On Polynomials over a Finite Field of Even Characteristic with Maximum Absolute Value of the Trigonometric Sum

We study trigonometric sums in finite fields $F_Q$. The Weil estimate of such sums is well known: $|S(f)|\le (\deg f-1)\sqrt Q$, where $f $is a polynomial with coefficients from $F(Q)$. We construct two classes of polynomials $f$, $(Q,2)=2$, for which $|S(f)|$ attains the largest possible value and, in particular, $|S(f)|=(\deg f-1)\sqrt Q$.