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Let $p$ be an odd prime number. In this paper, we construct $2(2p-3)$ classesof codes over the ring $R=\Bbb F_p+u\Bbb F_p,u^2=0$, which are associated withdown sets. We compute the Lee weight distributions of the $2(2p-3)$ classes ofcodes 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 Griesmerbound.

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