It would be useful to me to have a result of the following kind (which I would need to generalize, but this case is already interesting). Let $r<n$ be positive integers and let $\delta>0$ be a fixed constant such as 1/100. Does there exist a subspace $V$ of $\mathbb F\_2^n$ that is a $\delta r$-separated $r$-net? That is, I would like every vector to have Hamming distance at most $r$ from some $v\in V$ and any two distinct vectors in $V$ to have distance at least $\delta r$ from each other. 


This feels like the kind of thing that coding theorists ought to know about, but what comes up when I do a Google search is more to do with the dimension of $V$ than with whether it is a good net. But perhaps it comes into the "known to be hard" category.


I am interested in more or less the full range of $r$ for given $n$: the existence of such a linear code for some specific $r$ would be interesting but not enough for my eventual purposes.



When can this condition on linear codes be satisfied?
Any maximal $r$-separated set is an $r$-net, otherwise you can augment it with a non-covered point.


But the same holds for linear codes! If a subspace $V$ is $r$-separated but not an $r$-net, you may take a far point $u$: then $\langle V,u\rangle$ is still $r$-separated. So a maximal $V$ fits.


