Cyclic Codes And Constacyclic Codes Over Finite Rings

With the development of the theory about cyclic codes and constacyclic codes over finite field, the theory of cyclic codes and constacyclic codes over finite chain rings also have obtained extensively research and development. At the same time, some finite non-chain rings also cause the interests of coding researchers. In this paper, we mainly study the structures and relative properties of cyclic codes and constacyclic codes over several classes of finite rings. The details are given as follows:(1) The negacyclic codes and dual codes of arbitrary lengths over rings Fpm+uFpm and Fpm+uFpm+u2Fpm are studied, the structures of these codes are given, and the Gray images of these negacyclic are studied. The distance distributions of negacyclic codes over ring Fpm+uFpmare also studied.(2) We study the cyclic codes and a class of constacyclic codes of arbitrary lengths over finite non-chain ring Zpm+vZpm, the structures and Gray images of these codes are given, the necessary and sufficient conditions for a linear code to be self-dual or self-dual constacyclic are also given.the research shows that the self-dual cyclic codes over R of length n do not exist when p> 2 and m is odd.(3) We give the structures of (1-2v)-constacyclic codes and Gray images of length ps over finite non-chain ring R+vR, R is a finite chain ring with maximal ideal <λ>, the relative properties of this class of constacyclic codes and these Gray images are also studied.(4) We study the cyclic codes and constacyclic codes over finite non-chain ring R+vR+v2R, the relative properties of these codes and their Gray images are given.