Consider unitary polynomials of degree $n$ over $GF(2)$. That is, polynomials of the form $p(x) = \sum_{i=0}^n a_i x^i$ where $a_i\in GF(2)$ and $a_n=1$.
Can we always find such an irreducible polynomial $p(x)$ of degree $n$ where $\textrm{degree}(p(x)-x^n)\leq n/2$?