  Mixed graphs can be seen as digraphs that have both arcs and edges (or
digons, that is, two opposite arcs). In this paper, we consider the case in
which such graphs are Cayley graphs of Abelian groups. These groups can be
constructed by using a generalization to $\mathbb{Z}^n$ of the concept of
congruence in $\mathbb{Z}$. Here we use this approach to present some families
of mixed graphs, which, for every fixed value of the degree, have an
asymptotically large number of vertices as the diameter increases. In some
cases, the results obtained are shown to be optimal.
