Lower bounds for corner-free sets

A corner is a set of three points in $\mathbf{Z}^2$ of the form $(x, y), (x +
d, y), (x, y + d)$ with $d \neq 0$. We show that for infinitely many $N$ there
is a set $A \subset [N]^2$ of size $2^{-(c + o(1)) \sqrt{\log_2 N}} N^2$ not
containing any corner, where $c = 2 \sqrt{2 \log_2 \frac{4}{3}} \approx
1.822\dots$.