Some constructions of spherical 5-designs
This paper deals with the construction of spherical 5-designs in \(\mathbb{R}^3\). A spherical \(t\)-design in \(\mathbb{R}^n\) is a finite set \(X\subset S^{n- 1}\) such that for every polynomial \(P\) of degree \(\leq t\), the average value of \(P\) on \(X\) is the same as the average of \(P\) on \(S^{n- 1}\). It is already known that if \(X\) is a spherical 5-design in \(\mathbb{R}^3\), then \(|X|\geq 12\), and \(X\) is made up of the vertices of a regular icosahedron if \(|X|= 12\). It is shown that spherical 5-designs in \(\mathbb{R}^3\) with cardinality \(12, 16, 18, 20, 22, 24\) and every integer \(\geq 26\) exist. Spherical 5-designs in \(\mathbb{R}^n\) are called tight if \(|X|= n(n+ 1)\), and these are known to exist only for \(n= 2, 3, 7\), and 23, and any such design must consist of \({n(n+ 1)\over 2}\) antipodal pairs of points. It is shown that for \(n\geq 3\), there exists no spherical 5-design in \(\mathbb{R}^n\) which consists of \({n(n+ 1)\over 2}+ 1\) antipodal pairs of points.
scientific article; zbMATH DE number 803736
scientific article