# Improved efficiency for covering codes matching the sphere-covering bound

## TL;DR

This paper presents a new construction of covering codes that approach the sphere-covering bound more efficiently, significantly reducing the required block length compared to previous methods.

## Contribution

The authors introduce a novel code construction method that achieves near-optimal covering properties with substantially smaller block length.

## Key findings

- Constructed codes with relative covering radius close to the sphere-covering bound.
- Achieved code block length of at most exp(O((1/epsilon) log(1/epsilon))) for any epsilon > 0.
- Improved upon folklore constructions with longer block lengths.

## 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.