A construction of spherical 3-designs

TL;DR

This paper presents a new method for constructing spherical 3-designs, extending previous work by Bondarenko, to facilitate better understanding and generation of these mathematical structures.

Contribution

It introduces a generalized construction technique for spherical 3-designs, expanding on Bondarenko's earlier methods.

Findings

Provides a new construction method for spherical 3-designs

Generalizes Bondarenko's previous work

Enables systematic generation of spherical 3-designs

Abstract

We give a construction for spherical 3-designs. This construction is a generalization of Bondarenko's work.

Equations

$$S^d=\{x=(x_1,\dots ,x_{d+1})\in {\mathbb{R}}^{d+1}\mid x_1^2+\cdots +x_{d+1}^2=1\}$$

$$\frac{1}{|X|}\sum _{x\in X}f(x)=\frac{1}{|S^d|}{\int }_{S^d}f(x)d\sigma (x),$$

$$\frac{1}{|X{|}^2}\sum _{x_i,x_j\in X}(x_i,x_j{)}^2=\frac{1}{d+1}.$$

$$\displaystyle\left\{\begin{array}[]{l}\displaystyle\frac{1}{|X|^{2}}\sum_{x_{i},x_{j}\in X}(x_{i},x_{j})^{2}=\frac{1}{d+1},\\ \displaystyle\frac{1}{|X|^{2}}\sum_{x_{i},x_{j}\in X}(x_{i},x_{j})^{4}=\frac{3}{(d+3)(d+1)}.\end{array}\right.$$

$$\Delta =\sum _{j=1}^{d+1}\frac{{\partial }^2}{\partial x_j^2}.$$

$$\dim \text{mbox}{Harm}_k(S^d)=\frac{2k+d-1}{k+d-1}\left(\genfrac{}{}{0ex}{}{d+k-1}{k}\right).$$

$$⟨P,Q⟩:={\int }_{S^d}P(x)Q(x)d\sigma (x),$$

$$⟨P_x,Q⟩=Q(x)\text{mbox}forallQ\in \text{mbox}{Harm}_k(S^d).$$

$$P_x(y)=g_{k,d}((x,y)),$$

$$G_x=\frac{P_x}{g_{k,d}(1{)}^{1/2}}.$$

$$⟨G_x,G_y⟩=\frac{g_{k,d}((x,y))}{g_{k,d}(1)}.$$

$$-G_X:=\{-G_x\mid G_x\in G_X\}.$$

$$\displaystyle\left\{\begin{array}[]{l}\displaystyle\frac{1}{|X|^{2}}\sum_{x_{i},x_{j}\in X}(x_{i},x_{j})^{2}=\frac{1}{d+1},\\ \displaystyle\frac{1}{|X|^{2}}\sum_{x_{i},x_{j}\in X}(x_{i},x_{j})^{4}=\frac{3}{(d+3)(d+1)},\end{array}\right.$$

$$g_{2,d}(x)=\frac{d+1}{d}{x}^2-\frac{1}{d}.$$

$$\frac{1}{|X{|}^2}\sum _{x_i,x_j\in X}⟨G_{x_i},G_{x_j}{⟩}^2=\frac{2}{d(d+3)}$$

$$\dim \text{mbox}{Harm}_2(S^d)=\frac{d+3}{d+1}\left(\genfrac{}{}{0ex}{}{d+1}{2}\right)=\frac{d(d+3)}{2}$$

$$\frac{1}{|X{|}^2}\sum _{x_i,x_j\in X}⟨G_{x_i},G_{x_j}{⟩}^2$$

$$=\frac{1}{|X{|}^2}\sum _{x_i,x_j\in X}\left(\frac{(d+1{)}^2}{d^2}(x_i,x_j{)}^4-2\frac{(d+1)}{d^2}(x_i,x_j{)}^2+\frac{1}{d^2}\right)$$

$$=\frac{(d+1{)}^2}{d^2}\frac{3}{(d+3)(d+1)}-2\frac{(d+1)}{d^2}\frac{1}{d+1}+\frac{1}{d^2}$$

$$=\frac{2}{d(d+3)}.$$

$$A_2,D_4,E_6,E_6^{♯},E_7,E_7^{♯},E_8,K_{10},K_{10}^{♯},C{T}_{12}.$$

$$\begin{array}[]{c||c|c|c}L&(d+1,N,a)\ {\rm code}&|(x_{i},x_{j})|&|\langle G_{x_{i}},G_{x_{j}}\rangle|\\ \hline\cr\hline\cr A_{2}&(2,6,1/2)&\{1/2\}&\{1/2\}\\ \hline\cr D_{4}&(9,24,1/3)&\{0,1/2\}&\{0,1/3\}\\ \hline\cr E_{6}&(20,72,1/5)&\{0,1/2\}&\{1/10,1/5\}\\ \hline\cr E_{6}^{\sharp}&(20,54,1/8)&\{1/4,1/2\}&\{1/10,1/8\}\\ \hline\cr E_{7}&(27,126,1/6)&\{0,1/2\}&\{1/8,1/6\}\\ \hline\cr E_{7}^{\sharp}&(27,56,1/27)&\{1/3\}&\{1/27\}\\ \hline\cr E_{8}&(35,240,1/7)\ \cite[cite]{[\@@bibref{}{B}{}{}]}&\{0,1/2\}&\{1/7\}\\ \hline\cr K_{10}&(54,276,1/6)&\{0,1/4,1/2\}&\{1/24,1/9,1/6\}\\ \hline\cr K_{10}^{\sharp}&(54,54,1/6)&\{1/8,1/4,1/2\}&\{1/24,3/32,1/6\}\\ \hline\cr CT_{12}&(77,756,2/11)&\{0,1/4,1/2\}&\{1/44,1/11,2/11\}\\ \end{array}$$

$$\begin{array}[]{c||c|c|c}L&(d+1,N,a)\ {\rm code}&|(x_{i},x_{j})|&|\langle G_{x_{i}},G_{x_{j}}\rangle|\\ \hline\cr\hline\cr{\rm BW16}&(135,4320,1/5)&\{0,1/4,1/2\}&\{0,1/15,1/5\}\\ \end{array}$$

$$\begin{array}[]{c||c|c|c}L&(d+1,N,a)\ {\rm code}&|(x_{i},x_{j})|&|\langle G_{x_{i}},G_{x_{j}}\rangle|\\ \hline\cr\hline\cr{\rm Leech}&(299,196560,5/23)&\{0,1/4,1/2\}&\{1/46,1/23,5/23\}\\ \end{array}$$

Full text

Key Words: Spherical designs, Lattices, Spherical harmonics.

2010 Mathematics Subject Classification. Primary 05B30; Secondary 11H06.

Classification codes according to UDC. 519.1

1 Introduction

This paper is inspired by [1], which gives an optimal antipodal spherical $(35,240,1/7)$ code whose vectors form a spherical $3$-design. To explain our results, we review the concept of spherical $t$-designs and [1].

First, we explain the concept of spherical $t$-designs.

Definition 1.1** ([3]).**

For a positive integer $t$, a finite nonempty set $X$ in the unit sphere

[TABLE]

is called a spherical $t$-design in $S^{d}$ if the following condition is satisfied:

[TABLE]

for all polynomials $f({x})=f(x_{1},\ldots,x_{d+1})$ of degree not exceeding $t$. Here, the righthand side involves the surface integral over the sphere and $|S^{d}|$, the volume of sphere $S^{d}$.

The meaning of spherical $t$-designs is that the average value of the integral of any polynomial of degree up to $t$ on the sphere can be replaced by its average value over a finite set on the sphere.

The following is an equivalent condition of the antipodal spherical designs:

Proposition 1.1** ([6]).**

An antipodal set $X=\{x_{1},\ldots,x_{N}\}$ in $S^{d}$ forms a spherical $3$-design if and only if

[TABLE]

An antipodal set $X=\{x_{1},\ldots,x_{N}\}$ in $S^{d}$ forms a spherical $5$-design if and only if

[TABLE]

Next, we review [1]. Let

[TABLE]

We say that a polynomial $P$ in $\mathbb{R}^{d+1}$ is harmonic if $\Delta P=0$. For integer $k\geq 1$, the restriction of a homogeneous harmonic polynomial of degree $k$ to $S^{d}$ is called a spherical harmonic of degree $k$. We denote by $\mbox{Harm}_{k}(S^{d})$ the vector space of the spherical harmonics of degree $k$. Note that (see for example [6])

[TABLE]

For $P,Q\in\mbox{Harm}_{k}(S^{d})$, we denote by $\langle P,Q\rangle$ the usual inner product

[TABLE]

where $d\sigma(x)$ is a normalized Lebesgue measure on the unit sphere $S^{d}$. For $x\in S^{d}$, there exists $P_{x}\in\mbox{Harm}_{k}(S^{d})$ such that

[TABLE]

It is known that

[TABLE]

where $g_{k,d}$ is a Gegenbauer polynomial. Let

[TABLE]

We remark that

[TABLE]

(For a detailed explanation of Gegenbauer polynomials, see [6].) Therefore, if we have a set $X=\{x_{1},\ldots,x_{N}\}$ in $S^{d}$, then we obtain the set $G_{X}=\{G_{x_{1}},\ldots,G_{x_{N}}\}$ in $S^{\dim{\rm Harm}_{k}(S^{d})-1}$.

Let $X=\{x_{1},\ldots,x_{120}\}$ be an arbitrary subset of $240$ normalized minimum vectors of the $E_{8}$ lattice such that no pair of antipodal vectors is present in $X$. Set $P_{x}(y)=g_{2,7}((x,y))$. [1] showed that $G_{X}\cup-G_{X}$ is an optimal antipodal spherical $(35,240,1/7)$ code whose vectors form a spherical $3$-design, where

[TABLE]

Furthermore, [1] showed that $G_{X}\cup-G_{X}$ is a spherical $3$-design, using the special properties of the $E_{8}$ lattice. However, this fact is an example that extends to a more general setting as follows. The spherical $3$-design obtained by Bondarenko in [1] is a special case of our main result, which is presented as the following theorem.

Theorem 1.1**.**

Let $X$ be a finite subset of sphere $S^{d}$ satisfying the condition (3). We set $P_{x}(y)=g_{2,d}((x,y))$. Then $G_{X}\cup-G_{X}$ is a spherical $3$-design in $S^{\dim{\rm Harm}_{2}({S^{d}})-1}$.

We denote by $\widetilde{G_{X}}$ the set $G_{X}\cup-G_{X}$ defined in Theorem 1.1.

Corollary 1.1**.**

Let $X$ be a spherical $4$-design in $S^{d}$. Then $\widetilde{G_{X}}$ is a spherical $3$-design in $S^{\dim{\rm Harm}_{2}({S^{d}})-1}$. 2. 2.

Let $X$ be a spherical $4$-design in $S^{d}$ and an antipodal set. Let $X^{\prime}$ be an arbitrary subset of $X$ with $|X^{\prime}|=|X|/2$ such that no pair of antipodal vectors is present in $X^{\prime}$. Then $\widetilde{G_{X^{\prime}}}$ is a spherical $3$-design in $S^{\dim{\rm Harm}_{2}({S^{d}})-1}$.

In section 2, we give a proof of Theorem 1.1. In section 3, we give some examples.

2 Proof of Theorem 1.1

Now, we give the proof of Theorem 1.1.

Proof of Theorem 1.1.

Let $X=\{x_{1},\ldots,x_{N}\}$ be in $S^{d}$ and $G_{X}=\{G_{x_{1}},\ldots,G_{x_{N}}\}$ be in $\mbox{Harm}_{2}(S^{d})$. By Proposition 1.1, we have

[TABLE]

since $X$ is a spherical $4$-design. We have the following Gegenbauer polynomial of degree $2$ on $S^{d}$:

[TABLE]

It is enough to show that

[TABLE]

since

[TABLE]

and $G_{X}\cup-G_{X}$ is an antipodal set. We remark that if $X$ is a spherical $t$-design, then $X\cup-X$ is also a spherical $t$-design.

In fact,

[TABLE]

Therefore, if $X=\{x_{1},\ldots,x_{N}\}$ is a spherical $4$-design, then $G_{X}\cup-G_{X}$ is a spherical $3$-design.

∎

3 Examples

In this section, we give some examples of using Theorem 1.1.

First we recall the concept of a strongly perfect and spherical $(d+1,N,a)$ code.

Definition 3.1** ([6]).**

A lattice $L$ is called strongly perfect if the minimum vectors of $L$ form a spherical $5$-design.

Definition 3.2** ([2]).**

An antipodal set $X=\{x_{1},\ldots,x_{N}\}$ in $S^{d}$ is called an antipodal spherical $(d+1,N,a)$ code if $|(x_{i},x_{j})|\leq a$ for some $a>0$ and all $x_{i},x_{i}\in X$, $i\neq j$, are not antipodal.

Next we give some examples.

Example 3.1**.**

The strongly perfect lattices whose ranks are less than $12$ have been classified [4, 5]. Such lattices whose ranks are greater than $1$ are as follows:

[TABLE]

(For a detailed explanation, see [4, 5].) Let $L$ be one of the above lattices and $X$ be the minimum vectors of $L$. Then, let $X^{\prime}$ be an arbitrary subset of $X$ with $|X^{\prime}|=|X|/2$ such that no pair of antipodal vectors is present in $X^{\prime}$.

By Corollary 1.1, $G_{X}\cup-G_{X}$ is a spherical $3$-design in $S^{d}$, where $d$ is as follows:

[TABLE]

Example 3.2**.**

Let $X$ be the minimum vectors of the Barnes–Wall lattice of rank $16$, and let $X^{\prime}$ be an arbitrary subset of $X$ with $|X^{\prime}|=|X|/2$ such that no pair of antipodal vectors is present in $X^{\prime}$. We remark that $X$ is a spherical $7$-design.

By Corollary 1.1, $G_{X}\cup-G_{X}$ is a spherical $3$-design in $S^{d}$, where $d$ is as follows:

[TABLE]

Example 3.3**.**

Let $X$ be the minimum vectors of the Leech lattice, and let $X^{\prime}$ be an arbitrary subset of $X$ with $|X^{\prime}|=|X|/2$ such that no pair of antipodal vectors is present in $X^{\prime}$. We remark that $X$ is a spherical $11$-design.

By Corollary 1.1, $G_{X}\cup-G_{X}$ is a spherical $3$-design in $S^{d}$, where $d$ is as follows:

[TABLE]

Acknowledgments

This work was supported by JSPS KAKENHI (18K03217).