A Counterexample to a Conjecture of Niho

A conjecture of Niho states that under certain assumptions the Fourier transform of the function Tr (x<sup>d</sup>) on F<sub>2</sub>n ,where d = (2<sup>tk</sup>1) /(2<sup>k</sup> +1), has a spectrum with at most five values. We present a counterexample to this conjecture, and the theory behind finding it. We use the theory of quadratic forms over F<sub>2</sub>.

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