Probability of Solvability of Random Systems of 2-Linear Equations over GF(2)

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 all the n(n-1)/2 possible edges is present in the graph independently of all others, with probability p. We prove that when G(n,p) is at the subcritical phase and |λ| >> n^1/39, |λ| = O(n^1/12-ε) with a fixed 0<ε<1/12-1/39, the probability of solvability of the random system corresponding to G(n,p) is asymptotic to e^3/8 |λ| ^1/4 n^-1/12 as n goes to 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_λ n^-1/12 as n goes to infinity, where the constant c_λ is expressed as a convergent double series depending on λ.