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.
翻译：一种新方法用于解决Niho长期的猜想,即关于最大线性递归序列2 ⁇ 2m}-1美元,其相对耗损值为$=2 ⁇ m+2}-3美元(美元平差)。结果显示,最多有5种不同的交叉关系值。同样,结果显示,在功率调整($f(x)=x*d$x*d$)的沃尔什频谱中,最多有5种不同的数值,超过2 ⁇ 2m}的限定字段,在2 ⁇ 2m}-1美元的周期代码中,最多有5种不同的非零重量,其长度为$2 ⁇ 2m}-1美元,其原始非零值为$\alpha$和$\alpha ⁇ d$。获得这一结果所使用的方法证明,某些七度的多元度对有限字段单位圆的根数存在限制。当美元为奇数时,该方法也起作用,在这种情况下,相关的交叉关系和沃尔什光谱仪的数值最多为6个不同的数值。
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