发表机构
The Hong Kong University of Science and Technology; Harbin Institute of Technology, Shenzhen（香港科技大学; 哈尔滨工业大学（深圳））
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文针对Niho型抽取d=4(2^m-1)+1，通过归一化根集、结合特征和与Carlitz定理求值，确定了有限域F_{2^{2m}}上的互相关分布并给出频率显式公式。
AI 中文摘要
互相关问题是序列设计中的经典问题。本文对每个正整数m，确定了有限域F_{2^{2m}}上Niho型抽取d=4(2^m-1)+1的互相关分布。令q=2^m，这等价于对所有a∈F_{q^2}，确定多项式f_a(x)=x^7+ax^4+\bar{a}x^3+1在集合U_{q+1}中的根的数量分布，其中U_{q+1}={x∈F_{q^2}:x^{q+1}=1}。主要难点在于计数U_{q+1}中可作为f_a根集的四元子集。我们通过乘积对这类子集进行归一化，利用相关预解式研究所得条件，将所需计数简化为F_q上的若干方程，这些方程通过特征和与Kloosterman和求解。剩余的混合Kloosterman和与F_{2^{2m}}上的Kloosterman和相关，利用Carlitz定理对其求值。最终，我们得到了互相关分布中所有频率的显式公式。
英文摘要
The cross-correlation problem is a classical problem in sequence design. In this paper, we determine the cross-correlation distribution of the Niho-type decimation $d=4(2^m-1)+1$ over $\mathbb F_{2^{2m}}$ for every positive integer $m$. With $q=2^m$, this is equivalent to determining the distribution of the number of roots in $U_{q+1}$ of $f_a(x)=x^7+ax^4+\bar a x^3+1$ for $a\in\mathbb F_{q^2}$, where $U_{q+1}=\{x\in\mathbb F_{q^2}:x^{q+1}=1\}$. The main difficulty is to count the four-element subsets of $U_{q+1}$ that may occur as root sets of $f_a$. We normalize such subsets by their product and study the resulting condition through the associated resolvent. This reduces the required enumeration to several equations over $\mathbb F_q$, which are evaluated by character sums and Kloosterman sums. The remaining mixed Kloosterman sum is related to a Kloosterman sum over $\mathbb F_{2^{2m}}$ and is evaluated using a theorem of Carlitz. Consequently, we obtain explicit formulas for all the frequencies in the cross-correlation distribution.
