2025/01/20 by Kenji Tanino, Tanino, Kenji, Tamaru, Tomoki +4
Engineering · Mathematics · #65D32 Secondary 11E76 #Combinatorics (math.CO) #FOS: Mathematics #Manufacturing Process and Optimization #Mathematical Approximation and Integration #Numerical Analysis (math.NA) #Primary 05E99
paper · pdf · doi:10.48550/arxiv.2501.11437
openalex publication_date 2025/01/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
This paper explores a full generalization of the classical corner-vector method for constructing weighted spherical designs, which we call the \it generalized corner-vector method. First we establish a uniform upper bound for the degree of designs obtained from the proposed method. Our proof is a hybrid argument that employs techniques in analysis and combinatorics, especially a famous result by Xu(1998) on the interrelation between spherical designs and simplical designs, and the cross-ratio comparison method for Hilbert identities introduced by Nozaki and Sawa(2013). We extensively study conditions for the existence of designs obtained from our method, and present many curious examples of degree 7 through 13, some of which are, to our surprise, characterized in terms of integral lattices.