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

Gyarmati, Mauduit and S'ark"ozy introduced the cross-correlation measure ${\mathrm{\Phi }}_k(G)$ 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.