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 pair maximum length linear recursive sequences 2 2m -1 with relative decimation d=2 xmlns:xlink="http://www.w3.org/1999/xlink">m+2 -3, where m even. The result indicates that there are at most five distinct values. Equivalently, values in Walsh power permutation f(x)=x xmlns:xlink="http://www.w3.org/1999/xlink">d over finite field order and nonzero weights cyclic code two primitive nonzeros ? . obtain this proves constraints on number roots certain seventh degree polynomials can have unit circle field. also works when odd, which case associated spectra six