  A difference matrix over a group is a discrete structure that is intimately
related to many other combinatorial designs, including mutually orthogonal
Latin squares, orthogonal arrays, and transversal designs. Interest in
constructing difference matrices over $2$-groups has been renewed by the recent
discovery that these matrices can be used to construct large linking systems of
difference sets, which in turn provide examples of systems of linked symmetric
designs and association schemes. We survey the main constructive and
nonexistence results for difference matrices, beginning with a classical
construction based on the properties of a finite field. We then introduce the
concept of a contracted difference matrix, which generates a much larger
difference matrix. We show that several of the main constructive results for
difference matrices over abelian $p$-groups can be substantially simplified and
extended using contracted difference matrices. In particular, we obtain new
linking systems of difference sets of size $7$ in infinite families of abelian
$2$-groups, whereas previously the largest known size was $3$.
