$k$-point semidefinite programming bounds for equiangular lines

TL;DR

This paper introduces a hierarchy of $k$-point semidefinite programming bounds that extend existing bounds for spherical codes, enabling the computation of bounds for the maximum number of equiangular lines with fixed angles.

Contribution

It develops a new hierarchy of $k$-point bounds that generalize previous bounds and provides optimized methods to compute these bounds for higher points.

Findings

Successfully computed 4, 5, and 6-point bounds for equiangular lines.

Extended the existing bounds from 2-point and 3-point to higher points.

Enhanced the accuracy of bounds for equiangular lines in Euclidean space.

Abstract

We give a hierarchy of $k$-point bounds extending the Delsarte-Goethals-Seidel linear programming $2$-point bound and the Bachoc-Vallentin semidefinite programming $3$-point bound for spherical codes. An optimized implementation of this hierarchy allows us to compute~$4$, $5$, and $6$-point bounds for the maximum number of equiangular lines in Euclidean space with a fixed common angle.

Figures

Tables

Table 1. Table 2. By considering  Δ k ​ ( G ) ∗ subscript Δ 𝑘 superscript 𝐺 \Delta_{k}(G)^{*} for k ≥ 4 𝑘 4 k\geq 4 we find out that the maximum dimension  n 𝑛 n for which the bound M a ​ ( n ) ≤ ( 1 / a 2 − 2 ) ​ ( 1 / a 2 − 1 ) / 2 subscript 𝑀 𝑎 𝑛 1 superscript 𝑎 2 2 1 superscript 𝑎 2 1 2 M_{a}(n)\leq(1/a^{2}-\nobreak 2)(1/a^{2}-1)/2 is valid is larger than  3 / a 2 − 16 3 superscript 𝑎 2 16 3/a^{2}-16 as given by Theorem  5.6 and  Δ 3 ​ ( G ) ∗ subscript Δ 3 superscript 𝐺 \Delta_{3}(G)^{*} ; the table shows the improved dimensions.

$𝒂$	$\left(\mathrm{𝟏}/{𝒂}^{\mathrm{𝟐}}-\mathrm{𝟐}\right)\left(\mathrm{𝟏}/{𝒂}^{\mathrm{𝟐}}-\mathrm{𝟏}\right)/\mathrm{𝟐}$	${𝚫}_{\mathrm{𝟑}}{\left(𝑮\right)}^{\ast }$ [5, 25]	${𝚫}_{\mathrm{𝟒}}{\left(𝑮\right)}^{\ast }$	${𝚫}_{\mathrm{𝟓}}{\left(𝑮\right)}^{\ast }$	${𝚫}_{\mathrm{𝟔}}{\left(𝑮\right)}^{\ast }$
$1/5$	$276$	$60$	$65$	$69$	$70$
$1/7$	$1128$	$131$	$145$	$158$	$169$
$1/9$	$3160$	$227$	$251$	$273$	$300$
$1/11$	$7140$	$347$	$381$	$413$	$448$

Equations

$$M(n)=\max \{A(n,\{a,-a\}):a\in [-1,1)\}.$$

$$\displaystyle\max\bigg{\{}\,\sum_{S\in I_{=1}}\nu_{S}:\nu\in\mathbb{R}^{I_{2t}}_{\geq 0},\ \nu_{\emptyset}=1,\text{ and }M(\nu)\succeq 0\,\bigg{\}},$$

$$M(\nu {)}_{J,J^{\prime }}=\{\begin{array}{ll}{\nu }_{J\cup J^{\prime }}& \text{if }J\cup J^{\prime }\text{ is independent};\\ 0& \text{otherwise}\end{array}$$

$$\displaystyle\max\bigg{\{}\,1+2\sum_{S\in I_{=2}}\nu_{S}:\nu\in\mathbb{R}^{I_{2t}\setminus\{\emptyset\}}_{\geq 0},\ \sum_{S\in I_{=1}}\nu_{S}=1,\text{ and }M(\nu)\succeq 0\,\bigg{\}},$$

$$M_Q(\nu )(x,y)=\{\begin{array}{ll}{\nu }_{Q\cup \{x,y\}}& \text{if Q∪{x,y}∈Ik};\\ 0& \text{otherwise}\end{array}$$

$$\displaystyle\max\bigg{\{}\,1+2\sum_{S\in I_{=2}}\nu_{S}:\nu\in\mathbb{R}^{I_{k}\setminus\{\emptyset\}}_{\geq 0},\ \sum_{S\in I_{=1}}\nu_{S}=1,\text{ and }M_{Q}(\nu)\succeq 0\text{ for }Q\in I_{k-2}\,\bigg{\}}.$$

$$\mathcal{M}(I_k\setminus \{\varnothing \}{)}_{\ge 0}=\{\nu \in \mathcal{M}(I_k\setminus \{\varnothing \}):\nu (f)\ge 0\text{ for all }f\in \mathcal{C}(I_k\setminus \{\varnothing \}{)}_{\ge 0}\};$$

$$B_{k}T(S)=\sum _{\begin{array}{c}Q\subseteq S\\ |Q|\le k-2\end{array}}\sum _{\begin{array}{c}x,y\in S\\ Q\cup \{x,y\}=S\end{array}}T(x,y,Q).$$

$$\sum _{Q\in I_{k-2}}\sum _{x,y\in V}{M}_Q(\nu )(x,y)T(x,y,Q)$$

$$=\sum _{S\in I_k\setminus \{\varnothing \}}{\nu }_S{B}_{k}T(S).$$

$$\begin{gathered} {\Delta }_k(G)=\sup \{1+2\nu (I_{=2}):\nu \in \mathcal{M}(I_k\setminus \{\varnothing \}{)}_{\ge 0}, \\ \nu (I_{=1})=1,\text{ and }{B}_k^{\ast }\nu \in \mathcal{M}(V^2\times I_{k-2}{)}_{⪰0}\}. \end{gathered}$$

$${\Delta }_k(G{)}^{\ast }=\inf \{1+\lambda :\lambda \in \mathbb{R},T\in \mathcal{C}(V^2\times I_{k-2}{)}_{⪰0},\text{ and }{B}_{k}T\le \lambda {\chi }_{I_{=1}}-2{\chi }_{I_{=2}}\},$$

$$\sum _{\begin{array}{c}S\subseteq C\\ |S|\le k,S\ne \varnothing \end{array}}{B}_{k}T(S)\le \left(\genfrac{}{}{0ex}{}{|C|}{1}\right)\lambda +\left(\genfrac{}{}{0ex}{}{|C|}{2}\right)(-2)=|C|(1+\lambda -|C|).$$

$$\sum _{\begin{array}{c}S\subseteq C\\ |S|\le k,S\ne \varnothing \end{array}}{B}_{k}T(S)$$

$$=\sum _{\begin{array}{c}Q\subseteq C\\ |Q|\le k-2\end{array}}\sum _{x,y\in C}T(x,y,Q)\ge 0$$

$$(\gamma K)(x,y)=K({\gamma }^{-1}x,{\gamma }^{-1}y).$$

$$\gamma \varnothing =\varnothing \text{and}\gamma \{x_1,\dots ,x_t\}=\{\gamma x_1,\dots ,\gamma x_t\},$$

$$(\gamma T)(x,y,S)=T({\gamma }^{-1}x,{\gamma }^{-1}y,{\gamma }^{-1}S).$$

$$\overline{T}(x,y,S)={\int }_{\Gamma }T({\gamma }^{-1}x,{\gamma }^{-1}y,{\gamma }^{-1}S)\mathrm{d}\gamma ,$$

$$\Psi :\underset{R\in {\mathcal{R}}_{k-2}}{⨁}\mathcal{C}(V^2{)}^{{\mathrm{Stab}}_{\Gamma }(R)}\to \mathcal{C}(V^2\times I_{k-2}{)}^{\Gamma }$$

$$\Psi ((K_R{)}_{R\in {\mathcal{R}}_{k-2}})(x,y,Q)=K_{{\gamma }_{Q}Q}({\gamma }_{Q}x,{\gamma }_{Q}y)$$

$$\coprod _{R\in {\mathcal{R}}_{k-2}}{V}^2/{\mathrm{Stab}}_{\Gamma }(R),$$

$$\psi :\coprod _{R\in {\mathcal{R}}_{k-2}}{V}^2/{\mathrm{Stab}}_{\Gamma }(R)\to (V^2\times I_{k-2})/\Gamma$$

$${\psi }^{-1}(\Gamma (x,y,Q))=({\mathrm{Stab}}_{\Gamma }({\gamma }_{Q}Q)({\gamma }_{Q}x,{\gamma }_{Q}y),{\gamma }_{Q}Q).$$

$$\psi_{R}\big{(}\mathrm{Stab}_{\Gamma}(R)(x,y)\big{)}=\Gamma(x,y,R)$$

$$\mathcal{C}\biggl{(}\coprod_{R\in\mathcal{R}_{k-2}}V^{2}/\mathrm{Stab}_{\Gamma}(R)\biggr{)}\simeq\bigoplus_{R\in\mathcal{R}_{k-2}}\mathcal{C}(V^{2})^{\mathrm{Stab}_{\Gamma}(R)}$$

$$\mathcal{C}((V^2\times I_{k-2})/\Gamma )\simeq \mathcal{C}(V^2\times I_{k-2}{)}^{\Gamma },$$

$$\mathcal{C}\biggl{(}\coprod_{R\in\mathcal{R}_{k-2}}V^{2}/\mathrm{Stab}_{\Gamma}(R)\biggr{)}\to\mathcal{C}((V^{2}\times I_{k-2})/\Gamma),\qquad f\mapsto f\circ\psi^{-1},$$

$$\mathcal{C}(S^{n-1}\times S^{n-1}{)}_{⪰0}^{{\mathrm{Stab}}_{\mathrm{O}(n)}(R)}\text{for }R\in {\mathcal{R}}_{k-2}$$

$${\mathrm{Stab}}_{\mathrm{O}(n)}(R)\simeq {\mathcal{S}}_R\times {\mathrm{Stab}}_{\mathrm{O}(n)}(\mathrm{span}(R)),$$

$$K(x,y)=\sum _{l=0}^{\infty }{a}_l{P}_l^n(x\cdot y)$$

Full text

D.d.L. was partially supported by NWO Rubicon grant 680-50-1528. F.C.M was supported by grant #2017/25237-4, from the São Paulo Research Foundation (FAPESP) and was financed in part by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior — Brasil (CAPES) — Finance Code 001. F.V. was partially supported by the SFB/TRR 191 “Symplectic Structures in Geometry, Algebra and Dynamics”, funded by the DFG. This project has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie agreement number 764759.

1. Introduction

Given $D\subseteq[-1,1)$, a subset $C$ of the unit sphere $S^{n-1}=\{\,x\in\mathbb{R}^{n}:\|x\|=1\,\}$ is a spherical $D$-code if $x\cdot y\in D$ for all distinct $x$, $y\in C$, where $x\cdot y$ is the Euclidean inner product between $x$ and $y$. The maximum cardinality of a spherical $D$-code in $S^{n-1}$ is denoted by $A(n,D)$.

A fundamental tool for computing upper bounds for $A(n,D)$ is the linear programming bound of Delsarte, Goethals, and Seidel [12]. If $D=[-1,\cos(\pi/3)]$, then $A(n,D)$ is the kissing number, the maximum number of pairwise nonoverlapping unit spheres that can touch a central unit sphere; the linear programming bound was one of the first nontrivial upper bounds for the kissing number.

The linear programming bound is the optimal value of a convex optimization problem. It is a $2$-point bound, because it takes into account interactions between pairs of points on the sphere: pairs $\{x,y\}$ with $x\cdot y\notin D$ correspond to constraints in the optimization problem. Bachoc and Vallentin [2] extended the linear programming bound to a $3$-point bound by taking into account interactions between triples of points. The resulting semidefinite programming bound gives the best known upper bounds for the kissing number in dimensions $n=5$, …, $23$, except for dimension $8$ where the linear programming bound is sharp.

In this paper, we give a hierarchy of $k$-point bounds that extend both the linear and semidefinite programming bounds. The parameter $A(n,D)$ is the independence number of a topological packing graph, namely the infinite graph with vertex set $S^{n-1}$ in which two vertices $x$ and $y$ are adjacent if $x\cdot y\notin D$. The linear programming bound corresponds to an extension of the Lovász theta number to this infinite graph [1]. In Section 2, we derive our hierarchy from a generalization [11] of Lasserre’s hierarchy to the independent set problem in these topological packing graphs. The first level of our hierarchy is the Lovász theta number, and is therefore equivalent to the linear programming bound; the second level is the semidefinite programming bound, as shown in Section 4.2.

Though our hierarchy is not as strong, in theory, as Lasserre’s hierarchy, it is computationally less expensive. This allows us to use it to compute $4$, $5$, and $6$-point bounds for the maximum number of equiangular lines with a certain angle, a problem that corresponds to the case $|D|=2$. Aside from a previous result of de Laat [10], which uses Lasserre’s hierarchy directly, this is the first successful use of $k$-point bounds for $k>3$ for geometrical problems; it yields improved bounds for the number of equiangular lines with given angles in several dimensions.

To perform computations, we transform the resulting problems into semidefinite programming problems. To this end, for a given $k\geq 2$ we use a characterization of kernels $K\colon S^{n-1}\times S^{n-1}\to\mathbb{R}$ on the sphere that are invariant under the action of the subgroup of the orthogonal group that stabilizes $k-2$ given points. For $k=2$, this characterization was given by Schoenberg [40] and for $k=3$, by Bachoc and Vallentin [2]; Musin [33] extended these two results for $k>3$; a similar result is given by Kuryatnikova and Vera [39].

Our hierarchy is particularly suited for problems like the equiangular lines problem, where $D$ is finite. Indeed, let $I_{k}$ be the set of all spherical $D$-codes of cardinality at most $k$, and let the orthogonal group act on $S\in I_{k}$ by rotating each point in $S$. The final semidefinite programming problem has one block corresponding to each orbit of $I_{k}$ under this action. So, if $k\leq 3$ or $D$ is finite, then the number of orbits is finite and we get a finite semidefinite programming problem.

Still, a naive implementation of our approach would be too slow even to generate the problems for $k=5$. The implementation available with the arXiv version of this paper was carefuly written to deal with the orbits of $I_{k}$ in an efficient way; this allows us to generate problems even for $k=6$. This implementation could be of interest to others working on similar problems.

1.1. Equiangular lines

A set of equiangular lines is a set of lines through the origin such that every pair of lines defines the same angle. If this angle is $\alpha$, then such a set of equiangular lines corresponds to a spherical $D$-code where $D=\{a,-a\}$ and $a=\cos\alpha$. So we are interested in finding $A(n,\{a,-a\})$ for a given $a\in[-1,1)$ and also in finding the maximum number of equiangular lines with any given angle, namely

[TABLE]

The study of $M(n)$ started with Haantjes [22] in 1948. He showed that $M(2)=3$ and that the optimal configuration is a set of lines in the plane having a common angle of $60$ degrees. He also showed that $M(3)=6$; the optimal configuration is given by the lines going through opposite vertices of a regular icosahedron, which have a common angle of $63.43\ldots$ degrees. These two constructions provide lower bounds; in both cases, Gerzon’s bound, which states that $M(n)\leq n(n+1)/2$ (see Theorem 5.1 below which is proven for example in Matoušek’s book [32, Miniature 9]), provides matching upper bounds.

In the setting of equiangular lines, the LP bound coincides with the relative bound (see Theorem 5.5). The $3$-point SDP bound was first specialized to this setting by Barg and Yu [4]. Gijswijt, Mittelmann, and Schrijver [16] computed $4$-point SDP bounds for binary codes and Litjens, Polak, and Schrijver [31] extended these $4$-point bounds for $q$-ary codes. For spherical codes, no $k$-point bounds have been computed or formulated for $k\geq 4$.

Next to being fundamental objects in discrete geometry, equiangular lines have applications, for example in the field of compressed sensing: Only measurement matrices whose columns are unit vectors determining a set of equiangular lines can minimize the coherence parameter [15, Chapter 5].

In general, it is a difficult problem to determine $M(n)$ for a given dimension $n$. Currently, the first open case is dimension $n=14$ where it is known that $M(14)$ is either $28$ or $29$; see Table 1. Sequence A002853 in The On-Line Encyclopedia of Integer Sequences [41] is $M(n)$.

2. Derivation of the hierarchy

Let $G=(V,E)$ be a graph. A subset of $V$ is independent if it does not contain a pair of adjacent vertices. The independence number of $G$, denoted by $\alpha(G)$, is the maximum cardinality of an independent set. For an integer $k\geq 0$, let $I_{k}$ be the set of independent sets in $G$ of size at most $k$ and $I_{=k}$ be the set of independent sets in $G$ of size exactly $k$.

Assume for now that $G$ is finite. We can obtain upper bounds for the independence number via the Lasserre hierarchy [27] for the independent set problem, whose $t$-th step can be formulated as

[TABLE]

where $M(\nu)$ is the matrix indexed by $I_{t}\times I_{t}$ such that

[TABLE]

and $M(\nu)\succeq 0$ means that $M(\nu)$ is positive semidefinite. To produce an optimization program where the variables are easier to parameterize, we construct in two stages a weaker hierarchy with matrices indexed only by the vertex set of the graph. First, we modify the problem to remove $\emptyset$ from the domain of $\nu$; this gives the possibly weaker problem

[TABLE]

where $M(\nu)$ is now considered as a matrix indexed by $(I_{t}\setminus\{\emptyset\})\times(I_{t}\setminus\{\emptyset\})$.

Second, we construct a weaker hierarchy by only requiring certain principal submatrices of $M(\nu)$ to be positive semidefinite, an approach similar to the one employed by Gvozdenović, Laurent, and Vallentin [21]. For this we fix $k\geq 2$ and, for each $Q\in I_{k-2}$, define the matrix $M_{Q}(\nu)\colon V\times V\to\mathbb{R}$ by

[TABLE]

and replace the condition ‘$M(\nu)\succeq 0$’ by ‘$M_{Q}(\nu)\succeq 0$ for all $Q\in I_{k-2}$’. With these conditions we can restrict the support of $\nu$ to the set $I_{k}\setminus\{\emptyset\}$, obtaining the relaxation

[TABLE]

We extend this problem to infinite graphs in the same way that the Lasserre hierarchy is extended by de Laat and Vallentin [11]. This extension can be carried out for compact topological packing graphs; these are graphs whose vertex sets are compact Hausdorff spaces and in which every finite clique is contained in an open clique. The independence number of a compact topological packing graph is finite and $I_{k}$, considered with the topology inherited from $V$, is the disjoint union of the compact and open sets $I_{=s}$ for $s=0$, …, $k$ [11, Section 2].

The extension relies on the theory of conic optimization over infinite-dimensional spaces presented e.g. by Barvinok [6]. The first step is to introduce the spaces for the variables of our problem; we will use both the space $\mathcal{C}(X)$ of continuous real-valued functions on a compact space $X$ and its topological dual (with respect to the supremum norm) $\mathcal{M}(X)$, the space of signed Radon measures.

In the infinite setting, the nonnegative variable $\nu$ from (1) becomes a measure in the dual of the cone $\mathcal{C}(I_{k}\setminus\{\emptyset\})_{\geq 0}$ of continuous and nonnegative functions, namely

[TABLE]

when $V$ is finite, $\mathcal{M}(I_{k}\setminus\{\emptyset\})_{\geq 0}$ can be identified with $\mathbb{R}^{I_{k}\setminus\{\emptyset\}}_{\geq 0}$.

Let $\mathcal{C}(V^{2}\times I_{k-2})_{\mathrm{sym}}$ be the set of continuous real-valued functions on $V^{2}\times I_{k-2}$ that are symmetric in the first two coordinates and let $\mathcal{M}(V^{2}\times I_{k-2})_{\mathrm{sym}}$ be the space of symmetric and signed Radon measures111A measure $\mu\in\mathcal{M}(V^{2}\times I_{k-2})$ is symmetric if $\mu(E\times E^{\prime}\times C)=\mu(E^{\prime}\times E\times C)$ for all Borel sets $E$, $E^{\prime}\subseteq V$ and $C\subseteq I_{k-2}$.. A kernel $K\in\mathcal{C}(V^{2})$ is positive if for every finite $U\subseteq V$ the matrix \bigl{(}K(x,y)\bigr{)}_{x,y\in U} is positive semidefinite. A function $T\in\mathcal{C}(V^{2}\times I_{k-2})$ is positive if for every $Q\in I_{k-2}$ the kernel $(x,y)\mapsto T(x,y,Q)$ is positive. The set of all positive functions in $\mathcal{C}(V^{2}\times I_{k-2})$ is a convex cone denoted by $\mathcal{C}(V^{2}\times I_{k-2})_{\succeq 0}$; its dual cone is denoted by $\mathcal{M}(V^{2}\times I_{k-2})_{\succeq 0}$.

Define the operator $B_{k}\colon\mathcal{C}(V^{2}\times I_{k-2})_{\mathrm{sym}}\to\mathcal{C}(I_{k}\setminus\{\emptyset\})$ by

[TABLE]

Note that, though the number of summands in (2) varies with the size of $S$, the function $B_{k}T$ is still continuous since, by the assumption that $G$ is a topological packing graph, $I_{k}\setminus\{\emptyset\}$ can be written as the disjoint union of the compact and open subsets $I_{=s}$ for $s=1$, …, $k$ and $B_{k}T$ is continuous in each of these parts. When $V$ is finite, $B_{k}$ is the adjoint of the operator that maps $\nu\in\mathbb{R}^{I_{k}\setminus\{\emptyset\}}_{\geq 0}$ to the tuple of matrices \big{(}M_{Q}(\nu)\big{)}_{Q\in I_{k-2}}, since the inner product between $(x,y,Q)\mapsto M_{Q}(\nu)(x,y)$ and $T$ is equal to the inner product between $\nu$ and $B_{k}T$:

[TABLE]

So, when $V$ is finite, we may rewrite the constraints ‘$M_{Q}(\nu)\succeq 0$ for all $Q\in I_{k-2}$’ from (1) as ‘$B_{k}^{*}\nu\in\mathcal{M}(V^{2}\times I_{k-2})_{\succeq 0}$’.

This last observation leads us naturally to the generalized $k$-point bound. Indeed, when $G=(V,E)$ is a compact topological packing graph, since the number of summands in (2) is bounded by a constant depending only on $k$, the operator $B_{k}$ is continuous. Thus it has an adjoint $B_{k}^{*}\colon\mathcal{M}(I_{k}\setminus\{\emptyset\})\to\mathcal{M}(V^{2}\times I_{k-2})_{\mathrm{sym}}$. Using the adjoint, we define the generalized $k$-point bound for $k\geq 2$:

[TABLE]

For a finite graph with the discrete topology this reduces to (1).

Using the duality theory of conic optimization as described e.g. by Barvinok [6, Chapter IV], we can derive the following dual problem for (3):

[TABLE]

where $\chi_{I_{=1}}$ and $\chi_{I_{=2}}$ are the characteristic functions of $I_{=1}$ and $I_{=2}$, which are continuous since $G$ is a topological packing graph. From now on, we will denote both the optimal value of (4) and the optimization problem itself by $\Delta_{k}(G)^{*}$.

It is a direct consequence of weak duality that $\Delta_{k}(G)^{*}$ is an upper bound for the independence number of $G$, but it is instructive to see a direct proof.

Proposition 2.1**.**

If $G=(V,E)$ is a compact topological packing graph, then $\alpha(G)\leq\Delta_{k}(G)^{*}$.

Proof.

Let $C\subseteq V$ be a nonempty independent set and let $(\lambda,T)$ be a feasible solution of $\Delta_{k}(G)^{*}$. On the one hand, since $B_{k}T\leq\lambda\chi_{I_{=1}}-2\chi_{I_{=2}}$, we have

[TABLE]

On the other hand, since $T\in\mathcal{C}(V^{2}\times I_{k-2})_{\succeq 0}$, we have

[TABLE]

since, by the definition of $\mathcal{C}(V^{2}\times I_{k-2})_{\succeq 0}$, the matrices \bigl{(}T(x,y,Q)\bigr{)}_{x,y\in C} are positive semidefinite for all $Q\in I_{k-2}$. Putting it all together we get $|C|\leq 1+\lambda$. ∎

3. Parameterizing the variables by positive semidefinite matrices

Symmetry reduction plays a key role in the computation of $\Delta_{k}(G)^{*}$ via semidefinite programming. We now see how to exploit symmetry to parameterize the variable $T$ of (4) in terms of positive semidefinite matrices, an essential step in the solution of these optimization problems by computer.

3.1. Symmetry reduction

Let $\Gamma$ be a compact group that acts continuously on $V$ and that is a subgroup of the automorphism group222The automorphism group $\mathrm{Aut}(G)$ of a graph $G=(V,E)$ is the group of permutations $\sigma\colon V\to V$ that respect the adjacency relation, that is, $\sigma(x)$ and $\sigma(y)$ are adjacent if and only if $x$ and $y\in V$ are adjacent. of the graph $G$. The group $\Gamma$ acts coordinatewise on $V^{2}$, and this action extends to an action on $\mathcal{C}(V^{2})$ by

[TABLE]

The group $\Gamma$ acts continuously on $I_{t}$ by

[TABLE]

and hence it also acts on $\mathcal{C}(V^{2}\times I_{k-2})_{\mathrm{sym}}$ by

[TABLE]

If $\Gamma$ acts on a set $X$, we denote by $X^{\Gamma}$ the set of elements of $X$ that are invariant under this action. In this way we write $\mathcal{C}(V^{2})^{\Gamma}$, $\mathcal{C}(V^{2}\times I_{k-2})_{\succeq 0}^{\Gamma}$, etc.

Given a feasible solution $(\lambda,T)$ of $\Delta_{k}(G)^{*}$, the pair $(\lambda,\overline{T})$ with

[TABLE]

where we integrate against the Haar measure on $\Gamma$ normalized so that the total measure is 1, is also feasible with the same objective value. So we may assume that $T$ is invariant under the action of $\Gamma$.

Let $\mathcal{R}_{k-2}$ be a complete set of representatives of the orbits of $I_{k-2}/\Gamma$. For $R\in\mathcal{R}_{k-2}$, let $\mathrm{Stab}_{\Gamma}(R)=\{\,\gamma\in\Gamma:\gamma R=R\,\}$ be the stabilizer of $R$ with respect to $\Gamma$ and, for $Q\in\Gamma R$, let $\gamma_{Q}\in\Gamma$ be a group element such that $\gamma_{Q}Q=R$. When $I_{k-2}/\Gamma$ is finite, we can decompose the space $\mathcal{C}(V^{2}\times I_{k-2})^{\Gamma}$ as a direct sum of simpler spaces.

Proposition 3.1**.**

If $I_{k-2}/\Gamma$ is finite, then

[TABLE]

given by

[TABLE]

is an isomorphism that moreover preserves positivity, that is, if $(K_{R})_{R\in\mathcal{R}_{k-2}}$ is such that $K_{R}$ is a positive kernel for each $R$, then $\Psi((K_{R})_{R\in\mathcal{R}_{k-2}})$ is positive.

Proof.

We first show that $(V^{2}\times I_{k-2})/\Gamma$ is homeomorphic to the coproduct

[TABLE]

which is the union of $(V^{2}/\mathrm{Stab}_{\Gamma}(R))\times\{R\}$ over all $R\in\mathcal{R}_{k-2}$, endowed with the disjoint union topology. More precisely, we show that

[TABLE]

given by $\psi(\mathrm{Stab}_{\Gamma}(R)(x,y),R)=\Gamma(x,y,R)$ is such a homeomorphism with inverse

[TABLE]

Indeed, the map $\psi$ is well defined because $\Gamma(x,y,R)=\Gamma(\gamma x,\gamma y,R)$ for all $\gamma$ in $\mathrm{Stab}_{\Gamma}(R)$. For each $R\in\mathcal{R}_{k-2}$, the map $\psi_{R}\colon V^{2}/\mathrm{Stab}_{\Gamma}(R)\to(V^{2}\times I_{k-2})/\Gamma$ given by

[TABLE]

is continuous, as follows from the definition of quotient topology. By the definition of disjoint union topology on the coproduct this implies $\psi$ is continuous.

The map (5) is well defined, for if we replace $\gamma_{Q}$ by $\xi\gamma_{Q}$, where $\xi\in\mathrm{Stab}_{\Gamma}(\gamma_{Q}Q)$, then the right-hand side of (5) does not change. Direct verification shows $\psi^{-1}\circ\psi$ and $\psi\circ\psi^{-1}$ are the identity maps.

Since $\mathcal{R}_{k-2}$ is finite, the domain of $\psi$ is compact. So $\psi$ is a continuous bijection between compact Hausdorff spaces, and hence a homeomorphism.

Now the proposition follows easily. Under the isomorphisms

[TABLE]

and

[TABLE]

the operator $\Psi$ is equal to

[TABLE]

which is a well-defined isomorphism since $\psi$ is a homeomorphism. Finally, it follows directly from the definitions of positive kernels and $\mathcal{C}(V^{2}\times I_{k-2})^{\Gamma}_{\succeq 0}$ that $\Psi$ preserves positivity. ∎

The above proposition shows that to characterize $\mathcal{C}(V^{2}\times I_{k-2})^{\Gamma}$ we need to characterize the sets $\mathcal{C}(V^{2})^{\mathrm{Stab}_{\Gamma}(R)}$ for $R\in\mathcal{R}_{k-2}$. In the next section we give this characterization for the case of spherical symmetry.

3.2. Parameterizing invariant kernels on the sphere

From now on we assume $G=(V,E)$ is the graph where $V=S^{n-1}$ and where two distinct vertices $x$, $y\in S^{n-1}$ are adjacent if $x\cdot y\notin D$ for some closed $D\subseteq[-1,1)$. Taking $\Gamma=\mathrm{O}(n)$, we are in the situation described above. Let us see how to parameterize the cones

[TABLE]

by positive semidefinite matrices. For notational simplicity, we make the assumption that every $R\in\mathcal{R}_{k-2}$ consists of linearly independent vectors, which is true for all $G$ and $k$ considered in this paper.

Let $\{e_{1},\ldots,e_{n}\}$ be the standard basis of $\mathbb{R}^{n}$ and fix $R\in\mathcal{R}_{k-2}$. By rotating a set $R\in\mathcal{R}_{k-2}$ if necessary, we may assume that $R$ is contained in $\mathrm{span}(\{e_{1},\ldots,e_{m}\})$, where $m=\dim(\mathrm{span}(R))$. The stabilizer subgroup of $\mathrm{O}(n)$ with respect to $R$ is isomorphic to the direct product of two groups, namely

[TABLE]

where $\mathcal{S}_{R}$ is isomorphic to a finite subgroup of $\mathrm{O}(m)$ that acts on the first $m$ coordinates and acts on $R$ as a permutation of its elements and $\mathrm{Stab}_{\mathrm{O}(n)}(\mathrm{span}(R))$ is a group isomorphic to $\mathrm{O}(n-m)$ that acts on the last $n-m$ coordinates.

If $k=2$, then $R=\emptyset$ and $\mathrm{Stab}_{\mathrm{O}(n)}(\mathrm{span}(R))=\mathrm{O}(n)$. By a classical result of Schoenberg [40], each positive $\mathrm{O}(n)$-invariant kernel $K\colon S^{n-1}\times S^{n-1}\to\mathbb{R}$ is of the form

[TABLE]

for some nonnegative numbers $a_{0}$, $a_{1}$, … with absolute and uniform convergence, where $P_{l}^{n}$ is the Gegenbauer polynomial of degree $l$ in dimension $n$ normalized so that $P_{l}^{n}(1)=1$ (equivalently, $P_{l}^{n}$ is the Jacobi polynomial with both parameters equal to $(n-3)/2$).

Kernels invariant under the stabilizer of one point were considered by Bachoc and Vallentin [2] and kernels invariant under the stabilizer of more points were considered by Musin [34]. The analogue of Schoenberg’s theorem for kernels invariant under the stabilizer of one or more points is stated in terms of multivariate Gegenbauer polynomials.

For $0\leq m\leq n-2$, $t\in\mathbb{R}$, and $u$, $v\in\mathbb{R}^{m}$, the multivariate Gegenbauer polynomial $P_{l}^{n,m}$ is the $(2m+1)$-variable polynomial

[TABLE]

where $\|v\|=\sqrt{v\cdot v}$. If we use the convention $\mathbb{R}^{0}=\{0\}$, then $P_{l}^{n}(t)=P_{l}^{n,0}(t,0,0)$. Since the Gegenbauer polynomials are odd or even according to the parity of $l$, the function $P_{l}^{n,m}(t,u,v)$ is indeed a polynomial in the variables $u$, $v$, and $t$. Musin [34] denotes $P_{l}^{n,m}$ by $G_{l}^{(n,m)}$ and Bachoc and Vallentin [2] denote $P_{l}^{n,1}$ by $Q_{l}^{n-1}$.

Fix $d\geq 0$, let $\mathcal{B}_{l}$ be a basis of the space of $m$-variable polynomials of degree at most $l$ (e.g. the monomial basis), and write $z_{l}(u)$ for the column vector containing the polynomials in $\mathcal{B}_{l}$ evaluated at $u\in\mathbb{R}^{m}$. Let $Y_{l}^{n,m}$ be the matrix of polynomials

[TABLE]

The choice of $d$ makes $Y_{l}^{n,m}$ a $\binom{d-l+m}{m}\times\binom{d-l+m}{m}$ matrix with $(2m+1)$-variable polynomials of degree at most $2d$ as its entries.

Given a matrix $X$ with linearly independent columns, set $L(X)=B^{-1}X^{\sf T}$, where $B$ is the matrix such that $BB^{\sf T}$ is the Cholesky factorization of $X^{\sf T}X$, which is unique since $X^{\sf T}X$ is positive definite. For each $R\in\mathcal{R}_{k-2}$, fix a matrix $A_{R}$ whose columns are the vectors of $R$ in some order. The rows of $L(A_{R})$ span the same space as the columns of $A_{R}$ because $B$ is invertible, and by construction the rows of $L(A_{R})$ are orthonormal:

[TABLE]

Therefore, for $x\in\mathbb{R}^{n}$, $L(A_{R})x$ is a vector with the coordinates of the projection of $x$ onto $\mathrm{span}(R)$ with respect to an orthonormal basis of the linear span.

The following theorem is a restatement of a result of Musin [34, Corollary 3.2] in terms of invariant kernels and with adapted notation. We will use only the sufficiency part of the statement, proved in Appendix A for completeness.

For square matrices $A$, $B$ of the same dimensions, write $\langle A,B\rangle=\operatorname{tr}(A^{\sf T}B)$ for their trace product.

Theorem 3.2**.**

Let $R\subseteq S^{n-1}$ with $m=\dim(\mathrm{span}(R))=|R|\leq n-2$ and let $A_{R}$ be a matrix whose columns are the vectors of $R$ in some order. Fix $d\geq 0$ and, for each $0\leq l\leq d$, let $F_{l}$ be a positive semidefinite matrix of size $\binom{d-l+m}{m}\times\binom{d-l+m}{m}$. Then $K\colon S^{n-1}\times S^{n-1}\to\mathbb{R}$ given by

[TABLE]

is a positive, continuous, and $\mathrm{Stab}_{\mathrm{O}(n)}(\mathrm{span}(R))$-invariant kernel. Conversely, every $\mathrm{Stab}_{\mathrm{O}(n)}(\mathrm{span}(R))$-invariant kernel $K\in\mathcal{C}(S^{n-1}\times S^{n-1})_{\succeq 0}$ can be uniformly approximated by kernels of the above form.

Theorem 3.2 gives us a parameterization of $\mathrm{Stab}_{\mathrm{O}(n)}(\mathrm{span}(R))$-invariant kernels. To get a parameterization of $\mathrm{Stab}_{\mathrm{O}(n)}(R)$-invariant kernels we still have to deal with the symmetries in $\mathcal{S}_{R}$. By construction, for an orthogonal matrix $\sigma\in\mathcal{S}_{R}$ there is a permutation matrix $P_{\sigma}$ such that $\sigma A_{R}=A_{R}P_{\sigma}$. Since $\sigma\in\mathrm{O}(n)$ and $A_{R}^{\sf T}A_{R}=A_{R}^{\sf T}\sigma^{\sf T}\sigma A_{R}=P_{\sigma}^{\sf T}A_{R}^{\sf T}A_{R}P_{\sigma}$, the elements of $\mathcal{S}_{R}$ correspond precisely to the symmetries of the Gram matrix $A_{R}^{\sf T}A_{R}$ under simultaneous permutations of rows and columns. Indeed, if the Gram matrix $A_{R}^{\sf T}A_{R}$ is invariant under a certain simultaneous permutation of rows and columns, then since $R$ is linearly independent, this permutation defines a linear transformation of $\mathrm{span}(R)$ that preserves all inner products between vectors of $R$, whence it is orthogonal and therefore corresponds to an element of $\mathcal{S}_{R}$. This observation leads to the following corollary.

Corollary 3.3**.**

Let $R\subseteq S^{n-1}$ with $m=\dim(\mathrm{span}(R))=|R|\leq n-2$ and let $A_{R}$ be a matrix whose columns are the vectors of $R$ in some order. Fix $d\geq 0$ and, for each $0\leq l\leq d$, let $F_{l}$ be a positive semidefinite matrix of size $\binom{d-l+m}{m}\times\binom{d-l+m}{m}$. Then $K\colon S^{n-1}\times S^{n-1}\to\mathbb{R}$ given by

[TABLE]

where

[TABLE]

is a positive, continuous, and $\mathrm{Stab}_{\mathrm{O}(n)}(R)$-invariant kernel.

Proof.

If $K$ is given by (7), then by writing

[TABLE]

we see using Theorem 3.2 that $K$ is a sum of $|\mathcal{S}_{R}|$ positive, continuous, and $\mathrm{Stab}_{\mathrm{O}(n)}\break(\mathrm{span}(R))$-invariant kernels, and hence it is itself positive, continuous, and $\mathrm{Stab}_{\mathrm{O}(n)}\break(\mathrm{span}(R))$-invariant.

Since, for every $\sigma\in\mathcal{S}_{R}$,

[TABLE]

(recall $BB^{\sf T}$ is the Cholesky decomposition of $A_{R}^{\sf T}A_{R}=(A_{R}P_{\sigma})^{\sf T}(A_{R}P_{\sigma})$), and since $x\cdot y=(\sigma^{\sf T}x)\cdot(\sigma^{\sf T}y)$, we have that

[TABLE]

where

[TABLE]

Now it follows directly from (8) that $K$ is $\mathrm{Stab}_{\mathrm{O}(n)}(R)$-invariant. ∎

4. Semidefinite programming formulations

Before giving the semidefinite programming formulations, let us discuss how the matrix-valued function $\mathcal{F}_{l}(R)(x,y)$ can be computed. We have

[TABLE]

where $BB^{\sf T}$ is the Cholesky decomposition of $A_{R}^{\sf T}A_{R}=(A_{R}P_{\sigma})^{\sf T}(A_{R}P_{\sigma})$. This shows that $L(A_{R}P_{\sigma})x$ depends only on the inner products between the vectors in the set $R\cup\{x\}$ and on the ordering of the columns of $A_{R}$. Since the size of $R$ is bounded by $k-2$, this also shows that all computations for setting up the problem can be done in a relatively small dimension depending on $k$ and not on $n$.

4.1. An SDP formulation for spherical finite-distance sets

To write the full semidefinite programming formulation corresponding to (4), we use Corollary 3.3 together with the isomorphism from Proposition 3.1. Let $\mathcal{S}_{\succeq 0}^{N}$ denote the cone of $N\times N$ positive semidefinite matrices. If for $R\in\mathcal{R}_{k-2}$ and $0\leq l\leq d$ we have $F_{R,l}\in\mathcal{S}_{\succeq 0}^{N}$, where $N=\binom{d-l+|R|}{|R|}$, then $T\colon S^{n-1}\times S^{n-1}\times I_{k-2}\to\mathbb{R}$ given by

[TABLE]

is a function in $\mathcal{C}(S^{n-1}\times S^{n-1}\times I_{k-2})_{\succeq 0}^{\mathrm{O}(n)}$ and hence, for $S\in\mathcal{R}_{k}\setminus\{\emptyset\}$, the expression for $B_{k}T(S)$ becomes

[TABLE]

Since the action of $\mathrm{O}(n)$ on $S^{n-1}\simeq I_{=1}$ is transitive, the quotient $I_{=1}/\mathrm{O}(n)$ has only one element. We set $\mathcal{R}_{1}\setminus\mathcal{R}_{0}=\{e_{1}\}$, where $e_{1}$ is the first canonical basis vector of $\mathbb{R}^{n}$. We replace the objective $1+\lambda$ in (4) by $1+B_{k}T(\{e_{1}\})$, which we can further simplify by noticing that $Y_{0}^{n,1}(1,1,1)$ is the all-ones matrix $J_{d+1}$ of size $(d+1)\times(d+1)$ and $Y_{l}^{n,1}(1,1,1)$ is the zero matrix for $l>0$. This gives the semidefinite programming formulation

[TABLE]

For each fixed $d$ this gives an upper bound for $\Delta_{k}(G)^{*}$ that converges to $\Delta_{k}(G)^{*}$ as $d$ tends to infinity.

4.2. A precise connection between the Bachoc-Vallentin bound

and the Lasserre hierarchy

The bound $\Delta_{2}(G)^{*}$ immediately reduces to the generalization of the Lovász $\vartheta$ number as given by Bachoc, Nebe, Oliveira, and Vallentin [1], which coincides with the LP bound [12] after symmetry reduction. Here we show that $\Delta_{3}(G)^{*}$ can be interpreted as a nonsymmetric version of the Bachoc-Vallentin $3$-point bound [2].

Suppose $T$ is feasible for $\Delta_{3}(G)^{*}$. If $S=\{a,b\}$ with $a\neq b$, then

[TABLE]

By using $T(x,y,\emptyset)=\sum_{l=0}^{d}F_{\emptyset,l}P_{l}^{n}(x\cdot y)$ and

[TABLE]

we see that

[TABLE]

where we use the notation $S_{l}^{n}=\frac{1}{6}\sum_{\sigma\in S_{3}}\sigma Y_{l}^{n,1}$, in which $\sigma$ runs through the group of all permutations of three variables and acts on $Y_{l}^{n,1}$ by permuting its arguments.

If $|S|=3$, say $S=\{a,b,c\}$, then

[TABLE]

Using the above expressions we see that the constraints $B_{3}T(S)\leq-2$ for $S\in I_{=2}$ and $B_{3}T(S)\leq 0$ for $S\in I_{=3}$ in $\Delta_{3}(G)^{*}$ are exactly the ones that appear in Theorem 4.2 of Bachoc and Vallentin [2]. Except for the presence of an ad hoc $2\times 2$ matrix variable $B$ that comes from a separate argument, both bounds are identical.

Remark 4.1*.*

Recall that for our method it is essential that $I_{k-2}/\mathrm{O}(n)$ be finite and that $I_{=m}/\mathrm{O}(n)$ can be represented by the set of $m\times m$ positive semidefinite matrices of rank at most $n$ with ones in the diagonal and elements of $D$ elsewhere, up to simultaneous permutations of the rows and columns. So $I_{k-2}/\mathrm{O}(n)$ is finite for $k=2$, $3$, but infinite whenever $D$ is infinite and $k\geq 4$. This explains why it is not clear how to compute a $4$-point bound generalization of the LP [12] and SDP [2] bounds for the size of spherical codes with given minimal angular distance. For the spherical finite-distance problem, however, the set $I_{k-2}/\mathrm{O}(n)$ is always finite, so that we can perform computations beyond $k=3$.

5. Two-distance sets and equiangular lines

If $D=\{a,-a\}$ for some $0<a<1$, then the vectors in a spherical $D$-code correspond to a set of equiangular lines with common angle $\arccos a$. We set

[TABLE]

and write

[TABLE]

for the maximum number of equiangular lines in $\mathbb{R}^{n}$ with any common angle.

A semidefinite programming bound based on the method of Bachoc and Vallentin [2], and hence equivalent to $\Delta_{3}(G)^{*}$, was applied to this problem by Barg and Yu [5] (see also the table computed by King and Tang [25]) which, together with previous results, determines $M(n)$ for most $n\leq 43$.

Barg and Yu present [4, Eqs. (14)–(17)] a semidefinite programming formulation that corresponds exactly to the formulation given in Section 4.1 when $k=3$ (except for an ad hoc $2\times 2$ matrix). In the other papers [5, 25, 38, 45] where this semidefinite program is considered, a primal version is given instead, which is less convenient from the perspective of rigorous verification of bounds.

In this paper we compute new upper bounds for $M_{a}(n)$ for $a=1/5$, $1/7$, $1/9$, and $1/11$ and many values of $n$ using $\Delta_{k}(G)^{*}$ with $k=4$, $5$, and $6$. The results do not improve the known bounds for $M(n)$ but greatly improve the known bounds for $M_{a}(n)$ for certain ranges of dimensions; these results are presented in Section 5.2.

5.1. Overview of the literature

The literature on equiangular lines is vast; here is a summary.

5.1.1. Bounds for $M(n)$

The interest in $M(n)$ started with Haantjes [22], who showed $M(3)=M(4)=6$ in 1948. Since then, much progress has been made using different techniques, and $M(n)$ has been determined for many values of $n\leq 43$. Table 1 presents the known values for $M(n)$ for small dimensions.

The most general bound for $M(n)$, called the absolute bound, is due to Gerzon:

Theorem 5.1** (Gerzon, cf. Lemmens and Seidel [28]).**

We have

[TABLE]

Moreover, if equality holds, then $n=2$, $n=3$, or $n=l^{2}-2$ for some odd integer $l$ and the cosine of the common angle is $a=1/l$.

The four cases where it is known that the bound is attained are $n=2$, $3$, $7$, and $23$. Delsarte, Goethals and Seidel [12, Example 8.3] show that equality holds if and only if the union of the code with its antipodal code is a tight spherical 5-design, and in this case Cohn and Kumar [9] show this union is a universally optimal code (which means it minimizes every completely monotonic potential function in the squared chordal distance). Bannai, Munemasa, and Venkov [3] and Nebe and Venkov [36] show that there are infinitely many odd integers $l$ for which no tight spherical 5-design exists in $S^{n-1}$ with $n=l^{2}-2$, so that Gerzon’s bound cannot be attained in those dimensions. This list starts with $l=7$, $9$, $13$, $21$, $25$, $45$, $57$, $61$, $69$, $85$, $93$, … (resp. $n=47$, $79$, $167$, $439$, $623$, $2023$, $3247$, $3719$, $4759$, $7223$, $8647$, …). For the remaining possible dimensions, attainability is an open problem.

For the dimensions that are not of the form $l^{2}-2$ for some odd integer $l$, the absolute bound can be improved:

Theorem 5.2** **(Glazyrin and Yu [17] and King and

Tang [25]).

Let $l$ be the unique odd integer such that $l^{2}-2\leq n\leq(l+2)^{2}-3$. Then,

[TABLE]

Furthermore, if the bound is attained, then the cosine of the angle between the lines is $a=1/(l+2)$ for the first case and $a=1/l$ for the second.

Glazyrin and Yu also proved another theorem [17, Theorem 4] about the codes that attain the bound from Theorem 5.2:

Theorem 5.3** (Glazyrin and Yu [17]).**

Suppose $l$ is a positive odd integer. If $X$ is a $\{1/l,-1/l\}$-spherical code of size $(l^{2}-2)(l^{2}-1)/2$ contained in $S^{n-1}$ with $n\leq 3l^{2}-16$, then $X$ must belong to a $(l^{2}-2)$-dimensional subspace.

Since $(l+2)^{2}-3\leq 3l^{2}-16$ for $l\geq 5$, this last theorem implies that if the second bound from Theorem 5.2 is attained, then Gerzon’s bound also has to be attained for $n=l^{2}-2$. For the first two cases where tight spherical 5-designs do not exist, this implies $M(n)\leq 1127$ for $47\leq n\leq 75$ and $M(n)\leq 3159$ for $79\leq n\leq 116$. The following theorem is adapted from Larman, Rogers, and Seidel [26, Theorem 2]:

Theorem 5.4** (Larman, Rogers, and Seidel [26]).**

We have

[TABLE]

where $l$ is the largest odd integer such that $l\leq\sqrt{2n}$.

Most of the results for $M(n)$ rely on Theorem 5.4, which shows that to bound $M(n)$ one just has to consider finitely many angles. This motivates the consideration of $M_{a}(n)$ when $1/a$ is an odd integer.

5.1.2. Bounds for $M_{a}(n)$

Bounds for fixed $a$ are known as relative bounds, as opposed to Gerzon’s absolute bound from Theorem 5.1. The first relative bound is due to van Lint and Seidel [43]:

Theorem 5.5** (van Lint and Seidel [43]).**

If $n<1/a^{2}$, then

[TABLE]

As shown by Glazyrin and Yu [17, Theorem 5], Theorem 5.5 can be derived from the positivity of the Gegenbauer polynomials $P_{2}^{n}$, and indeed this is the bound given by the semidefinite programming techniques when $n\leq 1/a^{2}-2$. This bound is also the first case of Theorem 5.2.

After computing the semidefinite programming bound for many values of $n$ and $a$, Barg and Yu [5] observed long ranges $1/a^{2}-2\leq n\leq 3/a^{2}-16$ where the bound remained stable, matching Gerzon’s bound (Theorem 5.1) at $n=1/a^{2}-2$. Based on this observation, Yu [45] proved the following theorem:

Theorem 5.6** (Yu [45]).**

If $n\leq 3/a^{2}-16$ and $a\leq 1/3$, then

[TABLE]

An alternative proof for the previous theorem is given by Glazyrin and Yu [17, Theorem 6], where the use of the positivity of the Gegenbauer polynomials $P_{1}^{n-1}$ and $P_{3}^{n-1}$ is made more explicit. The bounds given by the semidefinite programming method were also used to prove the following theorem:

Theorem 5.7** (Okuda and Yu [38]).**

If $3/a^{2}-16\leq n\leq 3/a^{2}+6/a+1$, then

[TABLE]

The bounds from Theorems 5.5, 5.6, and 5.7 coincide with the values given by the semidefinite programming formulation when $k=3$ (see the points labeled “$\Delta_{3}(G)^{*}$ [5, 25]” in Figures 1–4). Another source of relative bounds is a technique called pillar decomposition, introduced by Lemmens and Seidel [28] and used to determine $M_{1/3}(n)$:

Theorem 5.8** (Lemmens and Seidel [28]).**

If $n\geq 15$, then

[TABLE]

For $a=1/5$, they obtained results that lead to the following conjecture:

Conjecture 5.9** **(Lemmens and

Seidel [28]).

We have

[TABLE]

Note that $276$ is the bound given by Theorem 5.6 when $a=1/5$ and this shows (together with the fact that there exists a $\{-1/5,1/5\}$-code of size $276$ in dimension $n=23$) that the conjecture is true for $n\leq 59$. In fact, the semidefinite programming bound computed by Barg and Yu [5] also shows $M_{1/5}(60)=276$. Neumaier [37] (see also [19, Corollary 6.6]) proved that there exists a large $N$ such that $M_{1/5}(n)=\lfloor\frac{3}{2}(n-1)\rfloor$ for all $n>N$. Neumaier claimed, without a proof, that $N$ should be at most $30251$.

Recently, Lin and Yu [30] made progress in this conjecture by proving some claims from Lemmens and Seidel [28]. The only case still open is when the code has a set with $4$ unit vectors with mutual inner products $-1/5$ and no such set with $5$ unit vectors (up to replacement of some vectors by their antipodes).

Glazyrin and Yu [17] introduced a new method to derive upper bounds for spherical finite-distance sets. By using Gegenbauer polynomials together with the polynomial method, they proved a theorem that, specialized for two-distance sets, is:

Theorem 5.10** (Glazyrin and Yu [17]).**

For all $a$, $b$, and $n$, we have

[TABLE]

whenever the right-hand side is positive.

With this result, they proved the following relative bound, which provides the best bounds for moderately large values of $n$ (see Figures 2–4):

Theorem 5.11** (Glazyrin and Yu [17]).**

If $a\leq 1/3$, then

[TABLE]

King and Tang [25] improved the pillar decomposition technique and got a better bound for $M_{1/5}(n)$ [25, Theorem 7]. Recently, Lin and Yu [30] further improved parts of their argument; by combining [30, Proposition 4.5] with the proof of [25, Theorem 7] we get:

Theorem 5.12** (Lin and Yu [30]).**

If $n\geq 63$, then

[TABLE]

The previous results give three competing methods to bound $M_{1/5}(n)$, each one being the best for a different range of dimensions. One can either use semidefinite programming to bound $M_{1/5}(n)$ directly, use Theorem 5.12 together with semidefinite programming to bound A\big{(}n-4,\{1/13,-5/13\}\big{)}, or use Theorem 5.10. King and Tang [25] made this comparison, computing the semidefinite programming bound $\Delta_{3}(G)^{*}$. See in Table 3 and in Figure 1 the comparison with the new semidefinite programming bound $\Delta_{6}(G)^{*}$.

Regarding asymptotic results, while it is known that $M(n)$ is asymptotically quadratic in $n$, for fixed $a$ we have that $M_{a}(n)$ is linear in $n$. Bukh [8] was the first to show a bound for $M_{a}(n)$ of the form $M_{a}(n)\leq cn$, although with a large constant $c$. Theorem 5.11 has another linear bound good to give results for intermediate values of $n$, while the best asymptotic result is due to Jiang et al. [23]. They completely settled the value of $\lim_{n\to\infty}M_{a}(n)/n$ for every $a$ in terms of a parameter called the spectral radius order $r(\lambda)$, which is defined for $\lambda>0$ as the smallest integer $r$ so there exists a graph with $r$ vertices and adjacency matrix with largest eigenvalue exactly $\lambda$, and is defined $r(\lambda)=\infty$ in case no such graph exists.

Theorem 5.13** (Jiang et al. [23]).**

Fix $0<a<1$. Let $\lambda=(1-a)/(2a)$ and $r=r(\lambda)$ be its spectral radius order. The maximum number $M_{a}(n)$ of equiangular lines in $\mathbb{R}^{n}$ with common angle $\arccos a$ satisfies

1. (a)

$M_{a}(n)=\lfloor r(n-1)/(r-1)\rfloor$* for all sufficiently large $n>n_{0}(a)$ if $r<\infty$.* 2. (b)

$M_{a}(n)=n+o(n)$* as $n\to\infty$ if $r=\infty$.*

Jiang et al. remarks that the $n_{0}(a)$ from their theorem may be really big, though. When $a=1/(2r-1)$ for some positive integer $r$, then $\lambda=r-1$ and $r(\lambda)=r$ (since the complete graph on $r$ vertices has spectral radius $r-1$). Theorem 5.13 confirms a conjecture made by Bukh [8]:

Corollary 5.14** (Jiang et al. [23]).**

If $a=1/(2r-1)$ for some positive integer $r\geq 2$, then for all $n$ sufficiently large,

[TABLE]

There is a simple construction that achieves the value from Corollary 5.14. Let $a=1/(2r-1)$ for some positive integer $r$ and $t,s$ be arbitrary positive integers. Then one can show that a matrix with $t$ diagonal blocks, each of size $r$, and $s$ diagonal blocks of size $1$, with diagonal entries equal to 1, off-diagonal entries inside each block equal to $-a$, and all other entries equal to $a$ is the Gram matrix of a $\{-a,a\}$-code in $S^{(r-1)t+s}$ of size $rt+s$. Letting $t=\lfloor(n-1)/(r-1)\rfloor$ and $s=n-1-(r-1)\lfloor(n-1)/(r-1)\rfloor$ we get the desired size.

5.2. New semidefinite programming bounds

As observed in Section 5, the semidefinite programming bounds computed by Barg and Yu [5] and King and Tang [25] correspond to $\Delta_{3}(G)^{*}$. In this paper we compute new upper bounds for $M_{a}(n)$ for $a=1/5$, $1/7$, $1/9$, and $1/11$ and many values of $n$ using $k=4$, $5$, and $6$. We always use degree $d=5$ for the polynomials since, as reported by Barg and Yu [4], no improvement is observed for larger values of $d$ (but this may change if sets $D$ with cardinality greater than $2$ are considered). The semidefinite programs were produced using a script written in Julia [7] using Nemo [14], were solved with SDPA-GMP [35], and the results were rigorously verified using the interval arithmetic library Arb [24]. The rigorous verification procedure is much simpler than that for similar problems [13]. The scripts used to generate the programs and verify the results can be found with the arXiv version of this paper.

The results are presented in Figures 1–4 and Tables 3–6, where we compile the bounds for $M_{a}(n)$ for each $n$ that is a multiple of $5$; the best bounds are displayed in boldface. While it takes only a few seconds to generate and solve a single instance of the semidefinite programming problem for $k=3$, the process takes about 5 days using a single core of an Intel i7-8650U processor for $k=6$; that is why the tables have some missing values for $\Delta_{6}(G)^{*}$.

No improvements were obtained for $n\leq 3/a^{2}-16$; we observed in this case that $\Delta_{6}(G)^{*}=\Delta_{3}(G)^{*}$ which is equal to the values given by Theorems 5.5 and 5.6. Since this is the range of dimensions that influences $M(n)$, no improvements for $M(n)$ were obtained. We obtained great improvements for all dimensions $n>3/a^{2}-16$, making the semidefinite programming bound competitive with the other methods (like Theorem 5.11) for more dimensions. Asymptotically, the semidefinite programming bounds behave badly, loosing even to Gerzon’s bound.

In particular, we improved the range of dimensions for which the bound remains stable, showing that $n=3/a^{2}-16$ from Theorem 5.6 is not optimal. Table 2 shows how much this range is increased for the values of $a$ considered. This observation motivates the following two questions, where $a$ is such that $1/a$ is an odd integer:

1. (1)

What is the smallest $n$ such that $M_{a}(n)=(1/a^{2}-2)(1/a^{2}-1)/2$? 2. (2)

What is the smallest $n$ such that $M_{a}(n)>(1/a^{2}-2)(1/a^{2}-1)/2$?

Question (1) is the more interesting of the two since if the smallest $n$ is $1/a^{2}-2$, then Gerzon’s bound is attained. Theorem 5.3 makes progress in this direction, showing that Gerzon’s bound is also attained if the smallest $n$ is at most $3/a^{2}-16$; this is known not to be the case for many $a$ (due to the nonexistence of some tight spherical 5-designs, as mentioned after Theorem 5.1), which implies $M_{1/7}(n)\leq 1127$ for $n\leq 131$ and $M_{1/9}(n)\leq 3159$ for $n\leq 227$. Table 2 also suggests that the constraint $n\leq 3/a^{2}-16$ in Theorem 5.3 may not be optimal.

Question (2) seems interesting because Table 2 shows that $n=3/a^{2}-15$ is not a good candidate solution. In fact, the smallest $n$ is likely much larger, as suggested by Conjecture 5.9 for $M_{1/5}(n)$ and the construction described after Corollary 5.14. Using this construction, we know that $(1/a^{2}-2)(1/a^{2}-1)/2$ is achieved when $n=(1/a^{2}-2)(1/a-1)^{2}/2+1$, which corresponds to the dimensions $185$, $847$, $2529$, and $5951$ for $a=1/5$, $1/7$, $1/9$, and $1/11$ respectively.

We also improve the bounds computed by King and Tang [25] for $M_{1/5}(n)$ by replacing their theorem [25, Theorem 7] by Theorem 5.12 and by using $\Delta_{6}(G)^{*}$ to compute better bounds for $A(n,\{1/13,-5/13\})$. Lin and Yu [30] observed that $A(n,\{1/13,-5/13\})\geq 3n/2-3$ and therefore there is a limit to the power of this approach: it will never be able to prove Conjecture 5.9 no matter how much we increase $k$. In general, it is not clear how good the bound $\Delta_{k}(G)^{*}$ can be for $M_{a}(n)$ if one allows $k$ to increase; in contrast, de Laat and Vallentin [11, Theorem 2] show that their version of the Lasserre hierarchy for compact topological packing graphs converges to the independence number if enough steps are computed. Whether such a convergence result holds for $\Delta_{k}(G)^{*}$ is an open question; in any case, it takes days to compute $\Delta_{k}(G)^{*}$ for $k=6$, so one can expect that solving the resulting semidefinite programs for $k>6$ will be hard in practice.

Appendix A Proof of the sufficiency part of Theorem 3.2

We now prove, for the sake of completeness, a theorem that, together with the linear transformation $L(A_{R})$ used to compute the coordinates of the projection of a vector with respect to an orthonormal basis of $\mathrm{span}(R)$, amounts to the sufficiency part of Theorem 3.2, which is the direction used in this paper. It is a restatement of a proposition of Musin [34, Corollary 3.1].

Recall that $P_{l}^{n}$ is the Gegenbauer polynomial of degree $l$ in dimension $n$ normalized so that $P_{l}^{n}(1)=1$. For $0\leq m\leq n-2$, $t\in\mathbb{R}$, and $u$, $v\in\mathbb{R}^{m}$, the multivariate Gegenbauer polynomial $P_{l}^{n,m}$ is the $(2m+1)$-variable polynomial

[TABLE]

where $\|v\|=\sqrt{v\cdot v}$. If we use the convention $\mathbb{R}^{0}=\{0\}$, then $P_{l}^{n}(t)=P_{l}^{n,0}(t,0,0)$. Fix $d\geq 0$, let $\mathcal{B}_{l}$ be a basis of the space of $m$-variable polynomials of degree at most $l$ (e.g. the monomial basis), and write $z_{l}(u)$ for the column vector containing the polynomials of $\mathcal{B}_{l}$ evaluated at $u\in\mathbb{R}^{m}$. The matrix $Y_{l}^{n,m}$ is the matrix of polynomials

[TABLE]

Theorem A.1** (Musin [34]).**

Let $R\subseteq S^{n-1}$ with $m=\dim(\mathrm{span}(R))\leq n-2$ and let $E$ be an $m\times n$ matrix whose rows form an orthonormal basis for $\mathrm{span}(R)$. Fix $d\geq 0$ and, for each $0\leq l\leq d$, let $F_{l}$ be a positive semidefinite matrix of size $\binom{d-l+m}{m}\times\binom{d-l+m}{m}$. Then $K\colon S^{n-1}\times S^{n-1}\to\mathbb{R}$ given by

[TABLE]

is a positive, continuous, and $\mathrm{Stab}_{\mathrm{O}(n)}(\mathrm{span}(R))$-invariant kernel.

First we prove that the multivariate Gegenbauer polynomials satisfy the following positivity property [34, Theorem 3.1].

Proposition A.2** (Musin [34]).**

For $0\leq m\leq n-2$, let $E$ be an $m\times n$ matrix whose rows form an orthonormal set in $\mathbb{R}^{n}$ and $C$ be a finite subset of $S^{n-1}$. Then, for every nonnegative integer $l$, the matrix \big{(}P_{l}^{n,m}(x\cdot y,Ex,Ey)\big{)}_{x,y\in C} is positive semidefinite.

Proof.

If $l=0$ then all polynomials evaluate to $1$ and the proposition holds, so we assume $l\neq 0$. Let $L$ be the space spanned by the rows of $E$ and $z$ be a unit vector in $L^{\perp}$. For each $x\in C$, write $x=x_{1}+x_{2}$ with $x_{1}\in L$ and $x_{2}\in L^{\perp}$. If $\|x_{2}\|>0$, then let $\bar{x}=x_{2}/\|x_{2}\|$, otherwise write $\bar{x}=z$. If $\|x_{2}\|$, $\|y_{2}\|\neq 0$, then

[TABLE]

Since the rows of $E$ are orthonormal, we have $x_{1}\cdot y_{1}=(Ex)\cdot(Ey)$ and hence $\|x_{2}\|^{l}\|y_{2}\|^{l}P_{l}^{n-m}(\bar{x}\cdot\bar{y})=P_{l}^{n,m}(x\cdot y,Ex,Ey)$.

If, say, $\|x_{2}\|=0$, then $\|x_{2}\|^{l}\|y_{2}\|^{l}P_{l}^{n-m}(\bar{x}\cdot\bar{y})=0$, while $P_{l}^{n,m}(x\cdot y,Ex,Ey)$ is also [math] as can be seen from (9) since $x\cdot y-x_{1}\cdot y_{1}=x_{2}\cdot y_{2}=0$.

Now $\{\,\bar{x}:x\in C\,\}$ is contained in an embedding of $S^{n-m-1}$ into $S^{n-1}$ and by Schoenberg’s theorem [40] we have that \big{(}P_{l}^{n-m}(\bar{x}\cdot\bar{y})\big{)}_{x,y\in C} is positive semidefinite. Since \big{(}\|x_{2}\|^{l}\|y_{2}\|^{l}\big{)}_{x,y\in C} is positive semidefinite, so is \big{(}\|x_{2}\|^{l}\|y_{2}\|^{l}P_{l}^{n-m}(\bar{x}\cdot\bar{y})\big{)}_{x,y\in C}, and we are done. ∎

Proof of Theorem A.1.

Since all entries of $Y_{l}^{n,m}$ are polynomials, $K$ is continuous, and since $x\cdot y$, $Ex$, and $Ey$ are invariant under the action of $\mathrm{Stab}_{\mathrm{O}(n)}(\mathrm{span}(R))$ on $(x,y)$, $K$ is invariant. To prove positivity, let $C$ be a finite subset of $S^{n-1}$ and $w\colon C\to\mathbb{R}$ be a function. We have

[TABLE]

To show this quantity is nonnegative, we will show that for all $l=0$, …, $d$ the matrix \sum_{x,y\in C}w_{x}w_{y}Y_{l}^{n,m}\big{(}x\cdot y,Ex,Ey\big{)} is positive semidefinite. For this, write it as a product of matrices: if $B$ is the matrix whose columns are given by $z_{d-l}(Ex)$ for $x\in C$, then

[TABLE]

and, since the matrix \big{(}P_{l}^{n,m}(x\cdot y,Ex,Ey)\big{)}_{x,y\in C} is positive semidefinite by Proposition A.2, we are done. ∎