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

We study the properties of the number v of non-zero solutions of system of random equations over GF(2) with the left-hand sides which are products of expressions of the form &t\\ + · · · + tnXn 4&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 v and necessary and sufficient conditions of the validity of the Poisson limit theorem for v. The research was supported by the Russian Foundation for Basic Research, grants 99-01-00012 and 96-15-96092. We consider the system of random linear equations over GF(2) mt °> f = l,...,r, (1) where m, G {!,... ,n} and the coefficients a{y are independent random variables taking the values 0 and 1 with probabilities 1/2. The quantities a\ are assumed to be known. System (1) has the zero solution if and only if a/ . . . a?' = 0 for all t = 1 , . . . , Τ. In order to simplify the exposition, we study the number v of non-zero solutions of system (1). We denote by V the η-dimensional linear space over GF(2) and set (m)r = m(m-l). . .(m-r-t-l). Let