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 \(\sf{R}\)^{n} and letting x_{k} = e_{1} + ··· + e_{k} (k = 1, 2, ..., n), the set

$$
{\cal I}^n_k={\bf x}_{\bf k}^H=\{ {\bf x}_{\bf k}^g \mbox{} | \mbox{} g \in H \}$$

is the vertex set of a generalized regular hyperoctahedron in \(\sf{R}\)^{n}.

A finite set \({\cal X} \subset \sf{R}^n\) with a weight function \(w: {\cal X} \rightarrow \sf{R}^+\) 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 \({\cal X}\),W_{r} is the total weight of all elements of \({\cal X}\) with norm r, S_{r} is the n-dimensional sphere of radius r centered at the origin, and \(\bar{f}_{S_{r}}\) 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 \(|{\cal X}| \geq N(n,p,t)\) for the minimum size of a Euclidean t-design in \(\sf{R}\)^{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).

Keywords

Euclidean design Spherical design Tight design Harmonic polynomial Hyperoctahedral group