The Lemmens-Seidel conjecture for base size \(5\)

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 \(23\).