The complexity of subcube partition relates to the additive structure of the support

Abstract

The subcube partition of a Boolean function is a partition of ${\{0,1\}}^n$ into the union of subcubes ${\cup }_i{C}_i$, such that the value of the function f is the same on each vector of $C_i$, i.e. for every i and $x,y\in C_i$, $f\left(x\right)=f\left(y\right)$. The complexity of it denotes by $H_{SCP}\left(f\right)$ is the minimum number of subcubes in a subcube partition which computes the Boolean function f. We give a lower bound of the complexity of subcube partitions of Boolean function which relates the additive behaviour of the support and the influence of the function.