Z₂Z₄-Additive Cyclic Codes And Their Properties

Information and information science are two common nouns.Their contents are very extensive.Especially in today’s society,they have very important meanings and rich meanings.Coding is coming along with the development of technology,in which coding theory serves to convey information.As early in 1948,C.E.Shannon published the foundation of the"Mathematical Theory of Communication",which produced the information theory and pointed out the existence of error correcting codes^[1].In 1950,R.W.Hamming published"Error Code and Error Correcting Codes"^[2],and first proposed how to use error correcting codes to correct encoding errors.Subsequently,P.Delsart et al.On this basis of error correcting code in-depth study^[3]-[4],from then on the rapid development of the coding theory.Linear code,a class of important and widely used error correcting codes,is the basis for discussing various codes.In the past 60 years,people often use them to check them.The Hamming code was found independently by Marcel Golay in 1949 and Richard Hamming in 1950.Hamming code has many good properties,and it can be decode in a simple and efficient way.It is a kind of error correcting code which is widely used.In 1957,E.Prange first introduced linear codes,and then the concept of cyclic codes was introduced on the basis of linear codes.Cyclic code as error detection code,it can check out the interval error.For example,two yuan Hamming code is equivalent to a cyclic code.Subsequently,Taher Abualrub and others further extended the study of cyclic code to the ring^[5]-[6].The wide application of two element linear codes is the first time the code on the ring is put forward in T.Abualrub,especially the code on the ring.Then there are more ring codes^[7]-[10].In 1994,Hammons and others studied the structure of the code on Z₄,gave the concepts and properties of Hamming weight and Lee weight,and gave the important connection between binary code and quanternary code:Gray mapping^[8].In1999,Jacques Wolfmann also studied the negative cyclic code on theZ₄^[11].By twenty-first Century,researchers turned the research direction to theZ₂Z₄-additive code^{[1 2]-[14]}.In 2010,M.Bilai and J.Borges were extended on the basis of Z₄ and Z₄,that isZ₂Z₄-linear codes^[14],and the two element optimal code ofZ₂Z₄-cyclic code under Gray mapping is given.This article consists of four chapters:（1）The first chapter introduces the contents of coding theory,and introduces error correcting codes.Lists commonly used error correcting codes such as linear codes and cyclic codes,and gives their concepts and properties.（2）In the first section of the second chapter,the structure of the code on the Z₄ is first introduced,including theZ₄-linear negative cyclic code.The second section introduces the definition ofZ₂Z₄-additive cyclic code.In the third section,we introduce the definition and construction ofZ₂Z₄-additive negative cyclic codes based on the related results ofZ₂Z₄-additive cyclic codes,and construct their generator polynomials and spanning sets,and illustrate that they have the same parameters asZ₂Z₄-additive cyclic codes by examples.In the fourth section,through the orthogonal relationship underZ₂Z₄,get the duality is also aZ₂Z₄-negacyclic codes,and then through the establishment of isomorphic mapping betweenZ₂Z₄-codes and inZ₄[x]-submodule to describe the parameter types of negacyclic code structure and code,and by the method of construction launched its dual minimum spanning set;（3）In the third chapter,we first introduce the Gray mapping ofZ₂Z₄-additive negative cyclic codes,and then introduce the generation matrix ofZ₂Z₄-additive negative cyclic codes.