# Orbits of the hyperoctahedral group as Euclidean designs

Béla Bajnok
Gettysburg College
<EMAIL_ADDRESS>
###### Abstract

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 ${\bf e_{1}},\dots,{\bf e_{n}}$ denoting the standard basis
vectors of $\mathbb{R}^{n}$ and letting ${\bf x_{k}}={\bf e_{1}}+\cdots+{\bf
e_{k}}$ ($k=1,2,\dots,n$), the set

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

is the vertex set of a generalized regular hyperoctahedron in
$\mathbb{R}^{n}$.

A finite set ${\cal X}\subset\mathbb{R}^{n}$ with a weight function $w:{\cal
X}\rightarrow\mathbb{R}^{+}$ is called a Euclidean $t$-design, if

$\sum_{r\in R}W_{r}\overline{f}_{S_{r}}=\sum_{{\bf x}\in{\cal X}}w({\bf
x})f({\bf x})$

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 $\overline{f}_{S_{r}}$ 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 $\mathbb{R}^{n}$ 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).

2000 Mathematics Subject Classification:
Primary: 05B99;
Secondary: 05B30, 33C45, 41A55, 51M99, 65D32.

Key words and phrases:
Euclidean design, spherical design, tight design, harmonic polynomial,
hyperoctahedral group.

##  1 Introduction

A Euclidean design is a finite weighted set of points in the $n$-dimensional
real Euclidean space $\mathbb{R}^{n}$ with a certain approximation property,
as explained below. First we introduce a few notations and discuss some
background.

The norm of a point ${\bf x}\in\mathbb{R}^{n}$, denoted by $||{\bf x}||$, is
its distance from the origin; the collection of all points with given norm
$r>0$ is the sphere $S_{r}^{n-1}$. For a finite set ${\cal
X}\subset\mathbb{R}^{n}\setminus\\{{\bf 0}\\}$, we call $R=\\{||{\bf
x}||\mbox{ }|\mbox{ }{\bf x}\in{\cal X}\\}$ the _norm spectrum_ of ${\cal X}$.
(In this paper, for convenience, we exclude the possibility of ${\bf
0}\in{\cal X}$; see [4] for a discussion.) We can partition ${\cal X}$ into
_layers_ ${\cal X}=\cup_{r\in R}{\cal X}_{r}$ where ${\cal X}_{r}={\cal X}\cap
S_{r}^{n-1}$. A weight function on ${\cal X}$ is a function $w:{\cal
X}\rightarrow\mathbb{R}^{+}$; the _weight distribution of $w$ on $R$_ is the
function $W:R\rightarrow\mathbb{R}^{+}$ given by $W_{r}=\sum_{{\bf x}\in{\cal
X}_{r}}w({\bf x})$. (Throughout this paper, we only consider positive
weights.)

We denote the spaces of real polynomials, homogeneous polynomials, and
homogeneous harmonic polynomials on $n$ variables by
$\mathrm{Pol}(\mathbb{R}^{n})$, $\mathrm{Hom}(\mathbb{R}^{n})$, and
$\mathrm{Harm}(\mathbb{R}^{n})$, respectively. Often we will restrict the
domain of these polynomials to a subset ${\cal Y}$ of $\mathbb{R}^{n}$ or
their degrees to a fixed integer $s$; the corresponding polynomial spaces will
be denoted by $\mathrm{Pol}_{s}({\cal Y})$, etc.

Let $\sigma^{n}$ denote the regular surface measure on $S_{r}^{n-1}$. The
(continuous) _average value_ of a polynomial $f$ on $S_{r}^{n-1}$ is

$\overline{f}_{S_{r}^{n-1}}=\frac{1}{\sigma^{n}(S_{r}^{n-1})}\int_{S_{r}^{n-1}}f({\bf
x})d\sigma^{n}({\bf x}),$

where

$\sigma^{n}(S_{r}^{n-1})=\frac{2r^{n-1}\pi^{\frac{n}{2}}}{\Gamma({\frac{n}{2}})}$

is the surface area of $S_{r}^{n-1}$.

We are now ready to state our definition of Euclidean $t$-designs.

######  Definition 1 (Neumaier–Seidel, [17])

Let ${\cal X}\subset\mathbb{R}^{n}\setminus\\{{\bf 0}\\}$ be a finite set with
norm spectrum $R$, and suppose that a weight function $w$ is given on ${\cal
X}$ which has weight distribution $W$ on $R$. Let $t$ be non-negative integer.
We say that $({\cal X},w)$ is a Euclidean $t$-design if

$\sum_{r\in R}W_{r}\overline{f}_{S_{r}^{n-1}}=\sum_{{\bf x}\in{\cal X}}w({\bf
x})f({\bf x})$

holds for every $f\in\mathrm{Pol}_{t}(\mathbb{R}^{n})$. The largest value of
$t$ for which $({\cal X},w)$ is a Euclidean $t$-design is called the (maximum)
strength of the design.

An important special case of Euclidean designs is spherical designs: there we
assume that all points in the design have the same norm, and we do not allow
the points to have different weights. Spherical designs enjoy a vast and
rapidly growing literature, and have been studied from a variety of
perspectives, including algebra, combinatorics, functional analysis, geometry,
number theory, numerical analysis, and statistics. For general references,
please see [3], [6], [7], [9], [12], [13], [14], [15], [20], and [21]. The
concept of Euclidean designs was introduced by Neumaier and Seidel in 1988 in
[17] as a generalization of spherical designs, and was subsequently studied by
Delsarte and Seidel in [10], Seidel in [18] and in [19], and just recently by
Bannai and Bannai in [4].

In [2] we provided a recursive construction for Euclidean $t$-designs in
$\mathbb{R}^{n}$. Namely, we showed how to use certain Gauss–Jacobi quadrature
formulae to “lift” a Euclidean $t$-design $({\cal A}^{n-1},w^{n-1})$ in
$\mathbb{R}^{n-1}$ to a Euclidean $t$-design $({\cal A}^{n},w^{n})$ in
$\mathbb{R}^{n}$. Our recursion preserved both the norm spectrum $R$ and the
weight distribution $W$ on $R$; that is, we had

$R=\\{||{\bf x}||\mbox{ }|\mbox{ }{\bf x}\in{\cal A}^{n-1}\\}=\\{||{\bf
x}||\mbox{ }|\mbox{ }{\bf x}\in{\cal A}^{n}\\}$

and

$W_{r}=\sum_{{\bf x}\in{\cal A}^{n-1}_{r}}w({\bf x})=\sum_{{\bf x}\in{\cal
A}^{n}_{r}}w({\bf x})$

for each $r\in R$. Our recursion yields Euclidean $t$-designs in any
dimension; we note that this construction for Euclidean $t$-designs is “semi-
explicit” in the sense that the coordinates and the weights of the points in
the design are given in terms of the roots of certain Gegenbauer polynomials.
Additionally, in [2] we provided explicit constructions for Euclidean designs
in the plane ($n=2$); in particular, we analyzed Euclidean $t$-designs of
minimum size for every $t$. Therefore from now on we assume that $n\geq 3$
unless otherwise noted.

The subject of the present paper is a family of very explicit Euclidean
designs; namely, we consider point sets which are unions of orbits of single
points under the action of the hyperoctahedral group. We make this more
precise as follows.

Let $O(n)$ be the group of isometries of $S^{n-1}$. For $1\leq i\leq n$, let
$\rho_{i}\in O(n)$ be the reflection with respect to the coordinate hyperplane
$x_{i}=0$; for $1\leq i_{1}<i_{2}\leq n$, let $\sigma_{i_{1},i_{2}}$ be the
reflection with respect to the hyperplane $x_{i_{1}}=x_{i_{2}}$. Then the sets
$\\{\rho_{i}\mbox{ }|\mbox{ }1\leq i\leq n\\}$ and
$\\{\sigma_{i_{1},i_{2}}\mbox{ }|\mbox{ }1\leq i_{1}<i_{2}\leq n\\}$ generate
subgroups $H_{\rho}$ and $H_{\sigma}$ of $O(n)$ which are isomorphic to
$(\mathbb{Z}/2\mathbb{Z})^{n}$ and the symmetric group $S_{n}$, respectively.
The subgroup $H$ generated by all these reflections (the semidirect product of
$H_{\rho}$ and $H_{\sigma}$) is called the _hyperoctahedral group_ (or the
Weyl group of Lie type $B_{n}$). We see that $H$ has order $2^{n}\cdot n!$.

If $g\in H$, then ${\bf x}^{g}$ denotes the image of ${\bf
x}\in\mathbb{R}^{n}$ after applying $g$; we also set ${\bf x}^{H}=\\{{\bf
x}^{g}\mbox{ }|\mbox{ }g\in H\\}$. We make the following definition.

###### Definition 2

Let ${\bf e_{1}},\dots,{\bf e_{n}}$ be the standard basis vectors of
$\mathbb{R}^{n}$, and for $k=1,2,\dots,n$, set ${\bf x_{k}}={\bf
e_{1}}+\cdots+{\bf e_{k}}$. Let $H$ be the hyperoctahedral group of dimension
$n$. The set

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

is the vertex set of a generalized regular hyperoctahedron in
$\mathbb{R}^{n}$.

Note that the $n$ generalized hyperoctahedra in dimension $n$, together with
the origin, form a partition of the $3^{n}$ elements in ${\cal
I}^{n}=\\{-1,0,1\\}^{n}$ according to their norm. For example, for $n=3$, this
partition is the following:

${\cal I}^{3}=\\{{\bf
0}\\}\cup\\{\mbox{octahedron}\\}\cup\\{\mbox{cuboctahedron}\\}\cup\\{\mbox{cube}\\}.$

Furthermore, we have

$|{\cal I}^{n}_{k}|=2^{k}\cdot{{n}\choose{k}}.$

Our goal is to use such generalized hyperoctahedra to construct Euclidean
designs. We consider sets ${\cal X}$ of the form

${\cal X}={\cal X}(J)=\cup_{k\in J}\frac{r_{k}}{\sqrt{k}}{\cal I}^{n}_{k}$

where $J\subseteq\\{1,2,\dots,n\\}$ and $r_{k}>0$ for every $k\in J$. Note
that ${\cal X}$ has norm spectrum $R=\\{r_{k}\mbox{ }|\mbox{ }k\in J\\}$. The
weight function $w:{\cal X}\rightarrow\mathbb{R}^{+}$ on ${\cal X}$ will be
constant on each layer of ${\cal X}$ (see [4]); let us denote the weight of
${\bf x}\in\frac{r_{k}}{\sqrt{k}}{\cal I}^{n}_{k}$ by $w({\bf x})=w_{k}$. Then
$({\cal X(J)},w)$ is an _antipodal design_ , that is, for every $x\in{\cal
X}$, $-x\in{\cal X}$ and $w(-x)=w(x)$.

Our goal is to determine necessary and sufficient conditions for each
dimension $n$, norm spectrum $R$, strength $t$, and index set $J$, whether a
(positive) weight function exist for which $({\cal X(J)},w)$, as defined
above, is a Euclidean $t$-design. We carry this out here for $|R|\leq 3$ and
$|J|\leq 3$. Note that, since our sets are antipodal, the maximum strength of
each Euclidean design supported by them is necessarily odd.

It turns out that, in order to state our results precisely, the following
function is useful. For positive integers $k_{1},k_{2}\leq n$ we define

$G(k_{1},k_{2})=(n+2-3k_{1})(n+2-3k_{2})+6(k_{1}-1)(k_{2}-1)+2(n-1).$

We say that $n$ has _property $G$_ if there are positive integers
$k_{1},k_{2}\leq n$ for which $G(k_{1},k_{2})=0$; in this case we write $n\in
G$.

It is an interesting question to characterize integers with property $G$;
these values under a 100 are

1, 2, 5, 8, 10, 11, 14, 16, 17, 20, 23, 26, 28, 29, 31, 32, 35, 38, 41, 44,
46, 47, 50, 52, 53, 56, 59, 61, 62, 64, 65, 68, 71, 73, 74, 76, 77, 80, 82,
83, 86, 89, 91, 92, 94, 95, 98, 100.

Clearly, no $n$ value with $n\equiv 0$ mod 3 has property $G$. We can also see
that all $n\equiv 2$ mod 3 has property $G$: for example, we can take
$k_{1}=1$ and $k_{2}=\frac{n+4}{3}$. Among integers congruent to 1 mod 3,
there are 176 such integers under a thousand (slightly over half). We can also
find infinite subsequences of $n\equiv 1$ mod 3 with property $G$: for
example, each $n$ of the form $n=90c^{2}+45c+1$ has property $G$ (with
$k_{1}=3c+1$ and $k_{2}=30c^{2}+17c+2$).

We now state our characterization as follows.

###### Theorem 3

Let ${\cal X}(J)=\cup_{k\in J}\frac{r_{k}}{\sqrt{k}}{\cal I}^{n}_{k}$ be a
union of generalized hyperoctahedra, as defined above.

  1. 1.

The strength of $({\cal X}(J),w)$ is 3, 5, or 7 for all choices of $R$, $J$,
and $w$.

  2. 2.

Suppose that $|R|=1$.

$|J|=1;t=5$
If $J=\\{k\\}$, then there is a weight function $w$ for which $({\cal
X}(J),w)$ a Euclidean $5$-design, if and only if, $n\equiv 1$ mod 3 and
$k=\frac{n+2}{3}$.

$|J|=1;t=7$
If $|J|=1$, then $({\cal X}(J),w)$ is never a Euclidean $7$-design.

$|J|=2;t=5$
Let $J=\\{k_{1},k_{2}\\}$ where $k_{1}<k_{2}$. Then there is weight function
$w$ for which $({\cal X}(J),w)$ is a Euclidean $5$-design, if and only if,
$k_{1}<\frac{n+2}{3}<k_{2}$.

$|J|=2;t=7$
If $J=\\{k_{1},k_{2}\\}$, then there is a weight function $w$ for which
$({\cal X}(J),w)$ is a Euclidean $7$-design, if and only if,
$G(k_{1},k_{2})=0$.

$|J|=3;t=7$
Let $J=\\{k_{1},k_{2},k_{3}\\}$ where $k_{1}<k_{2}<k_{3}$. Then there is a
weight function $w$ for which $({\cal X}(J),w)$ is a Euclidean $7$-design, if
and only if, $G(k_{1},k_{2})>0$, $G(k_{2},k_{3})>0$, and $G(k_{1},k_{3})<0$.

  3. 3.

Suppose that $|R|=2$.

$|J|=2;t=5$
Let $J=\\{k_{1},k_{2}\\}$ where $k_{1}<k_{2}$. Then there is weight function
$w$ for which $({\cal X}(J),w)$ is a Euclidean $5$-design, if and only if,
$k_{1}<\frac{n+2}{3}<k_{2}$.

$|J|=2;t=7$
If $|J|=2$, then $({\cal X}(J),w)$ is never a Euclidean $7$-design.

$|J|=3;t=7$
Let $J=\\{k_{1},k_{2},k_{3}\\}$ where $k_{1}<k_{2}<k_{3}$. Then there is a
weight function $w$ for which $({\cal X}(J),w)$ is a Euclidean $7$-design, if
and only if, $n\equiv 1$ mod 3, $k_{2}=\frac{n+2}{3}$, $r_{k_{1}}=r_{k_{3}}$,
and $G(k_{1},k_{3})<0$.

  4. 4.

Suppose that $|R|=3$.

$|J|=3;t=7$
Let $J=\\{k_{1},k_{2},k_{3}\\}$ where $k_{1}<k_{2}<k_{3}$. Then there is a
weight function $w$ for which $({\cal X}(J),w)$ is a Euclidean $7$-design, if
and only if, $G(k_{1},k_{2})>0$, $G(k_{2},k_{3})>0$, $G(k_{1},k_{3})<0$, and

$\sum_{i=1,2,3}k_{i}\cdot(n+2-3k_{i})\cdot(k_{i+1}-k_{i+2})\cdot
G(k_{i+1},k_{i+2})\cdot\frac{1}{r_{k_{i}}^{2}}=0,$ (1)

holds where the indeces are to be taken mod 3 (i.e. $k_{4}=k_{1}$ and
$k_{5}=k_{2}$).

We prove Theorem 3 in Section 3. As a consequence, we have the following.

###### Corollary 4

Let $\tau(p,j)$ be the maximum achievable strength of a Euclidean design
formed by the union of $j$ generalized hyperoctahedra on $p$ concentric
spheres. We have the following.

$\begin{array}[]{lll}\tau(1,1)=\left\\{\begin{array}[]{l}3\mbox{ if
}n\not\equiv 1(3)\\\ \\\ 5\mbox{ if }n\equiv
1(3)\end{array}\right.\par\par&\mbox{ }&\tau(2,2)=5\mbox{ for all
}n\par\par\par\par\\\ \\\ \\\ \tau(1,2)=\left\\{\begin{array}[]{l}5\mbox{ if
}n\not\in G\\\ \\\ 7\mbox{ if }n\in G\end{array}\right.\par\par&\mbox{
}&\tau(2,3)=\left\\{\begin{array}[]{l}5\mbox{ if }n\not\equiv 1(3)\\\ \\\
7\mbox{ if }n\equiv 1(3)\end{array}\right.\par\par\\\ \\\ \\\
\tau(1,3)=7\mbox{ for all }n\par\par\par\par&\mbox{
}&\tau(3,3)=\left\\{\begin{array}[]{l}5\mbox{ if }n=4\\\ \\\ 7\mbox{ if
}n\not=4\end{array}\right.\par\end{array}$

In particular, we see that

  * •

for every $n\geq 3$ and for $p\in\\{1,2\\}$, we can find at most two
generalized hyperoctahedra whose union supports a Euclidean $5$-design;

  * •

for every $n\geq 3$ other than $n=4$ and for $p\in\\{1,3\\}$, we can find at
most three generalized hyperoctahedra whose union supports a Euclidean
$7$-design; and

  * •

for $p=2$, we can find three generalized hyperoctahedra whose union supports a
Euclidean $7$-design only if $n\equiv 1$ mod 3.

In order to establish our results, we introduce a necessary and sufficient
criterion for $({\cal X},w)$ to be a Euclidean $t$-design which, in our
situation, is much easier to use than Definition 1. First we make the
following definition.

###### Definition 5

Let $H$ be the hyperoctahedral group in dimension $n$. A subset ${\cal X}$ of
$\mathbb{R}^{n}$ is said to be fully symmetric if ${\cal X}^{g}={\cal X}$ for
every $g\in H$. Furthermore, a Euclidean design $({\cal X},w)$ is fully
symmetric if ${\cal X}$ is a fully symmetric set, and $w({\bf x})=w({\bf y})$
whenever ${\bf x}={\bf y}^{g}$ for some $g\in H$.

Note that the designs supported on generalized hyperoctahedra are fully
symmetric.

The following result proves to be quite useful.

###### Theorem 6

Suppose that $({\cal X},w)$ is fully symmetric with norm spectrum $R$, and
assume that its weight function is a constant $w_{r}$ on each layer ${\cal
X}_{r}$ of ${\cal X}$. Define polynomials $f_{4,2}$, $f_{6,3}$, $f_{8,2}$, and
$f_{8,4}$ as follows.

$\displaystyle f_{4,2}$ $\displaystyle=$ $\displaystyle
x_{1}^{4}-6x_{1}^{2}x_{2}^{2}+x_{2}^{4},$ $\displaystyle f_{6,3}$
$\displaystyle=$ $\displaystyle
2(x_{1}^{6}+x_{2}^{6}+x_{3}^{6})-15(x_{1}^{4}x_{2}^{2}+x_{1}^{2}x_{2}^{4}+x_{1}^{4}x_{3}^{2}+x_{1}^{2}x_{3}^{4}+x_{2}^{4}x_{3}^{2}+x_{2}^{2}x_{3}^{4})+180x_{1}^{2}x_{2}^{2}x_{3}^{2},$
$\displaystyle f_{8,2}$ $\displaystyle=$ $\displaystyle
x_{1}^{8}-28x_{1}^{6}x_{2}^{2}+70x_{1}^{4}x_{2}^{4}-28x_{1}^{2}x_{2}^{6}+x_{2}^{8},$
$\displaystyle f_{8,4}$ $\displaystyle=$ $\displaystyle
3(x_{1}^{8}+x_{2}^{8}+x_{3}^{8}+x_{4}^{8})-28(x_{1}^{6}x_{2}^{2}+\cdots+x_{3}^{2}x_{4}^{6})+210(x_{1}^{4}x_{2}^{2}x_{3}^{2}+\cdots+x_{2}^{2}x_{3}^{2}x_{4}^{4})-3780x_{1}^{2}x_{2}^{2}x_{3}^{2}x_{4}^{2}.$

Then we have the following.

  1. 1.

$({\cal X},w)$ is at least a Euclidean $3$-design.

  2. 2.

$({\cal X},w)$ is a Euclidean $5$-design, if and only if,

$\sum_{r\in R}\sum_{{\bf x}\in{\cal X}_{r}}w_{r}f_{4,2}({\bf x})=0.$ (2)

  3. 3.

$({\cal X},w)$ is a Euclidean $7$-design, if and only if,

$\left\\{\begin{array}[]{l}\sum_{r\in R}\sum_{{\bf x}\in{\cal
X}_{r}}w_{r}r^{2s_{1}}f_{4,2}({\bf x})=0\\\ \\\ \sum_{r\in R}\sum_{{\bf
x}\in{\cal X}_{r}}w_{r}f_{6,3}({\bf x})=0\\\ \\\ \end{array}\right.$ (3)

holds for $s_{1}=0,1$.

  4. 4.

$({\cal X},w)$ is a Euclidean $9$-design, if and only if,

$\left\\{\begin{array}[]{l}\sum_{r\in R}\sum_{{\bf x}\in{\cal
X}_{r}}w_{r}r^{2s_{1}}f_{4,2}({\bf x})=0\\\ \\\ \sum_{r\in R}\sum_{{\bf
x}\in{\cal X}_{r}}w_{r}r^{2s_{2}}f_{6,3}({\bf x})=0\\\ \\\ \sum_{r\in
R}\sum_{{\bf x}\in{\cal X}_{r}}w_{r}f_{8,2}({\bf x})=0\\\ \\\ \sum_{r\in
R}\sum_{{\bf x}\in{\cal X}_{r}}w_{r}f_{8,4}({\bf x})=0\end{array}\right.$ (4)

holds for $s_{1}=0,1,2$ and $s_{2}=0,1$.

We prove Theorem 6 in the next section.

Let us now examine Euclidean designs of minimum size. For a non-negative
integer $s$, let

$d(s):=\dim\mathrm{Hom}_{s}(\mathbb{R}^{n})={{s+n-1}\choose{n-1}}.$

We have the following _Fisher-type inequality_. The inequality provides the
minimum size for the case of an even $t$; for odd $t$ we only have a lower
bound if the design is antipodal.

######  Theorem 7 (Delsarte–Seidel, [10])

For a positive integer $k$, let

$N_{k}=d\left(\left\lfloor\frac{t}{2}\right\rfloor+2-2k\right)+d\left(\left\lfloor\frac{t-1}{2}\right\rfloor+2-2k\right)$

and

$N(n,p,t)=\sum_{k=1}^{p}N_{k}.$

Suppose that $({\cal X},w)$ is a Euclidean $t$-design on $p$ layers in
$\mathbb{R}^{n}$; if $t$ is odd, assume further that the design is antipodal.
Then we have $|{\cal X}|\geq N(n,p,t).$

###### Definition 8

We say that a Euclidean $t$-design on $p$ concentric spheres is tight if it
has size $N(n,p,t)$.

In [2] we explicitly constructed tight Euclidean designs in the plane for
every $t$ and every $p\leq\left\lfloor\frac{t+5}{4}\right\rfloor$. Our
construction was a generalization of the example for $p=2$ and $t=4$ which
already appeared in [4]; namely, we have shown how a union of concentric
regular polygons, with appropriate weights and rotations about the origin,
forms a tight Euclidean design.

Let us now turn to the case of tight Euclidean designs in higher dimensions
($n\geq 3$). Since here we restrict our search to fully symmetric designs
which are antipodal, we may assume that $t$ is odd.

It is easy to see that for $t=1$ a pair of two antipodal points, with equal
weights, forms an antipodal tight 1-design in $\mathbb{R}^{n}$; and that for
$t=3$, the $n$ pairs $\pm r_{i}\cdot{\bf e_{i}}$, with weights inversely
proportional to their squared norms, form antipodal tight 3-designs in
$\mathbb{R}^{n}$ (in particular, the $2n$ vertices of the octahedron ${\cal
I}^{n}_{1}$ form an antipodal tight Euclidean 3-design). Bannai [5] classified
all antipodal tight Euclidean 3-designs and all antipodal tight Euclidean
5-designs with $p=2$. Here we provide examples of antipodal tight Euclidean
designs for $(n,p,t)=$(3,2,5), (3,3,7), and (4,2,7). Namely, we will prove
that

  1. 1.

in $\mathbb{R}^{3}$, the union of an octahedron and a cube, with appropriate
weights, forms an antipodal tight 5-design;

  2. 2.

in $\mathbb{R}^{3}$, the union of an octahedron, a cuboctahedron, and a cube,
with appropriate weights, forms an antipodal tight 7-design; and

  3. 3.

in $\mathbb{R}^{4}$, the points of minimum non-zero norm in the lattice
$D_{4}$ together with the points of minimum non-zero norm in the dual lattice
$D_{4}^{*}$, with appropriate weights, form an antipodal tight 7-design.

It might be useful to summarize these examples more precisely, as follows.

###### Corollary 9

Let $({\cal X}(J)=\cup_{k\in J}\frac{r_{k}}{\sqrt{k}}{\cal I}^{n}_{k},w)$ be a
Euclidean design supported by a union of generalized hyperoctahedra, as
defined above. Let $r,\rho,w\in\mathbb{R}^{+}$, $\rho\not=r$.

  1. 1.

For $n=3$, $p=2$, and $t=5$, choose $({\cal X}(J),w)$ according to the
following table.

$\begin{array}[]{||c||c|c||}\hline\cr\hline\cr k\in J&1&3\\\
\hline\cr\hline\cr r_{k}&r&\rho\\\ \hline\cr
w_{k}&w&\frac{9}{8}\frac{r^{4}}{\rho^{4}}\cdot w\\\
\hline\cr\hline\cr\end{array}$

Then $({\cal X}(J),w)$ is an antipodal tight Euclidean 5-design of size 14 in
$\mathbb{R}^{3}$.

  2. 2.

For $n=3$, $p=3$, and $t=7$, choose $({\cal X}(J),w)$ according to the
following table.

$\begin{array}[]{||c||c|c|c||}\hline\cr\hline\cr k\in J&1&2&3\\\
\hline\cr\hline\cr
r_{k}&r&\sqrt{\frac{3r^{2}+2\rho^{2}}{5}}\cdot\frac{r}{\rho}&\sqrt{\frac{3r^{2}+2\rho^{2}}{5}}\\\
\hline\cr w_{k}&w&\frac{100\rho^{6}}{(3r^{2}+2\rho^{2})^{3}}\cdot
w&\frac{675r^{6}}{8(3r^{2}+2\rho^{2})^{3}}\cdot w\\\
\hline\cr\hline\cr\end{array}$

Then $({\cal X}(J),w)$ is an antipodal tight Euclidean 7-design of size 26 in
$\mathbb{R}^{3}$.

  3. 3.

For $n=4$, $p=2$, and $t=7$, choose $({\cal X}(J),w)$ according to the
following table.

$\begin{array}[]{||c||c|c|c||}\hline\cr\hline\cr k\in J&1&2&4\\\
\hline\cr\hline\cr r_{k}&r&\rho&r\\\ \hline\cr
w_{k}&w&\frac{r^{6}}{\rho^{6}}\cdot w&w\\\ \hline\cr\hline\cr\end{array}$

Then $({\cal X}(J),w)$ is an antipodal tight Euclidean 7-design of size 48 in
$\mathbb{R}^{4}$.

We prove Corollary 9 in Section 4. In particular, we will see that each design
has the same stated strength even when $\rho=r$; however, they are only tight
when $\rho\not=r$. Note also that in (2) above $\rho\not=r$ implies
$|\\{r_{1},r_{2},r_{3}\\}|=3$ (while $\rho=r$ would yield
$r_{1}=r_{2}=r_{3}$).

Finally, we point out that, like the results in [2], [4], and [5], Corollary 9
disproves the conjecture of Neumaier and Seidel in [17] (see also [10]) that
there are no tight Euclidean $t$-designs with $p\geq 2$ and $t\geq 4$. It
seems to be an interesting problem to classify all other tight Euclidean
designs.

##  2 Harmonic polynomials over $\mathbb{R}^{n}$

In this section we prove Theorem 6. In order to do this, we first review some
information on harmonic polynomials over $\mathbb{R}^{n}$; then develop some
very useful results about a special subspace of
$\mathrm{Harm}_{s}(\mathbb{R}^{n})$.

There are several equivalent definitions of Euclidean designs. For our
purposes in this paper, the following will be convenient. Recall that a
polynomial is _harmonic_ if it satisfies Laplace’s equation $\Delta f=0$. The
set of homogeneous harmonic polynomials of degree $s$ in $\mathbb{R}^{n}$
forms the vector space $\mathrm{Harm}_{s}(\mathbb{R}^{n})$ with

$\dim\mathrm{Harm}_{s}(\mathbb{R}^{n})={{n+s-1}\choose{n-1}}-{{n+s-3}\choose{n-1}}.$

######  Proposition 10 (Neumaier–Seidel, [17])

The weighted set $({\cal X},w)$ is a Euclidean $t$-design in $\mathbb{R}^{n}$,
if and only if,

$\sum_{{\bf x}\in{\cal X}}w({\bf x})||{\bf x}||^{2s_{1}}f({\bf x})=0$

for every $0\leq 2s_{1}\leq t$ and $f\in\mathrm{Harm}_{s}(\mathbb{R}^{n})$
with $1\leq s\leq t-2s_{1}$.

An explicit basis for $\mathrm{Harm}_{s}(\mathbb{R}^{n})$ can be found using
Gegenbauer polynomials, as follows. Recall that, for fixed $\alpha>-1$, the
_Gegenbauer polynomial_ (also known as the ultraspherical polynomial)
$P_{s}^{\alpha}:I=[-1,1]\rightarrow\mathbb{R}$ can be defined as

$P_{s}^{\alpha}(x)=\frac{(-1)^{s}}{2^{s}s!(1-x^{2})^{\alpha-1/2}}\frac{d^{s}(1-x^{2})^{\alpha+s-1/2}}{dx^{s}}.$

(This is one of the various normalizations used.) The Gegenbauer polynomial
$P_{s}^{\alpha}$ has degree $s$ and has $s$ distinct roots in the interval
$(-1,1)$; it is an even function when $s$ is even and an odd function when $s$
is odd.

In order to form a basis for $\mathrm{Harm}_{s}(\mathbb{R}^{n})$, choose
integers $m_{0},m_{1},\dots,m_{n-2}$ with $0\leq m_{n-2}\leq\cdots\leq
m_{1}\leq m_{0}=s$. For $k=0,1,\dots,n-3$, set

$r_{k}=r_{k}(x_{k+1},\dots,x_{n})=\sqrt{\sum_{l=k+1}^{n}x_{l}^{2}},$

and define

$g_{k}=g_{k}(x_{k+1},\dots,x_{n})=r_{k}^{m_{k}-m_{k+1}}P_{m_{k}-m_{k+1}}^{m_{k+1}+(n-k-2)/2}\left(\frac{x_{k+1}}{r_{k}}\right)$

where $P_{m_{k}-m_{k+1}}^{m_{k+1}+(n-k-2)/2}$ is the Gegenbauer polynomial
defined above. Note that $g_{k}$ is a homogeneous polynomial of degree
$m_{k}-m_{k+1}$.

Let also

$h_{1}(x_{n-1},x_{n})=\mathrm{Re}(x_{n-1}+ix_{n})^{m_{n-2}}$

and

$h_{2}(x_{n-1},x_{n})=\mathrm{Im}(x_{n-1}+ix_{n})^{m_{n-2}}.$

Finally, for integer(s) $1\leq\mu\leq\mathrm{min}\\{2,m_{n-2}+1\\}$, define

$f_{m_{0},m_{1},\dots,m_{n-2},\mu}(x_{1},\dots,x_{n})=h_{\mu}(x_{n-1},x_{n})\cdot\prod_{k=0}^{n-3}g_{k}(x_{k+1},\dots,x_{n})$

and the set

$\Phi_{s}^{n}=\\{f_{m_{0},m_{1},\dots,m_{n-2},\mu}(x_{1},\dots,x_{n})\\}.$

Note that $f_{m_{0},m_{1},\dots,m_{n-2},\mu}(x_{1},\dots,x_{n})$ is a
homogeneous polynomial of degree $s$ and that

$\displaystyle|\Phi_{s}^{n}|$ $\displaystyle=$
$\displaystyle|\\{(m_{n-2},\dots,m_{0},\mu)\in\mathbb{Z}^{n}\mbox{ }|$
$\displaystyle\mbox{ }\mbox{ }0\leq m_{n-2}\leq\cdots\leq
m_{0}=s,1\leq\mu\leq\mathrm{min}\\{2,m_{n-2}+1\\}\\}|$ $\displaystyle=$
$\displaystyle 2\cdot{{n+s-3}\choose{n-2}}+{{n+s-3}\choose{n-3}}$
$\displaystyle=$ $\displaystyle{{n+s-1}\choose{n-1}}-{{n+s-3}\choose{n-1}}$
$\displaystyle=$ $\displaystyle\dim\mathrm{Harm}_{s}(\mathbb{R}^{n}).$

Furthermore, we have the following.

######  Proposition 11 ([11])

With the above notations, the set $\Phi_{s}^{n}$ forms a basis for
$\mathrm{Harm}_{s}(\mathbb{R}^{n})$.

For larger values of $n$, Propositions 10 and 11 are not convenient due to the
large size of $\Phi_{s}^{n}$. However, if we consider only fully symmetric
designs, then the necessary criteria can be greatly reduced as explained
below.

Recall that, as defined by Definition 5, a Euclidean design $({\cal X},w)$ is
fully symmetric if ${\cal X}^{g}={\cal X}$ for every $g\in H$ and $w({\bf
x})=w({\bf y})$ whenever ${\bf x}={\bf y}^{g}$ for some $g\in H$. Here $H$ is
the hyperoctahedral group in dimension $n$, that is the subgroup of the
orthogonal group $O(n)$ generated by all transpositions and reflections with
respect to coordinate hyperplanes. For a polynomial
$f\in\mathrm{Pol}(\mathbb{R}^{n})$, we let $f^{g}$ denote the function $f({\bf
x}^{g})$. As before, for $1\leq i\leq n$, we let $\rho_{i}\in O(n)$ be the
reflection with respect to the coordinate hyperplane $x_{i}=0$, and for $1\leq
i_{1}<i_{2}\leq n$, let $\sigma_{i_{1},i_{2}}$ be the reflection with respect
to the hyperplane $x_{i_{1}}=x_{i_{2}}$. Note that $\rho_{i}$ acts on a
polynomial $f$ by replacing its variable $x_{i}$ be $-x_{i}$, and
$\sigma_{i_{1},i_{2}}$ acts by switching $x_{i_{1}}$ and $x_{i_{2}}$.

Let us first address symmetry with respect to coordinate hyperplanes. For this
purpose, we are specifically interested in _fully even_ harmonic polynomials.

###### Definition 12

A polynomial $f\in\mathbb{R}[x_{1},\dots,x_{n}]$ is said to be fully even if
$f^{\rho_{i}}=f$ for every $1\leq i\leq n$, and partially odd if
$f^{\rho_{i}}=-f$ for some $1\leq i\leq n$. We let
$\mathrm{FEvenHarm}_{s}(\mathbb{R}^{n})$ and
$\mathrm{POddHarm}_{s}(\mathbb{R}^{n})$ denote the set of fully even
polynomials and the set of partially odd polynomials in
$\mathrm{Harm}_{s}(\mathbb{R}^{n})$, respectively.

Note that in a polynomial which belongs to
$\mathrm{FEvenHarm}_{s}(\mathbb{R}^{n})$, in every term every variable has an
even degree; while in members of $\mathrm{POddHarm}_{s}(\mathbb{R}^{n})$, at
least one variable appears only with an odd degree in every term.

We let

$\mathrm{FEven}\Phi_{s}^{n}=\Phi_{s}^{n}\cap\mathrm{FEvenHarm}(\mathbb{R}^{n}),$

and

$\mathrm{POdd}\Phi_{s}^{n}=\Phi_{s}^{n}\cap\mathrm{POddHarm}(\mathbb{R}^{n}).$

Recall that a Gegenbauer polynomial of degree $s$ is an even function when $s$
is even and an odd function when $s$ is odd. Therefore the polynomial
$g_{k}=g_{k}(x_{k+1},\dots,x_{n})$, defined above, has its variable $x_{k+1}$
with even exponents only if its degree $m_{k}-m_{k+1}$ is even, and odd
exponents only if $m_{k}-m_{k+1}$ is odd; the variables $x_{k+2},\dots,x_{n}$
all appear with even exponents only. Consequently, we can determine easily
which members of $\Phi_{s}^{n}$ are in $\mathrm{FEven}\Phi_{s}^{n}$, and we
can count the elements of $\mathrm{FEven}\Phi_{s}^{n}$, as follows.

###### Proposition 13

With the notations above, we have the following.

  1. 1.

$\mathrm{\Phi}_{s}^{n}=\mathrm{FEven}\Phi_{s}^{n}\cup\mathrm{POdd}\Phi_{s}^{n};$

  2. 2.

$\mathrm{FEven}\Phi_{s}^{n}$ is a basis for
$\mathrm{FEvenHarm}_{s}(\mathbb{R}^{n})$;

  3. 3.

A polynomial
$f_{m_{0},m_{1},\dots,m_{n-2},\mu}(x_{1},\dots,x_{n})\in\Phi_{s}^{n}$ is in
$\mathrm{FEven}\Phi_{s}^{n}$, if and only if, $m_{0},\cdots,m_{n-2}$ are all
even and $\mu=1$;

  4. 4.

If $s$ is even, then

$\dim\mathrm{FEvenHarm}_{s}(\mathbb{R}^{n})={{n+\frac{s}{2}-2}\choose{n-2}}.$
(5)

We can now use Propositions 10 and 13 to restate Proposition 10 for fully
symmetric designs as follows.

###### Proposition 14

A fully symmetric weighted set $({\cal X},w)$ is a Euclidean $t$-design in
$\mathbb{R}^{n}$, if and only if,

$\sum_{{\bf x}\in{\cal X}}w({\bf x})||{\bf x}||^{2s_{1}}f({\bf x})=0$

holds for every $0\leq 2s_{1}\leq t$ and
$f\in\mathrm{FEvenHarm}_{s}(\mathbb{R}^{n})$ with $s$ even and $2\leq s\leq
t-2s_{1}$.

Note that $\dim\mathrm{FEvenHarm}_{s}(\mathbb{R}^{n})$ is substantially
smaller than $\dim\mathrm{Harm}_{s}(\mathbb{R}^{n})$. However, rather than
working with $\mathrm{FEven}\Phi_{s}^{n}$ resulting from the algorithm of
Proposition 11, we now attempt to choose a more convenient basis for
$\mathrm{FEvenHarm}_{s}(\mathbb{R}^{n})$ which further reduces the number of
equations to be checked.

For $s=2$, by (5) we have $\dim\mathrm{FEvenHarm}_{2}(\mathbb{R}^{n})=n-1$; we
find, for example, that

$F_{2}=\\{x_{i}^{2}-x_{n}^{2}\mbox{ }|\mbox{ }1\leq i\leq n-1\\}$

forms a basis for $\mathrm{FEvenHarm}_{2}(\mathbb{R}^{n})$. From this, we
immediately see that, if the weighted set $(X,w)$ is fully symmetric, then the
equation in Proposition 10 holds for each $f\in F_{2}$, and therefore we have
Theorem 6, 1: every fully symmetric weighted set $(X,w)$ is a Euclidean
3-design.

Before we turn to finding bases for $s\geq 4$, we introduce a notation. For a
fixed positive integer $j\leq n$, let $S_{n,j}$ denote the set of strictly
increasing functions from $\\{1,2,\dots,j\\}$ to $\\{1,2,\dots,n\\}$; in other
words, permutations of the form

$g=\left(\begin{array}[]{cccc}1&2&\dots&j\\\
i_{1}&i_{2}&\dots&i_{j}\end{array}\right)$

where $1\leq i_{1}<i_{2}<\cdots<i_{j}\leq n$. If
$f\in\mathbb{R}[x_{1},x_{2},\dots,x_{j}]$ is a polynomial on $j$ variables and
$g\in S_{n,j}$ is as above, then $f^{g}$ denotes the polynomial
$f(x_{i_{1}},x_{i_{2}},\dots,x_{i_{j}})\in\mathbb{R}[x_{1},x_{2},\dots,x_{n}]$.
We also write $f^{S_{n,j}}$ for the set $\\{f^{g}\mbox{ }|\mbox{ }g\in
S_{n,j}\\}$; note that this set has size ${{n}\choose{j}}$.

In this paper we focus on $t\leq 9$, thus we must find bases for
$\mathrm{FEvenHarm}_{s}(\mathbb{R}^{n})$ for $s=4$, $s=6$, and $s=8$.
According to (5), we have

$\displaystyle\dim\mathrm{FEvenHarm}_{4}(\mathbb{R}^{n})$ $\displaystyle=$
$\displaystyle{{n}\choose{2}},$
$\displaystyle\dim\mathrm{FEvenHarm}_{6}(\mathbb{R}^{n})$ $\displaystyle=$
$\displaystyle{{n+1}\choose{3}}={{n}\choose{2}}+{{n}\choose{3}},\mbox{ and}$
$\displaystyle\dim\mathrm{FEvenHarm}_{8}(\mathbb{R}^{n})$ $\displaystyle=$
$\displaystyle{{n+2}\choose{4}}={{n}\choose{2}}+2{{n}\choose{3}}+{{n}\choose{4}}.$

This suggests that we may be able to find polynomials
$f_{4,2},f_{6,2},f_{8,2}\in\mathbb{R}[x_{1},x_{2}]$,
$f_{6,3},f_{8,3,1},f_{8,3,2}\in\mathbb{R}[x_{1},x_{2},x_{3}]$, and
$f_{8,4}\in\mathbb{R}[x_{1},x_{2},x_{3},x_{4}]$ for which

$\displaystyle F_{4}$ $\displaystyle=$ $\displaystyle f_{4,2}^{S_{n,2}},$
$\displaystyle F_{6}$ $\displaystyle=$ $\displaystyle f_{6,2}^{S_{n,2}}\cup
f_{6,3}^{S_{n,3}},\mbox{ and}$ $\displaystyle F_{8}$ $\displaystyle=$
$\displaystyle f_{8,2}^{S_{n,2}}\cup f_{8,3,1}^{S_{n,3}}\cup
f_{8,3,2}^{S_{n,3}}\cup f_{8,4}^{S_{n,4}}$

are bases for $\mathrm{FEvenHarm}_{s}(\mathbb{R}^{n})$ for $s=4$, $s=6$, and
$s=8$, respectively.

Indeed, with

$\displaystyle f_{4,2}$ $\displaystyle=$ $\displaystyle
x_{1}^{4}-6x_{1}^{2}x_{2}^{2}+x_{2}^{4},$ $\displaystyle f_{6,2}$
$\displaystyle=$ $\displaystyle
x_{1}^{6}-15x_{1}^{4}x_{2}^{2}+15x_{1}^{2}x_{2}^{4}-x_{2}^{6},$ $\displaystyle
f_{6,3}$ $\displaystyle=$ $\displaystyle
2(x_{1}^{6}+x_{2}^{6}+x_{3}^{6})-15(x_{1}^{4}x_{2}^{2}+x_{1}^{2}x_{2}^{4}+x_{1}^{4}x_{3}^{2}+x_{1}^{2}x_{3}^{4}+x_{2}^{4}x_{3}^{2}+x_{2}^{2}x_{3}^{4})+180x_{1}^{2}x_{2}^{2}x_{3}^{2},$
$\displaystyle f_{8,2}$ $\displaystyle=$ $\displaystyle
x_{1}^{8}-28x_{1}^{6}x_{2}^{2}+70x_{1}^{4}x_{2}^{4}-28x_{1}^{2}x_{2}^{6}+x_{2}^{8},$
$\displaystyle f_{8,3,1}$ $\displaystyle=$ $\displaystyle
x_{1}^{8}-x_{2}^{8}+14(x_{1}^{2}x_{2}^{6}+x_{2}^{6}x_{3}^{2})-14(x_{1}^{6}x_{2}^{2}+x_{1}^{6}x_{3}^{2})+210x_{1}^{4}x_{2}^{2}x_{3}^{2}-210x_{1}^{2}x_{2}^{4}x_{3}^{2},$
$\displaystyle f_{8,3,2}$ $\displaystyle=$ $\displaystyle
x_{1}^{8}-x_{3}^{8}+14(x_{1}^{2}x_{3}^{6}+x_{2}^{2}x_{3}^{6})-14(x_{1}^{6}x_{3}^{2}+x_{1}^{6}x_{2}^{2})+210x_{1}^{4}x_{2}^{2}x_{3}^{2}-210x_{1}^{2}x_{2}^{2}x_{3}^{4},$
$\displaystyle f_{8,4}$ $\displaystyle=$ $\displaystyle
3(x_{1}^{8}+x_{2}^{8}+x_{3}^{8}+x_{4}^{8})-28(x_{1}^{6}x_{2}^{2}+\cdots+x_{3}^{2}x_{4}^{6})+210(x_{1}^{4}x_{2}^{2}x_{3}^{2}+\cdots+x_{2}^{2}x_{3}^{2}x_{4}^{4})$
$\displaystyle-3780x_{1}^{2}x_{2}^{2}x_{3}^{2}x_{4}^{2},$

we see that the sets $F_{4}$, $F_{6}$, and $F_{8}$ defined above are linearly
independent; and, since they have the right cardinality, they form bases for
$\mathrm{FEvenHarm}_{s}(\mathbb{R}^{n})$ for $s=4$, $s=6$, and $s=8$,
respectively.

Note that if $({\cal X},w)$ is fully symmetric, then the equation of
Proposition 14 always holds for _skew-symmetric_ polynomials, that is those
polynomials $f$ for which $f^{\sigma_{i_{1},i_{2}}}=-f$ for some transposition
$\sigma_{i_{1},i_{2}}$; this is the case for $f_{6,2}$, $f_{8,3,1}$, and
$f_{8,3,2}$. This completes the proof of Theorem 6.

_Remarks._ While we do not attempt to generalize Theorem 6 for higher values
of $s$, we can indicate what one can expect. Consider first the following
combinatorial identity, often referred to as Vandermonde’s Convolution,

${{n+\frac{s}{2}-2}\choose{n-2}}=\sum_{j=2}^{\frac{s}{2}}{{\frac{s}{2}-2}\choose{j-2}}{{n}\choose{j}},$
(6)

which holds for every even $s$ with $s\geq 4$. (In this paper we think of $n$
as large, in particular, larger than $s/2$ and thus carry the summation to
$s/2$; although (6) is of course valid without this assumption.)

This identity suggests that for each even $s\geq 4$, one might find
polynomials $f_{s,j,i}$, where $2\leq j\leq\frac{s}{2}$ and $1\leq
i\leq{{\frac{s}{2}-2}\choose{j-2}}$, so that

$F_{s}=\bigcup_{i,j}f_{s,j,i}^{S_{n,j}}$

is a basis for $\mathrm{FEvenHarm}_{s}(\mathbb{R}^{n})$. This implies that the
system of equations in Proposition 14 needs to be checked only for less than
$2^{\lfloor t/2\rfloor}$ equations, a quantity independent of $n$ and
substantially smaller than $\dim\mathrm{FEvenHarm}_{s}(\mathbb{R}^{n})$. We
intend to pursue this interesting question in a future study.

##  3 Designs supported by generalized hyperoctahedra

In this section we prove Theorem 3.

First, let us recall our notations. Let $H$ be the hyperoctahedral group in
$n$ dimensions. Let ${\bf e_{1}},\dots,{\bf e_{n}}$ be the standard basis
vectors of $\mathbb{R}^{n}$, and for $k=1,2,\dots,n$, set ${\bf x_{k}}={\bf
e_{1}}+\cdots+{\bf e_{k}}$. The set

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

is the vertex set of a generalized regular hyperoctahedron in
$\mathbb{R}^{n}$. We consider sets ${\cal X}$ of the form

${\cal X}={\cal X}(J)=\cup_{k\in J}\frac{r_{k}}{\sqrt{k}}{\cal I}^{n}_{k}$

where $J\subset\\{1,2,\dots,n\\}$ and $r_{k}>0$ for every $k\in J$. We see
that ${\cal X}$ is antipodal and has norm spectrum $R=\\{r_{k}\mbox{ }|\mbox{
}k\in J\\}$. We choose a weight function $w:{\cal X}\rightarrow\mathbb{R}^{+}$
on ${\cal X}$ which is constant on each layer of ${\cal X}$; let us denote the
weight of ${\bf x}\in\frac{r_{k}}{\sqrt{k}}{\cal I}^{n}_{k}$ by $w({\bf
x})=w_{k}$.

Our goal is to determine for each strength $t$ whether an index set $J$, a
norm spectrum $R$, and a weight function $w$ exist for which $({\cal X},w)$ is
a Euclidean $t$-design. Note that $({\cal X(J)},w)$ is fully symmetric, and
therefore its strength $t$ is an odd positive integer. We first prove the
following.

###### Proposition 15

Let $({\cal X(J)},w)$ be the union of weighted generalized hyperoctahedra, as
defined above. Then

  1. 1.

$({\cal X(J)},w)$ is at least a Euclidean $3$-design for any $J$, $R$, and
$w$.

  2. 2.

There are no choices of $J$, $R$, and $w$ for which $({\cal X(J)},w)$ is a
Euclidean $9$-design.

Proof. The first statement follows immediately from Theorem 6, 1. For the
second statement, we will use (4) of Theorem 6. In particular, the equation
involving $f_{8,2}$ implies that, if $({\cal X(J)},w)$ is a Euclidean
9-design, then

$\sum_{k\in J}\sum_{{\bf x}\in{\cal
I}_{k}^{n}}w_{r}\cdot\frac{r_{k}^{8}}{k^{4}}\cdot f_{8,2}({\bf x})=0.$

However, we find that

$\sum_{{\bf x}\in{\cal I}^{n}_{k}}f_{8,2}({\bf
x})=2^{k}\cdot\left[2\cdot{{n-1}\choose{k-1}}-28\cdot
2\cdot{{n-2}\choose{k-2}}+70\cdot 2\cdot{{n-2}\choose{k-2}}\right],$

a value which is positive for every $k$, and this is a contradiction.

According to Proposition 15, the maximum strength of $({\cal X(J)},w)$ is 3,
5, or 7. Next we determine cases when the strength is 5.

###### Proposition 16

Let ${\cal X}={\cal X}(J)=\cup_{k\in J}\frac{r_{k}}{\sqrt{k}}{\cal I}^{n}_{k}$
be a union of generalized hyperoctahedra, as defined above.

  1. 1.

Suppose that $|J|=1$. If $n\not\equiv 1$ mod 3, then ${\cal X}(J)$ is never a
Euclidean $5$-design. Suppose that $n\equiv 1$ mod 3 and $J=\\{k\\}$; then
$({\cal X}(J),w)$ is a Euclidean $5$-design, if and only if,
$k=\frac{n+2}{3}$.
In particular, if $n\equiv 1$ mod 3 and $k=\frac{n+2}{3}$, then $({\cal
X}(\\{k\\}),w)$ is a Euclidean $5$-design for any positive $r_{k}$ and
$w_{k}$.

  2. 2.

Suppose that $|J|\geq 2$. Then there is weight function $w$ for which $({\cal
X}(J),w)$ is a Euclidean $5$-design, if and only if, there exist $k_{1}\in J$
and $k_{2}\in J$ for which $k_{1}<\frac{n+2}{3}<k_{2}$.
In particular, $({\cal X}(\\{1,n\\}),w)$ is a Euclidean $5$-design for any
positive $r_{1}$, $r_{n}$, $w_{1}$, and $w_{n}$ which satisfy
$\frac{w_{1}}{w_{n}}\cdot\frac{r_{1}^{4}}{r_{n}^{4}}=\frac{n^{2}}{2^{n}}$.

_Proof._ To see whether a union of generalized hyperoctahedra is a Euclidean
$5$-design, we use Theorem 6, 2. We compute that

$\sum_{{\bf x}\in{\cal I}^{n}_{k}}f_{4,2}({\bf x})=2\cdot
2^{k}\cdot{{n-1}\choose{k-1}}-6\cdot 2^{k}\cdot{{n-2}\choose{k-2}},$

and so $({\cal X}(J),w)$ is a 5-design, if and only if,

$\sum_{k\in J}w_{k}\cdot\frac{r_{k}^{4}}{k^{2}}\cdot
2^{k+1}\cdot{{n-1}\choose{k-1}}\cdot\left[1-3\cdot\frac{k-1}{n-1}\right]=0.$
(7)

From this our Proposition easily follows.

Now we determine when our construction yields Euclidean $t$-designs for $t=7$.
Recall that for positive integers $k_{1},k_{2}\leq n$ we have defined the
function

$G(k_{1},k_{2})=(n+2-3k_{1})(n+2-3k_{2})+6(k_{1}-1)(k_{2}-1)+2(n-1).$

###### Proposition 17

Let ${\cal X}={\cal X}(J)=\cup_{k\in J}\frac{r_{k}}{\sqrt{k}}{\cal I}^{n}_{k}$
be a union of generalized hyperoctahedra, as defined above.

  1. 1.

If $|J|=1$, then ${\cal X}(J)$ is never a Euclidean $7$-design.

  2. 2.

Suppose that $|J|=2$ with $J=\\{k_{1},k_{2}\\}$. Then there is a weight
function $w$ for which $({\cal X}(J),w)$ is a Euclidean $7$-design, if and
only if, $r_{k_{1}}=r_{k_{2}}$ and $G(k_{1},k_{2})=0$.

  3. 3.

Suppose that $|J|=3$ with $J=\\{k_{1},k_{2},k_{3}\\}$; assume that
$k_{1}<k_{2}<k_{3}$.

    1. (a)

If $|R|=1$, then there is a weight function $w$ for which $({\cal X}(J),w)$ is
a Euclidean $7$-design, if and only if, $G(k_{1},k_{2})>0$,
$G(k_{2},k_{3})>0$, and $G(k_{1},k_{3})<0$.

    2. (b)

If $|R|=2$, then there is a weight function $w$ for which $({\cal X}(J),w)$ is
a Euclidean $7$-design, if and only if, $n\equiv 1$ mod 3,
$k_{2}=\frac{n+2}{3}$, $r_{k_{1}}=r_{k_{3}}$, and $G(k_{1},k_{3})<0$.

    3. (c)

If $|R|=3$, then there is a weight function $w$ for which $({\cal X}(J),w)$ is
a Euclidean $7$-design, if and only if, $G(k_{1},k_{2})>0$,
$G(k_{2},k_{3})>0$, $G(k_{1},k_{3})<0$, and

$\sum_{i=1,2,3}k_{i}\cdot(n+2-3k_{i})\cdot(k_{i+1}-k_{i+2})\cdot
G(k_{i+1},k_{i+2})\cdot\frac{1}{r_{k_{i}}^{2}}=0,$ (8)

holds where the indeces are to be taken mod 3 (i.e. $k_{4}=k_{1}$ and
$k_{5}=k_{2}$).

To prove Proposition 17, we will need the following Lemma whose proof is
straight forward and will be omitted.

###### Lemma 18

For a positive integer $k$ we introduce the following notations:

$p_{k}=k\cdot\left(1-3\cdot\frac{k-1}{n-1}\right)\mbox{ and
}q_{k}=3\cdot\left(1-15\cdot\frac{k-1}{n-1}+30\cdot\frac{(k-1)(k-2)}{(n-1)(n-2)}\right).$

We then have the following.

  1. 1.

If $1\leq k_{1}<k_{2}\leq n$, then

$p_{k_{1}}q_{k_{2}}-p_{k_{2}}q_{k_{1}}=\frac{3(n+8)}{(n-1)^{2}(n-2)}\cdot(k_{1}-k_{2})\cdot
G(k_{1},k_{2})$

  2. 2.

If $1\leq k_{1}<k_{2}<k_{3}\leq n$, then

$\sum_{i=1,2,3}k_{i}\cdot(n+2-3k_{i})\cdot(k_{i+1}-k_{i+2})\cdot
G(k_{i+1},k_{i+2})=0,$ (9)

where the indeces are to be taken mod 3 (i.e. $k_{4}=k_{1}$ and
$k_{5}=k_{2}$).

  3. 3.

If $1\leq k_{1}<k_{2}<k_{3}<k_{4}\leq n$ and $G(k_{2},k_{3})\leq 0$, then we
have

    1. (a)

$k_{2}<\frac{n+2}{3}$ and $k_{3}>\frac{n+2}{3}$

    2. (b)

$p_{2}>0$ and $p_{3}<0$; and

    3. (c)

$G(k_{1},k_{2})>0$ and $G(k_{3},k_{4})>0$.

_Proof of Proposition 17._ Referring to Theorem 6, 3, we compute that

$\sum_{{\bf x}\in{\cal I}^{n}_{k}}f_{6,3}({\bf x})=6\cdot
2^{k}\cdot{{n-1}\choose{k-1}}-90\cdot 2^{k}\cdot{{n-2}\choose{k-2}}+180\cdot
2^{k}\cdot{{n-3}\choose{k-3}},$

and so $({\cal X}(J),w)$ is a 7-design, if and only if,

$\left\\{\begin{array}[]{l}\sum_{k\in J}w_{k}\cdot\frac{r_{k}^{4}}{k^{2}}\cdot
2^{k+1}\cdot{{n-1}\choose{k-1}}\cdot\left[1-3\cdot\frac{k-1}{n-1}\right]=0\\\
\\\ \sum_{k\in J}w_{k}\cdot\frac{r_{k}^{6}}{k^{3}}\cdot
2^{k+1}\cdot{{n-1}\choose{k-1}}\cdot
3\cdot\left[1-15\cdot\frac{k-1}{n-1}+30\cdot\frac{(k-1)(k-2)}{(n-1)(n-2)}\right]=0\\\
\\\ \sum_{k\in J}w_{k}\cdot\frac{r_{k}^{6}}{k^{2}}\cdot
2^{k+1}\cdot{{n-1}\choose{k-1}}\cdot\left[1-3\cdot\frac{k-1}{n-1}\right]=0\end{array}\right.$
(10)

With the notations introduced in Lemma 18 and with

$u_{k}=w_{k}\cdot\frac{2^{k+1}}{k^{3}}\cdot{{n-1}\choose{k-1}},$

(10) becomes

$\left\\{\begin{array}[]{l}\sum_{k\in J}u_{k}\cdot r_{k}^{4}\cdot p_{k}=0\\\
\\\ \sum_{k\in J}u_{k}\cdot r_{k}^{6}\cdot q_{k}=0\\\ \\\ \sum_{k\in
J}u_{k}\cdot r_{k}^{6}\cdot p_{k}=0\end{array}\right.$ (11)

We analyze (11), as follows. First, we see that ${\cal X}(J)$ is never a
Euclidean $7$-design for $|J|=1$, since there is no value of $k$ for which
$p_{k}=q_{k}=0$.

Suppose now that $|J|=2$ with $J=\\{k_{1},k_{2}\\}$. We first note that in
this case we must have $p_{k_{1}}\not=0$ and $p_{k_{2}}\not=0$, and therefore,
if (11) holds, then $r_{k_{1}}=r_{k_{2}}$, in which case (11) is equivalent to

$\left\\{\begin{array}[]{l}u_{k_{1}}\cdot p_{k_{1}}+u_{k_{2}}\cdot
p_{k_{2}}=0\\\ \\\ u_{k_{1}}\cdot q_{k_{1}}+u_{k_{2}}\cdot
q_{k_{2}}=0\end{array}\right.$ (12)

This system has non-zero solutions exactly when $p_{k_{1}}\cdot
q_{k_{2}}-p_{k_{2}}\cdot q_{k_{1}}=0$. By Lemma 18, 1, this is equivalent to
$G(k_{1},k_{2})=0$; by Lemma 18, 3 (b), we also see that in this case
$p_{k_{1}}$ and $p_{k_{2}}$ have opposite signs, and therefore we may have
$u_{k_{1}}>0$ and $u_{k_{2}}>0$.

Suppose next that $|J|=3$ with $J=\\{k_{1},k_{2},k_{3}\\}$; we also assume
that $k_{1}<k_{2}<k_{3}$.

We first show that, if (11) holds with non-zero $u_{k_{1}}$, $u_{k_{2}}$, and
$u_{k_{3}}$, then $G(k_{i},k_{j})\not=0$ for $1\leq i<j\leq 3$. Indeed, from
the last two equations of (11) we get

$\left\\{\begin{array}[]{l}u_{k_{1}}\cdot
r_{k_{1}}^{6}\cdot(p_{k_{3}}q_{k_{1}}-p_{k_{1}}q_{k_{3}})+u_{k_{2}}\cdot
r_{k_{2}}^{6}\cdot(p_{k_{3}}q_{k_{2}}-p_{k_{2}}q_{k_{3}})=0\\\ \\\
u_{k_{1}}\cdot
r_{k_{1}}^{6}\cdot(p_{k_{2}}q_{k_{1}}-p_{k_{1}}q_{k_{2}})+u_{k_{3}}\cdot
r_{k_{3}}^{6}\cdot(p_{k_{2}}q_{k_{3}}-p_{k_{3}}q_{k_{2}})=0\\\ \\\
u_{k_{2}}\cdot
r_{k_{2}}^{6}\cdot(p_{k_{1}}q_{k_{2}}-p_{k_{2}}q_{k_{1}})+u_{k_{3}}\cdot
r_{k_{3}}^{6}\cdot(p_{k_{1}}q_{k_{3}}-p_{k_{3}}q_{k_{1}})=0\end{array}\right.$
(13)

Therefore, if any $p_{k_{i}}q_{k_{j}}-p_{k_{j}}q_{k_{i}}$, $1\leq i<j\leq 3$,
equals zero, then they all equal zero; by Lemma 18, 1, this is equivalent to
$G(k_{1},k_{2})=G(k_{1},k_{3})=G(k_{2},k_{3})=0$, which would contradict Lemma
18, 3 (c).

We thus see that (11) is equivalent to

$\left\\{\begin{array}[]{l}u_{k_{2}}=\frac{p_{k_{1}}q_{k_{3}}-p_{k_{3}}q_{k_{1}}}{p_{k_{3}}q_{k_{2}}-p_{k_{2}}q_{k_{3}}}\cdot\frac{r_{k_{1}}^{6}}{r_{k_{2}}^{6}}\cdot
u_{k_{1}}\\\ \\\
u_{k_{3}}=\frac{p_{k_{2}}q_{k_{1}}-p_{k_{1}}q_{k_{2}}}{p_{k_{3}}q_{k_{2}}-p_{k_{2}}q_{k_{3}}}\cdot\frac{r_{k_{1}}^{6}}{r_{k_{3}}^{6}}\cdot
u_{k_{1}}\\\ \\\
\sum_{i=1,2,3}p_{k_{i}}\cdot(p_{k_{i+1}}q_{k_{i+2}}-p_{k_{i+2}}q_{k_{i+1}})\cdot\frac{1}{r_{k_{i}}^{2}}=0\end{array}\right.$
(14)

Using Lemma 18, 2, the last equation is equivalent to (8), and the first two
can be written as

$\left\\{\begin{array}[]{l}u_{k_{2}}=\frac{k_{1}-k_{3}}{k_{3}-k_{2}}\cdot\frac{G(k_{1},k_{3})}{G(k_{2},k_{3})}\cdot\frac{r_{k_{1}}^{6}}{r_{k_{2}}^{6}}\cdot
u_{k_{1}}\\\ \\\
u_{k_{3}}=\frac{k_{2}-k_{1}}{k_{3}-k_{2}}\cdot\frac{G(k_{1},k_{2})}{G(k_{2},k_{3})}\cdot\frac{r_{k_{1}}^{6}}{r_{k_{3}}^{6}}\cdot
u_{k_{1}}\end{array}\right.$ (15)

Our equations show that $u_{k_{1}}$, $u_{k_{2}}$, and $u_{k_{3}}$ are all
positive, if and only if, one of the following two possibilities holds: (i)
$G(k_{1},k_{2})>0$, $G(k_{1},k_{3})<0$, and $G(k_{2},k_{3})>0$, or (ii)
$G(k_{1},k_{2})<0$, $G(k_{1},k_{3})>0$, and $G(k_{2},k_{3})<0$. However, by
Lemma 18, 3 (c), the second possibility cannot occur.

It remains to see what happens when $|R|<3$. First note that, by Lemma 18, 2,
(8) always holds when $r_{k_{1}}=r_{k_{2}}=r_{k_{3}}$. If $|R|=2$, then we may
assume that for some $i\in\\{1,2,3\\}$, we have
$r_{k_{i}}=r_{k_{i+1}}\not=r_{k_{i+2}}$ (again, indices are to be taken mod
3). Comparing (8) and (9), we conclude that $k_{i+2}=\frac{n+2}{3}$. Since
$G(k_{1},k_{3})<0$, by Lemma 18, 3 (a), we see that this can only happen if
$i=3$. Finally, note that $k_{2}=\frac{n+2}{3}$ and $k_{1}<k_{2}<k_{3}$ imply,
by Lemma 18, 3 (c), that $G(k_{1},k_{2})>0$ and $G(k_{2},k_{3})>0$.

This completes our proof of Proposition 17 and hence Theorem 3.

##  4 Tight Euclidean designs and other examples

In this section we examine some examples and prove Corollaries 4 and 9. First
we prove the following.

###### Proposition 19

Let ${\cal X}(J)=\cup_{k\in J}\frac{r_{k}}{\sqrt{k}}{\cal I}^{n}_{k}$ be
defined as before.

  1. 1.

For each of the following sets $J$ and integers $p$, there exists a set $R$
with $|R|=p$ and a weight function $w$ for which $({\cal X}(J),w)$ is a
Euclidean $5$-design.

    1. (a)

$p=1$ and $J=\\{\frac{n+2}{3}\\}$, when $n\equiv 1$ mod 3;

    2. (b)

$p\in\\{1,2\\}$ and $J=\\{1,n\\}$.

  2. 2.

For each of the following sets $J$ and integers $p$, there exists a set $R$
with $|R|=p$ and a weight function $w$ for which $({\cal X}(J),w)$ is a
Euclidean $7$-design.

    1. (a)

$p=1$ and $J=\\{k_{1},k_{2}\\}$, when $G(k_{1},k_{2})=0$; for example,
$J=\\{1,\frac{n+4}{3}\\}$, when $n\equiv 2$ mod 3;

    2. (b)

$p=1$ and $J=\\{1,k,n\\}$, with $\frac{n+4}{6}<k<\frac{n+4}{3}$;

    3. (c)

$p=2$ and $J=\\{1,\frac{n+2}{3},n\\}$, when $n\equiv 1$ mod 3;

    4. (d)

$p=3$ and $J=\\{1,k,n\\}$, with $\frac{n+4}{6}<k<\frac{n+2}{3}$ or with
$k=\frac{n+3}{3}$.

_Proof._ Parts 1 (a), 1 (b), and 2 (a) are obvious from Theorem 3. Next, note
that $G(1,n)=-2(n-1)(n-2)$, $G(1,k)=(n-1)(n+4-3k)$, and
$G(k,n)=-2(n-1)(n+4-6k)$; so 2 (a) follows from Theorem 3, 2 (e). Part 2 (b)
follows similarly from Theorem 3, 3 (c). Finally, for part 2 (c), note that
with $k_{1}=1$, $k_{2}=k$, and $k_{3}=n$, since $G(1,k)>0$, $G(k,n)>0$, and
$G(1,n)<0$, in equation (8), the coefficient of $1/r_{k_{1}}^{2}$ is negative,
the coefficient of $1/{r_{k_{3}}^{2}}$ is positive, and the coefficient of
$1/{r_{k_{2}}^{2}}$ is positive or negative, depending on
$k_{2}>\frac{n+2}{3}$ or $k_{2}<\frac{n+2}{3}$. In the first case, an
appropriate $r_{1}$ exists for each choice of $r_{k}$ and $r_{n}$; in the
second case (which can only happen if $k=\frac{n+3}{3}$), an appropriate
$r_{n}$ exists for each choice of $r_{1}$ and $r_{k}$. This completes our
proof.

_Proof of Corollary 4._ First note that the open interval
$(\frac{n+4}{6},\frac{n+4}{3})$ contains an integer for every $n\geq 3$; and
the set $(\frac{n+4}{6},\frac{n+2}{3})\cup\\{\frac{n+3}{3}\\}$ contains an
integer for $n=3$ and every $n\geq 5$. Therefore, all cases of Corollary 4
follow from Proposition 19, except for $p=3$ and $|J|=3$ when $n=4$ – a case
which can easily be checked.

Our results can be conveniently summarized in the following table.

$\begin{array}[]{||c||c|c|c|c|c||}\hline\cr\hline\cr&n\equiv 0(3)&n=4&n\equiv
1(3),n\not\in G,n\not=4&n\equiv 1(3),n\in G&n\equiv 2(3)\\\
\hline\cr\mbox{First few such
}n&3,6,9,12,\dots&4&7,13,19,22,25,34,\dots&10,16,28,31,46,\dots&5,8,11,14,\dots\\\
\hline\cr\hline\cr\begin{array}[]{c|c}p=1&\begin{array}[]{c}|J|=1\\\ |J|=2\\\
|J|=3\\\ \end{array}\par\end{array}\par&\begin{array}[]{c}3\\\ 5\\\ 7\\\
\end{array}\par&\begin{array}[]{c}5\\\ 5\\\ 7\\\
\end{array}\par&\begin{array}[]{c}5\\\ 5\\\ 7\\\
\end{array}\par&\begin{array}[]{c}5\\\ 7\\\ 7\\\
\end{array}\par&\begin{array}[]{c}3\\\ 7\\\ 7\\\ \end{array}\par\\\
\hline\cr\begin{array}[]{c|c}p=2&\begin{array}[]{c}|J|=2\\\ |J|=3\\\
\end{array}\par\end{array}\par&\begin{array}[]{c}5\\\ 5\\\
\end{array}\par&\begin{array}[]{c}5\\\ 7\\\
\end{array}\par&\begin{array}[]{c}5\\\ 7\\\
\end{array}\par&\begin{array}[]{c}5\\\ 7\\\
\end{array}\par&\begin{array}[]{c}5\\\ 5\\\ \end{array}\par\\\
\hline\cr\begin{array}[]{c|c}p=3&\begin{array}[]{c}|J|=3\\\
\end{array}\par\end{array}\par&\begin{array}[]{c}7\\\
\end{array}\par&\begin{array}[]{c}5\\\ \end{array}\par&\begin{array}[]{c}7\\\
\end{array}\par&\begin{array}[]{c}7\\\ \end{array}\par&\begin{array}[]{c}7\\\
\end{array}\par\\\ \hline\cr\end{array}$

It is interesting to further analyze our results for $n=3$ and for $n=4$. The
following two tables show the maximum strength $\tau$ achievable for a
Euclidean design of type $({\cal X(J)},w)$ in 3 and in 4 dimensions on $p$
concentric spheres; we also list the sizes of the designs (we omit braces when
indicating the set $J$). All entries follow from previous results, except for
those in the last column for $n=4$; there we find Euclidean 7-designs for
$p=2$, if and only if, $r_{1}=r_{3}=r_{4}\not=r_{2}$ and for $p=3$, if and
only if, $r_{3}<r_{1}=r_{2}<r_{4}$ or $r_{4}<r_{1}=r_{2}<r_{3}$.

$\begin{array}[]{||c||c|c|c|c|c|c|c||}\hline\cr\hline\cr
J&1&2&3&1,2&1,3&2,3&1,2,3\\\ \hline\cr|{\cal X(J)}|&6&12&8&18&14&20&26\\\
\hline\cr\hline\cr p=1&3^{S}&3^{S}&3^{S}&5&5&3&7\\\ \hline\cr
p=2&\--&\--&\--&5&5^{T}&3&5\\\ \hline\cr p=3&\--&\--&\--&\--&\--&\--&7^{T}\\\
\hline\cr\hline\cr\end{array}$

$\begin{array}[]{||c||c|c|c|c|c|c|c|c|c|c|c|c|c|c|c||}\hline\cr\hline\cr
J&1&2&3&4&1,2&1,3&1,4&2,3&2,4&3,4&1,2,3&1,2,4&1,3,4&2,3,4&1,2,3,4\\\
\hline\cr|{\cal X(J)}|&8&24&32&16&32&40&24&56&40&48&64&48&56&72&80\\\
\hline\cr\hline\cr p=1&3^{S}&5^{S}&3^{S}&3^{S}&3&5&5^{S}&3&3&3&7&7&5&5&5\\\
\hline\cr p=2&\--&\--&\--&\--&3&5&5&3&3&3&7&7^{T}&5&5&7\\\ \hline\cr
p=3&\--&\--&\--&\--&\--&\--&\--&\--&\--&\--&5&5&5&5&7\\\
\hline\cr\hline\cr\end{array}$

Entries marked by an “S” indicate _spherical designs_ , that is, those with
$p=1$ and a constant weight function. This is obvious when $|J|=1$; for
example, for $n=4$, ${\cal I}^{4}_{2}$ is the spherical 5-design formed by
vectors with minimal non-zero norm in the lattice $D_{4}$. Additionally, for
$n=4$ we have ${\cal I}^{4}_{1}\cup\frac{1}{2}{\cal I}^{4}_{4}$ as a spherical
5-design: it is the set of vectors with minimal non-zero norm in the dual
lattice $D_{4}^{*}$. That this is the case follows from the last sentence of
the statement of Proposition 16 since for $n=4$ we have $n^{2}/2^{n}=1$.

Entries marked by a “T” indicate _antipodal tight Euclidean designs_ , that
is, those of size $N(n,p,t)$. Indeed, we have

$N(3,2,5)=14=|{\cal I}^{3}_{1}|+|{\cal I}^{3}_{3}|,$ $N(3,3,7)=26=|{\cal
I}^{3}_{1}|+|{\cal I}^{3}_{2}|+|{\cal I}^{3}_{3}|,$

and

$N(4,2,7)=48=|{\cal I}^{4}_{1}|+|{\cal I}^{4}_{2}|+|{\cal I}^{4}_{4}|.$

Our last task is to consider these tight cases.

_Proof of Corollary 9._ Let first $n=3$ and $t=5$. We then have

${\cal X}=r_{1}{\cal I}^{3}_{1}\cup\frac{r_{3}}{\sqrt{3}}{\cal I}^{3}_{3}.$

Equation (7) then becomes

$4r_{1}^{4}w_{1}-\frac{32}{9}r_{3}^{4}w_{3}=0,$

and this holds exactly when

$w_{3}=\frac{9}{8}\cdot\frac{r_{1}^{4}}{r_{3}^{4}}\cdot w_{1},$

proving Corollary 9, 1.

Next, let $n=3$ and $t=7$. Now we have

${\cal X}={\cal I}^{3}_{1}\cup\frac{r_{2}}{\sqrt{2}}{\cal
I}^{3}_{2}\cup\frac{r_{3}}{\sqrt{3}}{\cal I}^{3}_{3}.$

This time we use equations (10), which become

$\left\\{\begin{array}[]{l}4r_{1}^{4}w_{1}-2r_{2}^{4}w_{2}-\frac{32}{9}r_{3}^{4}w_{3}=0\\\
\\\ 12r_{1}^{6}w_{1}-39r_{2}^{6}w_{2}+\frac{256}{9}r_{3}^{6}w_{3}=0\\\ \\\
4r_{1}^{6}w_{1}-2r_{2}^{6}w_{2}-\frac{32}{9}r_{3}^{6}w_{3}=0\end{array}\right.$

All three equations hold, if and only if,

$r_{2}=\sqrt{\frac{2r_{3}^{2}}{5r_{3}^{2}-3r_{3}^{2}}}\cdot r_{1},\mbox{
}w_{2}=\frac{1}{10}\cdot\frac{(5r_{3}^{2}-3r_{1}^{2})^{3}}{r_{3}^{6}}w_{1},\mbox{
and }w_{3}=\frac{27}{40}\cdot\frac{r_{1}^{6}}{r_{3}^{6}}\cdot w_{1}.$

Letting $r=r_{1}$, $\rho=\sqrt{\frac{5r_{3}^{2}-3r_{1}^{2}}{2}}$, and
$w=w_{1}$ produces Corollary 9, 2.

Finally, let $n=4$ and $t=7$. In this case we have

${\cal X}={\cal I}^{4}_{1}\cup\frac{r_{2}}{\sqrt{2}}{\cal
I}^{4}_{2}\cup\frac{r_{4}}{2}{\cal I}^{4}_{4}.$

The system of equations (10) becomes

$\left\\{\begin{array}[]{l}4r_{1}^{4}w_{1}-4r_{4}^{4}w_{4}=0\\\ \\\
12r_{1}^{6}w_{1}-36r_{2}^{6}w_{2}+24r_{4}^{6}w_{4}=0\\\ \\\
4r_{1}^{6}w_{1}-4r_{4}^{6}w_{4}=0\end{array}\right.$

All three equations hold, if and only if, $r_{1}=r_{4}$, $w_{1}=w_{4}$, and
$w_{2}=\frac{r_{1}^{6}}{r_{2}^{6}}\cdot w_{1}$, as stated in Corollary 9, 3.
This completes the proof of Corollary 9.

_Remarks._ Bannai [5] proved that, up to similarity, the design in Corollary
9, 1 is one of exactly three different antipodal tight 5-designs for $p=2$ and
$n\geq 3$ (the other two are not fully symmetric). In Corollary 9, 2, two
particularly nice choices for the parameters are worth mentioning: with $r=1$
and $\rho=\sqrt{\frac{8}{3}}$, we have

${\cal X}={\cal I}^{3}_{1}\cup{\cal I}^{3}_{2}\cup\frac{1}{2}{\cal I}^{3}_{3}$

with weights $w_{2}=\frac{1}{10}w$ and $w_{3}=\frac{8}{5}w$; and with $r=1$
and $\rho=\sqrt{6}$, we have

${\cal X}={\cal I}^{3}_{1}\cup\frac{1}{2}{\cal I}^{3}_{2}\cup{\cal I}^{3}_{3}$

with weights $w_{2}=\frac{32}{5}w$ and $w_{3}=\frac{1}{40}w$. Finally, in
Corollary 9, 3, we see that with $r=1$ and $\rho=\sqrt{2}$ we have

${\cal X}=({\cal I}^{4}_{1}\cup\frac{1}{2}{\cal I}^{4}_{4})\cup{\cal
I}^{4}_{2},$

where ${\cal I}^{4}_{2}$ is the set of vectors with minimal non-zero norm in
the “checkerboard” lattice $D_{4}$, and ${\cal I}^{4}_{1}\cup\frac{1}{2}{\cal
I}^{4}_{4}$ is the set of vectors with minimal non-zero norm in its dual
lattice $D_{4}^{*}$. De la Harpe, Pache, and Venkov [8] used the root lattice
$D_{n}$ and its dual $D_{n}^{*}$ to construct (non-tight) Euclidean
$t$-designs for $p=1$ with several higher values of $n$ and $t$.

Acknowledgments. Work on this paper began during a workshop at the University
of Geneva, Switzerland, organized by Pierre de la Harpe during May 2004; and
it was continued at a second workshop at Kyushu University, in Fukuoka, Japan,
organized by Eiichi Bannai and Etsuko Bannai during January, 2005. I am very
grateful for their hospitality and the many helpful discussions.

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