Singularity of Random Matrices over Finite Fields

Abstract

Let A be an n times n random matrix with iid entries over a finite field of order q. Suppose that the entries do not take values in any additive coset of the field with probability greater than $1 - alpha for some fixed $0 0 are absolute. We also show that the determinant of A assumes each non-zero value with probability q -1 prodk=2 infty (1 - q -k) + O(e -calpha n), where the constants are absolute.