  In this paper, we establish the Singleton bound for pomset block codes
($(Pm,\pi)$-codes) of length $N$ over the ring $\mathbb{Z}_m$. We give a
necessary condition for a code to be MDS in the pomset (block) metric and prove
that every MDS $(Pm,\pi)$-code is an MDS $(P,\pi)$-code. Then we proceed on to
find $I$-perfect and $r$-perfect codes. Further, given an ideal with partial
and full counts, we look into how MDS and $I$-perfect codes relate to one
another. For chain pomset, we obtain the duality theorem for pomset block codes
of length $N$ over $\mathbb{Z}_m$; and, the weight distribution of MDS pomset
block codes is then determined.
