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

In 2005, Kuperberg proved that 2(s) points +/-root z(1) +/- root z(2) +/- . . . +/- root z(s)' 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 polynomial Q(x) = x(s) - x(s-1)/3 + x(s-2)/45 - . . . + (-1)(s)/1.3.15 ... (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 +/- a(1) +/- a(2) +/- . . . +/- a(s) in the Euclidean spaces of certain dimensions d >= 4. (C) 2014 Elsevier Inc. All rights reserved.