The Structure Of Algebraic Codes Over Some Rings And Their Applications In Quantum Codes Construction

Since the 1970s,scholars at home and abroad have not only discovered the linear structure of nonlinear codes hiding,but also made great breakthroughs in the field of linear codes over finite rings,such as the study of the structure and properties of Kerdock codes and Preparata codes over the ring Z4.In this paper,we mainly study the structures of several kinds of ring previous generation codes and their applications in the construction of quantum error correction codes.The structure of this paper is as follows:The first chapter briefly introduces the research background and significance,the main research status at home and abroad,as well as the main research content of this paper.The second chapter gives the basic knowledge of coding and algebra related to this paper.In Chapter 3,we study the algebraic structure of self-dual cyclic codes over ring F₂+uF₂+vF₂+uvF₂,where u²=0,v²=0 and uv=vu,and give the necessary and sufficient conditions for its existence.In addition,according to Gray map,we construct the self-dual codes over ring.In Chapter 4,we study the properties and structuures of cyclic codes over ring F_q+uFa+vFa+uvF_q.By establishing an effective Gray map,we obtain some new non-binary quantum codes.In Chapter 5,we study the structure and properties of constant cyclic codes over ring F_q+v1F_q+…+vrF_q,where q is the power of a prime number,v_i²=v_i,v_iv_j=v_jv_i=0,1?i,j?r and r?1.Under the inner product of the Euclidean and the Hermite,we get some good quantum codes based on Euclidean and Hermite dual inclusion ?-constacyclic codes.The sixth chapter summarizes the whole paper,lists our main research problems,and looks forward to our future research direction.