On the number of nodes in n -dimensional cubature formulae of degree 5 for integrals over the ball

In this note cubature formulae of degree 5 are studied for n-dimensional integrals over the ball with constant weight function. We apply the method of reproducing kernel to show that the existence of such formulae attaining the best known lower bound is equivalent to the existence of tight spherical 5-designs. The known results concerning spherical 5-designs show that the lower bound for the integral under consideration will not be attained in general. The bound will be attained for n = 2,3,7,23 and possibly for n = (2p+1)2-2, p ≥ 5. In all other cases the bound must be increased at least by 1, in particular, Stroud's formulae for n = 4,5,6,7 are minimal.