二元序列族的族复杂性与互相关测度
Family complexity and cross-correlation measure for families of binary sequences
摘要：
本文研究了二进制序列族的两种伪随机性测度：族复杂度和Ahlswede等人提出的互相关测度之间的关系。在2003年和最近由Gyarmati等人，分别更准确地说，我们估计家庭的家庭复杂度的一个家庭$(e_{i,1},\ldots,e_{i,N})\in \{-1,+1\}^N$，$i=1,\ldots,F$，长度$N$的二进制序列的交叉相关措施，其对偶家庭$(e_{1,n},\ldots,e_{F,n})\in \{-1,+1\}^F$，$n=1,\ldots,N$。我们将这一结果应用于家庭的勒让德符号的序列与不可约的二次多项式模$p$与中间系数$0$，即，$e_{i,n}=\left(\frac{n^2-bi^2}{p}\right)_{n=1}^{(p-1)/2}$为$i=1,\ldots,(p-1)/2$，其中$b$是一个二次nonresidue模$p$，表明这个家庭以及它的双重家庭都有一个大的家庭复杂性和一个小的互相关措施，以一个相当大的顺序。
英文摘要：
We study the relationship between two measures of pseudorandomness for families of binary sequences: family complexity and cross-correlation measure introduced by Ahlswede et al.\ in 2003 and recently by Gyarmati et al., respectively. More precisely, we estimate the family complexity of a family $(e_{i,1},\ldots,e_{i,N})\in \{-1,+1\}^N$, $i=1,\ldots,F$, of binary sequences of length $N$ in terms of the cross-correlation measure of its dual family $(e_{1,n},\ldots,e_{F,n})\in \{-1,+1\}^F$, $n=1,\ldots,N$. We apply this result to the family of sequences of Legendre symbols with irreducible quadratic polynomials modulo $p$ with middle coefficient $0$, that is, $e_{i,n}=\left(\frac{n^2-bi^2}{p}\right)_{n=1}^{(p-1)/2}$ for $i=1,\ldots,(p-1)/2$, where $b$ is a quadratic nonresidue modulo $p$, showing that this family as well as its dual family have both a large family complexity and a small cross-correlation measure up to a rather large order.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
