A symmetric conference matrix of order 86

2026-10-06

AI Summary

This study resolves the long-standing open problem concerning the existence of symmetric conference matrices of order 86. By employing a circulant matrix array construction combined with group automorphism analysis and combinatorial design theory, the authors successfully construct such a matrix, thereby filling the gap at the smallest previously unknown order. The principal contributions include establishing the existence of a symmetric conference matrix of order 86 and demonstrating its equivalence to regular two-graphs and equiangular tight frames. Furthermore, the work reveals deep connections between the matrix structure and strongly regular graphs. Specifically, the construction yields a matrix admitting an automorphism group of order 21, from which ten non-isomorphic strongly regular graphs are derived.

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 $12\times12$ array of $7\times7$ 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.

Problem

• symmetric conference matrix
• conference graph
• regular two-graph
• equiangular tight frame
• strongly regular graphs

Innovation

• symmetric conference matrix
• circulant array
• strongly regular graphs
• equiangular tight frame
• automorphism group