Geodesic
On Polynomials over a Finite Field of Even Characteristic with Maximum Absolute Value of the Trigonometric Sum
Matematičeskie zametki, Tome 72 (2002) no. 2, pp. 171-177

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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, $|S(f)|=(\deg f-1)\sqrt Q$. 
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     author = {L. A. Bassalygo and V. A. Zinov'ev},
     title = {On {Polynomials} over a {Finite} {Field} of {Even} {Characteristic} with {Maximum} {Absolute} {Value} of the {Trigonometric} {Sum}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {171--177},
     publisher = {mathdoc},
     volume = {72},
     number = {2},
     year = {2002},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2002_72_2_a1/}
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L. A. Bassalygo; V. A. Zinov'ev. On Polynomials over a Finite Field of Even Characteristic with Maximum Absolute Value of the Trigonometric Sum. Matematičeskie zametki, Tome 72 (2002) no. 2, pp. 171-177. http://geodesic.mathdoc.fr/item/MZM_2002_72_2_a1/