  The degree-diameter problem seeks to find the largest possible number of
vertices in a graph having given diameter and given maximum degree. Very often
the problem is studied for restricted families of graph such as
vertex-transitive or Cayley graphs, with the goal being to find a family of
graphs with good asymptotic properties. In this paper we restrict attention to
Cayley graphs, and study the asymptotics by fixing a small diameter and
constructing families of graphs of large order for all values of the maximum
degree. Much of the literature in this direction is focused on the diameter two
case. In this paper we consider larger diameters, and use a variety of
techniques to derive new best asymptotic constructions for diameters 3, 4 and 5
as well as an improvement to the general bound for all odd diameters. Our
diameter 3 construction is, as far as we know, the first to employ matrix
groups over finite fields in the degree-diameter problem.
