A Class Of Cyclic Codes Over Z_P^k+1

Content: Cyclic Codes are a class of important error-correcting codes.It is known that the numbers of cyclic codes over finite rings are more than those of linear codes over finite fields with the same parameters such as length and minimum distance.Consequently cyclic codes over finite rings have better information rates.Therefore cyclic codes over finite rings have been extensively studied.So far,there has been much achievement about cyclic codes over the ring Z₄. Structure and some properties of Z₄-cyclic codes under the Gray map are given. Although the Gray map has been defined for codes over Z_p^k+1 and some properties of linear codes over Z_p^k+1 are given in some reference literatures,there has been little research on the structure of linear cyclic codes over Z_p^k+1 of length n,where n and p are coprime, via the Nechaev-Gray map.On the basis of known results, this paper discusses a class of cyclic codes obtained by using p-adic linear cyclic codes.Their structures of the Nechaev-Gray images are determined and the relation between p-adic linear cyclic codes and Z_p^k+1 -cyclic codes is given.Firstly,we study the p-adic linear cyclic codes of length pn,they are Nechaev-Gray images of Z_p²-cyclic codes constructed by using p-adic linear cyclic codes of length n.Secondly, we discuss a family of cyclic codes obtained by using p-adic linear cyclic codes over Z_p^k+1.We determine their structures of the Nechaev-Gray images.By the method of inclusion we can conclude that p-adic cyclic codes of index p^k-1 and of length p^kn over Z_p are Nechaev-Gray images of Z_p^k+1 -cyclic codes.Finally, the condition that cyclic codes are linear codes is given.