In this paper, we establish the Singleton bound for pomset block codes ($(Pm,π)$-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 $(Pm,π)$-code is an MDS $(P,π)$-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.