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