The Lemmens-Seidel conjecture for base size $5$

TL;DR

This paper proves the Lemmens-Seidel conjecture for base size 5, addresses related questions about equiangular lines, and clarifies the structure of certain maximal sets in specific dimensions.

Contribution

It provides a complete proof of the Lemmens-Seidel conjecture for base size 5 and answers several open questions about equiangular lines in various dimensions.

Findings

Proof of the Lemmens-Seidel conjecture for base size 5.

Negative answer to the uniqueness of certain equiangular line sets.

Negative answer to the existence of strongly maximal equiangular line sets beyond known configurations.

Abstract

In 2020, Lin and Yu claimed to prove the so-called Lemmens-Seidel conjecture for base size $5$. However, their proof has a gap, and in fact, some set of equiangular lines found by Greaves et al. in 2021 is a counterexample to one of their claims. In this paper, we give a proof of the conjecture for base size $5$. Also, we answer in the negative a question of Greaves et al. in 2021 whether some sets of $57$ equiangular lines with common angle $\arccos(1/5)$ in dimension $18$ are contained in a unique set of $276$ equiangular lines with common angle $\arccos(1/5)$ in dimension $23$. In addition, we answer in the negative a question of Cao et al. in 2021 whether a strongly maximal set of equiangular lines with common angle $\arccos(1/5)$ exists except the set of $276$ equiangular lines with common angle $\arccos(1/5)$ in dimension $23$.