Hadamard partitioned difference families and their descendants

TL;DR

This paper introduces a doubling construction for Hadamard-type partitioned difference families, expanding the class of such families and providing new infinite families with specific parameters based on Hadamard difference sets.

Contribution

It presents a novel doubling construction method for Hadamard-type PDFs, enabling the generation of infinite classes of these families from existing ones.

Findings

Constructed an infinite class of PDFs from Hadamard difference sets.

Derived PDFs with specific parameters related to prime power divisors.

Extended the theory of difference families with new combinatorial structures.

Abstract

If $D$ is a $(4u^2,2u^2-u,u^2-u)$ Hadamard difference set (HDS) in $G$, then $\{G,G\setminus D\}$ is clearly a $(4u^2,[2u^2-u,2u^2+u],2u^2)$ partitioned difference family (PDF). Any $(v,K,\lambda)$-PDF will be said of Hadamard-type if $v=2\lambda$ as the one above. We present a doubling construction which, starting from any such PDF, leads to an infinite class of PDFs. As a special consequence, we get a PDF in a group of order $4u^2(2n+1)$ and three block-sizes $4u^2-2u$, $4u^2$ and $4u^2+2u$, whenever we have a $(4u^2,2u^2-u,u^2-u)$-HDS and the maximal prime power divisors of $2n+1$ are all greater than $4u^2+2u$.