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 ∈{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−H(δ) with block length at most $\exp (O((1/ \in )\log (1/ \in )))$ 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.