Globally optimizing small codes in real projective spaces

TL;DR

This paper classifies optimal point arrangements in real projective spaces for dimensions 5 and 6, using advanced matrix and convex analysis techniques to identify the best codes with equiangular lines and introducing a new certificate method.

Contribution

It provides a classification of optimal small codes in real projective spaces for specific dimensions and introduces approximate Positivstellensatz certificates for numerical validation.

Findings

Classified optimal arrangements of points in RP^4 and RP^5.

Developed a new approach for approximate certificates in polynomial optimization.

Identified configurations with maximal minimum distance in the specified spaces.

Abstract

For $d\in\{5,6\}$, we classify arrangements of $d + 2$ points in $\mathbf{RP}^{d-1}$ for which the minimum distance is as large as possible. To do so, we leverage ideas from matrix and convex analysis to determine the best possible codes that contain equiangular lines, and we introduce a notion of approximate Positivstellensatz certificates that promotes numerical approximations of Stengle's Positivstellensatz certificates to honest certificates.