# Orbits of the hyperoctahedral group as Euclidean designs

## Authors

- First Online:

- Received:
- Accepted:

DOI: 10.1007/s10801-006-0042-3

- Cite this article as:
- Bajnok, B. J Algebr Comb (2007) 25: 375. doi:10.1007/s10801-006-0042-3

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## Abstract

*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

^{n}.

*t*-design, if

*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).