Research On Several Classes Of Additive Cyclic Codes On Finite Commutative Chain Rings

As an intersection branch of computer science and mathematics,coding theory has undergone more than seventy years of development so far and has become a discipline with important applications.In recent years,the theory of coding over finite rings has received extensive attention and research from scholars at home and abroad.In this paper,we mainly study several types of additive cyclic codes over finite commutative chain rings,and discuss their algebraic structure,minimal generating sets,and their dual codes,respectively.The specific research content is as follows.In Chapter 3,we study FR-additive cyclic codes(where F is the finite field Fq,R=Fq+uFq,u2=0).Firstly,we give the definition of FR-additive cyclic codes,and then study the algebraic structure and minimal generating set of FR-additive cyclic codes.Then we define the Gray mapping and construct some examples of linear codes with good parameters using the Gray mapping.Finally,we study the dual codes of FR-additive cyclic codes and characterize the generators of the dual codes.In Chapter 4,we study FRR-additive cyclic codes(R=Fq+vFq+v2Fq,v3=0).Firstly,we give the definition of FRR-additive cyclic codes,and then study the algebraic structure and minimal generating set of FRR-additive cyclic codes.Subsequently,we define a new Gray map and use it to construct linear code examples with good parameters.Finally,we study dual codes of FRR-additive cyclic codes and characterize the generators of dual codes.In Chapter 5,we study R1R2R3-additive cyclic codes(R1=Fq+uFq+…+uk1Fq,uk1+1=0;R2=Fq+vFq+…+vk2Fq,vk2+1=0;R3=Fq+wFq+…+wk3Fq,wk3+1=0 with k1,k2,k3 ∈ N and k1≤k2≤k3).As an extension,we extend FRRadditive cyclic codes to more general R1R2R3-additive cyclic codes.We study the algebraic structure of R1R2R3-additive cyclic codes and further discuss the minimal generating set of R1R2R3-additive cyclic codes.