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Erschienen in:
25.04.2018
Hadamard partitioned difference families and their descendants
Erschienen in: Cryptography and Communications | Ausgabe 4/2019
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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 \(\phantom {\dot {i}\!}4u^{2}+2u\).