Cyclotomy and difference families in elementary abelian groups

AbstractBy a (v, k, λ)-difference family in an additive abelian group G of order v, we mean a family (Bi ∣ i ∈ I) of subsets of G, each of cardinality k, and such that among the differences (a − b ∣ a, b ∈ Bi; a ≠ b; i ∈ I) each nonzero g ∈ G occurs λ times. The existence of such a difference family implies the existence of a (v, k, λ)-BIBD with G as a regular group of automorphisms. The multiplicative structure of finite fields is used here to construct difference families in their additive groups. It is proven that if q is a prime power and q > {12k(k − 1)}k(k−1), then there exists a (q, k, λ)-difference family in the elementary abelian group of order q iff λ(q − 1) ≡ 0 (mod k(k − 1)). Some extensions of known constructions and partially balanced designs are also discussed