Invertibility of an $n$ by $n$ binary matrix in which each column has exactly $b$ ones

Question

In my research an important part has boiled down to the following linear algebra puzzle. For which $n, b \in \mathbb{R}$ can we construct a binary matrix $A$ such that $A$ is invertible and each column of $A$ has exactly $b$ ones in it. I would like to know if this holds for general $b \in (2, \dots, n -1)$.