A survey of partial difference sets

Abstract

Let G be a finite group of order ν. A k-element subset D of G is called a (ν, k, λ, μ)-partial difference set if the expressions gh-1, for g and h in D with g≠h, represent each nonidentity element in D exactly λ times and each nonidentity element not in D exactly μ times. If e∉D and g∈D iff g-1∈D, then D is essentially the same as a strongly regular Cayley graph. In this survey, we try to list all important existence and nonexistence results concerning partial difference sets. In particular, various construction methods are studied, e.g., constructions using partial congruence partitions, quadratic forms, cyclotomic classes and finite local rings. Also, the relations with Schur rings, two-weight codes, projective sets, difference sets, divisible difference sets and partial geometries are discussed in detail. © 1994 Kluwer Academic Publishers.