On Additive Codes And Their Applications

With the urgent need of technology development and the in-depth study of error correcting code theory,the theory of error correcting codes over finite rings has become one of the hot topics in recent years.Additive codes are a kind of important error correcting codes.They not only have good algebraic structure,but also have important applications in cryptography and quantum communication.Based on these,this thesis mainly studies the algebraic structure and application of Z₂Z₂(Z₂+uZ₂)-additive cyclic code,Z₂(Z₂+uZ₂)(Z₂+uZ₂+u2Z₂)-additive cyclic code and Z_p(Z_p+vZ_p+…+v^m-1Z_p)-additive cyclic code,and the specific contents are as follows.In Chapter 1,the thesis mainly introduces the research background and significance of error correction code and additive code,the main research status at home and abroad,and the main research contents.In Chapter 2,the thesis introduces the basic theory of additive codes,and gives the relatedknowledas of coding theory involved in the study of additive cyclic codes,which lays the foundation for the subsequent writing of this thesis.In Chapter 3,we study the definition of Z₂Z₂(Z₂+uZ₂)-additive cyclic codes,generating polynomials and minimal generator sets,and discuss the generating polynomials of its dual codes.In Chapter 4,we study the definition of Z₂(Z₂+uZ₂)(Z₂+uZ₂+u²Z₂)-additive cyclic codes,generator polynomials,generator matrices and minimal generator sets.As applications,we construct some binary linear codes with good performance by gray mapping from Z₂(Z₂+uZ₂)(Z₂+uZ₂+u²Z₂)to Z₂.In Chapter 5,the definition of Z_p(Z_p+vZ_p+…+v^m-1Z_p)-additive cyclic codes,generating polynomials and minimal generator sets are given,and the additive Euclidean selforthogonal codes are constructed.As applications,the CSS construction method based on quantum error correcting codes is used to construct quantum error correcting codes with good performance through Gray mapping from Z_p(Z_p+vZ_p+…+v^m-1Z_p)to Z_p which maintains self-orthogonality.In Chapter 6,the thesis summarizes the main results of the full text and looks forward to the future research content.