p-adic 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

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

(1)

Since

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

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 1.1.

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 $d=1$ ($n=2$), $d=2$ ($n=6$), $d=3$ ($n=12$), $d=4$ ($n=24$), $d=8$ ($n=240$), $d=24$ ($n=196560$)) [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 1.1 has connection even with sphere packing [39]. 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 1.2.

[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

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

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 1.2.

Theorem 1.3.

[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 

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

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 $n\le 1/c$.

In this paper, we introduce the notion of p-adic spherical codes. We show that Theorem 1.3, can be easily extended for p-adic Hilbert spaces.

2. p-adic Spherical Codes

We begin from the definition of p-adic Hilbert space.

Definition 2.1.

[40,41,42] Let $\mathbb{K}$ be a non-Archimedean complete valued field (with valuation $|·|$) and $\mathcal{X}$ be a non-Archimedean Banach space (with norm $\parallel ·\parallel$) over $\mathbb{K}$. We say that $\mathcal{X}$ is a p-adic Hilbert space if there is a map (called as p-adic inner product) $\langle ·,·\rangle :\mathcal{X}\times \mathcal{X}\to \mathbb{K}$ satisfying following.

i 

If $x\in \mathcal{X}$ is such that $\langle x,y\rangle =0$ for all $y\in \mathcal{X}$, then $x=0$.

ii 

$\langle x,y\rangle =\langle y,x\rangle$ for all $x,y\in \mathcal{X}$.

iii 

$\langle \alpha x+y,z\rangle =\alpha \langle x,z\rangle +\langle y,z\rangle$ for all $\alpha \in \mathbb{K}$, for all $x,y,z\in \mathcal{X}$.

iv 

$|\langle x,y\rangle |\le \parallel x\parallel \parallel y\parallel$ for all $x,y\in \mathcal{X}$.

Following is the standard example which we consider in the paper.

Example 2.2.

[41] Let p be a prime. For $d\in \mathbb{N}$, let ${\mathbb{Q}}_p^d$ be the standard p-adic Hilbert space equipped with the inner product

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

and norm

$$\begin{array}{c}\hfill \parallel {\left(x_j\right)}_{j=1}^d\parallel :=\underset{1\le j\le d}{max}\left|x_j\right|,\forall {\left(x_j\right)}_{j=1}^d\in {\mathbb{Q}}_p^d.\end{array}$$

We introduce p-adic spherical codes as follows.

Definition 2.3.

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

i 

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

ii 

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

iii 

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

(2)

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

Let ${\left\{{\tau }_j\right\}}_{j=1}^n$ be a p-adic $(d,n,\theta )$-spherical code in ${\mathbb{Q}}_p^d$. Since

$$\begin{array}{c}\hfill \parallel {\tau }_j-{\tau }_k{\parallel }^2\ge |\langle {\tau }_j-{\tau }_k,{\tau }_j-{\tau }_k\rangle |=|2-2\langle {\tau }_j,{\tau }_k\rangle |,\forall 1\le j,k\le n,\end{array}$$

Inequality (2) gives

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

(3)

However, note that Inequality (3) may not give Inequality (2). Note that we can formulate the definition of general p-adic $(d,n,\theta )$-spherical code by replacing Inequality (2) with Inequality (3) in Definition 2.3. It is also possible to consider Definition 2.3 by removing conditions (i) or (ii). Following is the p-adic version of Theorem 1.3.

Theorem 2.4.

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

i 

${\sum }_{1\le j,k\le n}\varphi \left(\right|2-2\langle {\tau }_j,{\tau }_k\rangle \left|\right)\ge 0$.

ii 

$\varphi \left(r\right)+c\le 0$ for all $r\in \left[2\right(1-cos\theta ),\infty )$.

Then

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

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

Proof. 

Define $\psi :[0,\infty )\ni r\mapsto \psi \left(r\right)\varphi \left(r\right)+c\in \mathbb{R}$. Then

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

We also have

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

Therefore

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

□

Corollary 2.5.

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

i 

${\sum }_{1\le j,k\le n}\varphi \left(\right|2-2\langle {\tau }_j,{\tau }_k\rangle \left|\right)\ge 0$.

ii 

$\varphi \left(r\right)+c\le 0$ for all $r\in [1,\infty )$.

Then

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

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

Following generalization of Theorem 2.4 is easy.

Theorem 2.6.

Let ${\left\{{\tau }_j\right\}}_{j=1}^n$ be a p-adic $(d,n,\theta )$-spherical code in ${\mathbb{Q}}_p^d$. Let $c>0$ and

$$\begin{array}{c}\hfill \varphi :\left\{\right|2-2\langle {\tau }_j,{\tau }_k\rangle |:1\le j,k\le n\}\to \mathbb{R}\end{array}$$

be a function satisfying following.

i 

${\sum }_{1\le j,k\le n}\varphi \left(\right|2-2\langle {\tau }_j,{\tau }_k\rangle \left|\right)\ge 0$.

ii 

$\varphi \left(r\right)+c\le 0$ for all $r\in \left\{\right|2-2\langle {\tau }_j,{\tau }_k\rangle |:1\le j,k\le n,j\ne k\}$.

Then

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

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

Note that, in the paper, we can replace ${\mathbb{Q}}_p^d$ by any p-adic Hilbert space.