Consider the vector space $V=\mathbb{Z}_2^n$ and take some linear subspace $U\subseteq V$ (we can assume we are given some basis for $U$). Now, for every basis $B$ of U, define $f(B)=\max_{b\in B}\{w(b)\}$ where $w(b)$ is the hamming weight of $b$ - the number of nonzero entries in it.
My goal is to compute $\min_Bf(B)$ where $B$ ranges over all the bases of $U$ (only a finite number of such bases exist since $V$ is a finite vector space).
I can use brute force, but prefer a better algorithm.