On the rate of convergence to the normal distribution of the number of false solutions of a system of nonlinear random Boolean equations

We prove a theorem on the limit normal distribution (as $n \to \infty$) of the number of false solutions of a system of nonlinear equations with independent random coefficients belonging to the field GF(2). We assume that every equation contains at least one coefficient for which the probability that it attains the value 1 is close to $\frac {1}{2}$; the number of equations $N$ and the number of unknowns $n$ are such that $n-N \to \infty$ as $n \to \infty$; the system has a solution containing $\rho (n)$ units and $\rho (n) \to \infty$ as $n \to \infty$.