Characteristics of random systems of linear equations over a finite field

We consider random and a priori consistent random systems of equations over a finite field with q elements in a unknowns. A random system consists of M = M(n) equations, each of which can depend on 2, 3, ... , m variables, which are obtained by sampling without replacement. We obtain limit distributions and estimates of moments for the numbers of solutions of random systems of equations provided that n -> infinity and the relation between the parameters n and M, the number of vertices and the number of hyperedges, falls into the subcritical domain of the evolution of random hypergraphs which describe the random systems of equations. The form and parameters of the limit distributions are determined by the characteristics of the limit distributions of the number of cycles of a special form in the corresponding random hypergraphs.