Irreducible polynomials with several prescribed coefficients

We study the number of irreducible polynomials over FqFq 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≤[(1/4−ϵ)n]r≤[(1/4−ϵ)n] for any ϵ>0ϵ>0 and q   is large; and when r≤δnr≤δn for some δ>0δ>0 and for any q  . The result improves earlier work of Pollack stating that a similar result holds for r≤[(1−ϵ)n].