Block Disjoint Difference Families for Steiner Triple Systems: v ≡ 3 mod 6

Abstract

A block disjoint (v,k,) difference family is a difference family with disjoint blocks. We show that disjoint (v; 3; 1) difference families exist for all v j 3 mod 6 with v 3. 1 Introduction Let G be a group of order v. A family of k-tuples of elements from G is a (v,k,) difference family if the collection of orbits of the k-tuples (disregarding repeated k-tuples) under the action of G form a balanced incomplete block design, BIBD(v; k; ). If the k-tuples are pairwise disjoint, call the family a block disjoint (v,k,) difference family. In this paper we show that there exists a block disjoint (v,3,1) difference family for all v j 3 mod 6. This is a companion paper to [2] which considered the case of block disjoint (v,3,1) difference family for v j 1 mod 6. The reader is refered to that paper for background information. 2 Constructions In this section we give constructions for block disjoint (v,3,1) difference families when v j 3 mod 6. We give a different construction for each congru..