Measures of pseudorandomness of binary lattices, III ( $Q_k$, correlation, normality, minimal values)

In an earlier paper Hubert, Mauduit and Sárk zy defined the notion of binary lattice, they introduced the measures of pseudorandomness of binary lattices, and they constructed a binary lattice with strong pseudorandom properties with respect to these measures. Later further constructions of this type have been given by different authors. In this series we study the measures of pseudorandomness of binary lattices. In particular, in this paper first we study the minimum of the measure $Q_k$ in one dimension. Then we introduce the correlation measure $C_k$ in $n$ dimensions, and we estimate the minima of $Q_k$, $C_k$ and the normality measures in two dimensions. The connection between the correlation measures of order two and three of binary lattices is also studied.