New asymptotic estimates for spherical designs

Let N(n,t)N(n,t) be the minimal number of points in a spherical t  -design on the unit sphere SnSn in Rn+1Rn+1. For each n⩾3n⩾3, we prove a new asymptotic upper boundN(n,t)⩽C(n)tan,N(n,t)⩽C(n)tan,where C(n)C(n) is a constant depending only on n  , a3⩽4a3⩽4, a4⩽7a4⩽7, a5⩽9a5⩽9, a6⩽11a6⩽11, a7⩽12a7⩽12, a8⩽16a8⩽16, a9⩽19a9⩽19, a10⩽22a10⩽22, andan10.