  In the degree-diameter problem for Abelian Cayley and circulant graphs of
diameter 2 and arbitrary degree d there is a wide gap between the best lower
and upper bounds valid for all d, being quadratic functions with leading
coefficient 1/4 and 1/2 respectively. Recent papers have presented
constructions which increase the coefficient of the lower bound to be at or
just below 3/8, but only for sparse sets of degree d related to primes of
specific congruence classes. By applying results from number theory these
constructions can be extended to be valid for every degree above some
threshold, establishing an improved asymptotic lower bound approaching 3/8. The
constructions use the direct product of the multiplicative and additive
subgroups of a Galois field and a specific coprime cyclic group. By
generalising this method an improved upper bound, with quadratic coefficient
3/8, is established for this class of construction of Abelian Cayley and
circulant graphs. Analysis of the order of the known extremal diameter 2
circulant graphs, up to degree 23, is shown to provide tentative support for a
quadratic coefficient of 3/8 for the asymptotic upper bound for the order of
general diameter 2 circulant graphs of arbitrary degree.
