Further results on (v, {5, w*}, 1)-PBDs

作者：Abel R Julian R; Ge Gennian; Greig Malcolm; Ling Alan C H

来源：2009, v 309, n 8.

DOI：10.1016/j.disc.2008.05.008

• Covering design
• Number of blocks
• Pairwise balanced design
• PBD

In this article we investigate the existence of pairwise balanced designs on v points having blocks of size five, with a distinguished block of size w, briefly (v, {5, w*}, 1)-PBDs. The necessary conditions for the existence of a (v, {5, w*}, 1)-PBD with a distinguished block of size w with v > w are that v ≥ 4 w + 1, v ≡ w ≡ 1 (mod 4) and either v ≡ w (mod 20) or v + w ≡ 6 (mod 20). Previously, Bennett et al. had shown that these conditions are sufficient for w > 2457 with the possible exception of v = 4 w + 9 when w ≡ 17 (mod 20), and had studied w &le 97 in detail, showing there that the necessary conditions are sufficient with 71 possible exceptions. In this article, we show sufficiency for w ≡ 1, 5, 13 (mod 20) and give a small list of possible exceptions containing 26 and 104 values for w ≡ 9, 17 (mod 20). For w ≡ 9 (mod 20), all possible exceptions satisfy either v = 4 w + 13 with w &le 489 or v &nequiv; w (mod 20) with v < 5 w and w &le 129; for w ≡ 17 (mod 20), all possible exceptions except (v, w) = (197, 37), (529, 37) satisfy either v = 4 w + 9 with w &le 1757 or v &nequiv; w (mod 20) with v < 5 w and w &le 257. As an application of our results for w = 97, we establish that, if v ≡ 9, 17 (mod 20), v ≥ 389 and v ≠ 429, then the smallest number of blocks in a pair covering design with k = 5 is ⌈ v (v - 1) / 20 ⌉, i.e., the Scho&die;nheim bound.

• 单位
  浙江大学; the University of New South Wales; university of new south wales