On the typical values of the cross-correlation measure

Gyarmati, Mauduit and S\'ark\"ozy 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 $2\leq |\mathcal{F}|<2^{N/12}$, $\Phi_k(\mathcal{F})$ is of order $\sqrt{N\log \binom{N}{k}+k\log |\mathcal{F}|}$ for any given $2\leq k \leq N/(6\log_2 |\mathcal{F}|)$.