Pitch tournament designs and other BIBDs—existence results for the case ν = 8n

Abstract

A pitch tournament is a resolvable or near resolvablevY 8Y 7 BIBD that satis®es certain criteria in addition to theusual condition that v 0 or 1 mod 8. Here we establish that for the case v 8n the necessary condition forpitch tournaments is suf®cient for all n b 1615, with at most 187 smaller exceptions. This complements our earlier study of the v 8n 1 case, where we established suf®ciency for all n b 224, with at most 28 smaller exceptions. The four missing cases for vY 8Y 7 BIBDs are provided, namely v Pf48Y 56Y 96Y 448g, thereby establishing that the necessary existence conditions are suf®cient without exception. Some constructions for resolvable designs are also provided, reducing the existence question for vY 8Y 7 RBIBDs to 21 possible exceptions.

Existence of (2, 8) GWhD(v) and (4, 8) GWhD(v) with $${v \equiv 0,1 (mod 8)}$$

“…For this study we were able to generate, for the case (t, k) = (2,8), the same "core-set" of specific designs (plus some new ones). Certainly, then, a repeat of the arguments contained in [5,6] would lead to the same conclusions regarding the existence of (2, 8) GWhD(v), v ≡ 0, 1 (mod 8). Rather than repeat the argumentation contained in [5,6] (which is lengthy) we present the "core-set" of specific designs in an Appendix and begin with the cases left open by the studies in [5,6].…”

“…If v ≡ 1 (mod k), then m = v and A = ∅. A (2, 4) GWhD(v) is the standard whist tournament (see e.g., [8]), and a (4,8) GWhD(v) is called a pitch tournament in [4][5][6]13,14]. Abel et al [4] contains many constructions for GWhDs, which we will cite as needed.…”

“…Existence of (4, 8) GWhD(v), v ≡ 1 (mod 8) and v ≡ 0 (mod 8) is discussed in [5] and [6], respectively. In each of these latter studies a certain "core-set" of specific designs (corresponding to (t, k) = (4, 8)) together with known results and constructions (both known and new) were combined to reduce the question of the existence of (4, 8) GWhD(v) to 28 values of v = 8n + 1 and 187 values of v = 8n.…”

Splitting balanced incomplete block designs with block size 3 × 2

“…For the case v ¼ 53, we begin with a 4-GDD of type 4 4 (for whose existence, see Lemma 2.2) and give three points in one block weight 0 and the remaining points weight 1 to obtain a f3; 4g-GDD of type 3 3 4 1 by applying Theorem 2.4. We then give the points of the resulting GDD weight 4 and apply Theorem 2.5 to obtain a 3 Â 2-splitting GDD of type 12 3 Proof. We begin with a GD½4; 1; t; 4t; t 1 ðmod 2Þ; t !…”

“…It is easy to see there is no ð9; 3 Â 2; 1Þ-splitting BIBD. [19]) and give one point in a group size 9 weight 0 and the remaining points weight 1 to obtain a f3; 4g-GDD of type 6 5 Proof. From Lemma 3.2.1 we need not consider v ¼ 85; 93.…”