  Let $p$ be an odd prime number. In this paper, we construct $2(2p-3)$ classes
of codes over the ring $R=\Bbb F_p+u\Bbb F_p,u^2=0$, which are associated with
down sets. We compute the Lee weight distributions of the $2(2p-3)$ classes of
codes when the down sets are generated by a single maximal element. Moreover,
by using the Gray map of the linear codes over $R$, we find out $2(p-1)$
classes of $p$-ary distance optimal linear codes. Two of them meet the Griesmer
bound.
