Approximate Method for Solving the Problem of Lattice Packing of Equal Spheres in N−Dimensional Euclidean Space

1. Introduction

The problem of the densest possible packing of equal spheres in Euclidean space is part of Hilbert’s eighteenth problem. If we limit the consideration and require the centers of the balls to form an additive subgroup (lattice) — we come to the classical problems of number theory about the minima of unimodular quadratic forms and their classification. This problem, in turn, is closely related to one of the most common problems of algebra — the classification of finite simple groups. All of these tasks are applicable to the problems of information transfer and storage theory and are closely related to the theory of error correcting codes. Very dense ball packages were discovered from constructions naturally arising from global fields (including modular curves over a finite field) and from elliptic curves over global fields [1].

The classical methods for solving the problem of the densest lattice packing of equal spheres include the works of A. N. Korkin and E. I. Zolotarev [2], G. F. Voronoi [3], G. Minkowski [4,5], G. Blichfeldt [6], B. A. Venkov [7], E. Barnes [8], G. Watson [9,10], S. S. Ryshkov and E. P. Baranovsky [11]. A wide range of problems close to the geometry of positive quadratic forms (PQF) is investigated in the book by L. Fejes Tot [12].

Somewhat different approaches to the theory of packings of equal spheres are given in the articles by G. Coxeter [13], E. Barnes [14], N. Hofreiter [15], J. Leech [16], N.K. Ignatiev [17], N. Sloane [18].

Estimation-asymptotic questions of packing theory in the book by Rogers [19], and also in the articles by V. M. Sidelnikov [20], G. A. Kabatyanskii and V. I. Levenshtein [21], B. F. Skubenko [22], S. N. Litsyn and M. A. Tsfasman [23], S. Shlosman and M. A. Tsfasman [24], M. Campos, M. Jenssen, M. Michelin and J. Sahasrabuddhe [25].

An important proof that no packing of unit balls in the Euclidean space $R^8$ has a density greater than the density of the Korkin–Zolotarev lattice packing $E_8$ is presented by M.S. Vyazovska in the work [26]. At the same time, H. Cohn, A. Kumar, S.D. Miller, D. Radchenko, and M.S. Vyazovskaya presented a proof that no packing of unit balls in the Euclidean space $R^{24}$ has a density greater than the density of the Leech lattice packing ${\Lambda }_{24}$ [27].

A large number of papers have been devoted to the study of the Hermite constant, which can be used to express the density of the closest lattice packing of equal spheres. Some of these papers include parts of the book by Fong K. Nguyen’s book [28], articles by R. Báez, R. Coulanjon, M. Ikas and M. O’Ryan [29], B. Meyer [30], M. Pohsta [31], L. Mechler and D. Nakkache [32].

2. Materials and Methods

In n-dimensional Euclidean space $E^n$, an orthonormal coordinate system is fixed and an n-dimensional basis is defined: $\epsilon =\{\overrightarrow{e_1},\overrightarrow{e_2},...,\overrightarrow{e_n}\}$, i.e. a system of n arbitrary independent vectors $\overrightarrow{e_1},...,\overrightarrow{e_n}$ with a common origin coinciding with the origin of coordinates. Let ${\xi }_{1j},{\xi }_{2j},...,{\xi }_{nj}$ be the column of coordinates of the vector $\overrightarrow{e_j}$ of the basis $\epsilon$ in this coordinate system. Thus, the basis $\epsilon$ is associated with its coordinate matrix $\Xi =\left({\xi }_{ij}\right)$, $|det\Xi |$ is equal to the volume V of the parallelepiped constructed on the vectors of the basis $\epsilon$.

Let us be given an n-dimensional lattice $\Gamma \in E^n$ and a set of n-dimensional balls of the same radius centered at the lattice points. These balls form a packing, i.e., pairwise have no common interior points. We call the maximum radius of balls in such a packing the packing radius corresponding to the lattice $\Gamma$ and denote it by $r\left(\Gamma \right)$:

$$r\left(\Gamma \right)={\displaystyle \frac{1}{2}}min\Gamma ,$$

(1)

where $min\Gamma$ is the length of the minimum lattice vector. The density of the lattice packing, i.e. the packing of spheres of radius $r\left(\Gamma \right)$ with centers at the lattice points $\Gamma$. Let us call the quantity

$$d\left(\Gamma \right)={\Omega }_n{\displaystyle \frac{r^n\left(\Gamma \right)}{V\left(\Gamma \right)}},$$

(2)

where $V\left(\Gamma \right)$ is the volume of the fundamental parallelepiped of the lattice $\Gamma$, and ${\Omega }_n$ is the volume of the n-dimensional unit ball,

$${\Omega }_n={\displaystyle \frac{{\pi }^{n/2}}{\Gamma (n/2+1)}},$$

(3)

where $\Gamma$ is the gamma function. The central density of the lattice packing is $\delta =1/V\left(\Gamma \right)$. The problem of densest lattice packings in the space $E^n$ consists of finding the value

$$d_n=supd\left(\Gamma \right)$$

(4)

and the lattices on which they are attained on a set of n-dimensional lattices. Since the density $d\left(\Gamma \right)$ does not change under similarity transformations of the space $E^n$, when solving the problem of densest lattice packings, the set of lattices under study can be normalized by requiring, for example, $V\left(\Gamma \right)=1$ or $r\left(\Gamma \right)=1$.

Let $\left\{f\right\}$ be the equivalence class of the PQF corresponding to the lattice $\Gamma$, and $f\in \left\{f\right\}$ be some representative of this class. Then we have

$$d\left(\Gamma \right)={\displaystyle \frac{{\Omega }_n}{2^n}}{[minf/{(detf)}^{1/n}]}^{n/2}.$$

(5)

The problem of the densest lattice packing in the space $E^n$ is equivalent to the problem of finding the upper bound of the ratio $minf/{(detf)}^{1/n}$ on the set of PQF in n variables:

$${\gamma }_n=sup[minf/{(detf)}^{1/n}]=sup[{(min{\Gamma }_f)}^2/{\left(V\left({\Gamma }_f\right)\right)}^{2/n}].$$

(6)

The quantity ${\gamma }_n$ is called the Hermite constant. The constant ${\gamma }_n$ and the density of the densest lattice packing are related by the formula [11]

$$d_n=2^{-n}{\Omega }_n{\gamma }^{n/2}.$$

(7)

When finding the Hermite constant, one need not consider the entire set of PQF in n variables; instead, one can take one representative from each equivalence class and normalize it by setting $minf=1$ or $detf=1$.

The theory of reduction of the PQF of an arbitrary number of variables was presented in the work of Korkin and Zolotarev and was used there as a method for obtaining values and estimates of the Hermite constant, and thus for solving the problem of the densest packings [2].

Let us consider the Lagrange expansion for an arbitrary PQF:

$$\begin{array}{c}\hfill f=A_1{(x_1-{\displaystyle \sum _{k=2}^n}{\alpha }_{1k}{x}_k)}^2+A_2{(x_2-{\displaystyle \sum _{k=3}^n}{\alpha }_{2k}{x}_k)}^2+...\\ \hfill +A_l{(x_l-{\displaystyle \sum _{k=l+1}^n}{\alpha }_{lk}{x}_k)}^2+...+A_n{(x_{n-1}-{\alpha }_{n-1,n}{x}_n)}^2+A_n{x}_n^2.\end{array}$$

(8)

A PQF is called reduced according to Korkin–Zolotarev if in its expansion for $l=1,2,...,n-1$ and $k=l+1,l+2,...,n$ the coefficients $A_l$ are the values of the minima of the forms:

$$\begin{array}{c}\hfill {\varphi }_1=f(x_1,x_2,...,x_n),{\varphi }_{l+1}(x_{l+1},x_{l+2},...,x_n)=\\ \hfill {\varphi }_l(x_l,x_{l+1},...,x_n)-A_l{(x_l-{\displaystyle \sum _{k=l+1}^n}{\alpha }_{lk}{x}_k)}^2,\end{array}$$

(9)

and the coefficients ${\alpha }_{l,l+1},{\alpha }_{lk}$ satisfy the inequalities $0\le {\alpha }_{l,l+1}\le 1/2,-1/2\le {\alpha }_{lk}\le 1/2.$ For each Korkin–Zolotarev expansion the following equality holds:

$$detf=A_1{A}_2...A_n,$$

(10)

and also “The first Korkin–Zolotarev inequality”: $A_{k+1}\ge (3/4)A_k$, where $k=1,2,...,n-1$ and “The second Korkin–Zolotarev inequality”: $A_{i+1}\ge (2/3)A_i$, where $i=1,2,...,n-2$.

According to the definition of the Hermite constant ${\gamma }_n$ and equality (10), representing each equivalence class of the PQF by a Korkine–Zolotarev decomposition, we have

$${\gamma }_n=sup{A}_1{(A_1{A}_2...A_n)}^{-1/n}.$$

(11)

Having determined the minimum lattice vector $min\Gamma =2$, the values of the heights of the fundamental parallelepipeds $h_n$ of the lattice $\Gamma$ correspond to the values of the coefficients $A_n$ of the positive quadratic forms reduced according to Korkine–Zolotarev and are related by the following equality [33], [p. 3, formula (3)]:

$$h_n=2\sqrt{A_n}.$$

(12)

In this same work, equalities are established for the values of the heights of the fundamental parallelepipeds $h_n$ of the lattice $\Gamma$ for groups of dimensions 1 — 8, 9 — 16, 17 — 24.

$$\begin{array}{c}\hfill h_1*h_8=h_2*h_7=h_3*h_6=h_4*h_5=2,\\ \hfill h_9*h_{16}=h_{10}*h_{15}=h_{11}*h_{14}=h_{12}*h_{13}=1,\\ \hfill h_{17}*h_{24}=h_{18}*h_{23}=h_{19}*h_{22}=h_{20}*h_{21}=1/2.\end{array}$$

(13)

Using the obtained results, a method for constructing lattice packings of equal spheres corresponding to the density of layered lattices of the “Lambda” series was developed. In the paper [34], a method for constructing such packings in dimensions 1-24 is proposed, using a series of coefficients to the height of a fundamental parallelepiped of dimension $(n-1)$.

In the course of extending the obtained regularities of lattice packings corresponding to layered lattices of the “Lambda” series in dimensions 1 — 24 to groups of higher dimensions, a hypothesis was formulated about the density function of the densest lattice packing of equal spheres in n-dimensional Euclidean space.

3. Results

The formulation of the hypothesis about the density function of the closest lattice packing of equal spheres is based on three propositions.

Before formulating the first proposition, let us clarify that the values of the heights of the fundamental parallelepipeds in the first group of dimensions 1 — 24 have an even stronger symmetry and the equalities (13) can be expressed as follows way

$$\begin{array}{c}\hfill h_1*h_{24}=h_2*h_{23}=h_3*h_{22}=h_4*h_{21}=\\ \hfill h_5*h_{20}=h_6*h_{19}=h_7*h_{18}=h_8*h_{17}=\\ \hfill h_9*h_{16}=h_{10}*h_{15}=h_{11}*h_{14}=h_{12}*h_{13}=1.\end{array}$$

(14)

Let us denote the product of the heights of the fundamental parallelepipeds of the first group of dimensions by ${\Pi }_{G_1}$. Proposition 1 will be justified by the results of B.B. Venkov’s work on spherical $t-$block-schemes [35].

Proposition 1. 

The symmetric group regularities established in the first group of dimensions 1 — 24 are preserved in the following groups of dimensions with period 24.

Then, in the group of dimensions 25 — 48, the following equalities will hold:

$$\begin{array}{c}\hfill h_{25}*h_{48}=h_{26}*h_{47}=h_{27}*h_{46}=h_{28}*h_{45}=\\ \hfill h_{29}*h_{44}=h_{30}*h_{43}=h_{31}*h_{42}=h_{32}*h_{41}=\\ \hfill h_{33}*h_{40}=h_{34}*h_{39}=h_{35}*h_{38}=h_{36}*h_{37}.\end{array}$$

(15)

Let us denote the product of the heights of the fundamental parallelepipeds of the second group of dimensions by ${\Pi }_{G_2}$. In the next group of dimensions 49 — 72, the equalities will hold

$$\begin{array}{c}\hfill h_{49}*h_{72}=h_{50}*h_{71}=h_{51}*h_{70}=h_{52}*h_{69}=\\ \hfill h_{53}*h_{68}=h_{54}*h_{67}=h_{55}*h_{66}=h_{56}*h_{65}=\\ \hfill h_{57}*h_{64}=h_{58}*h_{63}=h_{59}*h_{62}=h_{60}*h_{61}.\end{array}$$

(16)

Let us denote the product of the heights of the fundamental parallelepipeds of the third group of dimensions by ${\Pi }_{G_3}$ and so on up to ∞.

Let us denote the product of the heights of the fundamental parallelepipeds of the following groups of dimensions with period 24 by ${\Pi }_{G_k}$, where $k=1,2,...,n$, where $n\to \infty$.

Proposition 2. 

For each group of dimensions with period 24, the products of the heights of the fundamental parallelepipeds ${\Pi }_{G_k}$ take values approximately equal to the inverses of the consecutive numbers of the natural numbers.

We obtain the following equalities: ${\Pi }_{G_1}=1$, ${\Pi }_{G_2}\approx 1/2$, ${\Pi }_{G_3}\approx 1/3$, … , ${\Pi }_{G_k}\approx 1/k$. Using the equalities from propositions 1 and 2, and also taking into account the fact that the exact values of the heights of the fundamental parallelepipeds of the closest lattice packings with the exception of dimensions 1 — 8 and 24 are not known, we formulate the third proposition.

Proposition 3. 

As $n\to \infty$, the heights of the fundamental parallelepiped $h_n$, depending on the group of dimensions, take the values: $h_{1-24}=\sqrt{{\Pi }_{G_1}}$, $h_{25-48}=\sqrt{{\Pi }_{G_2}}$, $h_{49-72}=\sqrt{{\Pi }_{G_3}}$, …, $h_{(24n+1)-(24n+24)}=\sqrt{{\Pi }_{G_k}}$.

We obtain an infinite sequence of values for the heights of the fundamental parallelepiped $h_n$, which is given by the following formula

$$h_n={⌊{\displaystyle \frac{n+23}{24}}⌋}^{-1/2}.$$

(17)

The sequence given by formula (17) is analogous to sequence A194698 [36] after minor transformations.

Applying (11), (12), and (17), we obtain a formula for the Hermite constant

$${\gamma }_n={\left({\displaystyle \prod _{n=1}^n}4*⌊{\displaystyle \frac{n+23}{24}}⌋\right)}^{1/n}.$$

(18)

Then the density function of the densest lattice packing of equal spheres in n-dimensional Euclidean space takes the form

$$f\left(d_n\right)=2^{-n}*{\Omega }_n*{\left({\displaystyle \prod _{n=1}^n}4*⌊{\displaystyle \frac{n+23}{24}}⌋\right)}^{1/2}.$$

(19)

In what follows, to construct the most illustrative graph of the function $f\left(d_n\right)$, we will use the representation of the function in the form $f{\left(d_n\right)}^{1/n}$.

$$f{\left(d_n\right)}^{1/n}={\displaystyle \frac{1}{2}}*{\Omega }_n^{1/n}*{\left({\displaystyle \prod _{n=1}^n}4*⌊{\displaystyle \frac{n+23}{24}}⌋\right)}^{1/2n}.$$

(20)

4. Discussion

Verification of the main results. As a first upper bound for the density, we use the estimate for the constant obtained by C. Hermite in his work [37]

$${\gamma }_n\le {(4/3)}^{(n-1)/2}.$$

(21)

Then, from 7 and 21, we obtain

$${\left(d_n\right)}^{1/n}\le {\displaystyle \frac{1}{2}}*{\Omega }_n^{1/n}*{(4/3)}^{(n-1)/4}.$$

(22)

As the next upper bound, we use the estimate for the Hermite constant proposed by H. F. Blichfeldt in [38]

$${\gamma }_n\le {\displaystyle \frac{2}{\pi }}*\Gamma {(2+n/2)}^{2/n}.$$

(23)

Then, from (7) and (23), we obtain

$${\left(d_n\right)}^{1/n}\le {\displaystyle \frac{1}{2}}*{\Omega }_n^{1/n}*\sqrt{{\displaystyle \frac{2}{\pi }}}*\Gamma {(2+n/2)}^{1/n}.$$

(24)

As the next upper bound, we use the estimate proposed by V. M. Sidelnikov in his work [20]

$${\left(d_n\right)}^{1/n}\lesssim 2^{-0.509619\dots },n\to \infty .$$

(25)

As the best upper bound, we use the estimate by G.A. Kabatyansky and V.I. Levenshtein from the work [21]

$${\left(d_n\right)}^{1/n}\le 2^{-0.599+o\left(1\right)},n\to \infty .$$

(26)

As the best lower bound, we use the result of M. Campos, M. Jenssen, M. Michelin, and J. Sahasrabuddhe from the paper [25]

$${\left(d_n\right)}^{1/n}>{\left((1-o\left(1\right))*{\displaystyle \frac{n*log\left(n\right)}{2^{n+1}}}\right)}^{1/n}.$$

(27)

A comparison of the density values of known densest lattice packings of equal spheres in n-dimensional Euclidean space with the values of the function $f{\left(d_n\right)}^{1/n}$ from $n=24$ to $n=1048584$ is presented in Table A1.

The values of the density function of the closest lattice packing of equal spheres in n-dimensional Euclidean space $f{\left(d_n\right)}^{1/n}$ with the best known upper and lower bounds for large values of n are presented in Table A2.

Let us compare the values of the density function of the densest lattice packing of equal spheres in n-dimensional Euclidean space $f{\left(d_n\right)}^{1/n}$ with the density of lattices arising from number and function fields for higher dimensions. The best known logarithmic density $\lambda \left(\left\{L_n\right\}\right)$ for such lattices is known from the work of M.Yu. Rosenblum and M.A. Tsfasman [39], where $\lambda \left(\left\{L_n\right\}\right)\approx 1.39$. The logarithmic density for the values of the density function of the densest lattice packing of equal spheres in n-dimensional Euclidean space $f{\left(d_n\right)}^{1/n}$, $\lambda \left(\left\{L_n\right\}\right)\lesssim 1$. So, for $n=3*{10}^6$, $\lambda \left(\left\{L_n\right\}\right)\approx 0.9667\dots$

Further research aims to obtain precise values for the products in Proposition 2, as well as to increase perturbations of the density function of the densest lattice packing of equal spheres in n-dimensional Euclidean space to obtain precise density values for all dimensions.

Author thanks M.A. Tsfasman for helpful comments.