Limit theorems for the number of nonzero solutions of a system of random equations over GF(2)

The asymptotic behavior of a number of solutions of a system of random equations of a particular form over GF(2) is investigated. The left-hand sides of the equations of the system are products of independent equiprobable linear functions in $n$ variables for GF(2), whereas the right-hand sides are equal to zero. Under the natural restrictions on the way of changing the parameters of the scheme (the number of unknowns, the number of equations, and the number of multipliers in the left-hand side of each equation) it is shown that the distribution of the number of nonzero solutions converges to a Poisson distribution. Sufficient conditions are given for the number of nonzero solutions to be asymptotically normal. The proofs are based on the moment method.