Singularity of Random Matrices over Finite Fields

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 $0 < \alpha < 1$. We show that the singularity probability converges to the uniform limit with an exponentially small error depending only on $\alpha$. We also show that the distribution of the determinant of $A$ converges to its limiting distribution at an exponential rate.