Relation between covering radius and columns of parity check matrix 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}$?
Here is the idea. Suppose that $C$ has covering radius $\rho$. Pick a vector $s \in \mathbb{F}^{n-k}$. Since $H$ is full rank, there is a vector $x$ such that $Hx = s$. There is a vector $y \in C$ with $|x-y| \leq \rho$ (here $|\cdot|$ is Hamming distance). Then $H(x-y) = s$, and so $s$ is a sum of at most $\rho$ columns of $H$. The opposite direction is very similar.