On the typical values of the cross-correlation measure

Abstract

Gyarmati, Mauduit and Sárközy introduced the \textit{cross-correlation measure} ${\Phi }_k(\mathcal{F})$ to measure the randomness of families of binary sequences $\mathcal{F}\subset \{-1,1{\}}^N$. In this paper we study the order of magnitude of the cross-correlation measure ${\Phi }_k(\mathcal{F})$ for typical families. We prove that, for most families $\mathcal{F}\subset \{-1,1{\}}^N$ of size $2\le |\mathcal{F}|<2^{N/12}$, ${\Phi }_k(\mathcal{F})$ is of order $\sqrt{N\log \left(\genfrac{}{}{0ex}{}{N}{k}\right)+k\log |\mathcal{F}|}$ for any given $2\le k\le N/(6{\log }_2|\mathcal{F}|)$.

Cited by in corpus (1)

• The cross-correlation measure of families of finite binary sequences: limiting distributions and minimal values