The (1+p^k)-Cyclic Codes And The Gray Map Over Z_p^k+1

Since 1970s, some researchers, such as Blake[1] and Speigel[2], began to discuss error-correcting codes over the rings Zm of integers modulo m instead of studying codes over finite fields. In the beginning of 1990s, Forney etc[3] and Hammous etc[4] proved that some nonlinear codes, such as Kerdock code, Preparata code and Delsarte-Goethals code are the images of some linear codes over Z4, where the map is the Gray map. Since these non-linear codes have more codewords than those linear codes with the same length and the same Hamming distance, the research interest in codes over finite rings has grown rapidly. In 1998, Carlet[6] defined Gray map over Z2k by using Boolean function, then mapped the linear codes over Z2k to nonlinear codes over Z2 by the Gray map in this paper, and finally obtained generalized Kerdock code and generalized Goethals code. Ling[9] extended the definition of Gray map to Zpk+1 and proved that the Gray image of (1-pk)- cyclic codes are quasi-cyclic codes over Zp. In [9], they also proved that cyclic codes over the rings Zpk+1 are equivalent to quasi-cyclic codes by formulating a one-to-one correspondence between (1-pk)-cyclic codes and general cyclic codes, they also provided a sufficient condition for the images of codes over Zpk+1 to be linear.In this thesis, we shall continue the study on codes over Zpk+1. We obtain the following results.In chapter 2, by using the Gray map (see [9]) from Zpk+1n to Zppkn, and the unique p-adic expression of each element in Zpk+1, the Gray images of (1+pk)- cyclic codes of length n over the rings Zpk+1, general cyclic codes and negacyclic codes are obtained, where (n,p) = 1.In chapter 3, when (n,p) = 1, the generator of constacyclic codes with length n over Zpk+1 are obtained. By using the one-to-one correspondence between cyclic codes over Zpk+1 and (1+pk)- cyclic codes over Zpk+1, the generator of (1+pk)-cyclic codes are obtained.In chapter 4, we focus on codes over Zp2. First we note that the three class of codes-cyclic codes, constacyclic codes and quasi-cyclic codes, which are discussed in this thesis, may be nonlinear. In this chapter, the sufficient conditions for the Gray image of (1+p)- cyclic codes and the cyclic codes over Zp2 to be linear axe also given.