Orbits of the hyperoctahedral group as Euclidean designs

The hyperoctahedral group H in n dimensions (the Weyl group of Lie type Bₙ) is the subgroup of the orthogonal group generated by all transpositions of coordinates and reflections with respect to coordinate hyperplanes. A finite set X Rⁿ with a weight function w: X R^+ is called a Euclidean t-design, if ₑ ₑ Wᵣ fₒ_ₑ = ₗ { X} w (x) f (x) holds for every polynomial f of total degree at most t; here R is the set of norms of the points in X, Wᵣ is the total weight of all elements of X with norm r, Sᵣ is the n-dimensional sphere of radius r centered at the origin, and fₒ_ₑ is the average of f over Sₑ. Here we consider Euclidean designs which are supported by orbits of the hyperoctahedral group. Namely, we prove that any Euclidean design on a union of generalized hyperoctahedra has strength (maximum t for which it is a Euclidean design) equal to 3, 5, or 7. We find explicit necessary and sufficient conditions for when this strength is 5 and for when it is 7. In order to establish our classification, we translate the above definition of Euclidean designs to a single equation for t=5, a set of three equations for t=7, and a set of seven equations for t=9. Neumaier and Seidel (1988), as well as Delsarte and Seidel (1989), proved a Fisher-type inequality | X| N (n, p, t) for the minimum size of a Euclidean t-design in Rⁿ on p=|R| concentric spheres (assuming that the design is antipodal if t is odd). A Euclidean design with exactly N (n, p, t) points is called tight. We exhibit new examples of antipodal tight Euclidean designs, supported by orbits of the hyperoctahedral group, for N (n, p, t) = (3, 2, 5), (3, 3, 7), and (4, 2, 7).