Improved efficiency for covering codes matching the sphere-covering bound

Abstract

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.