Packings in real projective spaces

TL;DR

This paper uses advanced geometric techniques to analyze optimal point packings in real projective spaces, providing proofs, algorithms, and certificates for various packing configurations and their near-optimality.

Contribution

It introduces a linear-time approximation algorithm for packings, proves optimality of specific packings, and offers local optimality certificates for infinite families.

Findings

Optimal 6-packing in $ ext{RP}^3$ proven via computer-assisted methods

New linear-time approximation algorithm for packings in the Gerzon range

Local optimality certificates for two infinite families of packings

Abstract

This paper applies techniques from algebraic and differential geometry to determine how to best pack points in real projective spaces. We present a computer-assisted proof of the optimality of a particular 6-packing in $\mathbb{R}\mathbf{P}^3$, we introduce a linear-time constant-factor approximation algorithm for packing in the so-called Gerzon range, and we provide local optimality certificates for two infinite families of packings. Finally, we present perfected versions of various putatively optimal packings from Sloane's online database, along with a handful of infinite families they suggest, and we prove that these packings enjoy a certain weak notion of optimality.