Group divisible designs with block size four and two groups

Abstract

We give some constructions of new infinite families of group divisible designs, GDD(n, 2, 4; 1 , 2 ), including one which uses the existence of Bhaskar Rao designs. We show the necessary conditions are sufficient for 3 n 8. For n = 10 there is one missing critical design. If 1 > 2 , then the necessary conditions are sufficient for n ≡ 4, 5, 8 (mod 12). For each of n=10, 15, 16, 17, 18, 19, and 20 we indicate a small minimal set of critical designs which, if they exist, would allow construction of all possible designs for that n. The indices of each of these designs are also among those critical indices for every n in the same congruence class mod 12.

Group divisible designs of four groups and block size five with configuration (1; 1; 1; 2)

“…There is a more technical proof given in the book "Triple System" [2]. Similar results were established for GDDs with block size four in [6,8,9,14,19]. In [7,16], results about GDDs with two groups and block size five with fixed block configuration were presented.…”

The Use of Treatment Concurrences to Assess Robustness of Binary Block Designs Against the Loss of Whole Blocks

“…Sinha & Kageyama () and Arasu & Harris () give constructions for families of semi‐regular and regular group divisible designs, although it is difficult to draw general conclusions about the robustness status of designs belonging to these families. Constructions of new families of group divisible designs with m =2 and k =4 are given by Hurd & Sarvate (), some of which are not maximally robust as demonstrated in Example . Rodger & Rogers () generalise the three Clatworthy designs S 2, S 4 and R 96 to provide necessary and sufficient conditions for the existence of families of regular designs with the parameter sets (3 n , b , r ,4;(4,2)), (3 n , b , r ,4;(8,4)) and (3 n , b , r ,4;(4,5)) for certain values of n ; all designs in the family generalising R 96 are maximally robust by Theorem and all designs in the other two families are maximally robust by Theorem , as shown in Example for the case of the family generalising S 2.…”

“…Example A family of regular designs is given from a construction of Lemma of Hurd & Sarvate (). Ten treatments are allocated to 10 t +15 blocks, which consist of the blocks from 2 t +1 small designs with block‐size k =4.…”

A note on the robustness of PBIBD(2)s against breakdown in the event of observation loss

“…Since the publication of these tables, constructions for numerous additional PBIBD(2)s have appeared in the literature. See for example: Cheng, Constantine & Hedayat (1984), Sinha (1987), Greig, Kreher & Ling (2002), Henson, Sarvate & Hurd (2007) and Hurd & Sarvate (2008).…”