Functional Delsarte-Goethals-Seidel-Kabatianskii-Levenshtein-Pfender Bound

1. Introduction

Let $d\in \mathbb{N}$ and $\theta \in [0,2\pi )$. A set ${\left\{{\tau }_j\right\}}_{j=1}^n$ of unit vectors in ${\mathbb{R}}^d$ is said to be $(d,n,\theta )$-spherical code [1] in ${\mathbb{R}}^d$ if

$$\begin{array}{c}\hfill \langle {\tau }_j,{\tau }_k\rangle \le cos\theta ,\forall 1\le j,k\le n,j\ne k.\end{array}$$

Fundamental problem associated with spherical codes is the following.

Problem 1. 

Given d and θ, what is the maximum n such that there exists a $(d,n,\theta )$-spherical code ${\left\{{\tau }_j\right\}}_{j=1}^n$ in ${\mathbb{R}}^d$?

The case $\theta =\pi /3$ is known as the famous (Newton-Gregory) kissing number problem. With extensive efforts from many mathematicians, it is still not completely resolved in every dimension (but resolved in dimensions, 1, 2, 3, 4, 8 and 24) [2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21]. We refer [22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37] for more on spherical codes. Problem 1 has connection even with sphere packing [38]. Most used method for obtaining upper bounds on spherical codes is the Delsarte-Goethals-Seidel-Kabatianskii-Levenshtein bound which we recall. Let $n\in \mathbb{N}$ be fixed. The Gegenbauer polynomials are defined inductively as

$$\begin{array}{cc}\hfill G_0^{\left(n\right)}\left(r\right)& 1,\forall r\in [-1,1],\hfill \\ \hfill G_1^{\left(n\right)}\left(r\right)& r,\forall r\in [-1,1],\hfill \\ \hfill & ⋮\hfill \\ \hfill G_k^{\left(n\right)}\left(r\right)& {\displaystyle \frac{(2k+n-4)r{G}_{k-1}^{\left(n\right)}\left(r\right)-(k-1)G_{k-2}^{\left(n\right)}\left(r\right)}{k+n-3}},\forall r\in [-1,1],\forall k\ge 2.\hfill \end{array}$$

Then the family ${\left\{G_k^{\left(n\right)}\right\}}_{k=0}^{\infty }$ is orthogonal on the interval $[-1,1]$ with respect to the weight

$$\begin{array}{c}\hfill \rho \left(r\right){(1-r^2)}^{{\displaystyle \frac{n-3}{2}}},\forall r\in [-1,1].\end{array}$$

Theorem 1. 

[32,33] (Delsarte-Goethals-Seidel-Kabatianskii-Levenshtein Linear Programming Bound) Let ${\left\{{\tau }_j\right\}}_{j=1}^n$ be a $(d,n,\theta )$-spherical code in ${\mathbb{R}}^d$. Let P be a real polynomial satisfying following conditions.

(i) 

$P\left(r\right)\le 0$ for all $-1\le r\le cos\theta$.

(ii) 

Coefficients in the Gegenbauer expansion

$$\begin{array}{c}\hfill P=\sum _{k=0}^m{a}_k{G}_k^{\left(n\right)}\end{array}$$

satisfy

$$\begin{array}{c}\hfill a_0>0,a_k\ge 0,\forall 1\le k\le m.\end{array}$$

Then

$$\begin{array}{c}\hfill n\le {\displaystyle \frac{P\left(1\right)}{a_0}}.\end{array}$$

In 2007, Pfender gave a one-line proof for a variant of Theorem 1.

Theorem 2. 

[19] (Delsarte-Goethals-Seidel-Kabatianskii-Levenshtein-Pfender Bound) Let ${\left\{{\tau }_j\right\}}_{j=1}^n$ be a $(d,n,\theta )$-spherical code in ${\mathbb{R}}^d$. Let $c>0$ and $\varphi :[-1,1]\to \mathbb{R}$ be a function satisfying following.

(i) 

$$\begin{array}{c}\hfill \sum _{j=1}^n\sum _{k=1}^n\varphi \left(\langle {\tau }_j,{\tau }_k\rangle \right)\ge 0.\end{array}$$

(ii) 

$\varphi \left(r\right)+c\le 0$ for all $-1\le r\le cos\theta$.

Then

$$\begin{array}{c}\hfill n\le {\displaystyle \frac{\varphi \left(1\right)+c}{c}}.\end{array}$$

In particular, if $\varphi \left(1\right)+c\le 1$, then $n\le 1/c$.

Motivated from Theorem 2, we formulate the notion of codes in pointed metric spaces. We show that bound of Delsarte-Goethals-Seidel-Kabatianskii-Levenshtein-Pfender can be extended for pointed metric spaces (in particular, for Banach spaces).

2. Metric Codes

Let $(\mathcal{M},0)$ be a pointed metric space. The collection ${Lip}_0(\mathcal{M},\mathbb{R})$ is defined as ${Lip}_0(\mathcal{M},\mathbb{R})\{f:\mathcal{M}\to \mathbb{R}is~Lipschitz~andf\left(0\right)=0\}.$ For $f\in {Lip}_0(\mathcal{M},\mathbb{R})$, the Lipschitz norm is defined as

$$\begin{array}{c}\hfill {\parallel f\parallel }_{{Lip}_0}\underset{x,y\in \mathcal{M},x\ne y}{sup}{\displaystyle \frac{\left|f\right(x)-f(y\left)\right|}{d(x,y)}}.\end{array}$$

We introduce metric codes as follows.

Definition 1. 

Let $(\mathcal{M},0)$ be a pointed metric space with metric m. For $1\le j\le n$, let $f_j\in {Lip}_0(\mathcal{M},\mathbb{R})$ and ${\tau }_j\in \mathcal{M}$. The pair $({\left\{f_j\right\}}_{j=1}^n,{\left\{{\tau }_j\right\}}_{j=1}^n)$ is said to be a $(n,\theta )$-metric code or $(n,\theta )$-nonlinear code or $(n,\theta )$-Lipschitz code in $\mathcal{M}$ if following conditions hold.

(i) 

$\parallel f_j{\parallel }_{{Lip}_0}=1$ for all $1\le j\le n$.

(ii) 

$m({\tau }_j,0)=1$ for all $1\le j\le n$.

(iii) 

$f_j\left({\tau }_j\right)=1$ for all $1\le j\le n$.

(iv) 

$f_j\left({\tau }_k\right)\le cos\theta$ for all $1\le j,k\le n,j\ne k$.

We call the case $\theta =\pi /3$ as the nonlinear kissing number problem .

For Banach spaces, we define (linear) functional codes as follows.

Definition 2. 

Let $\mathcal{X}$ be a real Banach space. For $1\le j\le n$, let $f_j\in {\mathcal{X}}^{*}$ and ${\tau }_j\in \mathcal{X}$. The pair $({\left\{f_j\right\}}_{j=1}^n,{\left\{{\tau }_j\right\}}_{j=1}^n)$ is said to be a$(n,\theta )$-functional code in $\mathcal{X}$ if following conditions hold.

(i) 

$\parallel f_j\parallel =1$ for all $1\le j\le n$.

(ii) 

$\parallel {\tau }_j\parallel =1$ for all $1\le j\le n$.

(iii) 

$f_j\left({\tau }_j\right)=1$ for all $1\le j\le n$.

(iv) 

$f_j\left({\tau }_k\right)\le cos\theta$ for all $1\le j,k\le n,j\ne k$.

We call the case $\theta =\pi /3$ as thefunctional kissing number problem .

Proposition 1. 

For the space ${\mathbb{R}}^d$, Definition 2 matches with the spherical codes (in particular with the kissing-number problem).

Proof.Let $({\left\{f_j\right\}}_{j=1}^n,{\left\{{\tau }_j\right\}}_{j=1}^n)$ be a $(n,\theta )$-functional code in ${\mathbb{R}}^d$. We need to show that $f_j\left(x\right)={\tau }_j$ for all $x\in {\mathbb{R}}^d$ and for all $1\le j\le n$. Let $1\le j\le n$. From Riesz representation theorem, there exists a unique $w_j\in {\mathbb{R}}^d$ such that $f_j\left(x\right)=\langle x,{\omega }_j\rangle$ for all $x\in {\mathbb{R}}^d$ and $\parallel f_j\parallel =\parallel {\omega }_j\parallel$. Now we need to show that ${\omega }_j={\tau }_j$. Since $\parallel f_j\parallel =1$, we must have $\parallel {\omega }_j\parallel =1$. But then

$$\begin{array}{c}\hfill 1=f_j\left({\tau }_j\right)=\langle {\tau }_j,{\omega }_j\rangle \le \parallel {\tau }_j\parallel \parallel {\omega }_j\parallel =1.\end{array}$$

Therefore ${\omega }_j=\alpha {\tau }_j$ for some $\alpha \in \mathbb{R}$. The conditions $\parallel {\omega }_j\parallel =\parallel {\tau }_j\parallel =1$ and $\langle {\tau }_j,{\omega }_j\rangle =1$ then force ${\omega }_j={\tau }_j$. □

Following is a nonlinear generalization of Theorem 2.

Theorem 3. (Functional Delsarte-Goethals-Seidel-Kabatianskii-Levenshtein-Pfender Bound) Let $({\left\{f_j\right\}}_{j=1}^n,{\left\{{\tau }_j\right\}}_{j=1}^n)$ be a $(n,\theta )$-metric code in a pointed metric space $\mathcal{M}$. Let $c>0$ and $\varphi :[-1,1]\to \mathbb{R}$ be a function satisfying following.

(i) 

$$\begin{array}{c}\hfill \sum _{j=1}^n\sum _{k=1}^n\varphi \left(f_j\left({\tau }_k\right)\right)\ge 0.\end{array}$$

(ii) 

$\varphi \left(r\right)+c\le 0$ for all $-1\le r\le cos\theta$.

Then

$$\begin{array}{c}\hfill n\le {\displaystyle \frac{\varphi \left(1\right)+c}{c}}.\end{array}$$

In particular, if $\varphi \left(1\right)+c\le 1$, then $n\le 1/c$.

Proof. 

Define $\psi :[-1,1]\ni r\mapsto \psi \left(r\right)\varphi \left(r\right)+c\in \mathbb{R}$. Then

$$\begin{array}{cc}\hfill \sum _{j=1}^n\sum _{k=1}^n\psi \left(f_j\left({\tau }_k\right)\right)& =\sum _{j=1}^n\psi \left(f_j\left({\tau }_j\right)\right)+\sum _{1\le j,k\le n,j\ne k}\psi \left(f_j\left({\tau }_k\right)\right)\hfill \\ \hfill & =\sum _{j=1}^n\psi \left(1\right)+\sum _{1\le j,k\le n,j\ne k}\psi \left(f_j\left({\tau }_k\right)\right)\hfill \\ \hfill & =n(\varphi \left(1\right)+c)+\sum _{1\le j,k\le n,j\ne k}(\varphi \left(f_j\left({\tau }_k\right)\right)+c)\hfill \\ \hfill & \le n\left(\varphi \right(1)+c)+0=n\left(\varphi \right(1)+c).\hfill \end{array}$$

We also have

$$\begin{array}{c}\hfill \sum _{j=1}^n\sum _{k=1}^n\psi \left(f_j\left({\tau }_k\right)\right)=\sum _{j=1}^n\sum _{k=1}^n(\varphi \left(f_j\left({\tau }_k\right)\right)+c)=\sum _{j=1}^n\sum _{k=1}^n\varphi \left(f_j\left({\tau }_k\right)\right)+c{n}^2.\end{array}$$

Therefore

$$\begin{array}{c}\hfill c{n}^2\le \sum _{j=1}^n\sum _{k=1}^n\varphi \left(f_j\left({\tau }_k\right)\right)+c{n}^2\le n(\varphi \left(1\right)+c).\end{array}$$

□

Following generalization of Theorem 3 is easy.

Theorem 4. 

Let $({\left\{f_j\right\}}_{j=1}^n,{\left\{{\tau }_j\right\}}_{j=1}^n)$ be a $(n,\theta )$-metric code in a pointed metric space $\mathcal{M}$. Let $c>0$ and

$$\begin{array}{c}\hfill \varphi :\{f_j\left({\tau }_k\right):1\le j,k\le n\}\to \mathbb{R}\end{array}$$

be a function satisfying following.

(i) 

$$\begin{array}{c}\hfill \sum _{j=1}^n\sum _{k=1}^n\varphi \left(f_j\left({\tau }_k\right)\right)\ge 0.\end{array}$$

(ii) 

$\varphi \left(r\right)+c\le 0$ for all $r\in \{f_j\left({\tau }_k\right):1\le j,k\le n,j\ne k\}$.

Then

$$\begin{array}{c}\hfill n\le {\displaystyle \frac{\varphi \left(1\right)+c}{c}}.\end{array}$$

In particular, if $\varphi \left(1\right)+c\le 1$, then $n\le 1/c$.