On \((64, 28, 12)\) Difference Sets

Abstract

There are \(267\) nonisomorphic groups of order \(64\). It was known that \(259\) of these groups admit \((64, 28, 12)\) difference sets and the other eight groups do not admit \((64, 28, 12)\) difference sets. Despite this result, no research investigates the problem of finding all \((64, 28, 12)\) difference sets in a certain group of order \(64\).In this paper, we find all \((64, 28, 12)\) difference sets in \(111\) groups of order \(64. 106\) of these groups are nonabelian. The other five are \(\mathbb{Z}_{16} \times \mathbb{Z}_4\), \(\mathbb{Z}_{16} \times \mathbb{Z}_2^2\), \(\mathbb{Z}_8 \times \mathbb{Z}_8\), \(\mathbb{Z}_8 \times \mathbb{Z}_4 \times \mathbb{Z}_2\), and \(\mathbb{Z}_8 \times \mathbb{Z}_2^3\).In these \(111\) groups, we obtain \(74,922\) non-equivalent \((64, 28, 12)\) difference sets. These difference sets provide at least \(105\) nonisomorphic symmetric \((64, 28, 12)\) designs. Most of our work was done using programs with the software \(GAP\).