# Constructing Optimal L∞ Star Discrepancy Sets

arXiv: https://arxiv.org/abs/2311.17463 · HTML: https://mlsanity.com/paper/2311.17463/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 29 Nov 2023.

## Built on by (most cited first, 3 of 3)

- [Message-Passing Monte Carlo: Generating low-discrepancy point sets via Graph Neural Networks](https://mlsanity.com/paper/2405.15059/) (2024)
- [carps: A Framework for Comparing N Hyperparameter Optimizers on M Benchmarks](https://mlsanity.com/paper/2506.06143/) (2025)
- [LLM-Guided Evolutionary Program Synthesis for Quasi-Monte Carlo Design](https://mlsanity.com/paper/2510.03650/) (2025)

## Abstract (arXiv)

The L∞ star discrepancy is a very well-studied measure used to quantify the uniformity of a point set distribution. Constructing optimal point sets for this measure is seen as a very hard problem in the discrepancy community. Indeed, optimal point sets are, up to now, known only for n≤ 6 in dimension 2 and n ≤ 2 for higher dimensions. We introduce in this paper mathematical programming formulations to construct point sets with as low L∞ star discrepancy as possible. Firstly, we present two models to construct optimal sets and show that there always exist optimal sets with the property that no two points share a coordinate. Then, we provide possible extensions of our models to other measures, such as the extreme and periodic discrepancies. For the L∞ star discrepancy, we are able to compute optimal point sets for up to 21 points in dimension 2 and for up to 8 points in dimension 3. For d=2 and n≥ 7 points, these point sets have around a 50% lower discrepancy than the current best point sets, and show a very different structure.