On the number of solutions of systems of linear Boolean equations in a set of vectors with a given number of ones

We consider the distribution of the number of solutions of systems of random Boolean equations in the set of vectors with a given number of ones (or of a given weight). Both for systems with independent left-hand and right-hand sides and for a fortiori consistent systems, we give sufficient conditions for the distributions to converge to the Poisson law and to the standard normal law.