We study the relationship between two measures of pseudorandomness forfamilies of binary sequences: family complexity and cross-correlation measureintroduced 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 sequencesof 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 withirreducible quadratic polynomials modulo $p$ with middle coefficient $0$, thatis, $e_{i,n}=\left(\frac{n^2-bi^2}{p}\right)_{n=1}^{(p-1)/2}$ for$i=1,\ldots,(p-1)/2$, where $b$ is a quadratic nonresidue modulo $p$, showingthat this family as well as its dual family have both a large family complexityand a small cross-correlation measure up to a rather large order.
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机译：我们研究了二元序列家族的两种伪随机度量之间的关系：家族复杂性和互相关度量，分别由Ahlswede等人于2003年和Gyarmati等人于2003年提出。更准确地说，我们估算\ {-1，+ 1 \} ^ N $，$ i = 1，中的家庭$（e_ {i，1}，\ ldots，e_ {i，N}）\ \ ldots，F $，长度为$ N $的二进制序列，根据其双族$（e_ {1，n}，\ ldots，e_ {F，n}）的互相关度量，在\ {-1 ，+ 1 \} ^ F $，$ n = 1，\ ldots，N $。我们将此结果应用于具有不可约二次多项式以中间系数$ 0 $为模的$ p $模的Legendre符号序列族，即$ e_ {i，n} = \ left（\ frac {n ^ 2-bi ^ 2} { p} \ right）_ {n = 1} ^ {（p-1）/ 2} $ for $ i = 1，\ ldots，（p-1）/ 2 $，其中$ b $是二次非残差模$ p $，表明这个家庭及其双重家庭具有较大的家庭复杂性和较小的互相关度量，直至相当大的阶数。
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