Negative Latin square type partial difference sets and amorphic association schemes with Galois rings

A partial difference set (PDS) having parameters (n², r(n?1), n+r²?3r, r²?r) is called a Latin square type PDS, while a PDS having parameters (n², r(n+1), ?n+r²+3r, r² +r) is called a negative Latin square type PDS. There are relatively few known constructions of negative Latin square type PDSs, and nearly all of these are in elementary abelian groups. We show that there are three different groups of order 256 that have all possible negative Latin square type parameters. We then give generalized constructions of negative Latin square type PDSs in 2‐groups. We conclude by discussing how these results fit into the context of amorphic association schemes and by stating some open problems. © 2009 Wiley Periodicals, Inc. J Combin Designs 17: 266‐282, 2009