Optimal line packings from finite group actions

TL;DR

This paper presents a unified framework using finite group actions to construct optimal point arrangements in projective spaces, leading to new infinite families of equiangular lines with specific symmetries.

Contribution

It introduces a general program based on Schurian association schemes and Gelfand pairs for constructing optimal packings, unifying various existing methods.

Findings

Provides a systematic approach to optimal point arrangements

Constructs the first infinite family of equiangular lines with Heisenberg symmetry

Unifies multiple existing packing constructions

Abstract

We provide a general program for finding nice arrangements of points in real or complex projective space from transitive actions of finite groups. In many cases, these arrangements are optimal in the sense of maximizing the minimum distance. We introduce our program in terms of general Schurian association schemes before focusing on the special case of Gelfand pairs. Notably, our program unifies a variety of existing packings with heretofore disparate constructions. In addition, we leverage our program to construct the first known infinite family of equiangular lines with Heisenberg symmetry.