Factoring $(16, 6, 2)$ Hadamard Difference Sets

Abstract

We describe a "factoring" method which constructs all twenty-seven Hadamard $(16,6,2)$ difference sets. The method involves identifying perfect ternary arrays of energy 4 (PTA(4)) in homomorphic images of a group $G$, studying the image of difference sets under such homomorphisms and using the preimages of the PTA(4)s to find the "factors" of difference sets in $G$. This "factoring" technique generalizes to other parameters, offering a general mechanism for creating Hadamard difference sets.