Few-weight codes over $\Bbb F_p+u\Bbb F_p$ associated with down sets and their distance optimal Gray image.

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 classes of them are attained the Griesmer bound with equality as well.