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.