  Bennett, Hart, Iosevich, Pakianathan, and Rudnev found an exponent $s<d$ such
that any set $E\subset \mathbb{F}_q^d$ with $|E|\gtrsim q^s$ determines
$\gtrsim q^{\binom{k+1}{2}}$ congruence classes of $(k+1)$-point configurations
for $k\leq d$. Because congruence classes can be identified with tuples of
distances between distinct points when $k\leq d$, and because there are
$\binom{k+1}{2}$ such pairs, this means any such $E$ determines a positive
proportion of all congruence classes. In the $k>d$ case, fixing all pairs of
distnaces leads to an overdetermined system, so $q^{\binom{k+1}{2}}$ is no
longer the correct number of congruence classes. We determine the correct
number, and prove that $|E|\gtrsim q^s$ still determines a positive proportion
of all congruence classes, for the same $s$ as in the $k\leq d$ case.
