  The gonality of a smooth geometrically connected curve over a field $k$ is
the smallest degree of a nonconstant $k$-morphism from the curve to the
projective line. In general, the gonality of a curve of genus $g \ge 2$ is at
most $2g - 2$. Over finite fields, a result of F.K. Schmidt from the 1930s can
be used to prove that the gonality is at most $g+1$. Via a mixture of geometry
and computation, we improve this bound: for a curve of genus $g \ge 5$ over a
finite field, the gonality is at most $g$. For genus $g = 3$ and $g = 4$, the
same result holds with exactly $217$ exceptions: There are two curves of genus
$4$ and gonality $5$, and $215$ curves of genus $3$ and gonality $4$. The
genus-$4$ examples were found in other papers, and we reproduce their equations
here; in supplementary material, we provide equations for the genus-$3$
examples.
