{"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-high-order-spherical-designs-as","title":"Constructing high order spherical designs as a union of two of lower order","arxiv_id":"1912.07151","date":"2019-12-16","proceeding":null,"authors":["Mozhgan Mohammadpour","Shayne Waldron"],"abstract":"We show how the variational characterisation of spherical designs can be used to take a union of spherical designs to obtain a spherical design of higher order (degree, precision, exactness) with a small number of points. The examples that we consider involve taking the orbits of two vectors under the action of a complex reflection group to obtain a weighted spherical $(t,t)$-design. These designs have a high degree of symmetry (compared to the number of points), and many are the first known construction of such a design, e.g., a $32$ point $(9,9)$-design for $\\mathbb{C}^2$, a $48$ point $(4,4)$-design for $\\mathbb{C}^3$, and a $400$ point $(5,5)$-design for $\\mathbb{C}^4$.From a real reflection group, we construct a $360$ point $(9,9)$-design for $\\mathbb{R}^4$ (spherical half-design of order $18$), i.e., a $720$ point spherical $19$-design for $\\mathbb{R}^4$.","url_abs":"http://arxiv.org/abs/1912.07151v1","url_pdf":"http://arxiv.org/pdf/1912.07151v1.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-high-order-spherical-designs-as","repo_url":"https://github.com/aelzenaar/tightframes","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"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}