Partitioned difference families and harmonious linear spaces

Abstract

We say that a linear space is harmonious if it is resolvable and admits an automorphism group acting sharply transitively on the points and transitively on the parallel classes. Generalizing old results by the first author et al. we present some difference methods to construct harmonious linear spaces. We prove, in particular, that for any finite non-singleton subset $K$ of $\mathbb{Z}^+$ there are infinitely many values of $v$ for which there exists a partitioned difference family that is the base parallel class of a harmonious linear space with $v$ points whose block sizes are precisely the elements of $K$.