Codes and modules associated with designs and t-uniform hypergraphs

The first part of these lectures introduces the Smith normal form and the invariant factors of an integer matrix, and the relation of Smith form to systems of linear diophantine equations. We give selected examples of how invariant factors appear and may be applied to the theory of combinatorial designs. Importantly, we may sometimes construct sefl-dual p-ary codes from hypothetical designs and sometimes deduce the non-existence of designs from a theorem of Witt. In the second part of the notes, we are concernedwith diagonal forms of various incidence matrices arising from t-designs and t-uniform hypergraphs. Applications are given to a certain zero-sum Ramsey-type problem involving t-uniform hypergraphs.