  Let $M_n$ be a random $n\times n$ matrix with i.i.d. $\text{Bernoulli}(1/2)$
entries. We show that for fixed $k\ge 1$, \[\lim_{n\to
\infty}\frac{1}{n}\log_2\mathbb{P}[\text{corank }M_n\ge k] = -k.\]
