Further results on large sets of disjoint group‐divisible designs with block size three and type 2n41

Large sets of disjoint group‐divisible designs with block size three and type 2ⁿ4¹ were first studied by Schellenberg and Stinson because of their connection with perfect threshold schemes. It is known that such large sets can exist only for n ≡0 (mod 3) and do exist for all odd n ≡ (mod 3) and for even n=24m, where m odd ≥ 1. In this paper, we show that such large sets exist also for n=2^k(3m), where m odd≥ 1 and k≥ 5. To accomplish this, we present two quadrupling constructions and two tripling constructions for a special large set called ^*LS(2ⁿ). © 2002 Wiley Periodicals, Inc. J Combin Designs 11: 24–35, 2003; Published online in Wiley InterScience ( www.interscience.wiley.com ). DOI 10.1002/jcd.10032