The resolution of Niho's last conjecture concerning sequences, codes, and Boolean functions

A new method is used to resolve a long-standing conjecture of Niho concerning the crosscorrelation spectrum of a pair of maximum length linear recursive sequences of length $2^{2 m}-1$ with relative decimation $d=2^{m+2}-3$, where $m$ is even. The result indicates that there are only five possible crosscorrelation values. Equivalently, the result indicates that there are five possible values in the Walsh spectrum of the power permutation $f(x)=x^d$ over the finite field of order $2^{2 m}$ and five possible nonzero weights in the cyclic code of length $2^{2 m}-1$ with two primitive nonzeros $\alpha$ and $\alpha^d$. The method used to obtain this result proves constraints on the number of roots that certain seventh degree polynomials can have on the unit circle of a finite field. The method also works when $m$ is odd, in which case the associated crosscorrelation and Walsh spectra have only six possible values.