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A covering code is a subset ${\mathcal{C}} \subseteq {\{ 0,1\} ^n}$ with the property that any z &#x2208;{0,1}<sup>n</sup> is close to some $c \in {\mathcal{C}}$ in Hamming distance. For every &#x03F5;,&#x03B4; &#x003E; 0, we show a construction of a family of codes with relative covering radius &#x03B4;+ &#x03F5; and rate 1&#x2212;H(&#x03B4;) with block length at most $\exp (O((1/ \in )\log (1/ \in )))$ for every &#x03F5; &#x003E;0. This improves upon a folklore construction which only guaranteed codes of block length exp(1/ &#x03F5; <sup>2</sup>). 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.

Improved efficiency for covering codes matching the sphere-covering bound