  The Waring function $g(k,q)$ measures the difficulty of Waring's problem for
$k$th powers in the field of $q$ elements. Its calculation seems to be
difficult, and many partial results have been published, notably upper bounds
for certain regions of the $k$-$q$-plane. In this paper, we compute the exact
value of $g(k,q)$ for two infinite families of exponent-field pairs. In these,
$k$ is large compared to $q$. We use a new method of proof that is mainly
combinatorial in nature.
