Family complexity and cross-correlation measure for families of binary sequences

Abstract

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 $(e_{i,1},\dots ,e_{i,N})\in \{-1,+1{\}}^N$, $i=1,\dots ,F$, of binary sequences of length $N$ in terms of the cross-correlation measure of its dual family $(e_{1,n},\dots ,e_{F,n})\in \{-1,+1{\}}^F$, $n=1,\dots ,N$. We apply this result to the family of sequences of Legendre symbols with irreducible quadratic polynomials modulo $p$ with middle coefficient $0$, that is, $e_{i,n}={\left(\frac{n^2-b{i}^2}{p}\right)}_{n=1}^{(p-1)/2}$ for $i=1,\dots ,(p-1)/2$, 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.