There happens to be this NP-complete question,
Minimum-Distance-Over-$\mathbb{F}_{2^m}$
Given $w \in \mathbb{Z}^+$ and a $r \times n$ matrix $H$ over $\mathbb{F}_{2^m}$, is there a $x \in \mathbb{F}_{2^m}^n$ of weight $\leq w$ s.t $Hx=0$?
Is there some "easy" proof of this? (as opposed to say what I can partly fish out of this paper, http://ieeexplore.ieee.org/xpls/abs_all.jsp?arnumber=641542)
Is there any variant of it where the hardness is true even if I assume that $H$ has at least $r$ linearly independent columns? (and such a condition gets automatically satisfied if the rows of $H$ define a code?)