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 ∪iCi, such that the value of the function f is the same on each vector of Ci, i.e. for every i and x,y∈Ci, f(x)=f(y). The complexity of it denotes by HSCP(f) 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.