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

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