On the typical values of the cross-correlation measure

Gyarmati, Mauduit and S'ark"ozy introduced the \textit{cross-correlation measure} ${\mathrm{\Phi }}_k(\mathcal{F})$ 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 ${\mathrm{\Phi }}_k(\mathcal{F})$ 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}$)$isoforder$\sqrt{N\log \binom{N}{k}+k\log |$\mathcal{F}$|}$foranygiven$2\leq k \leq N/(6\log_2 |$\mathcal{F}$|)$.