On the rank of random matrices

Let M = mij‘ be a random n × n matrix over GF(2). Each matrix entry mij is independently and identically distributed, with Prmij = 0‘ = 1 − pn‘ and Prmij = 1‘ = pn‘. The probability that the matrix M is nonsingular tends to c2 ≈ 0:28879 provided minp; 1− p‘ ≥ log n+ dn‘‘/n for any dn‘ → ∞. Sharp thresholds are also obtained for constant dn‘. This answers a question posed in a paper by J. Blömer, R. Karp, and E. Welzl (Random Struct Alg, 10(4) (1997)). © 2000 John Wiley & Sons, Inc. Random Struct. Alg., 16, 209–232, 2000