Date of Award

6-2025

Document Type

Dissertation

Degree Name

Doctor of Philosophy (PhD)

Department

Department of Mathematics and Statistics

First Advisor

John D. Jasper, PhD

Abstract

An equiangular tight frame (ETF) is an equal norm sequence of vectors in a Hilbert space whose coherence achieves equality in the Welch bound. Such sequences necessarily have minimal coherence and thus are, in some sense, as "spread out" in space as possible. ETFs have a variety of applications, such as compressed sensing and waveform design. The main problem in the study of ETFs is determining the pairs (D, N) for which an ETF with N vectors in a D-dimensional space exists. Real ETFs are moreover equivalent to a special subset of a well-studied class of graphs known as strongly regular graphs (SRGs). The main topic of this dissertation is a new method for constructing ETFs. We introduce the concept of compatible orthobiangular tight frames (COBTF). We show how to construct an ETF from any COBTF. This method unifies and generalizes several known constructions of ETFs. We identify equivalences between certain classes of COBTFs and Steiner systems as well as difference sets. Using this new framework and the connections to difference set literature, we produce several new infinite families of ETFs. Infinitely often, these ETFs are real and give rise to new SRG.

AFIT Designator

FY25-AFIT-ENC-DS-25-J-001

Comments

An embargo was observed for posting this thesis on AFIT Scholar.
Approved for public release, distribution unlimited. PA case number 88ABW-2025-0538

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