Let $n > 2$ and let $S \subsetneq [n-1] = \{1, \ldots, n-1\}$. Has it been proven whether or not we can always find an irreducible polynomial $f(x)$ of degree $n$ over the binary field $GF(2)$ such that it has exactly an odd (even) number of $s \in S$ for which $[x^s] f(x) = 1$?
I am aware there are open conjectures regarding existence of primitive (irreducible) polynomials with several prescribed coefficients over GF(2), but I haven't seen anything regarding the parity of the prescribed set. In any case this should be a much weaker conjecture than these, but I wonder if anyone could refer me to previous work on this?