Research On The Structure Of Cyclic Codes Over Z_m

In this paper, two way of research on cyclic code and quasi-cyclic code over Z_p^m with lengths n which aire not prime to p; for cyclic codes over Z_p² some properties beween cyclic codes will be given by the aid of the idempotents which is in Z_p² [x]/(xⁿ -1):One way of research on quasi-cyclic codes over Z_p^m with lengths n which are not prime to p, is find the Gray images of the code. We achive their images which is over Z_p.Another way of research on cycic codes over Z_p^m with lengths n = p^e, is take the code as an ideal of Z_p²[x]/(xⁿ -1). We get some nilpotents of the ring Z_p²[x]/(xⁿ -1), then we achivesome of its inverse elements from the properties of the nilpotent in the ring , then we give out all of the cyclic codes on Z_p². If we get all the cyclic codes over Z_p^k with lengths p^e, then wegive a way of getting all the cyclic codes with lengths p^e over Z_p^k+1.For the cyclic codes over Z . with lengths n which are prime to p, in the case m = 2, we give a way of finding all cyclic codes over Z_p², then we give the idempotent generators of cyclic codes over Z_p² with lengths n which are prime to p, then we get some properties between cyclic codes over Z_p² with lengths n which are prime to p.