Saturated configuration and new large construction of equiangular lines

TL;DR

This paper investigates the maximum number of equiangular lines in Euclidean spaces, proving limitations of existing constructions in certain dimensions and introducing new large configurations in specific dimensions.

Contribution

It provides new large constructions of equiangular lines in various dimensions and establishes that current maximum configurations cannot be extended in certain cases.

Findings

Current known constructions cannot be extended for 14 ≤ d ≤ 20, d ≠ 15.

New large equiangular line sets: 248 in R^42, 200 in R^41, 168 in R^40, 152 in R^39, 56 in R^18.

Established limitations on augmenting existing maximum equiangular line sets.

Abstract

A set of lines through the origin in Euclidean space is called equiangular when any pair of lines from the set intersects with each other at a common angle. We study the maximum size of equiangular lines in Euclidean space and use graph theoretic approach to prove that all the currently known construction for maximum equiangular lines in $\mathbb R^d$ cannot add another line to form a larger equiangular set of lines if $14 \leq d \leq 20$ and $d \neq 15$. We give new constructions of large equiangular lines which are 248 equiangular lines in $\mathbb R^{42}$, 200 equiangular lines in $\mathbb{R}^{41}$, 168 equiangular lines in $\mathbb{R}^{40}$, 152 equiangular lines in $\mathbb R^{39}$ with angle $1/7$, and 56 equiangular lines in $\mathbb R^{18}$ with angle $1/5$.