Minimize the maximum Hamming weight of basis vectors spanning a binary subspace
This is background, rather than a complete solution of your specific question.

What you're looking at are called linear anticodes in the literature, while Ahlswede uses the term "diameter" for the maximum distance of a code. Like normal codes, there are some bounds on maximum size, given length and distance.

The problem may also be stated in terms of the minimum number of coordinates where two codewords agree. If the alphabet is nonbinary, it gets much more complicated, and [1] has an upper bound for that case, but also discusses the result from [2] which is the following:

Given a binary anticode with diameter (maximum distance) $n-t$ over the binary alphabet (which need not be linear), letting $N_2(n,t)$ be the maximum cardinality of such a code we have
$$N_2(n,t)=\left\{
\begin{array}{lrr}
\sum_{i=0}^{(n-t)/2} \binom{n}{i},&amp;\quad&amp; if~n-t~is~even,\\
 &amp; &amp; \\
2 \sum_{i=0}^{(n-t-1)/2} \binom{n-1}{i},&amp;\quad&amp; if~n-t~is~odd.
\end{array}
\right.
$$

See the reference [1] for details. Apparently the binary diametric theorem above was proved by Kleitman in [2]. This bound applies to all anticodes and would apply to your linear anticode as well.

Farrell had an earlier article on anticodes which considered linear anticodes obtained by deleting columns from a simplex code, which may also be of interest, it shouldn't be hard to locate the reference.

[1] Ahlswede and Khachtarian, The Diametric Theorem in Hamming
Spaces:Optimal Anticodes, Adv. Appl. Math:20, 429:449 1998.

[2] D. J. Kleitman, On a combinatorial conjecture of Erdos, J. Combin. Theory, Volume 1, pp.209-214, 1966.