Hadamard partitioned difference families and their descendants

Abstract

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