I am looking for a class of matrices $M(n(m), m, k(m), \phi)$ with the following properties:

- M is $n \times m$ where $n(m) > m$.
- Every subset of rows of size $k$ has (maximal) rank $m$.
- $n(m)$ grows slowly.
- $m \leq k(m) \leq (1 - \phi) n(m)$ and $k(m)$ is as close as possible to $m$.

and preferably, the matrix is binary.

Has this been studied before? I appreciate any pointers.

I know that a random uniform binary matrix with $n(m) \approx \frac{2}{1-\phi}(m + log 1/\delta)$ and $k(m) = m$ satisfies the properties above with probability $1 - \delta$ (for large enough $m$).

Also, any probabilistic suggestions that perform better than the uniform random binary matrix are welcome.

(I apologize if my question turns out to be elementary, I also appreciate any help with properly tagging the question.)