Frames over finite fields: Equiangular lines in orthogonal geometry

TL;DR

This paper explores equiangular tight frames in finite orthogonal geometries, revealing their relation to modular strongly regular graphs and establishing bounds and conditions for their existence.

Contribution

It introduces a modular generalization of SRGs for ETFs in finite geometries and derives new bounds and necessary conditions, extending known results from real ETFs.

Findings

Gerzon's bound is attained in infinitely many dimensions in finite geometries.

Most known SRG parameters satisfy the new constraints in finite fields.

Gerzon's bound cannot be achieved in a 5-dimensional finite orthogonal geometry.

Abstract

We investigate equiangular lines in finite orthogonal geometries, focusing specifically on equiangular tight frames (ETFs). In parallel with the known correspondence between real ETFs and strongly regular graphs (SRGs) that satisfy certain parameter constraints, we prove that ETFs in finite orthogonal geometries are closely aligned with a modular generalization of SRGs. The constraints in our finite field setting are weaker, and all but~18 known SRG parameters on $v \leq 1300$ vertices satisfy at least one of them. Applying our results to triangular graphs, we deduce that Gerzon's bound is attained in finite orthogonal geometries of infinitely many dimensions. We also demonstrate connections with real ETFs, and derive necessary conditions for ETFs in finite orthogonal geometries. As an application, we show that Gerzon's bound cannot be attained in a finite orthogonal geometry of dimension~5.