All \((64, 28, 12)\) Difference Sets and Related Structures

Abstract

There are \(267\) nonisomorphic groups of order \(64\). It was known that \(259\) of these groups admit \((64, 28, 12)\) difference sets. In \([4]\), the author found all \((64, 28, 12)\) difference sets in \(111\) groups. In this paper, we find all \((64, 28, 12)\) difference sets in all the remaining groups of order \(64\) that admit \((64, 28, 12)\) difference sets. Also, we find all nonisomorphic symmetric \((64, 28, 12)\) designs that arise from these difference sets. We use these \((64, 28, 12)\) difference sets to construct all \((64, 27, 10, 12)\) and \((64, 28, 12, 12)\) partial difference sets. Finally, we examine the corresponding strongly regular graphs with parameters \((64, 27, 10, 12)\) and \((64, 28, 12, 12)\).