{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/constructing-optimal-l-infty-star-discrepancy","title":"Constructing Optimal $L_{\\infty}$ Star Discrepancy Sets","arxiv_id":"2311.17463","date":"2023-11-29","proceeding":null,"authors":["François Clément","Carola Doerr","Kathrin Klamroth","Luís Paquete"],"abstract":"The $L_{\\infty}$ 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 $n\\leq 6$ in dimension 2 and $n \\leq 2$ for higher dimensions. We introduce in this paper mathematical programming formulations to construct point sets with as low $L_{\\infty}$ 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 $L_{\\infty}$ 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 $n\\ge 7$ points, these point sets have around a 50% lower discrepancy than the current best point sets, and show a very different structure.","url_abs":"https://arxiv.org/abs/2311.17463v2","url_pdf":"https://arxiv.org/pdf/2311.17463v2.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"constructing-optimal-l-infty-star-discrepancy","repo_url":"https://github.com/frclement/optisetsdiscrepancy","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":0,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}