The Smith normal form and the invariant factors of an integer matrix are introduced. We give selected examples of how invariant factors, a chain of linear codes, and self-dual codes have appeared and been applied in the theory of combinatorial designs. In the latter part of these notes, we are concerned with diagonal forms of various incidence matrices arising from designs and uniform hypergraphs. Results on diagonal forms of such matrices can be applied to a certain zero-sum Ramsey-type problem.
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