A theorem on the limiting distribution for the number of false solutions of a system of nonlinear random Boolean equations

We prove that the distribution of the number of false solutions of a consistent system of nonlinear random Boolean equations with stochastically independent coefficients is asymptotically Poisson with parameter $2^m$ as the number $n$ of unknowns tends to infinity. Our principal assumptions are: the distributions of the coefficients vary in a vicinity of the point $\frac 12$, $n$ and the number $N$ of equations of the system differ by a constant $m$ as $n\to\infty$; the system has a solution which contains $\rho(n)$ units, where $\rho(n)\to\infty$ as $n\to\infty$.