Orbits of the hyperoctahedral group as Euclidean designs

The hyperoctahedral group H in n dimensions (the Weyl group of Lie type B _n ) is the subgroup of the orthogonal group generated by all transpositions of coordinates and reflections with respect to coordinate hyperplanes.With e ₁ , ..., e _n denoting the standard basis vectors of ⁿ and letting x _k = e ₁ + ··· + e _k (k = 1, 2, ..., n), the set is the vertex set of a generalized regular hyperoctahedron in ⁿ . A finite set with a weight function is called a Euclidean t-design, if holds for every polynomial f of total degree at most t; here R is the set of norms of the points in ,W _r is the total weight of all elements of with norm r, S _r is the n-dimensional sphere of radius r centered at the origin, and is the average of f over S _r . 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 for the minimum size of a Euclidean t-design in ⁿ 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).