A symmetric conference matrix of order 86

Abstract

We construct a symmetric conference matrix of order $86$, the smallest order for which existence was open. Equivalently, there exist a conference graph with parameters $(85,42,20,21)$, a regular two-graph on $86$ points, and a real equiangular tight frame of $86$ vectors in ${\mathbb{R}}^{43}$. The matrix is a $12\times 12$ array of $7\times 7$ circulants with two border rows, and it admits a group of automorphisms of order $21$. Its $86$ descendants give exactly ten nonisomorphic strongly regular graphs.