# Deterministic constructions of high-dimensional sets with small dispersion

## TL;DR

This paper presents explicit deterministic algorithms for constructing low-dispersion point sets in high dimensions, leveraging coding theory, with applications in numerical integration and discrepancy theory.

## Contribution

It introduces a novel deterministic construction method for low-dispersion sets based on coding theory, improving over randomized approaches.

## Key findings

- Constructs low-dispersion point sets deterministically
- Algorithms run in polynomial time in dimension d
- Time complexity is super-exponential in 1/ε

## Abstract

The dispersion of a point set $P\subset[0,1]^d$ is the volume of the largest box with sides parallel to the coordinate axes, which does not intersect $P$. Here, we show a construction of low-dispersion point sets, which can be deduced from solutions of certain $k$-restriction problems, which are well-known in coding theory. It was observed only recently that, for any $\varepsilon>0$, certain randomized constructions provide point sets with dispersion smaller than $\varepsilon$ and number of elements growing only logarithmically in $d$. Based on deep results from coding theory, we present explicit, deterministic algorithms to construct such point sets in time that is only polynomial in $d$. Note that, however, the running-time will be super-exponential in $\varepsilon^{-1}$.