Spherical code

Builds on Dot product Angle between two vectors

Definition

Let $n \ge 1$ and let $S^{n-1} = \set{x \in \R^n : x \cdot x = 1}$ be the unit sphere, where $x \cdot y$ is the dot product. A spherical code in $\R^n$ is a finite nonempty subset $C \subset S^{n-1}$ . Let $0 < \theta \le \pi$ . The code $C$ has minimal angle at least $\theta$ if $x \cdot y \le \cos \theta$ for all distinct $x, y \in C$ . Equivalently, every two distinct points of $C$ make an angle at least $\theta$ at the origin. The largest size of a spherical code in $\R^n$ with minimal angle at least $\theta$ is denoted $A(n, \theta)$ .

Remarks

Example: the $2n$ points $\pm e_1, \dots, \pm e_n$ , where $e_i$ are the standard basis vectors, form a spherical code with minimal angle $\pi/2$ . Two distinct points among them have dot product $0$ or $-1$ , both at most $\cos(\pi/2) = 0$ . Non-example: in $\R^2$ , the points $(1, 0)$ , $(0, 1)$ and $(1/\sqrt 2, 1/\sqrt 2)$ do not have minimal angle at least $\pi/2$ , since the dot product of the first and the third is $1/\sqrt 2 > 0$ . The kissing number is $\tau(n) = A(n, \pi/3)$ .

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Tag	`VY2TTB854G`
URI	`m2://VY2TTB854G`
Kind	definition
Writing	W0 stub
MSC	52C17, 94B65, 05B40
Defines	Spherical code
Uses	Dot product, Angle between two vectors

Spherical code

Definition

Remarks

Reading path

Further reading

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