| We consider the random system of 2-linear equations over the finite field GF(2) whose left hand side corresponds to the random graph G(n, p) and whose right hand side consists of independent Bernoulli random variables with success probability 1/2, assuming that the right hand side is independent of the left hand side.;G(n, p) is the random graph with n labeled vertices such that each of the n2 possible edges is present in the graph independently of all others, with probability p. The structure of G( n, p) undergoes abrupt changes called "the phase transition", as n goes to infinity. When p = (1+lambda n-1/3)/n, where lambda → -infinity as n → infinity and 0 ≤ p ≤ 1, it is called the subcritical phase. At the subcritical phase, with high probability, G(n, p) has only trees and few unicyclic components. When p = (1 + lambda n-1/3)/n, where lambda is a fixed number, it is called the critical phase. At the critical phase, with high probability, large trees and unicyclic components of G(n, p) start to be joined together, giving rise to larger components with more than one cycle that soon merge to form a single giant component.;We prove that when G(n, p) is at the subcritical phase and |lambda| >> n1/39, |lambda| = O(n1/12-epsilon ) with a fixed 0 < epsilon < 1/12 - 1/39, the probability of solvability of the random system corresponding to G( n, p) is asymptotic to e3/8|lambda| 1/4n-1/12 as n → infinity. Also, we prove that when G(n, p) is at the critical phase, the probability of solvability of the random system corresponding to G(n, p) is asymptotic to c lambdan-1/12 as n → infinity, where the constant clambda is expressed as a convergent double series depending on lambda. |
