# Regular Simplices Within Doubly Transitive Equiangular Tight Frames

3-2023

Thesis

## Degree Name

Master of Science in Applied Mathematics

## Department

Department of Mathematics and Statistics

Matthew C. Fickus, PhD

## Abstract

An equiangular tight frame (ETF) yields an optimal way to pack a given number of lines into a given space of lesser dimension. Every ETF has minimal coherence, and this makes it potentially useful for compressed sensing. But, its usefulness also depends on its spark: the size of the smallest linearly dependent subsequence of the ETF. When formed into a sensing matrix, a larger spark means a lower chance that information is lost when sensing a sparse vector. Spark is difficult to compute in general, but if an ETF contains a regular simplex, then every such simplex is a linearly dependent subsequence of the ETF of minimal size, and so its spark is known. The “binder” of an ETF indicates all of the regular simplices that the ETF contains. If the binder of an ETF is empty, then it contains no regular simplex, and so its spark is larger than otherwise guaranteed. Proving that either holds, namely proving that a particular ETF’s binder is empty or instead proving that it is not, provides useful information about an ETF’s suitability for compressed sensing. In this thesis, we focus on doubly transitive equiangular tight frames (DTETFs), namely ETFs whose symmetry group acts in a doubly transitive way. We show that the binder of any DTETF is either empty or forms a balanced incomplete block design (BIBD), a fact that applies to several known infinite families of ETFs. We then fully characterize the binders of every member of one such infinite family, symplectic ETFs, and their Naimark complements. We show that all but a finite number of symplectic ETFs have an empty binder. We also provide a general, closed-form expression for the binder of the symplectic ETF’s Naimark complement and the number of blocks that it contains; this disproves a conjecture posed in the recent literature.

## AFIT Designator

AFIT-ENC-MS-23-M-004