关于序列、码和布尔函数的Niho最后一个猜想的解决
The resolution of Niho's last conjecture concerning sequences, codes, and Boolean functions
摘要：
用一种新的方法解决了Niho关于一对最大长度线性递归序列的互相关谱的一个长期猜想，该序列的长度为$2^{2 m}-1$，具有相对抽取$d=2^{m+2}-3$，其中$m$是偶数.结果表明，最多有五个不同的互相关值。等价地，结果表明：在$2^{2 m}$阶有限域上的幂置换$f(x)=x^d$的Walsh谱中最多有5个不同的值，在长度为$2^{2 m}-1$且具有两个本原非零值$α$和$α^d$的循环码中最多有5个不同的非零权值.用于获得这一结果的方法证明了某些七次多项式在有限域的单位圆上可以具有的根的数量的约束。当$m$是奇数时，该方法也起作用，在这种情况下，相关联的互相关和沃尔什谱具有至多六个不同的值。
英文摘要：
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 $α$ 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 at most six distinct values.
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