On the Propagation Criterion for Boolean Functions and on Bent Functions

We consider the parameters of a Boolean function that characterize its position relative to the first-order Reed-Muller code $R(1,n)$. We establish simple criteria of a given vector being, for a given Boolean function, unessential, a linear structure, or belonging to $P\mathbb C(f)$. We find conditions under which the set $P\mathbb C(f)$ contains some linear subspace (without zero). We show that the greater the dimension of the subspace, the more distant such functions are from $R(1,n)$. We obtain a new description of the class of bent functions most remote from $R(1,n)$.