论文标题
NIHO关于序列，代码和布尔功能的最后一个猜想的分辨率
The resolution of Niho's last conjecture concerning sequences, codes, and Boolean functions
论文作者
论文摘要
一种新方法用于解决NIHO的长期猜想，以涉及一对最大长度长度线性递归序列长度$ 2^{2 m} -1 $的互相关光谱，相对删除$ d = 2 = 2^{m+2} -3 $，$ m $均匀。结果表明最多有五个不同的互相关值。等效地，结果表明，在有限的订单$ 2^{2 m} $的有限范围内，功率排列的WALSH频谱中最多有五个不同的值，最多有五个不同的非零重量在长度为$ 2^{2 m} -1 $ 2^{2 m} -1 primitive nonzeros $ a $ a $的环状代码中。用于获得此结果的方法证明了某些七度多项式可以在有限场的单位圆上具有的根数的限制。当$ m $奇数时，该方法也可以工作，在这种情况下，相关的互相关和沃尔什光谱最多具有六个不同的值。
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.
