Classification of poset-block spaces admitting MacWilliams-type identity

In this work we prove that a poset-block space admits a MacWilliams-type identity if and only if the poset is hierarchical and at any level of the poset, all the blocks have the same dimension. When the poset-block admits the MacWilliams-type identity we explicit the relation between the weight enumerators of a code and its dual.