  We study the number of irreducible polynomials over $\mathbf{F}_{q}$ with
some coefficients prescribed. Using the technique developed by Bourgain, we
show that there is an irreducible polynomial of degree $n$ with $r$
coefficients prescribed in any location when $r \leq \left[\left(1/4 -
\epsilon\right)n \right]$ for any $\epsilon>0$ and $q$ is large; and when
$r\leq\delta n$ for some $\delta>0$ and for any $q$. The result is improved
from the earlier work of Pollack that the similar result holds for
$r\leq\left[(1-\epsilon)\sqrt{n}\right]$.
