Exponential number of inequivalent difference sets in ℤ

Kantor [5] proved an exponential lower bound on the number of pairwise inequivalent difference sets in the elementary abelian group of order 2 2s+2 . Dillon [3] generalized a technique of McFarland [6] to provide a framework for determining the number of inequivalent difference sets in 2‐groups with a large elementary abelian direct factor. In this paper, we consider the opposite end of the spectrum, the rank 2 group ℤ, and compute an exponential lower bound on the number of pairwise inequivalent difference sets in this group. In the process, we demonstrate that Dillon difference sets in groups ℤ can be constructed via the recursive construction from [2] and we show that there are exponentially many pairwise inequivalent difference sets that are inequivalent to any Dillon difference set. © 2003 Wiley Periodicals, Inc. J Combin Designs 11: 249–259, 2003; Published online in Wiley InterScience ( www.interscience.wiley.com ). DOI 10.1002/jcd.10046