New spherical (2s+1)-designs from Kuperberg's set: An experimental result

In 2005, Kuperberg proved that 2s points ±z1±z2±⋯±zs′ form a Chebyshev-type (2s+1)-quadrature formula on [-1,1] with constant weight if and only if the zi's are the zeros of polynomialQ(x)=xs-xs-13+xs-245-⋯+(-1)s1·3·15⋯(4s-1).The Kuperberg's construction on Chebyshev-type quadrature formula above may be regarded as giving an explicit construction of spherical (2s+1)-designs in the Euclidean space of dimension 3.Motivated by the Kuperberg's result, in this paper, we observe an experimental construction of spherical (2s+1)-designs, for certain s, from the Kuperberg set of the form ±a1±a2±⋯±as in the Euclidean spaces of certain dimensions d⩾4.