Threefold triple systems with nonsingular N2

There are various results connecting ranks of incidence matrices of graphs and hypergraphs with their combinatorial structure. Here, we consider the generalized incidence matrix N 2 (defined by inclusion of pairs in edges) for one natural class of hypergraphs: the triple systems with index three. Such systems with nonsingular N 2 (over the rationals) appear to be quite rare, yet they can be constructed with PBD closure. In fact, a range of ranks near ( v 2 ) is obtained for large orders v .