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 ∑r ∈ R Wᵣ f̄_Sᵣ = ∑_ x ∈ 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 f̄_Sᵣ is the average of f over Sᵣ. 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).