Let p be a prime number and q = psfor a positive integer s. For any positive divisor e of q - 1, we construct infinite families of codes C of size q2mwith few Lee-weight. These codes are defined as trace codes over the ring R = Fq+ uFq, u2= 0. Using Gaussian sums, their Lee weight distributions are provided. In particular, when gcd(e, m) = 1, under the Gray map, the images of all codes in C are of two-weight over the finite field Fq, which meet the Griesmer bound. Moreover, when gcd(e, m) = 2, 3, or 4, all codes in C are of most five-weight codes.

Codes over Galois rings have been studied extensively during the last three decades. Negacyclic codes over GR(2a, m) of length 2shave been characterized: the ring R2(a, m, -1) = GR(2a, m)[x]/(x2st 1) is a chain ring. Furthermore, these results have been generalized to λ-constacyclic codes for any unit λ of the form 4z - 1, z ∈ GR(2a, m). In this paper, we study more general cases and investigate all cases, where Rp(a, m, y) = GR(pa, m)[x]/(xps - y) is a chain ring. In particular, the necessary and sufficient conditions for the ring Rp(a, m, y) to be a chain ring are obtained. In addition, by using this structure we investigate all y-constacyclic codes over GR(pa, m) when Rp(a, m, y) is a chain ring. The necessary and sufficient conditions for the existence of self-orthogonal and self-dual y-constacyclic codes are also provided. Among others, for any prime p, the structure of Rp(a, m, y) = GR(pa, m)[x]/(xps- y) is used to establish the Hamming and homogeneous distances of y-constacyclic codes.