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

TL;DR

This paper studies the cross-correlation measure of binary sequence families, establishing its limiting distribution for random families and providing sharp bounds for minimal values, advancing understanding of pseudorandomness.

Contribution

It proves the convergence and limiting distribution of the cross-correlation measure for random families and nearly fully resolves the problem of bounding its minimal values.

Findings

Cross-correlation measure converges strongly for random families.

Limiting distribution of the cross-correlation measure is established.

Sharp bounds for minimal values of the measure are provided.

Abstract

Gyarmati, Mauduit and S\'ark\"ozy introduced the cross-correlation measure $\Phi_k(G)$ of order $k$ to measure the level of pseudorandom properties of families of finite binary sequences. In an earlier paper we estimated the cross-correlation measure of a random family of binary sequences. In this paper, we sharpen these earlier results by showing that for random families, the cross-correlation measure converges strongly, and so has limiting distribution. We also give sharp bounds to the minimum values of the cross-correlation measure, which settles a problem of Gyarmati, Mauduit and S\'ark\"ozy nearly completely.