Generalized packing designs

Abstract

Generalized t-designs, which form a common generalization of objects such as t-designs, resolvable designs and orthogonal arrays, were defined by Cameron [P.J. Cameron, A generalisation of t-designs, Discrete Math. 309 (2009), 4835-4842]. In this paper, we define a related class of combinatorial designs which simultaneously generalize packing designs and packing arrays. We describe the sometimes surprising connections which these generalized designs have with various known classes of combinatorial designs, including Howell designs, partial Latin squares and several classes of triple systems, and also concepts such as resolvability and block colouring of ordinary designs and packings, and orthogonal resolutions and colourings. Moreover, we derive bounds on the size of a generalized packing design and construct optimal generalized packings in certain cases. In particular, we provide methods for constructing maximum generalized packings with t = 2 and block size k = 3 or 4.

On generalized Howell designs with block size three

“…6 8 1 and 7 6 9 1 . Since there exist GHD (8,21), GHD (9,24), GHD (10,27) and GHD (12,33) by Lemma 4.2, the result follows by Corollary 3.9.…”

“…A Kirkman square of order 3 is trivial to construct. Mathon and Vanstone [37] showed the non-existence of a KS (9) or KS (15), while the non-existence of a DRNKTS (6) and DRNKTS (12) follows from Kotzig and Rosa [32]. For many years, the smallest known example of a GHD(s, v) with s = (other than the trivial case of s = 1, v = 3) was a DRNKTS (24), found by Smith in 1977 [43].…”

“…, T m ) (where T i ⊆ X i and |T i | = t i for all i) is contained in at most λ members of P. We say a generalized packing is optimal if it contains the largest possible number of blocks. (See [9] for further details, and for numerous examples. )…”

“…The notion of generalized t-designs introduced by Cameron in 2009 [14], and the broader notion of generalized packing designs introduced by Bailey and Burgess [9], provide a common framework for studying many classes of combinatorial designs. As described below, generalized Howell designs fit neatly into this framework.…”

“…For n = 7, x = 2, the following is an intransitive starter-adder over (Z 7 × {0, 1, 2}) for a GHD * (9,21). (In square brackets we either give the adder for the corrresponding starter block, or indicate whether it belongs to R or C.)…”