  In this paper we study random linear systems with $k$ variables per equation
over the finite field GF(2), or equivalently $k$-XOR-CNF formulas. In a
previous paper Creignou and Daud\'e proved that the phase transition for the
consistency (satisfiability) of such systems (formulas) exhibits a sharp
threshold. Here we prove that the phase transition occurs as the number of
equations (clauses) is proportional to the number of variables. For any $k\ge
3$ we establish first estimates for the critical ratio. For $k=3$ we get 0.93
as an upper bound, 0.89 as a lower bound, whereas experiments suggest that the
critical ratio is approximately 0.92.
