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

TL;DR

This paper resolves Niho's long-standing conjecture by showing that certain sequences, codes, and Boolean functions have at most five or six distinct spectral values, using a novel polynomial root constraint method.

Contribution

It introduces a new method to determine the maximum number of spectral values for sequences, codes, and Boolean functions related to Niho's conjecture, resolving a long-standing open problem.

Findings

At most five distinct crosscorrelation values for even m

At most six distinct spectral values for odd m

Method constrains roots of seventh degree polynomials on finite field unit circle

Abstract

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 at most five distinct crosscorrelation values. Equivalently, the result indicates that there are at most five distinct values in the Walsh spectrum of the power permutation $f(x)=x^d$ over a finite field of order $2^{2 m}$ and at most five distinct 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 at most six distinct values.