Bounds on antipodal spherical designs with few angles

A finite subset $X$ on the unit sphere $\mathbb{S}^{d-1}$ is called an $s$-distance set with strength $t$ if its angle set $A(X):=\{\langle \mathbf{x},\mathbf{y}\rangle : \mathbf{x},\mathbf{y}\in X,\mathbf{x}\neq\mathbf{y} \}$ has size $s$, and $X$ is a spherical $t$-design but not a spherical $(t+1)$-design. In this paper, we consider to estimate the maximum size of such antipodal set for small $s$. First, we improve the known bound on $|X|$ for each even integer $s\in[\frac{t+5}{2}, t+1]$ when $t\geq 3$. We next focus on two special cases: $s=3,\ t=3$ and $s=4,\ t=5$. Estimating the size of $X$ for these two cases is equivalent to estimating the size of real equiangular tight frames (ETFs) and Levenstein-equality packings, respectively. We first improve the previous estimate on the size of real ETFs and Levenstein-equality packings. This in turn gives a bound on $|X|$ when $s=3,\ t=3$ and $s=4,\ t=5$, respectively.

1. Introduction 1.1. Spherical designs with few angles. A finite set X ⊂ S d−1 is called an s-distance set if its angle set A(X) := { x, y : x, y ∈ X, x = y} contains s distinct values, and we say X has strength t if t is the largest integer such that X is a spherical t-design. We say that a finite set X ⊂ S d−1 is a spherical t-design if the following equality holds for any polynomial f of degree at most t (see [DGS77]). Here, µ d is the Lebesgue measure on S d−1 normalized by µ d (S d−1 ) = 1. In this paper we focus on the following problem which originally arises in design theory: Problem 1.1. Given s, t ∈ Z + , what is the maximum size of an s-distance set X ⊂ S d−1 with strength t?
Spherical designs with few angles usually display beautiful symmetry and optimality [CK07, BGMPV19,HS96], e.g., the universal optimality of the 600-cell on S 3 [CK07], which have been studied for several decades [DGS77,BD79,BMV04]. Estimating the size of these designs provides a necessary condition on their existence. See [BB09a,NS11] and the references for the recent work.
In this paper we devote our attention to the antipodal case of this problem, i.e., X = −X. We aim to bound the size of antipodal s-distance sets in S d−1 with strength t. Recall that the strength of an antipodal set must be an odd integer [DGS77,Theorem 5.2]. According to [DGS77,Theorem 6.8], we always have (1) |X| ≤ 2 d + s − 2 s − 1 and 2s ≥ t + 1 provided X ⊂ S d−1 is an antipodal s-distance set with strength t. The upper bound in (1) is called the Delsarte-Goethals-Seidel bound for an antipodal spherical s-distance set. Furthermore, the equality in (1) holds if and only if the s-distance set X forms a tight spherical (2s − 1)-design, i.e., t + 1 = 2s. In this paper we will focus on estimating the size of X when 2s is slightly greater than t + 1.
1.2. The optimal line packing problem. It is particularly interesting to consider two special cases, i.e., s = 3, t = 3 and s = 4, t = 5. These two cases are closely related to the optimal line packing problem, which aims to find a finite set Φ = {ϕ i } n i=1 ⊂ S d−1 with fixed size n > d and the minimal coherence µ(Φ) := max i =j | ϕ i , ϕ j | (see [CHS96,FJM18,HHM17,JKM19]). The followings are two well-known lower bounds on the coherence: The (2a) is called the Welch bound [Wel74] and the (2b) is called the Levenstein bound [Lev92,Lev98]. It is well known that the equality in (2a) occurs when Φ ∪ −Φ forms an antipodal 3-distance 3-strength set or an antipodal 3-distance 5-strength set with size d(d + 1) [DGS77, Example 8.3]; and the equality in (2b) occurs when Φ ∪ −Φ forms an antipodal 4-distance 5-strength set or an antipodal 4-distance 7-strength set with size d(d+1)(d+2) 3 [DGS77, Example 8.4]. Hence, estimating the size of the antipodal 3-distance 3-strength sets and of the antipodal 4-distance 5-strength sets is helpful to know the existence of these two kinds of optimal packings. In the context of frame theory, a set achieving the Welch bound in (2a) is known as a real equiangular tight frame (ETF). Hence, bounding the size of an antipodal 3-distance set with strength 3 is equivalent to bounding the size of a real ETF whose size is strictly smaller than d(d+1) 2 . This is particularly interesting since the existence of real ETFs is a long-standing open problem for most pairs (d, n) [FM15,STDH07]. For the nontrivial case where n > d + 1 > 2, an ETF may exist only if its size n satisfies the Gerzon bound [LS73,FM15,FJM18]: 1.3. Related work. We overview the known upper bounds on antipodal spherical designs with few angles. Let X ⊂ S d−1 be an antipodal s-distance set with strength t. As said before, t must be an odd integer [DGS77, Theorem 5.2]. Set If s ≥ 3 is odd and t ∈ [s − 2, 2s − 5], it is easy to see that the bound in (6) lowers the Delsarte-Goethals-Seidel bound by 2h t−s+2 . If s ≥ 2 is even and t ∈ [s−1, 2s−3], the upper bound in (6) becomes 2 d+s−1 s − 2h t−s+1 . When s is fixed, a simple calculation shows that Hence, if s is an fixed even integer and d is large enough, the upper bound in (6) is larger than the Delsarte-Goethals-Seidel bound.
1.4. Our contributions. Assume that X ⊂ S d−1 is an antipodal s-distance set with strength t. The aim of this paper is to present a better upper bound on |X|.
1.4.1. The general case. Motivated by the methods developed in [NS11], we present an upper bound for |X| which lowers the Delsarte-Goethals-Seidel bound when s ∈ [ t+5 2 , t + 1] is an even integer and t ≥ 3.
Theorem 1.1. Let d ≥ 2 be an integer. Assume that X ⊂ S d−1 is an antipodal s-distance set with strength t ≥ 3, where s ∈ [ t+5 2 , t + 1] is an even integer. Then, we have where h k is defined in (4) for each k ≥ 0.
We next consider to estimate |X| for the case when s = t+3 2 . We mainly focus on two special cases : s = 3, t = 3 and s = 4, t = 5. ).
Recall that X ⊂ S d−1 is an antipodal 3-distance set with strength 3 if and only if .
We next introduce another result on the existence of real ETFs. It is well known that the existence of a real ETF for R d with size n > d + 1 > 2 is equivalent to the existence of a strongly regular graph with parameters (n − 1, a, 3a−n 2 , a 2 ) [STDH07, Wal09, FM15], where Since every strongly regular graph satisfies the Krein conditions (see Lemma 4.2 for details), one is interested in whether the Krein conditions are covered by the Gerzon bound (3) or other known necessary conditions (see [Wal09,FM15]). In Proposition 4.1 we will give a positive answer to this question showing that the Krein conditions for strongly regular graphs with parameters (n − 1, a, 3a−n 2 , a 2 ) are equivalent to the Gerzon bound (3). if and only if Φ ∪ −Φ is an antipodal 4-distance set with strength 5 [DGS77,Example 8.4]. Thus, we mainly focus on estimating the size of Levenstein-equality packings.
We begin with providing an estimate on the size of Levenstein-equality packings, i.e., ].

9
. Hence, if we have a tight spherical 7-design in R d+1 , we can obtain a Levenstein-equality packings in R d with size d(d+2) 2

9
. We are interested in knowing whether each Levenstein-equality packing with size d(d+2) 2 9 comes from a tight spherical 7-design.
In Table 1 we summarize the best known upper bounds on the antipodal s-distance t-strength sets so far.
1.5. Organization. The paper is organized as follows. In Section 2, we introduce some definitions and lemmas. After presenting the proof of Theorem 1.1 in Section 3, we prove Theorem 1.2 in Section 4. We also show the equivalence between the Gerzon bound and the necessary conditions on the existence of real ETFs obtained from the Krein conditions in Section 4. Finally we prove Theorem 1.3 in Section 5.

Preliminaries
In this section, we introduce some definitions and lemmas which will be used in later sections.
2.1. Notations. Let Harm k (R d ) be the vector space of all real homogeneous harmonic polynomials of degree k on d variables, equipped with the standard inner product k (x) denote the Gegenbauer polynomial of degree k with the normalization G (d) k (1) = h k , which can be defined recursively as follows (see also [DGS77, Definition 2.1]): The following formulation is well-known [DGS77, Theorem 3.3]: We also need the following notations.
Definition 2.1. For a finite non-empty set X ⊂ S d−1 , we use the following notations: : (v) When X is antipodal, we say the subsetX ⊂ X is a half of X ifX satisfiesX∩−X = ∅ andX ∪ −X = X.
Note that H 0 (X) is exactly the all-ones vector of size |X|. According to (12), we have k ( x, y )) x,y∈X .
Throughout this paper, we use I, J to denote the identity matrix and all-ones matrix of appropriate size, respectively. We also set ∆ k,l := I, if k = l, 0, otherwise.
2.2. Spherical designs. By the notion of characteristic matrices, the following lemma provides two equivalent definitions of spherical t-designs.
We next prove some properties of antipodal spherical designs which will be used in Section 3.
Corollary 2.1. Assume X ⊂ S d−1 is an antipodal set and letX be a half of X. Then, hold when 0 ≤ k + l ≤ t and k ≡ l (mod 2).
(i) According to (14) we have Based on Lemma 2.1 we obtain that X is a spherical t-design if and only if H k (X) T H 0 (X) = 0 h k ×1 for each positive even integer k ≤ t.
(ii) Let k and l be two integers satisfying 0 ≤ k + l ≤ t and k ≡ l (mod 2). Equation (14) implies H k (X) T H l (X) = 2 · H k (X) T H l (X). Thus, according to Lemma 2.1 we obtain that H k (X) T H l (X) = |X| 2 · ∆ k,l if X is a spherical t-design. According to (ii) in Definition 2.1, we have The following lemma played a key role in Nozaki and Suda's framework [NS11]. Its main idea is to identify the size of an s-distance set X with the dimension of a sum of subspaces V k (X) defined in Definition 2.1.  V k (X)).
2.3. Spherical embeddings of strongly regular graphs. In this subsection we briefly introduce the spherical embeddings of strongly regular graphs, which will be used in our analysis of Levenstein-equality packings in Section 5. A regular graph Γ with v vertices and degree k is called strongly regular if every two adjacent vertices have λ common neighbors and every two non-adjacent vertices have µ common neighbors. Let Γ be a strongly regular graph with parameters (v, k, λ, µ). Denote its vertex set by {1, 2, . . . , v} for simplicity. The adjacency matrix A of Γ has three eigenvalues k, r 1 and r 2 , with multiplicities 1, n 1 and n 2 , respectively. The values of r 1 , r 2 , n 1 , n 2 can be calculated as follows [BPR17,Cam04] (15) r 1 = 1 2 ).
For each i ∈ {1, 2}, let E i denote the eigenspace of A with respect to the eigenvalue r i . Then a spherical embedding of Γ with respect to E i is a collection of unit vectors in R n i , obtained by orthogonally projecting a standard basis of R v onto the eigenspace E i and rescaling the projections to have unit norm. It is known that the obtained set is a twodistance spherical 2-design [BPR17,Cam04] In Section 5 we will introduce that each Levenstein-equality packing gives rise to a strongly regular graph. Then we will use one of the spherical embeddings of this strongly regular graph to provide a lower bound on the size of Levenstein-equality packings.

Proof of Theorem 1.1
In this section, motivated by the method developed in [NS11], we present a proof of Theorem 1.1.
Assume s is an even integer. Let X ⊂ S d−1 be an antipodal s-distance set with strength t and letX be a half of X (see (v) in Definition 2.1). Now we focus on estimating the maximum size ofX. Noting that X is antipodal and s is even, we assume

It follows that FX (x) is an odd function. Assume that FX (x) has the Gegenbauer expansion
k (x). It is well known that the Gegenbauer polynomial G is an odd function if k is odd and an even function if k is even [Sze39,Page 59]. This means that f k = 0 provided k ≤ s − 1 is even. Hence, by Lemma 2.2 we obtain Now we aim to prove that V t−s+2 (X) is contained in the sum of some other subspaces V 2k+1 (X) when s ∈ [ t+5 2 , t + 1]. The following lemma is analogous to [NS11, Lemma 3.3].
Lemma 3.1. Suppose X ⊂ S d−1 is an antipodal s-distance set with strength t, where s ∈ [ t+5 2 , t + 1] is an even integer and t ≥ 3 is an odd integer. LetX be a half of X. Assume the annihilator polynomial ofX has the Gegenbauer expansion Proof. Set F := (FX ( x, y )) x,y∈X . Noting that FX (1) = 1 and FX (α) = 0 for α ∈ A(X), we obtain that F is exactly the identity matrix of size |X|. On the other hand, by the Gegenbauer expansion of FX (x) and (13), we have For each integer i satisfying t − s + 2 ≤ 2i + 1 ≤ t−1 2 , we multiply D 2i+1 (X) on both sides of (19) and obtain (20) Since X is an antipodal spherical t-design, by Corollary 2.1, we have Noting that s ∈ [ t+5 2 , t + 1] is an even integer and t ≥ 3 is an odd integer, rearranging equation (21) gives Assume v is an eigenvector of D 2i+1 (X) with respect to an eigenvalue λ = 0. Then, we have which implies Note that a real symmetric matrix of size |X| always has |X| linear independent eigenvectors.
is a set of linear independent eigenvectors of D 2k+1 (X). Assume that λ Thus, if f 2i+1 = 1 |X| , then v can be written as a linear combination of vectors in Since v can be any vector in V 2i+1 (X), we arrive at our conclusion.
It remains to show that the coefficient f t−s+2 in the Gegenbauer expansion of FX (x) is not 1 |X| . We need the following lemma.
With the help of the above lemma, we now show that the coefficient f t−s+2 in the Gegenbauer expansion of FX (x) is not 1 |X| . Actually, we prove that f t−s+2 is the first coefficient with this property.
Lemma 3.3. Suppose X ⊂ S d−1 is an antipodal s-distance set with strength t, where s ∈ [ t+3 2 , t + 1] is an even integer and t ≥ 3 is an odd integer. LetX be a half of X. Assume the annihilator polynomial ofX has the Gegenbauer expansion FX (x) = . Then, f t−s+2 = 1 |X| and f l−s+2 = 1 |X| for each odd integer l satisfying s − 1 ≤ l < t.
h l−s+2 · FX (x) for each odd integer l satisfying s − 1 ≤ l ≤ t. Since FX (x) is a polynomial of degree s − 1, we see that Q l (x) is a polynomial of degree l + 1.
Noting that both G (d) l−s+2 (x) and FX (x) are odd functions, we obtain that Q l (x) is an even function. Thus we can assume Q l (x) has the Gegenbauer expansion Q l ( 2i (x).
On the other hand, we have 2i ( x, y )) Note that H 0 (X) is the all-ones vector of size |X|. According to (13), for each i ∈ {1, 2, . . . , l+1 2 }, we have Combining (24) and (27), we arrive at Since X has strength t, by Corollary 2.1 we have Then, by equation (28) we obtain The (31) implies q We next present a proof of Theorem 1.1.
Proof of Theorem 1.1. Recall that X is an antipodal s-distance set with strength t. Let X be a half of X. Combining Lemma 3.1 and Lemma 3.3, we know that V t−s+2 (X) is contained in V s−1 (X). Then by (18) we have The last inequality in (33) follows from dim V 2k+1 (X) = rank (D 2k+1 (X)) ≤ rank (H 2k+1 (X)) ≤ h 2k+1 .
Noting that |X| = 2|X|, we obtain |X| ≤ 2 d+s−2 Remark 3.1. Lemma 3.1 and Lemma 3.3 can be easily extended to the case when s is an odd integer. Using these extended results one can obtain an upper bound on |X| for odd s ∈ [ t+5 2 , t + 2], which is actually the same with the bound in (6). Hence, for clarity and convenience we only consider the case when s is an even integer in Lemma 3.1 and Lemma 3.3 .

Proof of Theorem 1.2
The aim of this section is to present a proof of Theorem 1.2. We need the following necessary condition on the existence of real ETFs. Using the above lemma, we present a proof of Theorem 1.2.
Hence, if we apply the above lemma to strongly regular graphs with parameters (n − 1, a, 3a−n 2 , a 2 ), then (36a) and (36b) provide two necessary conditions on the existence of nontrivial ETFs. The authors of [Wal09] and [FM15] wondered whether these two necessary conditions are covered by the Gerzon bound (3) or other known necessary conditions. In what follows we show that they are actually equivalent to the Gerzon bound (3).  Proof. Substituting v = n − 1, k = a, λ = 3a−n 2 , µ = a 2 into equation (15), we can represent r 1 and r 2 as follows: Next, substituting equation (37), (38a), (38b) into (36a) and (36b), we obtain LSEC, Inst. Comp. Math., Academy of Mathematics and System Science, Chinese Academy