Limit theorems for the number of nonzero solutions of a system of random equations over the field $\mathrm{GF}(2)$

We study the properties of the number $\nu$ of non-zero solutions of system of random equations over $\mathrm{GF}(2)$ with the left-hand sides which are products of expressions of the form $a_{t1}x_1+\ldots+a_{tn}x_n+a_t$ with independent equiprobable coefficients. The right-hand sides of the system are zeros. We derive inequalities for the factorial moments of the random variable $\nu$ and necessary and sufficient conditions of the validity of the Poisson limit theorem for $\nu$.
The research was supported by the Russian Foundation for Basic Research, grants 99–01–00012 and 96–15–96092.