The incidence matrix of a design is a (0,1)-matrix with rows representing blocks, columns representing points, and a one indicating incidence. The Smith normal form generalizes the idea of p-rank. We determine the Smith normal form of the incidence matrices of classical designs, those arising from the incidence of points and some other dimensional subspace of a finite geometry. The techniques involve the use of p-adic character sums and some representation theory.

We also obtain partial results in determining the Smith normal form for the incidence between sets of subspaces, neither one of which is the set of points. The p-part remains largely unknown in this case.

In the case of the designs associated with two families of difference sets with classical parameters, the p-ranks are the same when the parameters are the same. We show these difference sets are inequivalent by showing a difference in the Smith normal forms of the designs.