On the typical values of the cross-correlation measure

TL;DR

This paper analyzes the typical magnitude of the cross-correlation measure for random families of binary sequences, providing bounds that depend on sequence length, family size, and correlation order.

Contribution

It establishes the order of the cross-correlation measure for most families of binary sequences, extending understanding of their randomness properties.

Findings

For most families of size less than 2^{N/12}, the measure is approximately sqrt(N log choose(N,k) + k log |F|).

The measure's order is characterized for all k within a specified range.

Provides probabilistic bounds on the typical cross-correlation measure values.

Abstract

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 $2\leq k \leq N/(6\log_2 |\mathcal{F}|)$.