Hadamard matrices are used as experimental plans in factor screening in test and evaluation. Finding a Hadamard matrix in general is a difficult open problem. A Legendre pair of length n can be used to construct a Hadamard matrix of order 2n+2 for each odd integer n. Understanding the existence patterns of Legendre pairs will lead to better algorithms for finding Hadamard matrices. This reseach opportunity focuses on applying cyclotomic field extensions for developing a theory to better understand the existence patterns of Legendre pairs. This theory will be used for devoloping improved algorithms for finding Legendre pairs.  

I.S. Kotsireas, C. Koutschan, D.A. Bulutoglu, D.M. Arquette, J.S. Turner, K.J. Ryan: Legendre paits of length 0 mod 5: Special matrices 11: 2023

J.S. Turner, I.S. Kotsireas, DA Bulutoglu, A.J. Geyer: A Legendre pair of length 77 using complementary binary matriceswith fixed marginals:  Designs, Codes , and Cryptography 89:2021

K.T. Arasu, D.A. Bulutoglu, J. Hollon; Legendre G-array pairs and the theorethical unification of several G-array families: Journal of Combinatorial Designs 28:2020

key words

Hadamard matrices; Legendre pairs; Algebraic number theory; Algebraic combinatorics; Cyclotomic field extensions

Citizenship:  Open to U.S. citizens

Level:  Open to Postdoctoral and Senior applicants