On the typical values of the cross-correlation measure

Abstract

Gyarmati, Mauduit and Sárközy introduced the 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 $\sqrt{N\log \left( {\begin{array}{c}N\\ k\end{array}}\right) +k\log |\mathcal {F}|}$ for any given $2\le k \le N/(6\log _2 |\mathcal {F}|)$.