Threefold triple systems with nonsingular N2N2

There are various results connecting ranks of incidence matrices of graphs and hypergraphs with their combinatorial structure. Here, we consider the generalized incidence matrix N2N2 (defined by inclusion of pairs in edges) for one natural class of hypergraphs: the triple systems with index three. Such systems with nonsingular N2N2 (over the rationals) appear to be quite rare, yet they can be constructed with PBD closure. In fact, a range of ranks near (v2) is obtained for large orders vv.