Constructions of spherical 3-designs
type: Item
instance of: scholarly article
title: Constructions of spherical 3-designs
zbMATH Open document ID: 0899.05010
DOI: 10.1007/PL00007220
published in: Graphs and Combinatorics
publication date: 1998-10-20T00:00:00Z
review text: A finite set of points \(X\) on the \(d\)-dimensional Euclidean sphere \(\mathbb S^d\subset\mathbb R^{d+1}\) is called spherical \(t\)-design if \[ \frac{1}{\sigma_d(\mathbb S^d)}\int_{\mathbb S^d} f(x)\,d\sigma_d(x)=\frac{1}{| X|}\sum_{x\in X}f(x) \] for all polynomials \(f(x)=f(x_0,x_1,\ldots,x_d)\) of total degree at most \(t\) (\(\sigma_d\) is the surface measure on \(\mathbb S^d\)). In the paper under review, new spherical 3-designs are constructed in the following cases: \(d=1\) and \(| X|\geq 4\); \(d=2\) and \(| X|=6,8\) or \(| X|\geq 10\); \(d=3\) and \(| X|=8\) or \(| X|\geq 10\); \(d=4\) and \(| X|=10,12\) or \(| X|\geq 14\); \(d=5\) and \(| X|\geq 5(d+1)/2\) odd and \(| X|\geq 2d+2\) even. On the other hand, \textit{D. Danev}, \textit{S. Nikova} and the reviewer [Discrete Comput. Geom. 21, No. 1, 143--156 (1999; Zbl 0921.05016)] have shown many nonexistence results which, in particular, add evidence for the author's conjecture that the above list of cardinalities is complete. The tools used for constructions are also interesting. Sets of integers, which generalize the classical Sidon sequences are introduced and investigated. It can be proved that Conjecture 4.2(i) is true.
Mathematics Subject Classification ID: 05B30
zbMATH DE Number: 1182269
zbMATH Keywords: spherical designs
zbMATH Keywords: Sidon-type sets
author: Bajnok, Béla
reviewed by: Peter Boyvalenkov
MaRDI profile type: Publication
full work available at URL: pl00007220
OpenAlex ID: W2054504007
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description: scientific article; zbMATH DE number 1182269