# Codes and Expansions (CodEx) Seminar

## Gary Greaves (Nanyang Technological University)Real equiangular line systems in low dimensions

A system of lines through the origin of $\mathbb R^d$ for which the angle between any pair of lines is a constant is called equiangular. A Seidel matrix, which can be interpreted as a variation of the adjacency matrix of a graph, is a tool for studying equiangular line systems. In this talk, we present our recent improvements to the upper bounds for the cardinalities of equiangular line systems in low dimensions. In particular, we show that the largest cardinality of an equiangular line system in dimensions 14, 16, and 17, is 28, 40, and 48, respectively. A crucial ingredient for these results is a certain modular restriction on the coefficients of the characteristic polynomial of a Seidel matrix. We also present our construction of 57 equiangular lines in 18 dimensions, which improves the lower bound for the maximum cardinality of an equiangular line system in 18 dimensions.

This talk is based on joint work with Pavlo Yatsyna and Jeven Syatriadi.