Noncommutative 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 a $(d,n,\theta )$-spherical code[1] in ${\mathbb{R}}^d$ if

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

(1)

Since

$$\begin{array}{c}\hfill \langle \tau ,\omega \rangle =\frac{{2-\parallel \tau -\omega \parallel }^2}{2},\forall \tau ,\omega \in {\mathbb{R}}^d,\end{array}$$

we can rewrite Inequality (1) as

$$\begin{array}{c}\hfill \parallel {\tau }_j-{\tau }_k\parallel \ge \sqrt{2(1-cos\theta )},\forall 1\le j,k\le n,j\ne k.\end{array}$$

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 (KNP). KNP is still not completely resolved in every dimension (but resolved in dimensions $d=1$ ($n=2$), $d=2$ ($n=6$), $d=3$ ($n=12$), $d=4$ ($n=24$), $d=8$ ($n=240$), $d=24$ ($n=196560$)) [2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21]. We refer [22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38] for more on spherical codes. Problem 1 has connection even with important sphere packing problem [39]. Repeatedly used method for obtaining upper bounds on spherical codes is the Delsarte-Goethals-Seidel-Kabatianskii-Levenshtein bound. It is obtained using Gegenbauer polynomials. Let $n\in \mathbb{N}$ be fixed. The Gegenbauer polynomials are defined inductively as

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

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

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

Theorem 2.

[22,26] (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

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

satisfy

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

Then

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

In 2007, Pfender made a breakthrough by giving a one-line proof for a variant of Theorem 2.

Theorem 3.

[3] (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) 

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

(ii) 

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

Then

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

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

In this paper, we introduce the notion of noncommutative spherical codes. We show that Theorem 3 has an extension to Hilbert C*-modules.

2. Noncommutative Spherical Codes

Let $\mathcal{A}$ be a unital C*-algebra. For $d\in \mathbb{N}$, let ${\mathcal{A}}^d$ be the standard (left) Hilbert C*-module [40] equipped with the inner product

$$\begin{array}{c}\hfill \langle {\left(a_j\right)}_{j=1}^d,{\left(b_j\right)}_{j=1}^d\rangle :={\displaystyle \sum _{j=1}^d}{a}_j{b}_j^{*},\forall {\left(a_j\right)}_{j=1}^d,{\left(b_j\right)}_{j=1}^d\in {\mathcal{A}}^d\end{array}$$

and the norm

$$\begin{array}{c}\hfill \parallel {\left(a_j\right)}_{j=1}^d\parallel :={∥{\displaystyle \sum _{j=1}^d}{a}_j{a}_j^{*}∥}^{\frac{1}{2}},\forall {\left(a_j\right)}_{j=1}^d\in {\mathcal{A}}^d.\end{array}$$

We introduce noncommutative spherical codes as follows.

Definition 1.

Let $d\in \mathbb{N}$ and $\theta \in [0,2\pi )$. Let $\mathcal{A}$ be a unital C*-algebra. A set ${\left\{{\tau }_j\right\}}_{j=1}^n$ of vectors in ${\mathcal{A}}^d$ is said to be anoncommutative $(d,n,\theta )$-spherical codeor$(d,n,\theta )$-modular codein ${\mathcal{A}}^d$ if following conditions hold.

(i) 

$\langle {\tau }_j,{\tau }_j\rangle =1$ for all $1\le j\le n$.

(ii) 

$$2-\langle {\tau }_j,{\tau }_k\rangle -\langle {\tau }_k,{\tau }_j\rangle =\langle {\tau }_j-{\tau }_k,{\tau }_j-{\tau }_k\rangle \ge 2(1-cos\theta ),\forall 1\le j,k\le n,j\ne k.$$

(2)

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

Let ${\left\{{\tau }_j\right\}}_{j=1}^n$ be a noncommutative $(d,n,\theta )$-spherical code in ${\mathcal{A}}^d$. Since square root respects the order of positive elements in a C*-algebra, we have

$$\begin{array}{c}\hfill \parallel {\tau }_j-{\tau }_k\parallel \ge \sqrt{2(1-cos\theta )},\forall 1\le j,k\le n,j\ne k.\end{array}$$

(3)

However, note that Inequality (3) may not give Inequality (2). Following is the noncommutative version of Theorem 3.

Theorem 4.

(Noncommutative Delsarte-Goethals-Seidel-Kabatianskii-Levenshtein-Pfender Spherical Codes Bound) Let $\mathcal{A}$ be a unital C*-algebra and ${\mathcal{A}}^{+}\{a^{*}a:a\in \mathcal{A}\}$ be the set of all positive elements in $\mathcal{A}$. Let ${\left\{{\tau }_j\right\}}_{j=1}^n$ be a noncommutative $(d,n,\theta )$-spherical code in ${\mathcal{A}}^d$. Let $c\in (0,\infty )$ and $\varphi :{\mathcal{A}}^{+}\to \mathbb{R}$ be a function satisfying following.

(i) 

${\sum }_{1\le j,k\le n}\varphi \left(\langle {\tau }_j-{\tau }_k,{\tau }_j-{\tau }_k\rangle \right)\ge 0$.

(ii) 

$\varphi \left(a\right)+c\le 0$ for all $a\in {\mathcal{A}}^{+}$ with $a\ge 2(1-cos\theta )$.

Then

$$\begin{array}{c}\hfill n\le \frac{\varphi \left(0\right)+c}{c}.\end{array}$$

In particular, if $\varphi \left(0\right)+c\le 1$, then $n\le 1/c$.

Proof. 

Define $\psi :{\mathcal{A}}^{+}\ni a\mapsto \psi \left(a\right)\varphi \left(a\right)+c\in \mathbb{R}$. Then

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

We also have

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

Therefore

$$\begin{array}{c}\hfill c{n}^2\le \sum _{1\le j,k\le n}\varphi \left(\langle {\tau }_j-{\tau }_k,{\tau }_j-{\tau }_k\rangle \right)+c{n}^2=\sum _{1\le j,k\le n}\psi \left(\varphi \left(\langle {\tau }_j-{\tau }_k,{\tau }_j-{\tau }_k\rangle \right)\right)\le n(\varphi \left(0\right)+c).\end{array}$$

□

Corollary 1.

(Noncommutative Delsarte-Goethals-Seidel-Kabatianskii-Levenshtein-Pfender Kissing Number Bound) Let ${\left\{{\tau }_j\right\}}_{j=1}^n$ be a noncommutative $(d,n,\pi /3)$-spherical code in ${\mathcal{A}}^d$. Let $c\in (0,\infty )$ and $\varphi :{\mathcal{A}}^{+}\to \mathbb{R}$ be a function satisfying following.

(i) 

${\sum }_{1\le j,k\le n}\varphi \left(\langle {\tau }_j-{\tau }_k,{\tau }_j-{\tau }_k\rangle \right)\ge 0$.

(ii) 

$\varphi \left(a\right)+c\le 0$ for all $a\in {\mathcal{A}}^{+}$ with $a\ge 1$.

Then

$$\begin{array}{c}\hfill n\le \frac{\varphi \left(0\right)+c}{c}.\end{array}$$

In particular, if $\varphi \left(0\right)+c\le 1$, then $n\le 1/c$.

Following generalization of Theorem 4 is clear.

Theorem 5.

Let ${\left\{{\tau }_j\right\}}_{j=1}^n$ be a noncommutative $(d,n,\theta )$-spherical code in ${\mathcal{A}}^d$. Let $c\in (0,\infty )$ and

$$\begin{array}{c}\hfill \varphi :\{\langle {\tau }_j-{\tau }_k,{\tau }_j-{\tau }_k\rangle :1\le j,k\le n\}\to \mathbb{R}\end{array}$$

be a function satisfying following.

(i) 

${\sum }_{1\le j,k\le n}\varphi \left(\langle {\tau }_j-{\tau }_k,{\tau }_j-{\tau }_k\rangle \right)\ge 0$.

(ii) 

$\varphi \left(a\right)+c\le 0$ for all $a\in \{\langle {\tau }_j-{\tau }_k,{\tau }_j-{\tau }_k\rangle :1\le j,k\le n,j\ne k\}$.

Then

$$\begin{array}{c}\hfill n\le \frac{\varphi \left(0\right)+c}{c}.\end{array}$$

In particular, if $\varphi \left(0\right)+c\le 1$, then $n\le 1/c$.