This question may be very trivial, I apologize if it is so.

I have subspace $V\subset \mathbb Z_2^r$ with the property that for every choice of a subset $I$ of $k$ elements in $\{1,2,\dots r\}$, the projection of $V$ onto the corresponding $k$ coordinates is surjective.

QUESTION: Can one find a lower bound on $\dim V$, apart from the obvious $\dim V\ge k$?

In the case I'm interested in I have $k\ge \frac r 2$.