In the appendix of the paper by Tolhuizen ( http://ieeexplore.ieee.org/stamp/stamp.jsp?arnumber=841182) there is a very fast and easy probabilistic proof that for $k=cn$ where $c \in [0,1]$ is fixed, there is a matrix $A$ over $\mathbb{F}_2$ of size $k \times n$ such that the number of invertble $k \times k$ submatrices is at least $0.28 \binom{n}{k}$.
Question: Is there an explicit construction for any $c\in [0,1]$ of a $k \times n$ matrix $A$ over $\mathbb{F}_2$ where $k=cn$ such that there are at least $$\frac{1}{Poly(n)}\binom{n}{k} $$ invertible $k \times k$ submatrices of $A$? (here $Poly(n)$ denotes a fixed polynomial)
Motivation: This paper settles the maximal asymptotic size of a cancellative set system. I am studying set systems that are very similar to cancellative ones so I would like to build them from cancellative ones.