A Counterexample to a Conjecture of Niho

A conjecture of Niho states that under certain assumptions the Fourier transform of the function ${\rm Tr}(x^{d})$ on $\BBF _{2^{n}}$, where $d=(2^{tk}+1)/(2^{k}+1)$, 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 $\BBF _{2}$ .