Odd strength spherical designs attaining the Fazekas–Levenshtein bound for covering and universal minima of potentials

Abstract

We characterize the cases of existence of spherical designs of an odd strength attaining the Fazekas–Levenshtein bound for covering and prove some of their properties. We also find all universal minima of the potential of regular spherical configurations in two new cases: the demihypercube on $S^d$, $d\ge 4$, and the $2_{41}$ polytope on $S^7$ (which is dual to the $E_8$ lattice).

1 Statement of the problem and review of known results

Let $S^d:=\{(x_1,\ldots ,x_{d+1})\in \mathbb {R}^{d+1}: x_1^2+\ldots +x_{d+1}^2=1\}$ be the unit sphere in $\mathbb {R}^{d+1}$. Throughout the text, $\omega _N$ will denote a configuration of N pairwise distinct points on $S^d$ and ${\textbf{x}}_1,\ldots ,{\textbf{x}}_N$ will denote the points in $\omega _N$. We will also call $\omega _N$ a spherical code. We call a function $g:[-1,1]\rightarrow (-\infty ,\infty ]$ an admissible potential function if g is continuous on $[-1,1)$ with $g(1)=\lim \limits _{t\rightarrow 1^-}g(t)$ and differentiable in $(-1,1)$. Additional assumption(s) on the derivative(s) of g will be further specified in each case. Define the g-potential of $\omega _N$ by

$$ p^g({\textbf{x}},\omega _N):=\sum \limits _{i=1}^{N}g({\textbf{x}}\cdot {\textbf{x}}_i),\ \ \ {\textbf{x}}\in S^d. $$

We consider the following extremal problem over the sphere.

Problem 1.1

Find the quantity

$$\begin{aligned} P^g(\omega _N,S^d):=\min \limits _{{\textbf{x}}\in S^d}p^g({\textbf{x}},\omega _N) \end{aligned}$$

(1.1)

and points ${\textbf{x}}^*\in S^d$ attaining the minimum in (1.1).

We have an important special case when the potential function g is absolutely monotone on $[-1,1)$. Then $g(t)=f(2-2t)$ for some completely monotone function f on (0, 4]. Recall that a function g is called absolutely monotone or completely monotone on an interval I if $g^{(k)}\ge 0$ or $(-1)^kg^{(k)}\ge 0$ on I, respectively, for every $k\ge 0$. The function g is strictly absolutely or strictly completely monotone on I if the corresponding inequality is strict in the interior of I for all $k\ge 0$. The kernel

$$\begin{aligned} g({\textbf{x}}\cdot {\textbf{y}})=f\left( \left| {\textbf{x}}-{\textbf{y}}\right| ^2\right) \end{aligned}$$

(1.2)

with a strictly absolutely monotone g on $[-1,1)$ includes the Riesz s-kernel for $s>0$ and the Gaussian kernel. After adding an appropriate positive constant, it also includes the logarithmic kernel and the Riesz s-kernel for $-2<s<0$.

The problem about absolute extrema on the sphere of potentials of spherical codes was earlier solved by Stolarsky [23, 24] and Nikolov and Rafailov [20, 21] for Riesz s-kernels, $s\ne 0$, and sets of vertices of a regular N-gon on $S^1$ and of a regular simplex, regular cross-polytope, and cube inscribed in $S^d$. Hardin et al. [16] proved that absolute minima of the potential of a regular N-gon on $S^1$ with respect to a decreasing and convex function of the geodesic distance are attained at points of the dual regular N-gon.

Recently, results from [20, 21, 23, 24] for $s>-2$, $s\ne 0$ (except for absolute maxima for the cube), were extended to kernels (1.2) with an absolutely monotone function g and spherical designs of the highest (in a certain sense) strength. In particular, for a regular simplex on $S^d$, absolute maxima are at its vertices and absolute minima are at the antipods of its vertices, see [7]. Absolute maxima with respect to kernel (1.2) were found in [5] for sharp spherical codes that are antipodal or are designs of an even strength (called by some authors “strongly sharp”). Absolute maxima appear to be independent of the potential function g when g is strictly absolutely monotone on $[-1,1)$ (they are at points of the code itself). Such absolute maxima are called universal maxima.

The set of universal minima (defined in a similar way) of any spherical code that, for some $m\in \mathbb {N}$, is a $(2m-1)$-design forming m distinct dot products with some point ${\textbf{z}}\in S^d$, is also known (we will call such codes m-stiff). It is exactly the set of all such points ${\textbf{z}}$, see talk [6]¹ by the author for the proof or Lemma 3.5 in [5],² from which this result follows. We will call the set of such points ${\textbf{z}}$ the dual configuration. We restate this result here as Theorem 2.5. Immediate consequences of this result are universal minima of a regular 2m-gon on $S^1$, of a regular cross-polytope and cube on $S^d$, and of the 24-cell on $S^3$, since finding the dual configuration is elementary for these codes. Stiff spherical codes are exactly the ones attaining the Fazekas–Levenshtein bound for covering [13, Theorem 2] in the case of an odd strength. One of the goals of this paper is a further study of such spherical codes.

Universal minima of any strongly sharp code are at the antipods of points of the code. This also follows from [5, Lemma 3.5]. Immediate consequences of this fact (other than the regular simplex) are universal minima of a regular $(2m+1)$-gon on $S^1$, the Schläffi configuration on $S^5$, and the McLaughlin configuration on $S^{21}$. Strongly sharp spherical codes attain the Fazekas–Levenshtein bound for covering [13, Theorem 2] for even strength designs (this case has been well-studied before).

Boyvalenkov et al. [9, Theorems 3.4 and 3.7] (see also [10, Theorem 1.4])³ proved universal upper and lower bounds for the potential of a general spherical design. These bounds become sharp in the cases mentioned above: in the case of a minimum for stiff and strongly sharp configurations and in the case of a maximum for sharp antipodal and strongly sharp ones. The lower bound is an analogue of the Fazekas–Levenshtein bound for covering [13, Theorem 2].

Paper [9] also showed that the universal maxima of the 600-cell on $S^3$ are vertices of the 600-cell itself. The work [10] proved that a number of known sharp codes are also stiff. Then the result from [6] implies that their universal minima are at points of the dual configuration. Paper [10] further studies the dual configuration for each code and antipods of the two strongly sharp codes on $S^5$ and $S^{21}$ mentioned above. The author in paper [2]⁴ found explicitly the sets of all universal minima for five more stiff configurations (which are not sharp) on spheres of different dimensions as well as for the 56-point kissing configuration on $S^6$, which is a known sharp code (paper [10] gives one universal minimum of this code).

Certain remarkable spherical configurations are not stiff or strongly sharp. Universal minima of the regular icosahedron and regular dodecahedron on $S^2$ were characterized in [3]⁵ as well as universal minima of the $E_8$ lattice on $S^7$. Furthermore, one universal minimum and the corresponding absolute minimum value of the potential were found in [10]⁶ for the Leech lattice on $S^{23}$. Papers [3]⁷ and [10] establish general theorems (different to a certain extent) for the so-called “skip one add two” case and use them to establish the above mentioned results.

Critical points of the total potential of finite configurations of charges were also analysed (see [1, 14] and references therein). This work is related to the known Maxwell’s conjecture. A more detailed review of known results on extrema of potentials of spherical codes can be found, for example, in [2].

In this paper, we characterize the cases of existence of odd strength spherical designs attaining the Fazekas–Levenshtein bound for covering [13, Theorem 2] and prove certain properties of such designs and of sets of their universal minima (see Sect. 6). We also characterize universal minima of regular spherical configurations in two new cases: the demihypercube on $S^d$, $d\ge 4$ (including the 16-point sharp code⁸ on $S^4$) and the $2_{41}$ polytope on $S^7$ (it is dual to the $E_8$ lattice). For stiff spherical codes that have no antipodal pairs, we show that their universal minima are the same as the ones of their symmetrizations about the origin.

One important application of Problem 1.1 is the polarization problem on the sphere. Papers [5, 7, 9, 16, 20, 23] that we mentioned when reviewing results on extrema of potentials solve some of its cases. A more comprehensive review of known work on polarization can be found, for example, in book [8, Chapter 14] with most recent results reviewed in, e.g., [5].

The paper is structured as follows. Section 2 contains the necessary preliminaries. In Sect. 3, we characterize universal minima of the d-demicube for $d\ge 5$. In Sect. 4, we find all universal minima of the $2_{41}$ polytope on $S^7$. Section 5 extends known results on universal minima for some sharp configurations that are non-tight designs to their symmetrizations. In Sect. 6, we characterize the cases of existence of odd strength spherical designs attaining the Fazekas–Levenshtein bound for covering (Theorem 6.8) and establish certain properties of such designs and of sets of their universal minima.

2 Preliminaries

In this section, we state definitions and known facts used further in the paper. Define

$$ w_{d}(t):=\gamma _{d}(1-t^2)^{d/2-1}, $$

where the constant $\gamma _{d}$ is such that $w_{d}$ is a probability density on $[-1,1]$. The Gegenbauer orthogonal polynomials corresponding to the sphere $S^{d}$ in $\mathbb {R}^{d+1}$ are terms of the sequence $\{P_n^{(d)}\}_{n=0}^{\infty }$ of univariate polynomials such that ${\textrm{deg}}\ \! P_n^{(d)}=n$, $n\ge 0$, and

$$\begin{aligned} \int _{-1}^{1}P_i^{(d)}(t)P_j^{(d)}(t)w_{d}(t)\ \! dt =0,\ \ \ i\ne j, \end{aligned}$$

normalized so that $P_n^{(d)}(1)=1$, $n\ge 0$ (see [25, Chapter 4] or, e.g., [8, Chapter 5]).

For a configuration $\omega _N=\{{\textbf{x}}_1,\ldots ,{\textbf{x}}_N\}\subset S^d$, let ${\mathcal {I}}(\omega _N)$ be the set of all $n\in \mathbb {N}$ such that

$$\begin{aligned} \sum \limits _{i=1}^{N}\sum \limits _{j=1}^{N}P_n^{(d)}({\textbf{x}}_i\cdot {\textbf{x}}_j)=0. \end{aligned}$$

(2.1)

We call ${\mathcal {I}}(\omega _N)$ the index set of $\omega _N$. Let $\sigma _{d}$ be the d-dimensional area measure on the sphere $S^d$ normalized to be a probability measure. A configuration $\omega _N$ is called a spherical n-design if, for every polynomial p on $\mathbb {R}^{d+1}$ of degree at most n,

$$\begin{aligned} \frac{1}{N}\sum \limits _{i=1}^{N}p({\textbf{x}}_i)=\int _{S^{d}}p({\textbf{x}})\ \! d\sigma _{d}({\textbf{x}}), \end{aligned}$$

(2.2)

see the paper by Delsarte et al. [12]. The maximal number n in this definition is called the strength of the spherical design $\omega _N$.

We recall the following equivalent definitions of a spherical design. Let $\mathbb P_n$ denote the space of all univariate polynomials of degree at most n.

Theorem 2.1

(see [12, 13] or, e.g., [8, Lemma 5.2.2 and Theorem 5.4.2]) Let $d,n\ge 1$ and $\omega _N=\{{\textbf{x}}_1,\ldots ,{\textbf{x}}_N\}$ be a point configuration on $S^d$. The following are equivalent:

• (i) $\omega _N$ is a spherical n-design;

• (ii) $\{1,\ldots ,n\}\subset {\mathcal {I}}(\omega _N);$

• (iii) for every polynomial $q\in \mathbb P_n$, we have $p^q({\textbf{y}},\omega _N)=\sum _{i=1}^{N}q({\textbf{y}}\cdot {\textbf{x}}_i)=C$, ${\textbf{y}}\in S^d$, where C is a constant.

If item (iii) holds in the above theorem, then $C=a_0(q)N$, where

$$\begin{aligned} a_0(q):=\int _{-1}^{1}q(t)w_d(t)\ \! dt \end{aligned}$$

(2.3)

is the 0-th Gegenbauer coefficient of polynomial q. For a given $m\in \mathbb {N}$ and a given configuration $\omega _N\subset S^{d}$, denote by ${\mathcal {D}}_m(\omega _N)$ the set of all points ${\textbf{z}}\in S^{d}$ for which the set of dot products

$$ D({\textbf{z}},\omega _N):=\{{\textbf{z}}\cdot {\textbf{x}}_i: i=1,\ldots ,N\} $$

has at most m distinct elements.

Definition 2.2

We call a point configuration $\omega _N\subset S^{d}$ m-stiff, $d,m\ge 1$, if $\omega _N$ is a spherical $(2m-1)$-design and the set ${\mathcal {D}}_m(\omega _N)$ is non-empty. The set ${\mathcal {D}}_m(\omega _N)$ of a given m-stiff configuration $\omega _N$ is called the dual configuration for $\omega _N$.

For a given $m\in \mathbb {N}$, a spherical $(2m-1)$-design on $S^d$ attains the Fazekas–Levenshtein bound [13, Theorem 2] if and only if it is m-stiff. Following [11], we call a configuration $\omega _N\subset S^d$ sharp if, for some $m\in \mathbb N$, it is a $(2m-1)$-design and there are exactly m distinct values of the dot product between distinct points in $\omega _N$. If, in addition, $\omega _N$ is a 2m-design, we call it strongly sharp.

The next statement is part of the classification of quadratures in [12] corresponding to spherical designs of the highest strength; i.e., stiff, strongly sharp, or sharp antipodal codes. It is an immediate consequence of [13, Theorem 2]. For the reader’s convenience, we mention that a proof of Proposition 2.3 can also be found, for example, in [2, Proposition 7.2] or [9, Corollary 3.9]. Let $\{\varphi _1,\ldots ,\varphi _m\}$ be the fundamental polynomials for the set $-1<\kappa _1^{m}<\ldots<\kappa _{m}^{m}<1$ of zeros of the Gegenbauer polynomial $P_m^{(d)}$; that is, $\varphi _i\in \mathbb P_{m-1}$, $\varphi _i(\kappa _i^m)=1$, and $\varphi _i(\kappa _j^m)=0$, $j\ne i$, $i=1,\ldots ,m$.

Proposition 2.3

If $\omega _N$ is an m-stiff configuration on $S^d$, then for every ${\textbf{z}}\in {\mathcal {D}}_m(\omega _N)$, the set $D({\textbf{z}},\omega _N)$ contains exactly m distinct elements located in $(-1,1)$, which are $\kappa _1^{m},\ldots ,\kappa _{m}^{m}$. Furthermore, the number of indices i such that ${\textbf{z}}\cdot {\textbf{x}}_i=\kappa _j^{m}$ does not depend on ${\textbf{z}}$ and equals $a_0(\varphi _j) N$, $j=1,\ldots ,m$.

In particular, if $m=2$, then for every ${\textbf{z}}\in \mathcal D_2(\omega _N)$, we have $D({\textbf{z}},\omega _N)=\left\{ -\frac{1}{\sqrt{d+1}},\frac{1}{\sqrt{d+1}}\right\}$.

Remark 2.4

In view of Proposition 2.3, an m-stiff configuration may exist on $S^d$ for given $m,d\ge 1$, only if all the numbers $a_0(\varphi _i)$, $i=1,\ldots ,m$, are positive rationals. These numbers are the weights of the Gauss–Gegenbauer quadrature for integral (2.3) (the nodes are $\kappa _1^m,\ldots ,\kappa _m^m$). This is, in fact, an equivalent condition for the existence, see Theorem 6.8.

We next restate the result proved in talk [6]. Its complete proof can also be found in [2, Theorem 4.3].

Theorem 2.5

Let $m\ge 1$, $d\ge 1$, and g be an admissible potential function with a convex derivative $g^{(2m-2)}$ on $(-1,1)$. If $\omega _N=\{{\textbf{x}}_1,\ldots ,{\textbf{x}}_N\}$ is an m-stiff configuration on the sphere $S^{d}$, then the potential

$$ p^g({\textbf{x}},\omega _N)=\sum \limits _{i=1}^{N}g({\textbf{x}}\cdot {\textbf{x}}_i),\ \ \ {\textbf{x}}\in S^{d}, $$

attains its absolute minimum over $S^{d}$ at every point of the set ${\mathcal {D}}_m(\omega _N)$.

If, in addition, $g^{(2m-2)}$ is strictly convex on $(-1,1)$, then ${\mathcal {D}}_m(\omega _N)$ contains all points of absolute minimum of the potential $p^g(\cdot ,\omega _N)$ on $S^{d}$.

In fact, Theorem 2.5 is a special case of [5, Lemma 3.5] if one uses $-g$ as the potential function in Lemma 3.5 in [5] and lets q there be the Hermite interpolating polynomial for $-g$ at the m values of the dot product that a point ${\textbf{z}}$ from ${\mathcal {D}}_m(\omega _N)$ forms with points of $\omega _N$. Then q has degree at most $2m-1$, which is the strength of the spherical design $\omega _N$. Since $-g^{(2m-2)}$ is concave on $(-1,1)$, a standard argument counting sign changes for consecutive derivatives of the difference $-g(t)-q(t)$ shows that $q(t)\ge -g(t)$, $t\in [-1,1]$, with the inequality being strict at any non-interpolation point when $g^{(2m-2)}$ is strictly convex on $(-1,1)$. Now, the assumptions of Lemma 3.5 in [5] are satisfied and we obtain the assertion of Theorem 2.5.

We will also need the “skip one add two” result from [3, Theorem 3.1].

Theorem 2.6

Let $d,m\in \mathbb {N}$, $m\ge 2$, and $\omega _N=\{{\textbf{x}}_1,\ldots ,{\textbf{x}}_N\}$ be a point configuration on $S^d$ whose index set $\mathcal I(\omega _N)$ contains numbers $1,2,\ldots ,2\,m-3,2\,m-1,2\,m$. Assume that numbers $-1< t_1<t_2<\ldots<t_m< 1$ are such that

$$\begin{aligned} \sum _{i=1}^{m}t_i<t_m/2 \ \ \text { and }\ \ \sum \limits _{i=1}^{m}t_i^2-2\left( \sum \limits _{i=1}^{m}t_i\right) ^2<\frac{m(2m-1)}{4m+d-3}, \end{aligned}$$

(2.4)

and that the set ${\mathcal {D}}$ of points ${\textbf{x}}^*\in S^d$ with $D({\textbf{x}}^*,\omega _N)\subset \{t_1,\ldots ,t_m\}$ is non-empty. Let g be an admissible potential function with non-negative derivatives $g^{(2\,m-2)}$, $g^{(2\,m-1)}$, and $g^{(2\,m)}$ on $(-1,1)$. Then, for every point ${\textbf{x}}^*\in {\mathcal {D}}$,

$$\begin{aligned} \min \limits _{{\textbf{x}}\in S^d}\sum \limits _{i=1}^{N}g({\textbf{x}}\cdot {\textbf{x}}_i)=\sum \limits _{i=1}^{N}g({\textbf{x}}^*\cdot {\textbf{x}}_i). \end{aligned}$$

(2.5)

If, in addition, $g^{(2m)}>0$ on $(-1,1)$, then the absolute minimum in (2.5) is achieved only at points of the set ${\mathcal {D}}$.

We remark that proofs of Theorems 2.5 and 2.6 utilize the Delsarte-Yudin method (also known as the Delsarte or linear programming or polynomial method), see the work by Delsarte, Goethals, and Seidel [12] or by Yudin [26]. A detailed description of this approach and references to works using it can also be found, in particular, in [9,10,11, 17,18,19] and in [8, Chapter 5].

We next find universal minima of the potential of new cases of regular spherical configurations that we were able to handle using available tools. The proofs of these results are given in Sects. 3–5.

3 The 16-point sharp code on $S^4$ and the demihypercube

Denote by $\omega _{2d}^*:=\{\pm \textbf{e}_1,\ldots ,\pm \textbf{e}_d\}$, $d\ge 2$, where $\textbf{e}_1,\ldots ,\textbf{e}_d$ are vectors of the standard basis in $\mathbb {R}^d$, the set of vertices of the regular cross-polytope inscribed in $S^{d-1}$ and let $U_d$ be the set $\left\{ \left( \pm \frac{1}{\sqrt{d}},\ldots ,\pm \frac{1}{\sqrt{d}}\right) \right\} \subset \mathbb {R}^d$ of vertices of the cube inscribed in $S^{d-1}$. It is not difficult to see that $\mathcal D_2(\omega _{2d}^*)=U_d$ and ${\mathcal {D}}_2(U_d)=\omega _{2d}^*$.

Let ${{\overline{\omega }}}^d$, $d\ge 2$, be the set of $N=2^{d-1}$ points $\left( \pm \frac{1}{\sqrt{d}},\ldots ,\pm \frac{1}{\sqrt{d}}\right) \in U_d$ with an even number of minus signs. This configuration forms the set of vertices of a d-demicube (also called the demihypercube). The set ${{\widetilde{\omega }}}^d$ of vectors from $U_d$ with an odd number of minus signs is a reflection of ${{\overline{\omega }}}^d$ with respect to any of the coordinate hyperplanes; i.e., ${{\widetilde{\omega }}}^d$ is an isometric copy of ${{\overline{\omega }}}^d$. Therefore, it is sufficient to consider just ${{\overline{\omega }}}^d$. We have ${{\overline{\omega }}}^d\cup {{\widetilde{\omega }}}^d=U_d$ and the two sets are disjoint. For d odd, we have ${{\widetilde{\omega }}}^d=-{{\overline{\omega }}}^d$ with neither set containing antipodal pairs. For d even, each of the sets ${{\overline{\omega }}}^d$ and ${{\widetilde{\omega }}}^d$ is itself antipodal.

Observe that for $d=2$, both configurations consist of one antipodal pair; i.e., they are 1-stiff. For $d=3$, each one is a regular simplex inscribed in $S^2$ (strongly sharp and, hence, not stiff). For $d=4$, the set ${{\overline{\omega }}}^d$ consists of eight points $\left( \pm \frac{1}{2},\pm \frac{1}{2},\pm \frac{1}{2},\pm \frac{1}{2} \right)$ with an even number of minus signs. Each set ${{\overline{\omega }}}^4$ and ${{\widetilde{\omega }}}^4$ is an isometric copy of a regular cross-polytope in $\mathbb {R}^4$; i.e., it is 2-stiff. For $d=5$, configuration ${{\overline{\omega }}}^d$ consists of 16 points on $S^4$ of the form $\left( \pm \frac{1}{\sqrt{5}},\pm \frac{1}{\sqrt{5}},\pm \frac{1}{\sqrt{5}},\pm \frac{1}{\sqrt{5}},\pm \frac{1}{\sqrt{5}}\right)$ with an even number of minus signs. This is the well-known sharp (5, 16, 1/5)-code. It was described by Gossett [15]. The set ${{\widetilde{\omega }}}^5$ is the antipode of this code. The 2-stiffness property of ${{\overline{\omega }}}^5$ was observed in [10]. We show that the d-demicube is 2-stiff for any $d\ge 6$. We start with the following auxiliary statement.

Lemma 3.1

Let $\omega _N$ be a non-empty subset of $U_d=\left\{ -\frac{1}{\sqrt{d}},\frac{1}{\sqrt{d}}\right\} ^d$, $d\ge 3$. Then $\omega _N$ is a 3-design if, and only if, N is even and for every set I of one, two, or three pairwise distinct indices, exactly half of the vectors in $\omega _N$ have an even number of negative coordinates with indices in I and exactly half have an odd number of negative coordinates with indices in I.

Proof

Let $\omega _N\subset U_d$ be arbitrary. Using the notation ${\textbf{y}}=(y_1,\ldots ,y_d)$ for a point ${\textbf{y}}\in \omega _N$, define

$$ S_i:=\sum \limits _{{\textbf{y}}\in \omega _N}y_i,\ \ \ \ S_{i,j}:=\sum \limits _{{\textbf{y}}\in \omega _N}y_iy_j, \ \ \ \text {and}\ \ \ S_{i,j,k}:=\sum \limits _{{\textbf{y}}\in \omega _N}y_iy_jy_k. $$

Observe that $S_{i,j}$ and $S_{i,j,k}$ do not depend on permutations of indices and that $S_{i,i}=\frac{N}{d}$. When some two indices coincide, say $i=j$, we have

$$\begin{aligned} S_{i,j,k}=\sum \limits _{{\textbf{y}}\in \omega _N}\frac{y_k}{d}=\frac{1}{d}S_k. \end{aligned}$$

(3.1)

Formula (3.1) holds even if $i=j=k$. Let ${\textbf{x}}=(x_1,\ldots ,x_d)\in S^{d-1}$ be any vector. Then

$$\begin{aligned} \sum \limits _{{\textbf{y}}\in \omega _N}{\textbf{x}}\cdot {\textbf{y}}= & {} \sum \limits _{{\textbf{y}}\in \omega _N}\sum \limits _{i=1}^{d}x_iy_i=\sum \limits _{i=1}^{d}\sum \limits _{{\textbf{y}}\in \omega _N}x_iy_i=\sum \limits _{i=1}^{d}x_i\sum \limits _{{\textbf{y}}\in \omega _N}y_i=\sum \limits _{i=1}^{d}S_ix_i, \\ \sum \limits _{{\textbf{y}}\in \omega _N}\left( {\textbf{x}}\cdot {\textbf{y}}\right) ^2= & {} \sum \limits _{{\textbf{y}}\in \omega _N}\left( \sum \limits _{j=1}^{d}x_jy_j\right) ^2=\sum \limits _{{\textbf{y}}\in \omega _N}\sum \limits _{i=1}^{d}\sum \limits _{j=1} ^{d}x_ix_jy_iy_j\\= & {} \sum \limits _{i=1}^{d}\sum \limits _{j=1} ^{d}\sum \limits _{{\textbf{y}}\in \omega _N}x_ix_jy_iy_j=\sum \limits _{i=1}^{d}x_i^2\sum \limits _{{\textbf{y}}\in \omega _N}y_i^2+\sum \limits _{i,j=1\atop i\ne j}^{d}x_ix_j\sum \limits _{{\textbf{y}}\in \omega _N} y_iy_j\\= & {} \frac{N}{d}+\sum \limits _{i,j=1\atop i\ne j}^{d}S_{i,j}x_ix_j, \end{aligned}$$

and

$$\begin{aligned} \begin{aligned} \sum \limits _{{\textbf{y}}\in \omega _N}\left( {\textbf{x}}\cdot {\textbf{y}}\right) ^3&=\sum \limits _{{\textbf{y}}\in \omega _N}\left( \sum \limits _{j=1}^{d}x_jy_j\right) ^3=\sum \limits _{{\textbf{y}}\in \omega _N}\sum \limits _{i,j,k=1}^{d}x_ix_jx_ky_iy_jy_k \\&=\sum \limits _{i,j,k=1}^{d}x_ix_jx_k\sum \limits _{{\textbf{y}}\in \omega _N}y_iy_jy_k=\sum \limits _{i,j,k=1}^{d}S_{i,j,k}x_ix_jx_k. \end{aligned} \end{aligned}$$

The configuration $\omega _N$ will be a 3-design if, and only if, the three sums above are constant, see Theorem 2.1. If

$$\begin{aligned} S_i=0 \ \text {for all }i,\ \ S_{i,j}=0, \ \text {for }i\ne j, \ \ \text {and}\ \ S_{i,j,k}=0, \ \text {for}\ i,j,k \ \text {distinct} \end{aligned}$$

(3.2)

then three sums above will be constant (one should also use (3.1)). Conversely, if all three sums above are constant, then the first one has the same value for every vector $\pm \textbf{e}_i$, which is $\pm S_i$, $i=1,\ldots ,d$. Then $S_i=0$ for all i. For every vector ${\textbf{x}}\in S^{d-1}$ with the $\ell$-th coordinate being $1/\sqrt{2}$, the n-th coordinate being $\pm \frac{1}{\sqrt{2}}$, and the remaining coordinates being zero, $\ell \ne n$, the value of the second sum is $N/d\pm S_{\ell ,n}={\textrm{const}}$. This forces $S_{\ell ,n}=0$, $\ell \ne n$. For vector $\pm {\textbf{x}}$, where the $\ell$-th, n-th, and m-th coordinates of ${\textbf{x}}$ equal $1/\sqrt{3}$, $\ell ,n,m$ are pairwise distinct, and the remaining coordinates are zero, the third sum equals (use (3.1) and the fact that $S_i=0$ for all i)

$$ \pm \frac{1}{3\sqrt{3}}\sum _{i,j,k\in \{\ell ,n,m\}}S_{i,j,k}=\pm \frac{6}{3\sqrt{3}}S_{\ell ,n,m}={\textrm{const}}. $$

Then $S_{\ell ,n,m}=0$. Thus, $\omega _N$ is a 3-design if, and only if, relations (3.2) hold.

In each sum $S_i$, $S_{i,j}$, and $S_{i,j,k}$ in (3.2), all terms have the same absolute values. Then the value of each sum in (3.2) equals that common absolute value times the difference between the number of positive and negative terms. Therefore, relations (3.2) hold if, and only if, each sum in (3.2) has an equal number of positive and negative terms. This, in turn, will hold if, and only if, for any set of indices $I=\{i\}$ or $\{i,j\}$, where $i\ne j$, or $\{i,j,k\}$, where i, j, k are pairwise distinct, the number of vectors in $\omega _N$ with an even number of negative components with indices in I equals the number of vectors in $\omega _N$ with an odd number of negative components with indices in I. This also forces N to be even. $\square$

Lemma 3.2

The d-demicube ${{\overline{\omega }}}^d$, $d\ge 4$, is 2-stiff.

Proof

Since ${{\overline{\omega }}}^d$ is a subset of the set of vertices of a cube, it is contained in two parallel hyperplanes. Thus, it remains to show that ${{\overline{\omega }}}^d$ is a 3-design. Let I be any set of k pairwise distinct indices, where $k=1,2,3$. A combination of signs of coordinates corresponding to I with an even number of negative ones can be chosen in $2^{k-1}$ different ways. For each of these combinations, the remaining $d-k$ positions can have $2^{d-k}$ different combinations of signs with $2^{d-k-1}$ of them having an even number of minus signs. Then the total number of vectors in ${{\overline{\omega }}}^d$ with an even number of negative coordinates corresponding to I will be $2^{k-1}\cdot 2^{d-k-1}=2^{d-2}$. By a similar argument, the number of vectors in ${{\overline{\omega }}}^d$ with an odd number of negative coordinates corresponding to I will also be $2^{d-2}$. Lemma 3.1 now implies that ${{\overline{\omega }}}^d$ is a 3-design and, hence, is 2-stiff. $\square$

We next find the dual configuration of the d-demicube. For $d=2$, the d-demicube is a pair of antipodal vectors, which is 1-stiff. Its dual is the perpendicular pair of antipodal vectors. For $d=3$, the d-demicube is a regular simplex, which is not stiff, since it is strongly sharp. For $d=4$, the configuration ${{\overline{\omega }}}^d$ is a regular cross-polytope, and its dual is the corresponding cube inscribed in $S^3$. For $d\ge 5$, we have the following result.

Lemma 3.3

For every $d\ge 5$, we have $\mathcal D_2({{\overline{\omega }}}^d)=\omega ^*_{2d}$.

Since ${\mathcal {D}}_2(\omega _{2d}^*)=U_d$, $d\ge 2$, Lemma 3.3 shows that the inclusion $\omega _N\subset \mathcal D_m({\mathcal {D}}_m(\omega _N))$ can be strict for an m-stiff configuration $\omega _N$ with $m\ge 2$ even if $\omega _N$ is antipodal. Furthermore, for $d\ge 5$ odd, it provides another example of a non-antipodal m-stiff configuration with $m\ge 2$ (non-antipodal 1-stiff configurations are easy to construct).

Proof of Lemma 3.3

Every vector $\pm \textbf{e}_i\in \omega ^*_{2d}$ forms only dot products $\frac{1}{\sqrt{d}}$ and $-\frac{1}{\sqrt{d}}$ with points from ${{\overline{\omega }}}^d$; i.e., it belongs to $\mathcal D_2({{\overline{\omega }}}^d)$. Choose any ${\textbf{x}}=(x_1,\ldots ,x_d)\in {\mathcal {D}}_2({{\overline{\omega }}}^d)$. Assume to the contrary that ${\textbf{x}}$ has at least three non-zero coordinates. Let k be the number of strictly negative components in ${\textbf{x}}$. If k is even, we choose a vector ${\textbf{z}}=(z_1,\ldots ,z_d)\in {{\overline{\omega }}}^d$ with $-\frac{1}{\sqrt{d}}$ in all positions corresponding to strictly negative components in ${\textbf{x}}$ and $\frac{1}{\sqrt{d}}$ in all other positions. If k is odd, we choose ${\textbf{z}}\in {{\overline{\omega }}}^d$ with $-\frac{1}{\sqrt{d}}$ in all but one positions corresponding to strictly negative components of ${\textbf{x}}$ and $\frac{1}{\sqrt{d}}$ in all the other positions. Then the dot product ${\textbf{x}}\cdot {\textbf{z}}=x_1z_1+\ldots +x_dz_d$ has at most one strictly negative term and at least two other strictly positive terms, which we denote by $x_iz_i$ and $x_jz_j$. Since $d\ge 5$, we can choose two disjoint pairs of positions in ${\textbf{z}}$ one containing $z_i$ and the other one containing $z_j$ with both pairs avoiding the position corresponding to the possible negative term in ${\textbf{x}}\cdot {\textbf{z}}$. Changing the sign of the coordinates of ${\textbf{z}}$ in the first pair of positions, we keep ${\textbf{z}}$ in ${{\overline{\omega }}}^d$ and strictly decrease the dot product ${\textbf{x}}\cdot {\textbf{z}}$. Changing the sign of the coordinates of the new vector ${\textbf{z}}$ in the second pair of positions, we keep the resulting vector in ${{\overline{\omega }}}^d$ and further decrease the dot product ${\textbf{x}}\cdot {\textbf{z}}$. This shows that ${\textbf{x}}$ forms at least three distinct dor products with points of ${{\overline{\omega }}}^d$ contradicting its choice.

Therefore, ${\textbf{x}}$ has at most two non-zero components. Assume to the contrary that ${}\textbf{x}$ has exactly two non-zero components, say $x_\ell$ and $x_n$. Then ${\textbf{x}}$ forms dot products $\frac{\pm x_\ell \pm x_n}{\sqrt{d}}$ with vectors from ${{\overline{\omega }}}^d$ and at least three of them are distinct. Thus, ${\textbf{x}}$ has one non-zero component. Since ${\textbf{x}}$ is on $S^{d-1}$, this component must be $\pm 1$; that is, ${\textbf{x}}\in \omega ^*_{2d}$. Thus, ${\mathcal {D}}_2({{\overline{\omega }}}^d)=\omega ^*_{2d}$. $\square$

We are ready to characterize universal minima of the d-demicube for $d\ge 5$.

Theorem 3.4

Let $d\ge 5$ and g be an admissible potential function with a convex derivative $g''$ on $(-1,1)$. Then the potential $p^g(\cdot ,{{\overline{\omega }}}^d)$ of d-demicube ${{\overline{\omega }}}^d$ attains its absolute minimum over $S^{d-1}$ at every point of cross-polytope $\omega _{2d}^*$.

If, in addition, $g''$ is strictly convex on $(-1,1)$, then $\omega _{2d}^*$ contains all points of absolute minimum of the potential $p^g(\cdot ,{{\overline{\omega }}}^d)$ on $S^{d-1}$.

In the case $d=5$, the first paragraph of Theorem 3.4 follows from the results of [10].

Proof

Since ${{\overline{\omega }}}^d$ is 2-stiff, by Theorem 2.5, the potential $p^g(\cdot ,{{\overline{\omega }}}^d)$ attains its absolute minimum over $S^{d-1}$, $d\ge 5$, at points of the set $\mathcal D_2({{\overline{\omega }}}^d)$, which, by Lemma 3.3, equals $\omega ^*_{2d}$. If $g''$ is strictly convex on $(-1,1)$, then, by Theorem 2.5, the set $\mathcal D_2({{\overline{\omega }}}^d)=\omega ^*_{2d}$ contains all absolute minima of $p^g(\cdot ,{{\overline{\omega }}}^d)$ over $S^{d-1}$. $\square$

4 The $2_{41}$ polytope on $S^7$

Recall that the $E_8$ lattice is the set (lattice in $\mathbb {R}^8$) of vectors in $\mathbb Z^8 \cup (\mathbb Z+{1}/{2})^8$ whose coordinates sum to an even integer. Let ${{\overline{\omega }}}_{240}$ be the set of minimal length non-zero vectors of the $E_8$ lattice normalized to lie on $S^7$. The configuration $\overline{\omega }_{240}$ consists of $4\left( 8 \atop 2\right) =112$ vectors with 6 zero coordinates and two coordinates being $\pm 1/\sqrt{2}$ and $2^7=128$ vectors with all eight coordinates $\pm \frac{1}{2\sqrt{2}}$ and even number of “−” signs (this part is the 8-demicube). For brevity, we will also call $\overline{\omega }_{240}$ the $E_8$ lattice.

The $2_{41}$ polytope on $S^7$ (the name is due to Coxeter), denoted here by ${{\overline{\omega }}}_{2160}$, is the set of $N=2160$ vectors on $S^7$ that includes $16\left( 8 \atop 4\right) =1120$ vectors with 4 zero coordinates and 4 coordinates being $\pm 1/2$ (let us call them type I vectors), 16 vectors with 7 zero coordinates and one coordinate being $\pm 1$ (let us call them type II vectors), and $8\left( \left( 8 \atop 1\right) +\left( 8\atop 3\right) +\left( 8 \atop 5\right) +\left( 8\atop 7\right) \right) =1024$ vectors with 7 coordinates being $\pm 1/4$, one coordinate being $\pm 3/4$, and an odd number of negative coordinates (call them type III vectors). One can verify directly that equality (2.1) holds for $d=7$ and $n\in \{1,\ldots ,7,9,10\}$. Indeed, since ${{\overline{\omega }}}_{2160}$ is antipodal, (2.1) holds trivially for every n odd. For $n=2,4,6,10$, we have

$$\begin{aligned} \begin{aligned} \sum \limits _{{\textbf{x}}\in {{\overline{\omega }}}_{2160}}&\sum \limits _{{\textbf{y}}\in {{\overline{\omega }}}_{2160}}P_n^{(7)}({\textbf{x}}\cdot {\textbf{y}})=4320(P_n^{(7)}(1)+64P_n^{(7)}\left( 3/4\right) \\&+280P_n^{(7)}\left( 1/2\right) +448P_n^{(7)}(1/4)+287P_n^{(7)}\left( 0\right) )=0. \end{aligned} \end{aligned}$$

(4.1)

The code ${{\overline{\omega }}}_{2160}$ is a 7-design (and not an 8-design). However, it is not stiff, since $\mathcal D_4({{\overline{\omega }}}_{2160})=\emptyset$ as the following lemma suggests.

Lemma 4.1

For every vector ${\textbf{x}}\in S^7$, the set $D({\textbf{x}},{{\overline{\omega }}}_{2160})$ has at least five distinct elements. The only vectors ${\textbf{x}}\in S^7$ such that $D({\textbf{x}},{{\overline{\omega }}}_{2160})$ has exactly five distinct elements are those in ${{\overline{\omega }}}_{240}$. For each ${\textbf{x}}\in {{\overline{\omega }}}_{240}$, we have $D({\textbf{x}},{{\overline{\omega }}}_{2160})=\left\{ 0,\pm \frac{1}{2\sqrt{2}},\pm \frac{1}{\sqrt{2}}\right\}$.

Proof

Let ${\textbf{x}}=(x_1,\ldots ,x_8)\in \mathcal D_5({{\overline{\omega }}}_{2160})$ be arbitrary. If non-zero coordinates of ${\textbf{x}}$ had at least three distinct absolute values, then ${\textbf{x}}$ would form at least 6 distinct dot products with vectors of type II. Therefore, non-zero coordinates of ${\textbf{x}}$ have at most two distinct absolute values.

Assume to the contrary that non-zero coordinates of ${\textbf{x}}$ have exactly two distinct absolute values. Denote them by $0<b<c$. Then ${\textbf{x}}$ forms each of the dot products $\pm b,\pm c$ with vectors of type II. If ${\textbf{x}}$ formed with some vector ${\textbf{z}}\in {{\overline{\omega }}}_{2160}$ a positive dot product u distinct from b and c, since ${{\overline{\omega }}}_{2160}$ is antipodal, there would be a sixth dot product $-u$, contradicting the assumption that ${\textbf{x}}\in {\mathcal {D}}_5({{\overline{\omega }}}_{2160})$. Thus, $D({\textbf{x}},{{\overline{\omega }}}_{2160})$ contains only two positive dot products: b and c. Let k coordinates of ${\textbf{x}}$ have absolute value b and $\ell$ coordinates have absolute value c. If it were that $\ell \ge 2$, then ${\textbf{x}}$ would form positive dot products $\frac{2c+b+v}{2}$ and $\frac{b+v}{2}$ with two appropriately chosen vectors of type I, where $v\ge 0$ is the absolute value of one of the coordinates of ${\textbf{x}}$. Then $\frac{2c+b+v}{2}=c$ forcing $b\le 0$. Thus, $\ell =1$. If it were that $k\ge 3$, then ${\textbf{x}}$ would form positive dot products $\frac{c+3b}{2}$ and $\frac{c+b}{2}$ with two vectors of type I. This would force $\frac{c+b}{2}=b$; that is, $c=b$. Thus, $k\le 2$. We can now take vector ${\textbf{z}}$ to be of type III with the coordinate $\pm 3/4$ corresponding to a zero coordinate of ${\textbf{x}}$ such that ${\textbf{x}}$ forms positive dot products ${\textbf{x}}\cdot {\textbf{z}}=\frac{c+k b}{4}$ and $\frac{c+(k-2)b}{4}$. Then $c=\frac{c+k b}{4}\le \frac{c+2b}{4}<\frac{3c}{4}$, which is a contradiction.

Thus, all non-zero coordinates of ${\textbf{x}}$ have the same absolute value, which we denote by a. Let n be the number of non-zero coordinates of ${\textbf{x}}$. If $n=1$, then $a=1$ and ${\textbf{x}}$ forms nine dot products, $0,\pm 1/4,\pm 1/2,\pm 3/4,\pm 1$, with points of ${{\overline{\omega }}}_{2160}$. If $n=3$, then ${\textbf{x}}$ forms seven dot products, $0,\pm a/2,\pm a,\pm 3a/2$, with type I vectors. If now $4\le n\le 7$, then ${\textbf{x}}$ forms nine dot products $0,\pm a/2,\pm a,\pm 3a/2,\pm 2a$ with type I vectors. Therefore, $n=2$ or 8.

If $n=2$, then $a=1/\sqrt{2}$ and ${\textbf{x}}\in \overline{\omega }_{240}$. Finally, if $n=8$, then every coordinate of ${\textbf{x}}$ is $\pm \frac{1}{2\sqrt{2}}$. Assume to the contrary that ${\textbf{x}}$ has an odd number of negative coordinates. Then for every vector ${\textbf{z}}$ of type III, ${\textbf{x}}\cdot {\textbf{z}}$ is a sum of seven signed terms $\frac{1}{8\sqrt{2}}$ and one signed term $\frac{3}{8\sqrt{2}}$ with an even total number of minus signs. Then ${\textbf{x}}\cdot {\textbf{z}}$, in particular, has six values $\pm 2w,\pm 6w,\pm 10w$, where $w=\frac{1}{8\sqrt{2}}$. Therefore, the coordinates of ${\textbf{x}}$ have an even number of minus signs and ${\textbf{x}}\in {{\overline{\omega }}}_{240}$.

Thus, if ${\textbf{x}}\notin {{\overline{\omega }}}_{240}$ then ${\textbf{x}}$ forms more than five distinct dot products with points of ${{\overline{\omega }}}_{2160}$. One can also verify directly that every ${\textbf{x}}\in {{\overline{\omega }}}_{240}$ forms exactly five distinct dot products with points of ${{\overline{\omega }}}_{2160}$, which are $0,\pm \frac{1}{2\sqrt{2}},\pm \frac{1}{\sqrt{2}}$. $\square$

We are now ready to state the main result of this section.

Theorem 4.2

Let ${{\overline{\omega }}}_{2160}=\{{\textbf{x}}_1,\ldots ,{\textbf{x}}_{2160}\}$ be the $2_{41}$ polytope on $S^7$ and g be an admissible potential function with non-negative derivatives $g^{(8)}$, $g^{(9)}$, and $g^{(10)}$ on $(-1,1)$. Then, for every point ${\textbf{x}}^*\in {{\overline{\omega }}}_{240}$,

$$\begin{aligned} \min \limits _{{\textbf{x}}\in S^7}\sum \limits _{i=1}^{2160}g({\textbf{x}}\cdot {\textbf{x}}_i)=\sum \limits _{i=1}^{2160}g({\textbf{x}}^*\cdot {\textbf{x}}_i). \end{aligned}$$

(4.2)

If, in addition, $g^{(10)}>0$ on $(-1,1)$, then the absolute minimum in (4.2) is achieved only at points of the set ${{\overline{\omega }}}_{240}$.

Proof

We have $\{1,2,3,4,5,6,7,9,10\}\subset \mathcal I({{\overline{\omega }}}_{2160})$ in view of (4.1). Applying Theorem 2.6 with $d=7$, $m=5$, $\{t_1,\ldots ,t_5\}=\left\{ 0,\pm \frac{1}{2\sqrt{2}},\pm \frac{1}{\sqrt{2}}\right\}$ we have ${\mathcal {D}}={{\overline{\omega }}}_{240}$ and equality (4.2) holds for every ${\textbf{x}}^*\in {{\overline{\omega }}}_{240}$. If $g^{(10)}>0$ on $(-1,1)$, then, by Theorem 2.6, (4.2) holds only for ${\textbf{x}}^*\in {{\overline{\omega }}}_{240}$. $\square$

5 Symmetrizations of sharp codes that are non-tight designs

In this section, we discuss one simple method which allows us to construct some new stiff configurations and obtain their universal minima.

Lemma 5.1

Let $\omega _N\subset S^d$ be an m-stiff configuration, $m,d\ge 1$, which does not contain an antipodal pair. Let $\omega _N':=\omega _N\cup \left( -\omega _N\right)$ be its symmetrization. Then $\omega _N'$ is also m-stiff with $\mathcal D_m(\omega _N')={\mathcal {D}}_m(\omega _N)$.

Proof

The configurations $\omega _N$ and $-\omega _N$ are disjoint and both are $(2m-1)$-designs. Then their union $\omega _N'$ has 2N points and is also a $(2m-1)$-design. We immediately have $\mathcal D_m(\omega _N')\subset {\mathcal {D}}_m(\omega _N)$. If ${\textbf{x}}\in {\mathcal {D}}_m(\omega _N)$ then, by Proposition 2.3, ${\textbf{x}}$ forms one of the dot products $\kappa _1^m,\ldots ,\kappa _m^m$ with any point from $\omega _N$. Since $\kappa _1^m,\ldots ,\kappa _m^m$ are zeros of $P_m^{(d)}$, they are symmetric about the origin. Since $-{\textbf{x}}\in {\mathcal {D}}_m(\omega _N)$, for every ${\textbf{y}}\in -\omega _N$, we have ${\textbf{x}}\cdot {\textbf{y}}=-{\textbf{x}}\cdot (-{\textbf{y}})\in \{\kappa _1^m,\ldots ,\kappa _m^m\}$ because $-{\textbf{y}}\in \omega _N$. Then ${\textbf{x}}\in {\mathcal {D}}_m(\omega _N')$; that is, $\mathcal D_m(\omega _N)={\mathcal {D}}_m(\omega _N')$. Since $\mathcal D_m(\omega _N)\ne \emptyset$, we have ${\mathcal {D}}_m(\omega _N')\ne \emptyset$; that is, $\omega _N'$ is m-stiff. $\square$

We remark that Lemmas 5.1 and 3.2 immediately imply Lemma 3.3 for $d\ge 5$ odd, since, in this case, ${{\overline{\omega }}}^d$ has no antipodal pair, $U_d={{\overline{\omega }}}^d\cup (-{{\overline{\omega }}}^d)$, and $\mathcal D_2(U_d)=\omega _{2d}^*$. Lemma 5.1 also applies to the following two cases.

Table 1 in [11] contains known sharp codes on the sphere. Among them, the following codes are not tight designs. Vertices of an $(N-1)$-dimensional regular simplex in $\mathbb {R}^d$, $3\le N\le d$ (we denote this code by $T_N$), vertices of the demihypercube ${{\overline{\omega }}}^5$ on $S^4$, the 112- and 162-point sharp codes on $S^{20}$, the Higman–Sims configuration and the 891-point sharp code on $S^{21}$, as well as isotropic subspaces that appear for an infinite sequence of dimensions.

Since $T_N$ is 1-stiff, so is the code $T_{2N}:=T_N\cup (-T_N)$. By Proposition 2.3 and Theorem 2.5, the set of universal minima of $T_{2N}$ is the intersection of the sphere $S^{d-1}$ with the orthogonal complement of $T_N$. The symmetrization ${{\overline{\omega }}}_5\cup (-{{\overline{\omega }}}_5)$ is the cube $U_5$ whose universal minima are at the vertices of the cross-polytope in $\mathbb {R}^5$. We will not consider the 891-point sharp code⁹ or the sequence of isotropic subspaces, since the sets of their universal minima are not known.¹⁰ The remaining three codes will be split into two cases.

Case I. The Higman–Sims configuration, denoted by ${{\overline{\omega }}}_{100}$, is a 100-point 3-design on $S^{21}$, where distinct points form only dot products $-4/11$ and 1/11 with each other (see, e.g., [11, Table 1]). According to [10], this configuration is 2-stiff. Paper [10] finds 176 pairs of antipodal vectors on $S^{21}$, where each vector forms only dot products $\frac{1}{\sqrt{22}}$ and $-\frac{1}{\sqrt{22}}$ with vectors from ${{\overline{\omega }}}_{100}$. Denote this set of $2\cdot 176=352$ vectors by ${{\overline{\omega }}}_{352}$. We have ${{\overline{\omega }}}_{352}\subset {\mathcal {D}}_2({{\overline{\omega }}}_{100})$. Each of these vectors is a universal minimum of ${{\overline{\omega }}}_{100}$ (see Theorem 2.5). Since no dot product in ${{\overline{\omega }}}_{100}$ is $-1$, the set ${{\overline{\omega }}}_{100}$ does not contain an antipodal pair.

By Lemma 5.1, the symmetrized Higman–Sims configuration ${{\overline{\omega }}}_{200}:={{\overline{\omega }}}_{100}\cup (-{{\overline{\omega }}}_{100})$ is 2-stiff with ${\mathcal {D}}_2({{\overline{\omega }}}_{200})=\mathcal D_2({{\overline{\omega }}}_{100})\supset {{\overline{\omega }}}_{352}$.

Case II. Two sharp codes on $S^{20}$ can be derived from the McLaughlin configuration ${{\overline{\omega }}}_{275}\subset S^{21}$, which is strongly sharp (see, e.g., [11, Table 1]). Fix a point ${\textbf{x}}\in {{\overline{\omega }}}_{275}$. It forms dot product 1/6 with 162 points from ${{\overline{\omega }}}_{275}$. Let $\omega _{162}^{\textbf{x}}$ denote the set of these 162 points. Point ${\textbf{x}}$ forms dot product $-1/4$ with the remaining set of 112 points from ${{\overline{\omega }}}_{275}$, which we denote by $\omega _{112}^{\textbf{x}}$.

We apply homotheties to $\omega ^{\textbf{x}}_{162}$ and $\omega _{112}^{\textbf{x}}$ to scale them to $\widetilde{S}^{20}:=S^{21}\cap H$, where H is the 21-dimensional linear subspace of $\mathbb {R}^{22}$ orthogonal to ${\textbf{x}}$. Denote the resulting configurations by ${{\overline{\omega }}}_{162}$ and ${{\overline{\omega }}}_{112}$, respectively. Both ${{\overline{\omega }}}_{162}$ and ${{\overline{\omega }}}_{112}$ are 3-designs. They are known sharp configurations, see [11, Table 1] (${{\overline{\omega }}}_{112}$ is in fact the isotropic subspace with $q=3$). Since any vector from $\omega _{162}^{\textbf{x}}$ and any vector from $\omega ^{\textbf{x}}_{112}$ form only dot products 1/6 or $-1/4$ with each other, any vector from ${{\overline{\omega }}}_{162}$ and any vector from ${{\overline{\omega }}}_{112}$ form only dot products $\frac{1}{\sqrt{21}}$ or $-\frac{1}{\sqrt{21}}$. Then both configurations are 2-stiff with ${{\overline{\omega }}}_{112}\subset \mathcal D_2({{\overline{\omega }}}_{162})$ and ${{\overline{\omega }}}_{162}\subset {\mathcal {D}}_2({{\overline{\omega }}}_{112})$ (this was observed in [10]).

Since any two distinct points in ${{\overline{\omega }}}_{275}$ form dot products 1/6 or $-1/4$ with each other, any two distinct points in ${{\overline{\omega }}}_{162}$ form dot products 1/7 or $-2/7$ and any two distinct points in ${{\overline{\omega }}}_{112}$ form dot products 1/9 or $-1/3$; i.e., both configurations do not contain antipodal pairs. Let ${{\overline{\omega }}}_{324}:={{\overline{\omega }}}_{162}\cup (-{{\overline{\omega }}}_{162})$ and ${{\overline{\omega }}}_{224}:={{\overline{\omega }}}_{112}\cup (-{{\overline{\omega }}}_{112})$ be their symmetrization about the origin. It is not difficult to see that ${{\overline{\omega }}}_{224}\subset {\mathcal {D}}_2({{\overline{\omega }}}_{162})$ and ${{\overline{\omega }}}_{324}\subset \mathcal D_2({{\overline{\omega }}}_{112})$. Each point of ${{\overline{\omega }}}_{224}$ is a universal minimum of ${{\overline{\omega }}}_{162}$ and each point of ${{\overline{\omega }}}_{324}$ is a universal minimum of ${{\overline{\omega }}}_{112}$ (see Theorem 2.5).

By Lemma 5.1, both ${{\overline{\omega }}}_{324}$ and ${{\overline{\omega }}}_{224}$ are 2-stiff with

$$ {\mathcal {D}}_2({{\overline{\omega }}}_{324})=\mathcal D_2({{\overline{\omega }}}_{162})\supset {{\overline{\omega }}}_{224}\ \ \text {and} \ \ {\mathcal {D}}_2({{\overline{\omega }}}_{224})=\mathcal D_2({{\overline{\omega }}}_{112})\supset {{\overline{\omega }}}_{324}. $$

The conclusions in Cases I and II and Theorem 2.5 imply the following.

Proposition 5.2

Let g be an admissible potential function with a convex derivative $g''$ on $(-1,1)$. Then

• (i) every point of the configuration ${{\overline{\omega }}}_{352}$ is a point of absolute minimum over $S^{21}$ of the potential $p^g(\cdot ,{{\overline{\omega }}}_{200})$;

• (ii) every point of the configuration ${{\overline{\omega }}}_{224}$ is a point of absolute minimum over ${{\widetilde{S}}}^{20}$ of the potential $p^g(\cdot ,{{\overline{\omega }}}_{324})$;

• (iii) every point of the configuration ${{\overline{\omega }}}_{324}$ is a point of absolute minimum over ${{\widetilde{S}}}^{20}$ of the potential $p^g(\cdot ,{{\overline{\omega }}}_{224})$.

6 Certain properties of general stiff configurations

Every time we have a stiff configuration, in view of Theorem 2.5, we automatically have its universal minima (the dual configuration). Moreover, every stiff configuration attains the Fazekas–Levenshtein bound for covering [13, Theorem 2]. Therefore, it is important to study stiff codes and their duals in general. In this section, we characterize 1-stiff configurations on $S^d$, m-stiff configurations on $S^1$, and their duals and also prove some basic properties of stiff configurations and their duals. We call the point $\textbf{c}=\frac{1}{N}\sum _{i=1}^{N}{\textbf{x}}_i$ the center of mass of a configuration $\omega _N=\{{\textbf{x}}_1,\ldots ,{\textbf{x}}_N\}$.

Proposition 6.1

Let $d\ge 1$. A configuration $\omega _N\subset S^d$, $N\ge 1$, is 1-stiff if and only if its center of mass is at the origin and $\omega _N$ is contained in a d-dimensional linear subspace of $\mathbb {R}^{d+1}$.

Proof

The proposition follows from the fact that a point configuration is a spherical 1-design if and only if its center of mass is located at the origin and the fact that a hyperplane containing $\omega _N$ also contains its center of mass. $\square$

We next describe the dual of a 1-stiff configuration.

Proposition 6.2

Let $\omega _N\subset S^d$, $d\ge 1$, be a 1-stiff configuration. Then ${\mathcal {D}}_1(\omega _N)=L^\bot \cap S^d$, where L is the linear subspace of $\mathbb {R}^{d+1}$ spanned by $\omega _N$. If $k:={\textrm{dim}}\ \! L\le d-1$, then ${\mathcal {D}}_1(\omega _N)$ is a sphere in a $(d+1-k)$-dimensional subspace of $\mathbb {R}^{d+1}$. If $k=d$, then ${\mathcal {D}}_1(\omega _N)=\{{\textbf{a}},-{\textbf{a}}\}$ for some ${\textbf{a}}\in S^d$, which is a 1-stiff configuration.

Proof

Let ${\textbf{z}}$ be any vector in ${\mathcal {D}}_1(\omega _N)$. By Proposition 2.3, we have ${\textbf{z}}\cdot {\textbf{y}}=0$ for every ${\textbf{y}}\in \omega _N$, since 0 is the only root of $P_1^{(d)}$. Then ${\textbf{z}}\bot L$; i.e., ${\textbf{z}}\in L^\bot \cap S^d$. If ${\textbf{z}}$ is any vector in $L^\bot \cap S^d$, then it forms only one dot product (which is 0) with any point from $\omega _N$; that is, ${\textbf{z}}\in {\mathcal {D}}_1(\omega _N)$. The rest of Proposition 6.2 follows immediately. $\square$

We now charactirize stiff configurations on $S^1$.

Proposition 6.3

For every $m\ge 1$, a configuration on $S^1$ is m-stiff if and only if it is a regular 2m-gon.

We remark that the regular 2m-gon ${{\widetilde{\omega }}}_{2m}$ on $S^1$ is antipodal and its dual ${\mathcal {D}}_m({{\widetilde{\omega }}}_{2m})$ is another regular 2m-gon with ${\mathcal {D}}_m(\mathcal D_m({{\widetilde{\omega }}}_{2\,m}))={{\widetilde{\omega }}}_{2\,m}$.

Proof of Proposition 6.3

If $\omega _N={{\widetilde{\omega }}}_{2m}$ then it is a $(2m-1)$-design and the midpoint ${\textbf{y}}$ of the arc joining any two neighboring vertices forms m distinct values of dot products with points from $\omega _N$; i.e., $\omega _N$ is m-stiff.

By Proposition 2.3, the point ${\textbf{y}}$ forms dot products $\kappa _1^m,\ldots ,\kappa _m^m$ (zeros of $P_m^{(1)}$) with points of ${{\widetilde{\omega }}}_{2m}$, each with frequency $2=2ma_0(\varphi _i)$, $i=1,\ldots ,m$, where $\{\varphi _1,\ldots ,\varphi _m\}$ is the fundamental system of polynomials for the nodes $\kappa _i^m$. We have

$$\begin{aligned} a_0(\varphi _i)=\frac{1}{m} \ \ \text {and}\ \ \kappa _i^m=\cos \frac{(2i-1)\pi }{2m}, \ \ \ \ i=1,\ldots ,m. \end{aligned}$$

(6.1)

Assume that $\omega _N$ is m-stiff. Let ${\textbf{z}}$ be any point in ${\mathcal {D}}_m(\omega _N)$. By Proposition 2.3, point ${\textbf{z}}$ forms dot products $\kappa _1^m,\ldots ,\kappa _m^m$ with points of $\omega _N$, where $\kappa ^m_i$, $i=1,\ldots ,m$, are the zeros of $P_m^{(1)}$. Then $\omega _N$ is contained in the set of points of intersection of $S^1$ with m parallel lines; that is, $\#\omega _N\le 2m$. By Proposition 2.3 and (6.1), the frequency $M_i$ of the dot product $\kappa ^m_i$ is $M_i=Na_0(\varphi _i)=N/m\le 2$, $i=1,\ldots ,m$. Hence, frequencies $M_i$ are equal and each of the m parallel lines contains the same number of points from $\omega _N$ (one or two). Assume to the contrary that each $M_i$ equals 1. Then $\omega _N$ has only m points, which means that any point ${\textbf{y}}\in \omega _N$ forms at most m distinct values of the dot product with points of $\omega _N$; i.e., ${\textbf{y}}\in {\mathcal {D}}_m(\omega _N)$. One of the dot products is 1, while by Proposition 2.3, these dot products must be zeros $\kappa _i^m$ of $P_m^{(1)}$ none of which is 1. This contradiction shows that $M_i=2$, $i=1,\ldots ,m$, and, hence $\#\omega _N=2m$. Vector ${\textbf{z}}$ forms each of the angles $\frac{(2i-1)\pi }{2m}=\arccos \kappa _i^m$ with exactly two points from $\omega _N$, $i=1,\ldots ,m$. Then $\omega _N$ is a regular 2m-gon. $\square$

We next prove the following basic statement.

Proposition 6.4

For any configuration $\omega _N\subset S^d$, $d\ge 1$, with ${\mathcal {D}}_m(\omega _N)\ne \emptyset$, $m\ge 1$, the set $\mathcal D_m(\omega _N)$ is antipodal.

If $\omega _N$ is m-stiff, then $\omega _N\subset \mathcal D_m({\mathcal {D}}_m(\omega _N))$. If, in addition, $\mathcal D_m(\omega _N)$ is finite and m-stiff, then ${\mathcal {D}}_m(\mathcal D_m({\mathcal {D}}_m(\omega _N)))={\mathcal {D}}_m(\omega _N)$.

Though the dual ${\mathcal {D}}_m(\omega _N)$ is antipodal, this is not always true for the m-stiff configuration $\omega _N$ itself. For example, the d-demicube ${{\overline{\omega }}}^d$ on $S^{d-1}$ is not antipodal for any $d\ge 3$ odd.

In view of Proposition 6.4, the equality $\omega _N=\mathcal D_m({\mathcal {D}}_m(\omega _N))$ with an m-stiff configuration $\omega _N$ implies that $\omega _N$ is antipodal. However, the inclusion $\omega _N\subset {\mathcal {D}}_m({\mathcal {D}}_m(\omega _N))$ can be sometimes strict (even when both $\omega _N$ and $\mathcal D_m(\omega _N)$ are antipodal and m-stiff). This is the case, for example, for $\omega _N={{\overline{\omega }}}^d$ and any $d\ge 5$ in view of Lemma 3.3 and the fact that the dual of the cross-polytope $\omega _{2d}^*$ is the whole cube $U_d$ (for $d\ge 5$ even, both ${{\overline{\omega }}}^d$ and $\mathcal D_2({{\overline{\omega }}}^d)=\omega ^*_{2d}$ are antipodal and 2-stiff). At the same time, every stiff configuration from [2, Table 3] coincides with the dual of its dual.

Some other examples of non-antipodal stiff configurations $\omega _N$ are given in [10]. Since, by Proposition 6.4, the duals of their duals are antipodal, the inclusion $\omega _N\subset {\mathcal {D}}_m({\mathcal {D}}_m(\omega _N))$ is strict as well.

Proof of Proposition 6.4

Let $\omega _N\subset S^d$ be arbitrary with $\mathcal D_m(\omega _N)\ne \emptyset$. For any point ${\textbf{z}}\in \mathcal D_m(\omega _N)$, the point $-{\textbf{z}}$ also forms at most m distinct dot products with points of $\omega _N$; that is $-{\textbf{z}}\in \mathcal D_m(\omega _N)$ and the set ${\mathcal {D}}_m(\omega _N)$ is antipodal. Choose any point ${\textbf{z}}$ in an m-stiff $\omega _N$. For any point ${\textbf{y}}\in {\mathcal {D}}_m(\omega _N)$, by Proposition 2.3, we have ${\textbf{y}}\cdot {\textbf{z}}\in \{\kappa _1^m,\ldots ,\kappa _m^m\}$; that is, ${\textbf{z}}$ forms at most m distinct dot products with points of ${\mathcal {D}}_m(\omega _N)$. Then ${\textbf{z}}\in \mathcal D_m({\mathcal {D}}_m(\omega _N))$ and $\omega _N\subset \mathcal D_m({\mathcal {D}}_m(\omega _N))$. Assume additionally that $X:=\mathcal D_m(\omega _N)$ is finite and m-stiff. The inclusion $\omega _N\subset {\mathcal {D}}_m({\mathcal {D}}_m(\omega _N))$ implies that $X={\mathcal {D}}_m(\omega _N)\supset {\mathcal {D}}_m({\mathcal {D}}_m(X))$. Since X is m-stiff, we have the opposite inclusion. $\square$

We say that a point set in $\mathbb {R}^{d+1}$ is in general position if it is not contained in any hyperplane. One can construct plenty of examples of m-stiff configurations, $m\ge 2$, whose dual is not in general position. For instance, start with the cube $U_3$ inscribed in $S^2$ and let $\alpha _1$ and $\alpha _2$ be the parallel planes containing two paralel facets of $U_3$. For a given $n\ge 2$, we rotate the cube $U_3$ about the axis $\ell$ perpendicular to the planes $\alpha _1$ and $\alpha _2$ and passing through the origin at angles $\frac{\pi k}{2n}$, $k=0,1,\ldots ,n-1$, and let $\omega _N$ be the union of the resulting n cubes. Then $\omega _N$ is a 3-design as a disjoint union of finitely many 3-designs. Since $\omega _N$ is still contained in planes $\alpha _1$ and $\alpha _2$, it is 2-stiff. However, its dual is $\mathcal D_2(\omega _N)=\{{\textbf{a}},-{\textbf{a}}\}$, where ${\textbf{a}}$ is a unit vector parallel to the axis $\ell$. The dual is not in general position. It is also only 1-stiff. This example can, of course, be extended to other dimensions and other initial configurations.

Proposition 6.5

Let $\omega _N\subset S^d$, $d\ge 1$, be an m-stiff configuration, $m\ge 2$. Then

• (i) ${\mathcal {D}}_m(\omega _N)$ contains at most $m^{d+1}$ points;

• (ii) if ${\mathcal {D}}_m(\omega _N)$ is in general position, then $\omega _N$ contains at most $m^{d+1}$ points;

• (iii) if ${\mathcal {D}}_m(\omega _N)$ is not in general position, then ${\mathcal {D}}_m(\omega _N)$ is 1-stiff and, hence, not m-stiff.

Proposition 6.5 implies that an m-stiff configuration on $S^d$, $m\ge 2$, cannot have more than $m^{d+1}$ universal minima. This cardinality bound can be achieved: take $\omega _N$ to be the set of vertices of a regular cross-polytope inscribed in $S^d$. It is 2-stiff and its dual is a cube inscribed in $S^d$, which has exactly $2^{d+1}$ vertices. If the universal minima of an m-stiff configuration on $S^d$ are in general position, then the configuration itself contains at most $m^{d+1}$ points. This bound is attained by the set of $2^{d+1}$ vertices of a cube inscribed in $S^d$, which is 2-stiff. Its dual is the set of vertices of a $(d+1)$-dimensional regular cross-polytope.

For $m\ge 3$, there may exist an m-stiff configuration on $S^d$ with more than $2^{d+1}$ points in it or in its dual. For example, the 24-cell on $S^3$ has $N=24>2^4$ points (its dual is another 24-cell), or the dual of the symmetrized Schläffi configuration on $S^5$, which has $N=72>2^6$ points, see [2]. All these configurations are 3-stiff.

If the universal minima are not in general position, then no upper bound depending only on m and d can be written for the cardinality of an m-stiff configuration $\omega _N$, $d\ge 2$ (when $\omega _N$ exists for those m and d), see Proposition 6.7 below.

Another interesting question related to Proposition 6.5 is about the general assumptions under which the dual of a given m-stiff configuration, $m\ge 2$, is in general position and whether this is sufficient for the dual to be also m-stiff. The dual is m-stiff with the same m, for example, for every stiff configuration mentioned in [2, Table 3] and for the d-demicube, $d\ge 4$.

To establish Proposition 6.5, we need the following auxiliary statement.

Lemma 6.6

Suppose $m,d\ge 1$ and $X\subset S^d$ is a finite set in general position. Then for any m-element subset $\Lambda \subset [-1,1]$, there are at most $m^{d+1}$ points ${\textbf{z}}\in S^d$ such that $D({\textbf{z}},X)\subset \Lambda$.

Proof

Since X is in general position, it contains a linearly independent subset $\{{\textbf{y}}_1,\ldots ,{\textbf{y}}_{d+1}\}$. Let ${\textbf{z}}\in S^d$ be any point with $D({\textbf{z}},X)\subset \Lambda$. Then ${\textbf{z}}\cdot {\textbf{y}}_j=\alpha _j$, $j=1,\ldots ,d+1$, where $\alpha _1,\ldots ,\alpha _{d+1}\in \Lambda$. Consequently, ${\textbf{z}}$ is a solution to a linear system with a fixed non-singular coefficient matrix and vector of right-hand sides $(\alpha _1,\ldots ,\alpha _{d+1})\in \Lambda ^{d+1}$. Each such system has a unique solution. Since there are $m^{d+1}$ possible vectors of right-hand sides for these systems, there are at most $m^{d+1}$ points ${\textbf{z}}\in S^d$ with $D({\textbf{z}},X)\subset \Lambda$. $\square$

Proof of Proposition 6.5

Since $\omega _N$ is at least a 3-design, it is in general position. If it were not, then $\omega _N$ would not be a 2-design: for the polynomial $p({\textbf{x}})=({\textbf{x}}\cdot {\textbf{v}}-\alpha )^2$, where ${\textbf{x}}\cdot {\textbf{v}}=\alpha$ is an equation of the hyperplane containing $\omega _N$, the average of p over $\omega _N$ would be zero while the average of p over $S^d$ would be positive. By Proposition 2.3, for every ${\textbf{z}}\in \mathcal D_m(\omega _N)$, we have $D({\textbf{z}},\omega _N)\subset \Lambda _m:=\{\kappa _1^m,\ldots ,\kappa _m^m\}$, where $\kappa _j^m$’s are zeros of $P_m^{(d)}$. Then, by Lemma 6.6, we have $\# {\mathcal {D}}_m(\omega _N)\le m^{d+1}$.

Assume that ${\mathcal {D}}_m(\omega _N)$ is in general position. Since $\omega _N$ is m-stiff, by Proposition 2.3, for any ${\textbf{z}}\in \omega _N$ and ${\textbf{y}}\in {\mathcal {D}}_m(\omega _N)$, we have ${\textbf{z}}\cdot {\textbf{y}}\in \Lambda _m$. Then $D({\textbf{z}},\mathcal D_m(\omega _N))\subset \Lambda _m$, ${\textbf{z}}\in \omega _N$. By Lemma 6.6, we have $N\le m^{d+1}$.

Assume that ${\mathcal {D}}_m(\omega _N)$ is not in general position; i.e., contained in some hyperplane H. By Proposition 6.4, it is antipodal. Then its center of mass is at the origin; i.e., H is a d-dimensional linear subspace of $\mathbb {R}^{d+1}$. By Proposition 6.1, ${\mathcal {D}}_m(\omega _N)$ is 1-stiff. Since $\mathcal D_m(\omega _N)$ is contained in one hyperplane, by the above argument, it cannot be a 3-design. Then ${\mathcal {D}}_m(\omega _N)$ cannot be m-stiff for any $m\ge 2$. $\square$

The statement below implies that the set of possible cardinalities of m-stiff configurations on $S^d$, $d\ge 2$ (provided that an m-stiff configuration exists on $S^d$), forms an additive semigroup (in particular, it is not bounded above). This is not the case for $d=1$ in view of Proposition 6.3. In the case $m=1$ and $d\ge 2$, this semigroup is the set of all integers $N\ge 2$, see Proposition 6.1.

Proposition 6.7

Suppose that for given $m\ge 1$ and $d\ge 2$, there exist m-stiff configurations on $S^d$ of cardinalities $N_1$ and $N_2$. Then there is an m-stiff configuration on $S^d$ of cardinality $N_1+N_2$.

Proposition 6.7 and Bézout’s identity imply that if, for a given pair (m, d), $d\ge 2$, the cardinalities of two m-stiff configurations on $S^d$ have the greatest common divisor $\delta$, then for any sufficiently large multiple N of $\delta$, there exists an m-stiff configuration on $S^d$ of cardinality N.

Proof

Let $\omega _{N_i}\subset S^d$ be an m-stiff configuration of cardinality $N_i$, $i=1,2$, and let ${\textbf{z}}_i\in \mathcal D_m(\omega _{N_i})$, $i=1,2$, be chosen so that ${\textbf{z}}_1\ne {\textbf{z}}_2$ (if it happens that ${\textbf{z}}_1={\textbf{z}}_2$, we choose $-{\textbf{z}}_2$ instead of ${\textbf{z}}_2$). We will construct an m-stiff configuration of cardinality $N_1+N_2$. By Proposition 2.3, both vectors ${\textbf{z}}_1$ and ${\textbf{z}}_2$ form only dot products $\kappa _1^m,\ldots ,\kappa _m^m$ with points from the corresponding configuration $\omega _{N_i}$. Let H be the d-dimensional subspace of $\mathbb {R}^{d+1}$, which is the perpendicular bisector for the line segment $[{\textbf{z}}_1,{\textbf{z}}_2]$. Let $r_H:\mathbb {R}^{d+1}\rightarrow \mathbb {R}^{d+1}$ be the reflection transformation about the subspace H and let U be its matrix (U is orthogonal). Then for every ${\textbf{x}}\in \omega _{N_1}':=r_H(\omega _{N_1})$, there is ${\textbf{y}}\in \omega _{N_1}$ such that ${\textbf{x}}=U{\textbf{y}}$ and

$$\begin{aligned} {\textbf{x}}\cdot {\textbf{z}}_2=U{\textbf{y}}\cdot {\textbf{z}}_2={\textbf{y}}\cdot U^T{\textbf{z}}_2={\textbf{y}}\cdot {\textbf{z}}_1=\kappa _i^m \ \ \ \text {for some}\ \ \ i=1,\ldots ,m. \end{aligned}$$

(6.2)

The set $\omega _{N_1}'$ is m-stiff. Denote by ${\textbf{a}}\in S^d$ a vector perpendicular to ${\textbf{z}}_2$, not contained in any subspace ${\textrm{span}}\{{\textbf{w}}-{\textbf{v}}\}$, where ${\textbf{w}}\in \omega _{N_2}$ and ${\textbf{v}}\in \omega _{N_1}'$, and ${\textbf{a}}$ is not perpendicular to any vector from $\omega _{N_2}$. Such a vector ${\textbf{a}}$ exists, since all these conditions delete a set of the $(d-1)$-dimensional measure zero from $S^d\cap \{{\textbf{z}}_2\}^\bot$.

Then $L=\{{\textbf{a}}\}^\bot$ is disjoint with $\omega _{N_2}$. Furthermore, ${\textbf{z}}_2\in L$ and the configuration $\omega _{N_1}'':=r_L(\omega _{N_1}')$ is disjoint with $\omega _{N_2}$. If it were not, then L would be the perpendicular bisector for a line segment whose one endpoint ${\textbf{w}}_1$ is in $\omega _{N_2}$ and the other endpoint ${\textbf{v}}_1$ is in $\omega _{N_1}'$. We have ${\textbf{w}}_1\ne {\textbf{v}}_1$, since, otherwise, ${\textbf{w}}_1\in L$ and, hence, ${\textbf{a}}\bot {\textbf{w}}_1$. Then ${\textbf{w}}_1-{\textbf{v}}_1$ is a non-zero vector perpendicular to L and ${\textbf{a}}\in \textrm{span}\{{\textbf{w}}_1-{\textbf{v}}_1\}$ contradicting the choice of ${\textbf{a}}$.

Let V be the matrix of the reflection transformation $r_L$. Then for every ${\textbf{z}}\in \omega _{N_1}''$, there is ${\textbf{x}}\in \omega _{N_1}'$ such that ${\textbf{z}}=V{\textbf{x}}$ and using (6.2) we have

$$ {\textbf{z}}\cdot {\textbf{z}}_2=V{\textbf{x}}\cdot {\textbf{z}}_2={\textbf{x}}\cdot V^T{\textbf{z}}_2={\textbf{x}}\cdot {\textbf{z}}_2=\kappa _i^m\ \ \ \text {for some}\ \ \ i=1,\ldots ,m. $$

Thus, ${\textbf{z}}_2\in {\mathcal {D}}_m(\omega _{N_1}''\cup \omega _{N_2})$ and $\omega _{N_1}''\cup \omega _{N_2}$ is a disjoint union of two $(2m-1)$-designs. Then it is also a $(2m-1)$-design, and, hence is an m-stiff configuration of cardinality $N_1+N_2$. $\square$

We conclude this section with the following important existence result for codes attaining the Fazekas–Levenshtein bound for covering, see [13, Theorem 2]. Recall that for a given $m\in \mathbb {N}$, a spherical $(2m-1)$-design on $S^d$ attains the Fazekas–Levenshtein bound if and only if it is m-stiff. Any m-stiff configuration $\omega _N$ on $S^d$ is a set of nodes of a cubature (with equal weights) exact on multivariate polynomials of degree up to $2m-1$. It gives a rise to the unique univariate quadrature for integration with weight $w_d(t)$ with m nodes and algebraic degree of precision $2m-1$. It is called the Gauss–Gegenbauer quadrature. Its weights are rational and positive, since they equal the frequencies of the m dot products formed by each universal minimum point of $\omega _N$ with points of $\omega _N$. Here we establish the converse: if the Gauss–Gegenbauer quadrature with m nodes for integration with weight $w_d(t)$ has positive and rational weights, then one can construct an m-stiff configuration on $S^d$.

Recall that $\{\varphi _1,\ldots ,\varphi _m\}$ are the fundamental polynomials for the set of zeros $\kappa _1^m,\ldots ,\kappa _m^m$ of the m-th Gegenbauer polynomial $P_m^{(d)}$.

Theorem 6.8

For every $m,d\ge 1$, there exists an m-stiff configuration $\omega _N\subset S^d$ if and only if every 0-th Gegenbauer coefficient $a_0(\varphi _k)$, $k=1,\ldots ,m$, is a positive rational.

The Gauss–Gegenbauer quadrature is the quadrature

$$\begin{aligned} a_0(h)=\int _{-1}^{1}h(t)w_d(t)\ \! dt\approx \sum \limits _{i=1}^{m}\beta _ih(\kappa _i^m), \end{aligned}$$

(6.3)

where $\beta _i=a_0(\varphi _i)$, $i=1,\ldots ,m$. It is exact for all polynomials $h\in \mathbb P_{2m-1}$. Theorem 6.8 can be restated as follows: there extsis an m-stiff configuration on a given sphere $S^d$ if and only if all the coefficients of the corresponding Gauss–Gegenbauer quadrature are positive rationals.

Remark 6.9

The coefficients of the Gauss–Gegenbauer quadrature are positive rationals for $d=1$ and any $m\ge 1$. In this case, vertices of a regular 2m-gon on $S^1$ form an m-stiff configuration. The Gauss–Gegenbauer quadrature also has positive rational coefficients (for all $d\ge 1$) when $m=1$ (there is only one coefficient that equals 1), $m=2$ (both coefficients equal 1/2), and $m=3$. When $d\ge 2$ and $m\ge 4$, the Gauss–Gegenbauer quadrature may also have positive rational coefficients. For example, this happens for $d=22$ and $m=4$, when the 4600-point tight 7-design on $S^{22}$ appears to be 4-stiff, see [10].

We will need the following statement which immediately follows from the results of Seymour and Zaslavsky [22, Main Theorem].

Lemma 6.10

Let n and d be arbitrary positive integers. Then for every cardinality N sufficiently large, there exists an N-point spherical n-design on $S^d$.

Proof of Theorem 6.8

Assume that for a given pair $m,d\ge 1$, there exists an m-stiff configuration on $S^d$. Then by Proposition 2.3, each frequency $a_0(\varphi _i)N$, $i=1,\ldots ,m$, is a positive integer; that is, each $a_0(\varphi _i)$ is a positive rational.

Assume now that for a given pair $m,d\ge 1$, each coefficient $a_0(\varphi _i)$, $i=1,\ldots ,m$, is a positive rational. We will construct an m-stiff configuration on $S^d$.

Case I: $d=1$. An m-stiff configuration on $S^1$ is a regular 2m-gon.

Case II: $m=1$. There exists a 1-stiff configuration on $S^d$, $d\ge 1$, in view of Proposition 6.1. When $d=1$, it is an antipodal pair and, when $d\ge 2$, it is, for example, a regular d-simplex inscribed in the intersection of a d-dimensional linear subspace of $\mathbb {R}^{d+1}$ with $S^d.$

The remaining case ($d,m\ge 2$) will be handled using the following lemma. For a given $u\in [-1,1]$, define the mapping $\pi _u:S^{d-1}\rightarrow S^d$ by

$$ \pi _u({\textbf{z}}):=u\textbf{e}_{d+1}+\sqrt{1-u^2}({\textbf{z}},0),\ \ \ {\textbf{z}}\in S^{d-1}, $$

where we recall that $\textbf{e}_{d+1}=(0,\ldots ,0,1)\in \mathbb {R}^{d+1}$. The mapping $\pi _u$ is injective for $u\in (-1,1)$. $\square$

Lemma 6.11

Let $d\ge 2$ and $n\ge 1$. Suppose positive rationals $\gamma _1,\ldots ,\gamma _\nu$ and reals $-1<t_1<\ldots<t_\nu <1$ are such that

$$\begin{aligned} \int _{-1}^{1}h(t)w_d(t)\ \! dt=\sum \limits _{i=1}^{\nu }\gamma _i h(t_i) \end{aligned}$$

(6.4)

holds for every polynomial $h\in \mathbb P_n$. Suppose also that $X_i$, $i=1,\ldots ,\nu$, are spherical n-designs on $S^{d-1}$ such that $\frac{\# X_1}{\gamma _1}=\ldots =\frac{\# X_\nu }{\gamma _\nu }$. Denote $N:=\#X_1+\ldots +\# X_\nu$ and let

$$ \omega _N:=\pi _{t_1}(X_1)\cup \ldots \cup \pi _{t_\nu }(X_\nu ). $$

Then $\omega _N$ is an n-design on $S^d$ contained in the union of $\nu$ parallel hyperplanes.

Proof

By item (iii) of Theorem 2.1, it is sufficient to show that

$$\begin{aligned} p_\ell ({\textbf{x}},\omega _N):=\sum \limits _{i=1}^N ({\textbf{x}}\cdot {\textbf{x}}_i)^\ell =Nc_{\ell ,d},\ \ {\textbf{x}}\in S^d,\ \ \ \ell =1,\ldots ,n, \end{aligned}$$

(6.5)

where $c_{\ell ,d}:=\int _{-1}^{1}t^\ell w_d(t)\ \! dt$. Choose arbitrary $1\le \ell \le n$ and ${\textbf{x}}\in S^d$. Let u be the $(d+1)$-th coordinate of ${\textbf{x}}$. If $u\in (-1,1)$, let $\widetilde{\textbf{x}}\in \mathbb {R}^d$ be the vector obtained from ${\textbf{x}}$ by deleting the $(d+1)$-th coordinate and normalizing the resulting vector to lie on $S^{d-1}$. If $\left| u\right| =1$, we let $\widetilde{\textbf{x}}$ be any vector on $S^{d-1}$. Then

$$\begin{aligned} \begin{aligned} p_\ell ({\textbf{x}},\omega _N)&=\sum \limits _{i=1}^{\nu }\sum \limits _{{\textbf{y}}\in \pi _{t_i}(X_i)}({\textbf{x}}\cdot {\textbf{y}})^\ell \\&=\sum \limits _{i=1}^{\nu }\sum \limits _{{\textbf{z}}\in X_i}\left( (u\textbf{e}_{d+1}+\sqrt{1-u^2}(\widetilde{\textbf{x}},0))\cdot \left( t_i\textbf{e}_{d+1}+\sqrt{1-t_i^2}({\textbf{z}},0)\right) \right) ^\ell \\&=\sum \limits _{i=1}^{\nu }\sum \limits _{{\textbf{z}}\in X_i}\left( ut_i+\sqrt{1-u^2}\sqrt{1-t_i^2}(\widetilde{\textbf{x}}\cdot {\textbf{z}})\right) ^\ell . \end{aligned} \end{aligned}$$

Since quadrature (6.4) is exact for $h(t)=1$, we have $\sum _{i=1}^{\nu }\gamma _i=1$. Then $\# X_i=\gamma _i N$, $i=1,\ldots ,\nu$. Since $X_i$ is an n-design on $S^{d-1}$, by item (iii) of Theorem 2.1, for $u\in [-1,1]$ fixed, we have

$$ p_\ell ({\textbf{x}},\omega _N)=N\sum \limits _{i=1}^{\nu }\gamma _i\int \limits _{-1}^{1}\left( ut_i+t\sqrt{1-u^2}\sqrt{1-t_i^2}\right) ^\ell w_{d-1}(t)\ \! dt. $$

For $(u,v)\in [-1,1]^2$, we have

$$\begin{aligned} \begin{aligned} q(u,v)&:=\int \limits _{-1}^{1}\left( uv+t\sqrt{1-u^2}\sqrt{1-v^2}\right) ^\ell w_{d-1}(t)\ \! dt\\&=\int \limits _{-1}^{1}\sum \limits _{k=0}^{\ell }\left( \ell \atop k\right) u^{\ell -k}v^{\ell -k}(1-u^2)^{k/2}(1-v^2)^{k/2}t^k w_{d-1}(t)\ \! dt\\&=\sum \limits _{k=0\atop k \text { even}}^{\ell }c_{k,d-1}\left( \ell \atop k\right) u^{\ell -k}v^{\ell -k}(1-u^2)^{k/2}(1-v^2)^{k/2}, \end{aligned} \end{aligned}$$

(6.6)

where we omitted terms corresponding to k odd, since $c_{k,d-1}=0$ for odd k. Then q(u, v) is a polynomial defined on $[-1,1]^2$. For every $u\in [-1,1]$ fixed, q is a polynomial in v of degree $\ell \le n$ and, in view of (6.4),

$$\begin{aligned} p_\ell ({\textbf{x}},\omega _N)=N\sum \limits _{i=1}^{\nu }\gamma _iq(u,t_i)=N\int \limits _{-1}^{1}q(u,v)w_d(v)\ \! dv. \end{aligned}$$

(6.7)

If $\ell$ is odd, the polynomial q(u, v) is odd in v. Therefore,

$$ p_\ell ({\textbf{x}},\omega _N)=0=Nc_{\ell ,d}. $$

Assume that $\ell$ is even. From (6.6), we have

$$ \tau (u):=\int \limits _{-1}^{1}q(u,v)w_d(v)\ \! dv=\sum \limits _{k=0\atop k \text { even}}^{\ell }c_{k,d-1}\frac{\gamma _d}{\gamma _{d+k}}c_{\ell -k,d+k}\left( \ell \atop k\right) u^{\ell -k}(1-u^2)^{k/2}. $$

It remains to show that $\tau (u)=c_{\ell ,d}$, $u\in [-1,1]$. For $u\in (0,1]$, making the substitution $u=(w^2+1)^{-1/2}$, $w\in [0,\infty )$, we obtain

$$\begin{aligned} \begin{aligned} \tau (u)&=\frac{1}{(w^2+1)^{\ell /2}}\sum \limits _{k=0\atop k \text { even}}^{\ell }\theta ^\ell _{k,d}\left( \ell \atop k\right) w^k=\frac{1}{(w^2+1)^{\ell /2}}\sum \limits _{j=0}^{\ell /2}\theta _{2j,d}^\ell \left( \ell \atop 2j\right) w^{2j}, \end{aligned} \end{aligned}$$

(6.8)

where $\theta ^\ell _{k,d}:=c_{k,d-1}\frac{\gamma _d}{\gamma _{d+k}}c_{\ell -k,d+k}.$

We will show that the coefficient of $w^{2j}$ in (6.8) is proportional to the binomial coefficient $\left( \ell /2 \atop j\right)$. Then the sum in (6.8) will be a constant multiple of $(w^2+1)^{\ell /2}$ thus showing that $\tau (u)$ is constant. It is not difficult to verify that $c_{2j,d}=\frac{\Gamma \left( \frac{d+1}{2}\right) \Gamma \left( j+\frac{1}{2}\right) }{\Gamma \left( \frac{1}{2}\right) \Gamma \left( j+\frac{d+1}{2}\right) }$, $j\ge 0$, $\gamma _d=\frac{\Gamma \left( \frac{d+1}{2}\right) }{\Gamma \left( \frac{1}{2}\right) \Gamma \left( \frac{d}{2}\right) }$, and $\frac{\Gamma \left( k+\frac{1}{2}\right) }{\Gamma \left( \frac{1}{2}\right) (2k)!}=\frac{1}{k!\cdot 2^{2k}}$, $k\ge 0$. Then

$$\begin{aligned} \begin{aligned} \theta ^\ell _{2j,d}\left( \ell \atop 2j\right)&=\frac{\ell !\cdot \Gamma \left( \frac{d+1}{2}\right) }{\Gamma \left( \frac{\ell +d+1}{2}\right) }\cdot \frac{\Gamma \left( j+\frac{1}{2}\right) }{\Gamma \left( \frac{1}{2}\right) (2j)!}\cdot \frac{\Gamma \left( \frac{\ell -2j}{2}+\frac{1}{2}\right) }{\Gamma \left( \frac{1}{2}\right) (\ell -2j)!}\\&=\frac{\ell !\cdot \Gamma \left( \frac{d+1}{2}\right) }{\Gamma \left( \frac{\ell +d+1}{2}\right) }\cdot \frac{1}{j!\cdot 2^{2j}}\cdot \frac{1}{2^{\ell -2j}\left( \ell /2-j\right) !}\\&=\frac{\cdot \Gamma \left( \frac{d+1}{2}\right) }{\Gamma \left( \frac{\ell +d+1}{2}\right) }\cdot \frac{\ell !}{(\ell /2)!\cdot 2^\ell }\cdot \left( \ell /2\atop j\right) =\frac{\Gamma \left( \frac{d+1}{2}\right) \Gamma \left( \frac{\ell +1}{2}\right) }{\Gamma \left( \frac{\ell +d+1}{2}\right) \Gamma \left( \frac{1}{2}\right) }\cdot \left( \ell /2\atop j\right) \\&=c_{\ell ,d}\left( \ell /2\atop j\right) \end{aligned} \end{aligned}$$

holds for every $j=0,1,\ldots ,\ell /2$. From (6.8), we now have

$$ \tau (u)=\frac{c_{\ell ,d}}{(w^2+1)^{\ell /2}}\sum \limits _{j=0}^{\ell /2}\left( \ell /2\atop j\right) w^{2j}=c_{\ell ,d}, \ \ u\in (0,1]. $$

Since $\tau (u)$ is an even polynomial, we have $\tau (u)=c_{\ell ,d}$, $u\in [-1,1]$. From (6.7) we have

$$ p_\ell ({\textbf{x}},\omega _N)=N\tau (u)=Nc_{\ell ,d},\ \ u\in [-1,1]. $$

Thus, (6.5) holds for every $\ell =1,\ldots ,n$ and, trivially, for $\ell =0$. Then item (iii) of Theorem 2.1 implies that $\omega _N$ is an N-point n-design on $S^d$. It is contained in the union of $\nu$ parallel hyperplanes. $\square$

We are now ready to prove the remaining case of Theorem 6.8.

Case III: $m,d\ge 2$. In view of Lemma 6.10, there is a cardinality $N_0$ such that for every $N\ge N_0$, there exists a spherical $(2m-1)$-design on $S^{d-1}$. Denote $\rho :=\min \limits _{i=\overline{1,m}}a_0(\varphi _i)$ and let $\mu$ be the least common denominator of positive rationals $\frac{a_0(\varphi _i)}{\rho }N_0$, $i=1,\ldots ,m$. Let $X_i$ be a $(2m-1)$-design on $S^{d-1}$ of cardinality $N_i:=\frac{\mu a_0(\varphi _i)}{\rho }N_0$, $i=1,\ldots ,m$. Applying Lemma 6.11 with $\nu =m$, $\gamma _i=a_0(\varphi _i)$, $i=1,\ldots ,m$, and $t_i$’s being the zeros of $P_m^{(d)}$ and taking (6.3) into account, we obtain a spherical $(2m-1)$-design $\omega _N$ on $S^d$ of cardinality $N=\frac{\mu }{\rho }N_0$ contained in m parallel hyperplanes. Then $\omega _N$ is m-stiff. $\square$