The stability of sets of solutions for systems of equations with random distortions

Let in a system of equations left sides be functions from $\{0,1,\dots,N-1\}$ to $\{0,1\}$ and in the other system all equations be obtained from the equations of the first one by random distortions of the truth tables of these functions. We find conditions on the probability laws of distortions under which the set of solutions of the second system includes all, some or no solutions of the first system.