  Let $p$ be a prime number, $q=p^s$ for a positive integer $s$. For any
positive divisor $e$ of $q-1$, we construct an infinite family codes of size
$q^{2m}$ with few Lee-weight. These codes are defined as trace codes over the
ring $R=\mathbb{F}_q + u\mathbb{F}_q$, $u^2 = 0$. Using Gauss sums, their Lee
weight distributions are provided. When $\gcd(e,m)=1$, we obtain an infinite
family of two-weight codes over the finite field $\mathbb{F}_q$ which meet the
Griesmer bound. Moreover, when $\gcd(e,m)=2, 3$ or $4$ we construct new
infinite family codes with at most five-weight.
