What is the average minimum required number of independent $k$-sparse (having at most $k$ non-zero elements) random vectors belonging to $\mathbb{F}_2^n$ to span the whole space of $\mathbb{F}_2^n$? Any such vector is uniformly probable to be chosen among the total $\sum_{j=0}^k \binom{n}{j}$ vectors. 

Here are the two extreme cases: 

 1. If $k=n$, this average value is $n+1.6067$ as proved [here][1].
 2. If $k=1$, using coupon collector problem this average value is proved to be larger than $\Theta(n \log n)$.


Can we prove that if $k = \Theta(\log n)$, then this average value is $\Theta(n)$? or something similar? My simulation results show that for a pretty large range of $k$ this average value is $\Theta(n)$. 


[1]: http://math.stackexchange.com/questions/589725/expected-number-of-random-binary-vectors-to-make-matrix-of-order-n