Covering triples by quadruples: an asymptotic solution
Journal of Combinatorial Theory Series A
The existence of resolvable Steiner quadruple systems
Journal of Combinatorial Theory Series A
Resolvable Steiner Quadruple Systems for the Last 23 Orders
SIAM Journal on Discrete Mathematics
Handbook of Combinatorial Designs, Second Edition (Discrete Mathematics and Its Applications)
Handbook of Combinatorial Designs, Second Edition (Discrete Mathematics and Its Applications)
Constructions for generalized Steiner systems GS(3, 4, v, 2)
Designs, Codes and Cryptography
H-designs with the properties of resolvability or (1, 2)-resolvability
Designs, Codes and Cryptography
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In 1987, Hartman showed that the necessary condition v 驴 4 or 8 (mod 12) for the existence of a resolvable SQS(v) is also sufficient for all values of v, with 23 possible exceptions. These last 23 undecided orders were removed by Ji and Zhu in 2005 by introducing the concept of resolvable H-designs. In this paper, we first develop a simple but powerful construction for resolvable H-designs, i.e., a construction of an RH(g 2n ) from an RH((2g) n ), which we call group halving construction. Based on this construction, we provide an alternative existence proof for resolvable SQS(v)s by investigating the existence problem of resolvable H-designs with group size 2. We show that the necessary conditions for the existence of an RH(2 n ), namely, n 驴 2 or 4 (mod 6) and n 驴 4 are also sufficient. Meanwhile, we provide an alternative existence proof for resolvable H-designs with group size 6. These results are obtained by first establishing an existence result for resolvable H-designs with group size 4, that is, the necessary conditions n 驴 1 or 2 (mod 3) and n 驴 4 for the existence of an RH(4 n ) are also sufficient for all values of n except possibly n 驴 {73, 149}. As a consequence, the general existence problem of an RH(g n ) is solved leaving mainly the case of g 驴 0 (mod 12) open. Finally, we show that the necessary conditions for the existence of a resolvable G-design of type g n are also sufficient.