Sets of equiangular lines in dimension $18$ constructed from $A_5^3\oplus A_1^4$

Abstract

In 2023, Greaves, Syatriadi, and Yatsyna found a set of $57$ equiangular lines in ${\mathbb{R}}^{18}$, breaking the previous record. In 2025, Lin, Munemasa, Taniguchi, and Yoshino constructed a large number of sets of $57$ equiangular lines in ${\mathbb{R}}^{18}$ as affine equiangular sets in an integral overlattice of $A_9^2\oplus A_1$. In this paper, we construct further sets of $57$ equiangular lines in ${\mathbb{R}}^{18}$ from Latin squares of order $6$ and Pasch configurations, realized as affine equiangular sets in an integral overlattice of $A_5^3\oplus A_1^4$. Unlike the previously known examples, these sets are not strongly maximal. Moreover, some of them have only five distinct Seidel eigenvalues, fewer than any previously known examples.

16 pages