Lower bounds for projective designs, cubature formulas and related isometric embeddings

Yudin’s lower bound [V.A. Yudin, Lower bounds for spherical designs, Izv. Math. 61 (3) (1997) 673–683] for the spherical designs is generalized to the cubature formulas on the projective spaces over a field K∈{R,C,H}K∈{R,C,H}, and thus to isometric embeddings l2;Km→lp;Kn with p∈2Np∈2N. For large pp and in some other situations this result is substantially better than known before.