Improved efficiency for covering codes matching the sphere-covering bound

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 $ϵ,\delta >0$, we show a construction of a family of codes with relative covering radius $\delta +ϵ$ and rate $1 - $\mathrm{H}$(\delta) $withblocklengthatmost$\exp(O((1/\epsilon) \log (1/\epsilon)))$forevery$\epsilon > 0$. This improves upon a folklore construction which only guaranteed codes of block length $\exp (1\mathrm{/}{ϵ}^2)$. 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.