Complete sets of disjoint difference families and their applications (Q697993)

A \((v,k,\lambda)\) difference family in an abelian group \(G\) of order \(v\) is disjoint if its base blocks are pairwise disjoint. A collection of disjoint difference families is called complete if the base blocks of the families together form a partition of \(G-\{0\}\). Several constructions of complete sets of disjoint difference families (CDDF) are presented, and composition theorems are applied to produce new families. In particular, the cases \(k= 3\) or \(4\) and \(\lambda=1\) are considered; for example, it is shown that if \(|G|= p\) where \(p\) is prime, \(p\equiv 13\pmod{24}\), \(p> 13\), then a \((p,4,1)\)-CDDF exists in \(G\). The paper finishes with some applications to cyclically resolvable cyclic designs. For example, it is shown that if a \((6t+ 1,3,1)\)-CDDF exists in \(\mathbb{Z}_{6t+1}\) then there exists a cyclically resolvable cyclic \((18t+ 3,3,1)\) design in which the blocks in the regular short orbit are in different resolution classes.