Random Graphs and Systems of Linear Equations in Finite Fields

For a T × n matrix A = ‖ aij‖ in GF(2) we define a hypergraph GA with n vertices and T hyperedges et = {j: atj = 1}, t = 1,…, T. Denote at = (atn),t= 1,…, T. A set of row numbers {t1,…,tm} is called a critical set if the sum of vectors at1 + … + at m is the zero vector. In terms of hypergraph GA a critical set can be interpreted as a hypercycle. We can naturally definite the concept of independence for critical sets. Let s(A) be the maximal number of independent critical sets in A. The rank r(A) of the matrix A and s(A) are connected by the equality r(A) + s(A) = T. The total number of critical sets S(A) is equalt 2s(A) −1. Consider the folowing system of T random equations in GF(2): where i1(t.…,i(t), i = 1,…,T are indepndent identically distributed random variables which take values 1,…., n with equal probabilities. Denote Ar, n, T the matrix of this system. We prove that the number S(Ar,n,T) of critical sets in Ar, n, T or hypercycles in GAr,n,T has a threshold property. Let n, T→∞ and T/n→ α. Then for any fixed integer r ⩾ 3 there exists a constant α, such that MS(Ar,n,T)→0 if α α αr. © 1994 John Wiley & Sons, Inc.