Let $C$ be a $[n,k]$ linear Code over $\mathbb{F}_q$ .
I want to show that each vector of $\mathbb{F}_q^{n-k} $ is written as a linear combination of $m$ columns of $H$ iff $\rho \leq m$.
I have thought the following:
$$ d(C)=\min \{ d \in \mathbb{N} | \text{ there are d linearly dependent columns of H}\} $$
So $H$ has $ d-1 $ linearly independent columns , so each vector of $\mathbb{F}_q^n$ can be written as a linear combination of these columns.
But what can we say about the vectors of $ \mathbb{F}_q^{n-k}$?