Relation between covering radius and columns of parity check matrix
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.