The rank of sparse random matrices over finite fields

Let M be a random (n×n)-matrix over GF[q] such that for each entry Mij in M and for each nonzero field element α the probability Pr[Mij=α] is p/(q−1), where p=(log n−c)/n and c is an arbitrary but fixed positive constant. The probability for a matrix entry to be zero is 1−p. It is shown that the expected rank of M is n−𝒪(1). Furthermore, there is a constant A such that the probability that the rank is less than n−k is less than A/qk. It is also shown that if c grows depending on n and is unbounded as n goes to infinity, then the expected difference between the rank of M and n is unbounded.