Two or Few-Weight Trace Codes over Fq+uFq

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.