Z(pk+1)-linear codes

We characterize codes over Z_p which are the Gray images of (1-p^k)-cyclic codes or cyclic codes over Z(p^k+l) (k≥1). A necessary and sufficient condition for the Gray image of a Z(p²)-linear (1-p)-cyclic code to be linear is given. In many cases, this yields an explicit description of the Gray image of a linear (1-p)-cyclic code over Z(p²), of length relatively prime to p. Linear cyclic codes over Z(p²) whose Gray images are linear cyclic codes over Z_p have been characterized. Some generalizations of these results to the case of Z(p^k+1), where k≥2, are also obtained.