Research On Several Types Of Cyclic Codes Over Finite Rings

With the continuous development of coding theory,coding theory over finite rings has received extensive attention and research from scholars at home and abroad.The coding theory over finite rings has important research significance from both theoretical and practical aspects.Research on the theory of cyclic codes over finite rings has become a hot topic in recent years.This dissertation mainly studies the structural properties of several types of cyclic codes over finite rings,including self-dual and LCD?linear complementary dual?double circulant codes over finite non-chain rings,triple cyclic codes over finite chain rings,and the asymptotic properties of three families of additive cyclic codes.The specific research contents are as follows:1.The self-dual and LCD double circulant codes over F_q+vF_q+v²F_q are studied.By using the Chinese Remainder Theorem,we study the algebraic structure of the double circulant cyclic codes over the ring F_q+vF_q+v²F_q,and give the exact enumerations of self-dual and LCD double circulant codes over this ring.Under certain conditions,by Gray map,we prove that there are asymptotically good self-dual codes and LCD codes over F_q that satisfy the modified GV bound.2.The triple cyclic codes overq F_q+uF_q are studied.We present the generator polynomials and minimum generator sets of triple cyclic codes over this ring.We give the generator polynomials of the dual codes of triple cyclic codes,and discuss the relationship between the generator polynomials of triple cyclic codes and their duals.In addition,some linear codes with good parameters over finite field F_q are constructed by using the triple cyclic codes over the ring F_q+uF_q.3.The asymptotic properties of Z₂ Z₂?u?-additive cyclic codes and Z_pZ_p^s-additive cyclic codes are studied.We construct two families of additive cyclic codes with cyclic length m,and discuss the relationship of relative minimum distance between the two kinds of additive cyclic codes and quasi-cyclic codes with index 2.By constructing equal probability spaces,it is proved that both families of additive cyclic codes are asymptotically good codes satisfying the asymptotic GV bound.4.The asymptotic properties of Z_pZ_p^s-additive cyclic code are studied.We construct a class of Z_pZ_p^s-additive cyclic codes.Through the cumulative weight operator and Bernoulli variables,we calculate the asymptotic rate and the relative minimum distance of this kind of additive cyclic codes.It is proved that there exists a series of asymptotically good Z_pZ_p^s-additive cyclic codes that satisfy the asymptotic GV bound.