  Victoir (2004) developed a method to construct cubature formulae with various
combinatorial objects. Motivated by this, we generalize Victoir's method with
one more combinatorial object, called regular t-wise balanced designs. Many
cubature of small indices with few points are provided, which are used to
update Shatalov's table (2001) of isometric embeddings in small-dimensional
Banach spaces, as well as to improve some classical Hilbert identities. A
famous theorem of Bajnok (2007) on Euclidean designs invariant under the Weyl
group of Lie type B is extended to all finite irreducible reflection groups. A
short proof of the Bajnok theorem is presented in terms of Hilbert identities.
