There are no 76 equiangular lines in $R^{19}$

Abstract

Maximum size of equiangular lines in $\mathbb{R}^{19}$ has been known in the range between 72 to 76 since 1973. Acoording to the nonexistence of strongly regular graph $(75,32,10,16)$ \cite{aza15}, Larmen-Rogers-Seidel Theorem \cite{lar77} and Lemmen-Seidel bounds on equiangular lines with common angle $\frac 1 3$ \cite{lem73}, we can prove that there are no 76 equiangular lines in $\mathbb{R}^{19}$. As a corollary, there is no strongly regular graph $(76,35,18,14)$. Similar discussion can prove that there are no 96 equiangular lines in $\mathbb{R}^{20}$.