  A well-known result of von zur Gathen asserts that a non-exceptional
permutation polynomial of degree $n$ over $\mathbb{F}_{q}$ exists only if
$q<n^{4}$. With the help of the Weil bound for the number of
$\mathbb{F}_{q}$-points on an absolutely irreducible (possibly singular) affine
plane curve, Chahal and Ghorpade improved von zur Gathen's proof to replace
$n^{4}$ by a bound less than $n^{2}(n-2)^{2}$. Also based on the Weil bound, we
further refine the upper bound for $q$ with respect to $n$, by a more concise
and direct proof following Wan's arguments.
