A few more Hadamard Partitioned Difference Families

Abstract

A $(G,[k_1,\dots ,k_t],\lambda )$ {\it partitioned difference family} (PDF) is a partition $\mathcal{B}$ of an additive group $G$ into sets ({\it blocks}) of sizes $k_1$, \dots, $k_t$, such that the list of differences of $\mathcal{B}$ covers exactly $\lambda$ times every non-zero element of $G$. It is called {\it Hadamard} (HPDF) if the order of $G$ is $2\lambda$. The study of HPDFs is motivated by the fact that each of them gives rise, recursively, to infinitely many other PDFs. Apart from the {\it elementary} HPDFs consisting of a Hadamard difference set and its complement, only one HPDF was known. In this article we present three new examples in several groups and we start a general investigation on the possible existence of HPDFs with assigned parameters by means of simple arguments.