On algebraic and combinatorial properties of propagation criteria for Boolean functions

In this paper we investigate the algebraic and combinatorial properties of the propagation criterion for Boolean functions. We prove the necessary and sufficient conditions for the cardinality of the set of vectors satisfying the propagation criterion for Boolean functions in terms of the number of zeros un the special matrix associated with this function. We investigate the propagation criterion for Boolean functions supported by a subspace of the space $V_n$ of dimension $n-2$. We obtain the exact relations for the cardinality of the set of vectors satisfying the propagation criterion for a function representing the XOR of two Boolean functions from disjoint sets of variables and also for Boolean functions with nontrivial space of linear structures.