Date of Award
9-2023
Document Type
Dissertation
Degree Name
Doctor of Philosophy (PhD)
Department
Department of Mathematics and Statistics
First Advisor
Dursun Bulutoglu, PhD
Abstract
With applications spanning myriad disciplines, Hadamard matrices have tremendous utility. Infinitely many have been discovered, but there is no general proof of their existence. Proving the Hadamard conjecture would provide a significant technological edge. Hadamard matrices are difficult to construct in general; however, they can be created directly from Legendre pairs (LPs). While LPs are not trivial to find, a breakthrough in this area would be pivotal in the quest to prove the Hadamard conjecture. Most recent efforts rely upon refined search algorithms, and we seek to either develop a better such algorithm or discover an altogether new theoretical construction. By leveraging Galois theory, we are able to gain greater insight into the power spectral densities (PSDs) of LPs. We also develop a new theorem and conjecture about LP compressions. Finally, we provide a new standard for recording and reporting LPs which will benefit future efforts in this area.
AFIT Designator
AFIT-ENC-DS-23-S-001
Recommended Citation
Arquette, David M., "Leveraging Galois Theory and Computational Results in the Search for Legendre Pairs" (2023). Theses and Dissertations. 7658.
https://scholar.afit.edu/etd/7658
Comments
A 12-month embargo was observed for posting this dissertation on AFIT Scholar.
Approved for public release. PA case number on file.