Frames over finite fields: equiangular lines in orthogonal geometry

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≤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. Ministry of Education (MOE) GRWG was partially supported by the Singapore Ministry of Education Academic Research Fund (Tier 1); grant numbers: RG29/18 and RG21/20. JJ was supported by NSF DMS 1830066. DGM was partially supported by AFOSR FA9550-18-1-0107 and NSF DMS 1829955.