  A covering code is a subset $\mathcal{C} \subseteq \{0,1\}^n$ with the
property that any $z \in \{0,1\}^n$ is close to some $c \in \mathcal{C}$ in
Hamming distance. For every $\epsilon,\delta>0$, we show a construction of a
family of codes with relative covering radius $\delta + \epsilon$ and rate $1 -
\mathrm{H}(\delta) $ with block length at most $\exp(O((1/\epsilon) \log
(1/\epsilon)))$ for every $\epsilon > 0$. This improves upon a folklore
construction which only guaranteed codes of block length $\exp(1/\epsilon^2)$.
The main idea behind this proof is to find a distribution on codes with
relatively small support such that most of these codes have good covering
properties.
