic code, skew quasi-cyclic codes and skew constacyclic codes are also studied. We preliminary discuss the relationships between skew cyclic codes and quasi cyclic codes and cyclic codes. With the fast development of science and technology and actual requirement, the scope of research will expand gradually.In this thesis,we mainly study skew cyclic codes over two classes of rings based on the former and original results.The one is ringppF?v F)(2vv?,the other one is ringmmFv F22?)(2vv?.By exploring the structure characteristics of the ring itself,we provide new automorphisms.For the ringppF?v F,combing the definited new automorphism,we give the center of non-communicative skew polynomial rings xR],[?and discuss some important properties of skew cyclic codes with arbitrary length.For another ringmmFv F22?,we give another expression called direct sum decomposition by making use of its two maximal ideals.Furthermore,we obtain an conclusion:direct sum decomposition is right only if?satisfy certain conditions.According to above theories,we give another generated polynomial and related results of skew cyclic codes.Then,we study the existence the Euclidean and厄米特dual codes of skew cyclic codes and get the relationship between skew cyclic codes and quasi cyclic codes and cyclic codes.Furthermore,we choose the simple ring 44F?v F to give the generating polynomial and generating matrix of skew cyclic codes with length6 and 4.