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

TL;DR

This paper investigates the relationship between family complexity and cross-correlation measure in binary sequence families, providing bounds and applying results to sequences derived from quadratic residues, demonstrating their pseudorandom properties.

Contribution

It establishes a link between family complexity and cross-correlation measure for binary sequences and applies this to sequences based on quadratic residues, showing they have desirable pseudorandom properties.

Findings

Family complexity can be estimated using cross-correlation measure.

Sequences from quadratic residues have high family complexity.

Both the sequence family and its dual have low cross-correlation measures.

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 $(e_{i,1},\ldots,e_{i,N})\in \{-1,+1\}^N$, $i=1,\ldots,F$, 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.