New upper bounds for spherical codes and packings

Packing densities have been studied extensively, for purely mathematical reasons as well as for their connections to coding theory. The work of Conway and Sloane is a comprehensive reference for this subject [CS99]. We proceed by defining the basics of this subject. Consider $\mathbb{R}^{n}$ equipped with the Euclidean metric $|.|$ and the associated volume $\textup{vol}(.)$. For each real $r>0$ and each $x\in\mathbb{R}^{n}$, we denote by $B_{n}(x,r)$ the open ball in $\mathbb{R}^{n}$ centered at $x$ and of radius $r$. For each discrete set of points $S\subset\mathbb{R}^{n}$ such that any two distinct points $x,y\in S$ satisfy $|x-y|\geq 2$, we can consider

$$\cal{P}:=\cup_{x\in S}B_{n}(x,1),$$

the union of non-overlapping unit open balls centered at the points of $S$. This is called a sphere packing ($S$ may vary), and we may associate to it the function mapping each real $r>0$ to

$$\delta_{\cal{P}}(r):=\frac{\textup{vol}(\cal{P}\cap B_{n}(0,r))}{\textup{vol}(B_{n}(0,r))}.$$

The packing density of $\cal{P}$ is defined as

$$\delta_{\cal{P}}:=\limsup_{r\rightarrow\infty}\delta_{\cal{P}}(r).$$

Clearly, this is a finite number. The maximal sphere packing density in $\mathbb{R}^{n}$ is defined as

$$\delta_{n}:=\sup_{\cal{P}\subset\mathbb{R}^{n}}\delta_{\cal{P}},$$

a supremum over all sphere packings $\cal{P}$ of $\mathbb{R}^{n}$ by non-overlapping unit balls.

The linear programming method initiated by Delsarte is a powerful tool for giving upper bounds on sphere packing densities [Del72]. That being said, we only know the optimal sphere packing densities in dimensions 1,2,3,8 and 24 [FT43, Hal05, Via17, CKM⁺17]. Very recently, the first author proved an optimal upper bound on the sphere packing density of all but a tiny fraction of even unimodular lattices in high dimensions; see [Sar19, Theorem 1.1].

The best known linear programming upper bounds on sphere packing densities in low dimensions are based on the linear programming method developed by Cohn–Elkies [CE03] which itself was inspired by Delsarte’s linear programming method. As far as the exponent is concerned, in high dimensions, the best asymptotic upper bound goes back to Kabatyanskii–Levenshtein from 1978 [KL78] stating that $\delta_{n}\leq 2^{-(0.599+o(1))n}$ as $n\rightarrow\infty$. More recently, de Laat–de Oliveira Filho–Vallentin improved upper bounds in very low dimensions using the semi-definite programming method [dLdOFV14], partially based on the semi-definite programming method developed by Bachoc–Vallentin [BV08] for bounding kissing numbers. The work of Bachoc–Vallentin was further improved by Mittelmann–Vallentin [MV10], Machado–de Oliveira Filho [MdOF18], and very recently after the writing of our paper by de Laat–Leijenhorst [Ld22].

Another recent development is the discovery by Hartman–Mazác–Rastelli [HMR19] of a connection between the spinless modular bootstrap for two-dimensional conformal field theories and the linear programming bound for sphere packing densities. After the writing of our paper, Afkhami-Jeddi–Cohn–Hartman–de Laat–Tajdini [AJCH⁺20] numerically constructed solutions to the Cohn–Elkies linear programming problem and conjectured that the linear programming method is capable of producing an upper bound on sphere packing densities in high dimensions that is exponentially better than that of Kabatyanskii–Levenshtein.

A notion closely related to sphere packings in Euclidean spaces is that of spherical codes. By inequalities relating sphere packing densities to the sizes of spherical codes, Kabatyanskii–Levenshtein [KL78] obtained their bound on sphere packing densities stated above. The sizes of spherical codes are bounded from above using Delsarte’s linear programming method. In what follows, we define spherical codes and this linear programming method.

Given $S^{n-1}$, the unit sphere in $\mathbb{R}^{n}$, a $\theta$-spherical code is a finite subset $A\subset S^{n-1}$ such that no two distinct $x,y\in A$ are at an angular distance less than $\theta$. For each $0<\theta\leq\pi$, we define $M(n,\theta)$ to be the largest cardinality of a $\theta$-spherical code $A\subset S^{n-1}.$

The Delsarte linear programming method is applied to spherical codes as follows. Throughout this paper, we work with probability measures $\mu$ on $[-1,1].$ $\mu$ gives an inner product on the $\mathbb{R}$-vector space of real polynomials $\mathbb{R}[t]$, and let $\{p_{i}\}_{i=0}^{\infty}$ be an orthonormal basis with respect to $\mu$ such that the degree of $p_{i}$ is $i$ and $p_{i}(1)>0$ for every $i$. Note that $p_{0}=1$. Suppose that the basis elements $p_{k}$ define positive definite functions on $S^{n-1}$, that is,

(1)

$$\sum_{x_{i},x_{j}\in A}h_{i}h_{j}p_{k}\left(\left<x_{i},x_{j}\right>\right)\geq 0$$

for any finite subset $A\subset S^{n-1}$ and any real numbers $h_{i}\in\mathbb{R}.$ An example of a probability measure satisfying inequality (1) is

$$d\mu_{\alpha}=\frac{(1-t^{2})^{\alpha}}{\int_{-1}^{1}(1-t^{2})^{\alpha}dt}dt,$$

where $\alpha\geq\frac{n-3}{2}$ and $2\alpha\in\mathbb{Z}$. Given $s\in[-1,1]$, consider the space $D(\mu,s)$ of all functions $f(t)=\sum_{i=0}^{\infty}f_{i}p_{i}(t)$, $f_{i}\in\mathbb{R}$, such that

1. (1)

   $f_{i}\geq 0$ for every $i$, and $f_{0}>0,$

2. (2)

   $f(t)\leq 0$ for $-1\leq t\leq s.$

Suppose $0<\theta<\pi$, and $A=\{x_{1},\ldots,x_{N}\}$ is a $\theta$-spherical code in $S^{n-1}$. Given a function $f\in D(\mu,\cos\theta)$, we consider

$$\sum_{i,j}f(\left<x_{i},x_{j}\right>).$$

This may be written in two different ways as

$$Nf(1)+\sum_{i\neq j}f(\left<x_{i},x_{j}\right>)=f_{0}N^{2}+\sum_{k=1}^{\infty}f_{k}\sum_{i,j}p_{k}(\left<x_{i},x_{j}\right>).$$

Since $f\in D(\mu,\cos\theta)$ and $\left<x_{i},x_{j}\right>\leq\cos\theta$ for every $i\neq j$, this gives us the inequality

$$N\leq\frac{f(1)}{f_{0}}.$$

We define

(2)

$$\cal{L}(f):=\frac{f(1)}{f_{0}}.$$

In particular, this method leads to the linear programming bound

(3)

$$M(n,\theta)\leq\inf_{f\in D(d\mu_{\frac{n-3}{2}},\cos\theta)}\cal{L}(f).$$

One of the novelties of our work is the construction using triple points of new test functions in Section 3 satisfying conditions (1) and (2) of the Delsarte linear programming method. In fact, our functions are infinite linear combinations of coefficients of the matrices appearing in Theorem 3.2 of Bachoc–Vallentin [BV08]. Bachoc–Vallentin use semi-definite programming to obtain an upper bound on kissing numbers $M(n,\frac{\pi}{3})$ by summing over triples of points in spherical codes. On the other hand, we average one of the three points over the sphere, and take the other two points from the spherical code. Semi-definite programming is computationally feasible in very low dimensions, and improves upon linear programming bounds [dDV22]. After the writing of our paper, a semi-definite programming method using triple point correlations for sphere packings was developed by Cohn–de Laat–Salmon [CdS22], improving upon linear programming bounds on sphere packings in special low dimensions. In the semi-definite programming methods, the functions are numerically constructed. Furthermore, in high dimensions, there is no asymptotic bound using semi-definite programming which improves upon the linear programming bound of Kabatyanskii–Levenshtein [KL78], even up to a constant factor. The functions that we non-numerically construct improve upon [KL78] by a constant factor. In the same spirit, we also improve upon sphere packing density upper bounds in high dimensions.

Upper bounds on spherical codes are used to obtain upper bounds on sphere packing densities through inequalities proved using geometric methods. For example, for any $0<\theta\leq\pi/2$, Sidelnikov [Sid74] proved using an elementary argument that

(4)

$$\delta_{n}\leq\sin^{n}(\theta/2)M(n+1,\theta).$$

Let $0<\theta<\theta^{\prime}\leq\pi$. We write $\lambda_{n}(\theta,\theta^{\prime})$ for the ratio of volume of the spherical cap with radius $\frac{\sin(\theta/2)}{\sin(\theta^{\prime}/2)}$ on the unit sphere $S^{n-1}$ to the volume of the whole sphere. Sidelnikov [Sid74] used a similar argument to show that for $0<\theta<\theta^{\prime}\leq\pi$

(5)

$$M(n,\theta)\leq\frac{M(n+1,\theta^{\prime})}{\lambda_{n}(\theta,\theta^{\prime})}.$$

Kabatyanskii–Levenshstein used the Delsarte linear programming method and Jacobi polynomials to give an upper bound on $M(n,\theta)$ [KL78]. A year later, Levenshtein [Lev79] found optimal polynomials up to a certain degree and improved the Kabatyanskii–Levenshtein bound by a constant factor. Levenshtein obtained the upper bound

(6)

$$M(n,\theta)\leq M_{\textup{Lev}}(n,\theta),$$

where

(7)

$$M_{\textup{Lev}}(n,\theta):=\begin{cases}2{\binom{d+n-1}{n-1}}&\text{ if }t^{\alpha+1,\alpha}_{1,d}<\cos(\theta)\leq t^{\alpha+1,\alpha+1}_{1,d},\\ \binom{d+n-1}{n-1}+\binom{d+n-2}{n-1}&\text{ if }t^{\alpha+1,\alpha+1}_{1,d-1}<\cos(\theta)\leq t^{\alpha+1,\alpha}_{1,d}.\end{cases}$$

Here, $\alpha=\frac{n-3}{2}$ and $t^{\alpha,\beta}_{1,d}$ is the largest root of the Jacobi polynomial $p^{\alpha,\beta}_{d}$ of degree $d=d(n,\theta)$, a function of $n$ and $\theta$. We carefully define $d=d(n,\theta)$ and Levenshtein’s optimal polynomials in Subsection 5.1. Also see Subsection 5.1 for the definition and properties of Jacobi polynomials.

Throughout this paper, $\theta^{*}=62.997...^{\circ}$ is the unique root of the equation

$$\cos\theta\log(\frac{1+\sin\theta}{1-\sin\theta})-(1+\cos\theta)\sin\theta=0$$

in $(0,\pi/2)$. As demonstrated by Kabatyanskii–Levenshtein in  [KL78], for $0<\theta<\theta^{*}$, the bound $M(n,\theta)\leq M_{\textup{Lev}}(n,\theta)$ is asymptotically exponentially weaker in $n$ than

(8)

$$M(n,\theta)\leq\frac{M_{\textup{Lev}}(n+1,\theta^{*})}{\lambda_{n}(\theta,\theta^{*})}$$

obtained from inequality (5). Barg–Musin [BM07, p.11 (8)], based on the work [AVZ00] of Agrell–Vargy–Zeger, improved inequality (5) and showed that

(9)

$$M(n,\theta)\leq\frac{M(n-1,\theta^{\prime})}{\lambda_{n}(\theta,\theta^{\prime})}$$

whenever $\pi>\theta^{\prime}>2\arcsin\left(\frac{1}{2\cos(\theta/2)}\right)$. Cohn–Zhao [CZ14] improved sphere packing density upper bounds by combining the upper bound of Kabatyanskii–Levenshtein on $M(n,\theta)$ with their analogue [CZ14, Proposition 2.1] of (9) stating that for $\pi/3\leq\theta\leq\pi$,

(10)

$$\delta_{n}\leq\sin^{n}(\theta/2)M(n,\theta),$$

leading to better bounds than those obtained from (4). Aside from our main results discussed in the next subsection, in Proposition 2.2 of Section 2 we remove the angular restrictions on inequality (9) in the large $n$ regime by using a concentration of mass phenomenon. An analogous result removes the restriction $\theta\geq\pi/3$ on inequality (10) for large $n$.

Inequalities (9) and (10) give, respectively, the bounds

(11)

$$M(n,\theta)\leq\frac{M_{\textup{Lev}}(n-1,\theta^{*})}{\lambda_{n}(\theta,\theta^{*})},$$

for $\theta$ and $\theta^{*}$ restricted as in the conditions for inequality (9), and

(12)

$$\delta_{n}\leq\sin^{n}(\theta^{*}/2)M_{\textup{Lev}}(n,\theta^{*}).$$

Prior to our work, the above were the best bounds for large enough $n$. In Theorems 1.1 and 1.2, we improve both by a constant factor for large $n$, the first such improvement since the work of Levenshtein [Lev79] more than forty years ago. We also relax the angular condition in inequality (11) to $0<\theta<\theta^{*}$ for large $n$. For $\theta^{*}\leq\theta\leq\pi$, $M(n,\theta)\leq M_{\textup{Lev}}(n,\theta)$ is still the best bound. We also prove a number of other results, including the construction of general test functions in Section 3 that are of independent interest.

The geometric idea behind the construction of our new test functions is the following (see Figure 1). In [CZ14], for every $\pi\geq\theta\geq\frac{\pi}{3}$, Cohn and Zhao choose a ball of radius $r=\frac{1}{2\sin(\theta/2)}$ around each point of the sphere packing so that for every two points $\boldsymbol{a}$ and $\boldsymbol{b}$ that are centers of balls in the sphere packing, for every point $\boldsymbol{z}$ in the shaded region, $\boldsymbol{a}$ and $\boldsymbol{b}$ make an angle $\varphi\geq\theta$ with respect to $\boldsymbol{z}$. By averaging a function satisfying the Delsarte linear programming conditions for the angle $\theta$, a new function satisfying the conditions of the Cohn–Elkies linear programming method is produced. The negativity condition easily follows as $\varphi\geq\theta$ for every point $\boldsymbol{z}$ in the shaded region. Our insight is that since we are taking an average, we do not need pointwise negativity for each point $\boldsymbol{z}$ to ensure the negativity condition for the new averaged function. In fact, we can enlarge the radii of the balls by a quantity $\delta=O(1/n)$ so that the conditions of the Cohn–Elkies linear programming method continue to hold for this averaged function. Since the condition $\varphi\geq\theta$ is no longer satisfied for every point $\boldsymbol{z}$, determining how large $\delta$ may be chosen is delicate. We develop analytic methods for determining such $\delta$. This requires us to estimate triple density functions in Section 4, and estimating the Jacobi polynomials near their extreme roots in Subsection 5.1. It is known that the latter problem is difficult [Kra07, Conjecture 1]. In Subsection 5.1, we treat this difficulty by using the relation between the zeros of Jacobi polynomials; these ideas go back to the work of Stieltjes [Sti87]. More precisely, we use the underlying differential equations satisfied by Jacobi polynomials, and the fact that the roots of the family of Jacobi polynomials are interlacing. For Theorem 1.1, we use a similar idea but consider how much larger we can make certain appropriately chosen strips on a sphere. See Figure 3.

We use our geometrically constructed test functions to obtain the last column of Table 1. Table 1 is a comparison of upper bounds on sphere packing densities. This table does not include the computer-assisted mathematically rigorous bounds obtained using the Cohn–Elkies linear programming method or those of semi-definite programming. We now describe the different columns. The Rogers column corresponds to the bounds on sphere packing densities obtained by Rogers [Rog58]. The Levenshtein79 column corresponds to the bound obtained by Levenshtein in terms of roots of Bessel functions [Lev79]. The K.–L. column corresponds to the bound on $M(n,\theta)$ proved by Kabatyanskii and Levenshtein [KL78] combined with inequality (5). The Cohn–Zhao column corresponds to the column found in the work of Cohn and Zhao [CZ14]; they combined their inequality (10) with the bound on $M(n,\theta)$ proved by Kabatyanskii–Levenshtein [KL78]. We also include the column C.–Z.+L79 which corresponds to combining Cohn and Zhao’s inequality with improved bounds on $M(n,\theta)$ using Levenshtein’s optimal polynomials [Lev79]. The final column corresponds to the bounds on sphere packing densities obtained by our method. In Table 1, the highlighted entries are the best bounds obtained from these methods. Our bounds break most of the other bounds also in low dimensions. Our bounds are obtained from explicit geometrically constructed functions satisfying the Cohn–Elkies linear programming method. Our method only involves explicit integral calculations; in contrast to the numerical method in [CE03], we do not rely on any searching algorithm. Moreover, compared to the Cohn–Elkies linear programming method, in $n=120$ dimensions, we improve upon the sphere packing density upper bound of $1.164\times 10^{-17}$ obtained by forcing eight double roots.

In Section 2, we setup some of the notation used in this paper and prove Proposition 2.2. Section 3 concerns the general construction of our test functions that are used in conjunction with the Delsarte and Cohn–Elkies linear programming methods. In Section 5, we prove our main Theorems 1.1 and 1.2. In this section, we use our estimates on the triple density functions proved in Section 4. In Subsection 5.1, we describe Jacobi polynomials, Levenshtein’s optimal polynomials, and locally approximate Jacobi polynomials near their largest roots. In the final Section 6, we provide a table of improvement factors.

Acknowledgments. Both authors are thankful to Alexander Barg, Peter Boyvalenkov, Henry Cohn, Matthew de Courcy-Ireland, Mehrdad Khani Shirkoohi, Peter Sarnak, and Hamid Zargar. N.T.Sardari’s work is supported partially by the National Science Foundation under Grant No. DMS-2015305 N.T.S and is grateful to the Institute for Advanced Study and the Max Planck Institute for Mathematics in Bonn for their hospitality and financial support. M.Zargar was supported by the Max Planck Institute for Mathematics in Bonn, SFB1085 at the University of Regensburg, and the University of Southern California.

Since $h(\boldsymbol{a},\boldsymbol{b})$ is a point pair invariant function, it only depends on $t=\langle\boldsymbol{a},\boldsymbol{b}\rangle.$ For the rest of this paper, we abuse notation and consider $h$ as a real valued function of $t$ on $[-1,1],$ and write $h(\boldsymbol{a},\boldsymbol{b})=h(t).$