A few more Hadamard Partitioned Difference Families

TL;DR

This paper introduces three new examples of Hadamard partitioned difference families and explores their existence, expanding the limited known instances and initiating a broader investigation into their parameters.

Contribution

The paper presents three new Hadamard partitioned difference families and begins a systematic study of their existence conditions.

Findings

Three new HPDF examples in various groups.

Initial results on the existence of HPDFs with specific parameters.

Foundational steps for future classification of HPDFs.

Abstract

A $(G,[k_1,\dots,k_t],\lambda)$ {\it partitioned difference family} (PDF) is a partition $\cal B$ of an additive group $G$ into sets ({\it blocks}) of sizes $k_1$, \dots, $k_t$, such that the list of differences of ${\cal B}$ 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.