# From a 1-Rotational RBIBD to a Partitioned Difference Family

### Abstract

Generalizing the case of $\lambda=1$ given by Buratti and Zuanni [*Bull Belg. Math. Soc.* (1998)], we characterize the *$1$-rotational difference families* generating a 1-rotational $(v,k,\lambda)$-RBIBD, that is a $(v,k,\lambda)$ *resolvable balanced incomplete block design* admitting an automorphism group $G$ acting sharply transitively on all but one point $\infty$ and leaving invariant a resolution $\cal R$ of it. When $G$ is transitive on $\cal R$ we prove that removing $\infty$ from a parallel class of $\cal R$ one gets a *partitioned difference family*, a concept recently introduced by Ding and Yin [*IEEE Trans. Inform. Theory*, 2005] and used to construct *optimal constant composition codes*. In this way, by exploiting old and new results about the existence of 1-rotational RBIBDs we are able to derive a great bulk of previously unnoticed partitioned difference families. Among our RBIBDs we construct, in particular, a $(45,5,2)$-RBIBD whose existence was previously in doubt.