Some Notes On Cyclic Codes

With the development of coding theory,cyclic codes as a special kind of linear codes,have been widely studied due to their special algebraic structures.Constacyclic codes,quasi-cyclic codes and quasi-twisted codes are the gener-alizations of cyclic codes and they inherit the good performances of the cyclic codes.Meanwhile they also have some new features that the cyclic codes do not have.The quasi-cyclic codes,the constacyclic codes and the cyclic codes can be obtained from the quasi-twisted codes if the parameters are limited.Therefore it is meaningful to study the quasi-twisted codes,because it con-tains the cyclic codes,constacyclic codes and quasi-cyclic codes as its special cases.At the same time,the properties of some extended codes and the rela-tion between codes in finite field and correlation codes in ring are studied by analogy methods.The contents of the paper are as follows:1.The algebraic structures and basic properties of cyclic codes,consta-cyclic codes,quasi-cyclic codes,quasi-twisted codes are studied.We also study their dual codes.2.The relationship between the quasi-cyclic codes over finite field and the linear codes over ring,the relationship between the(1+u)-constacyclic codes over ring and the cyclic codes over finite field are studied.Moreover,the relationship between the(1+u)-l-quasi-cyclic codes over ring and the l-quasi-cyclic codes over finite field is also studied.3.The double cyclic codes,double constacyclic codes,double A-quasi-cyclic codes and double l-quasi-twisted codes are studied.The generator ma-trices,parity check matrices and dual codes of those extended codes are also discussed,the relation,between double(1+u)-constacyclic codes over ring and the cyclic codes over finite field,double(1+u)-l-quasi-cyclic codes over ring and the l-quasi-twisted codes over finite field are studied.