Signed hypergraph designs and diagonal forms for some incidence matrices (Q1963150)

Given a \(t\)-uniform hypergraph \(H\) on \(k\) vertices and an assignment of integers \(f(T)\) to the \(t\)-subsets \(T\) of a \(v\)-set \(X\) \((v\geq k+t)\), necessary and sufficient conditions are given for the existence of an assignment of integer multiplicities \(h(G)\) to those subhypergraphs \(G\) of the complete \(t\)-uniform hypergraph on \(v\) vertices that are isomorphic to \(H\), so that the sum of the integers \(h(G)\) over those \(G\) that contain \(T\) is \(f(T)\).