On the rank of random matrices

Abstract

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. Blomer, R. Karp, and E. Welzl (Random Struct Alg, 10(4) (1997)). ©2000 John Wiley & Sons, Inc. Random Struct. Alg., 16, 209–232, 2000