{"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/on-the-dependence-structure-and-quality-of","title":"On the dependence structure and quality of scrambled (t,m,s)-nets","arxiv_id":"1903.09877","date":"2019-03-23","proceeding":null,"authors":["Jaspar Wiart","Christiane Lemieux","Gracia Y. Dong"],"abstract":"In this paper we develop a framework to study the dependence structure of scrambled $(t,m,s)$-nets. It relies on values denoted by $C_b(\\mathbf{k};P_n)$, which are related to how many distinct pairs of points from $P_n$ lie in the same elementary $\\mathbf{k}-$interval in base $b$. These values quantify the equidistribution properties of $P_n$ in a more informative way than the parameter $t$. They also play a key role in determining if a scrambled set $\\tilde{P}_n$ is negative lower orthant dependent (NLOD). Indeed this property holds if and only if $C_b(\\mathbf{k};P_n) \\le 1$ for all $\\mathbf{k} \\in \\mathbb{N}^s$, which in turn implies that a scrambled digital $(t,m,s)-$net in base $b$ is NLOD if and only if $t=0$. Through numerical examples we demonstrate that these $C_b(\\mathbf{k};P_n)$ values are a powerful tool to compare the quality of different $(t,m,s)$-nets, and to enhance our understanding of how scrambling can improve the quality of deterministic point sets.","url_abs":"https://arxiv.org/abs/1903.09877v3","url_pdf":"https://arxiv.org/pdf/1903.09877v3.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":"on-the-dependence-structure-and-quality-of","repo_url":"https://github.com/wongey/digital-nets-walsh","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":null,"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}