  We study the distribution of the Galois group of a random $q$-additive
polynomial over a rational function field: For $q$ a power of a prime $p$, let
$f=X^{q^n}+a_{n-1}X^{q^{n-1}}+\ldots+a_1X^q+a_0X$ be a random polynomial chosen
uniformly from the set of $q$-additive polynomials of degree $n$ and height
$d$, that is, the coefficients are independent uniform polynomials of degree
${\rm deg}\, a_i\leq d$. The Galois group $G_f$ is a random subgroup of ${\rm
GL}_n(q)$. Our main result shows that $G_f$ is almost surely large as $d,q$ are
fixed and $n\to \infty$. For example, we give necessary and sufficient
conditions so that ${\rm SL}_n(q)\leq G_f$ asymptotically almost surely. Our
proof uses the classification of maximal subgroups of ${\rm GL}_n(q)$. We also
consider the limits: $q,n$ fixed, $d\to \infty$ and $d,n$ fixed, $q\to \infty$,
which are more elementary.
