I am interested in the following problem: I have a finite field $F_q$, two positive integers $n>m$ and elements $a_1,...,a_m\in F_q$. How many of the polynomials $x^n+a_1x^{n-1}+...+a_mx^{n-m}+c_{m+1}x^{n-m-1}+...+c_n,c_i\in F_q$ are irreducible? What are the best known estimates, esp. for $q$ fixed and $m,n\to\infty$?