Optimal Incomplete Block Designs

Abstract

Summary The class of regular graph (RG) designs, conjectured to be optimal among all incomplete block designs, is defined. The results of a systematic search of the class of RG designs are given for all cases in which v ≤ 12, r ≤ 10 and v ≤ b. In nearly every case, the optimum design is determined for each of the three optimality criteria: A, D and E. For v > b, we present the duals of the optimal designs for v < b. Many of the designs are new and these are tabulated in a back‐up report.

Designs on the web

“…(These optimality criteria are described in Section 4.6.) In particular, we are checking and extending the influential work of John and Mitchell [17] by considering those designs with v 12 and r 10. Usually, but not always, the designs we seek can be shown to be regular-graph designs.…”

Optimality and construction of some rectangular designs

“…In contradistinction to those of type 2, a balanced rectangular design of type 1 is an RGD (in fact a regular generalized line graph design) and hence it is E-optimal by Cheng and Constantine (1988) and Bagchi and Cheng (1991). John and Mitchell (1977) conjectured that whenever RGD's exist, optimal block designs are found among them. Since then, optimality property of a good many RGD's have been proved [see Shah and Sinha (1989) for a detailed survey].…”

“…However, examples of designs other than RGD's better than the best RGD have also been found in Jones and Eccleston (1980) (A-crireion) and in Constantine (1986) (E-criterion). In this paper we provide infinitely many counter examples to the John and Mitchell (1977) However, when v = 0 (rood 4), a balanced rectangular design of type 2 loses to a certain type of three class PBIBD which is an RGD. We call this a double cocktail party graph design.…”

“…We also observe that these designs, though themselves not regular graph designs (RGD's), are yet strictly E-better than every competing RGD, whenever v > 26 and v =-2 (mod 4). This provides an infinite series of counter examples to the conjecture of John and Mitchell (1977).We also present two methods of construction of the rectangular designs. Apart from providing infinitely many examples of the designs proved E-optimal in this paper and in Cheng and Constantine (1986), this construction also provides -as a special case -the first known infinite series of most balanced group divisible designs, which were proved optimal with respect to all type 1 criteria by Cheng (1978).…”

“…We also observe that these designs, though themselves not regular graph designs (RGD's), are yet strictly E-better than every competing RGD, whenever v > 26 and v =-2 (mod 4). This provides an infinite series of counter examples to the conjecture of John and Mitchell (1977).…”