The Resolution of Niho’s Last Conjecture Concerning Sequences, Codes, and Boolean Functions
The Resolution of Niho’s Last Conjecture Concerning Sequences, Codes, and Boolean Functions
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DOI：
10.1109/tit.2021.3098342
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发表时间：
2020-06
影响因子：
2.5
通讯作者：
T. Helleseth;D. Katz;Chunlei Li
中科院分区：
文献类型：
--
作者：
T. Helleseth;D. Katz;Chunlei Li
用一种新的方法解决了Niho关于长度为$2^{2m}-1$且相对抽取$d=2^{m+2}-3$的一对最大长度线性递推序列的互相关谱的长期猜想，其中$m$为偶。结果表明，最多存在5个不同的互相关值。同样地，结果表明在阶为$2^{2m}$的有限域上，幂置换$f(x)=x^{d}$的Walsh谱中最多有5个不同的值，并且在长度为$2^{2m}-1$的循环码中最多有5个不同的非零权值，其中$\alpha $和$\alpha ^{d}$为两个原始非零。得到这一结果的方法证明了有限域的单位圆上某些七次多项式的根数的约束。当$m$为奇数时，该方法也有效，在这种情况下，相关的互相关和沃尔什谱最多有六个不同的值。
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 $2^{2m}-1$ 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 $2^{2m}$ and at most five distinct nonzero weights in the cyclic code of length $2^{2m}-1$ 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.