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

In 2005, Kuperberg proved that 2^s 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-x^s-13+x^s-245-⋯+(-1) ^s1·3·15⋯(4^s-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±⋯± a_s in the Euclidean spaces of certain dimensions d≥4.