Orbital spherical 11-designs whose initial point is root of an invariant polynomial

In this paper we state the results discussed in a somewhat more general and convenient form. As an example, we consider the following well-known results. The orbit .0a of the Conway (1988) group .0 of all orthogonal transformations that fix the Leech lattice is an 11-design for any initial vector a, because the first .0-invariant polynomial with zero mean has degree 12. The main result of the paper is an explicit construction of an infinite family of 11-designs in the 2ⁿ-dimensional Euclidean space on the top of the groups φ _n,2 and Σ_n,2 n=1, 2,..., of orthogonal (2 ⁿ×2ⁿ)-matrices. The group φ_n,2 is a subgroup of the group Σ_n,2