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no. 5
More on the corner-vector construction for spherical designs
Tanino, Kenji1; Tamaru, Tomoki1; Hirao, Masatake2; Sawa, Masanori1
1 Graduate School of System Informatics Kobe University 1-1 Riokkodai Nada Kobe Hyogo 657-8501 Japan
2 School of Information and Science Technology Aichi Prefectural University 1522-3 Ibaragabasama Nagakute Aichi 480-1198 Japan
Algebraic Combinatorics, Volume 8 (2025) no. 5, pp. 1387-1414

This paper explores a full generalization of the classical corner-vector method for constructing weighted spherical designs, which we call the 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 simplicial 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.

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Accepted:
Published online:
DOI: 10.5802/alco.441
Classification: 05E99, 65D32, 11E76
Keywords: Cubature formula, explicit construction, Hilbert identity, integral lattice, simplicial design, spherical design

Tanino, Kenji 1; Tamaru, Tomoki 1; Hirao, Masatake 2; Sawa, Masanori 1

1 Graduate School of System Informatics Kobe University 1-1 Riokkodai Nada Kobe Hyogo 657-8501 Japan
2 School of Information and Science Technology Aichi Prefectural University 1522-3 Ibaragabasama Nagakute Aichi 480-1198 Japan
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights 
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Tanino, Kenji; Tamaru, Tomoki; Hirao, Masatake; Sawa, Masanori. More on the corner-vector construction for spherical designs. Algebraic Combinatorics, Volume 8 (2025) no. 5, pp. 1387-1414. doi: 10.5802/alco.441

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