Frames over finite fields: equiangular lines in orthogonal geometry

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 orthogon...

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Bibliographic Details
Main Authors: Greaves, Gary Royden Watson, Iverson, Joseph W., Jasper, John, Mixon, Dustin G.
Other Authors: School of Physical and Mathematical Sciences
Format: Article
Language:English
Published: 2022
Subjects:
Science::Mathematics
Equiangular Lines
Equiangular Tight Frames
Online Access:https://hdl.handle.net/10356/162564
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Institution: Nanyang Technological University
Language: English
Description
Summary: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.

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