Conjectures for Optimal Block Designs

Abstract

Summary In the study and construction of optimal block designs over the past 20 years a number of conjectures have been repeatedly stated and used. The purpose of this note is to bring together some of these conjectures. Although known to those actively engaged in research in design, it is hoped that this note will stimulate other statisticians and mathematicians to give their attention to these unsolved problems.

Construction of Row and Column Designs with Contiguous Replicates

“…We illustrate this further with an example: The generating array 0 0 0 0 1 2 3 4 4 3 2 1 3 1 4 2 for s = 5 satisfies condition (1). Although we have used the computer to obtain a suitable row and column design from a latinized a-design, effective progress can be made by hand using the intuitive idea that it is better to try and reduce the range of concurrences of varieties within rows (John & Williams, 1982). The resulting a-design is for r = 4, u = 20 and k = 4; the blocks are columns within replicates.…”

“…We have not worried about this aspect since from (l), variety concurrences in long columns can only be 0 or 1, and so the design will always be acceptable in terms of long columns as blocks (John & Williams, 1982). We want to choose a-arrays leading to efficient latinized a-designs; here we use as a criterion the average efficiency factor E,, taking columns within replicates as blocks.…”

Triangles and Efficiency Factors

“…Since the best general designs are believed to be regular-graph (John & Williams, 1982;Paterson, 1983), we would expect to find at least nearly optimal designs by this approach. The bounds in the present paper are modifications to those described by Jarrett (1983).…”

A Criterion for the Construction of Optimal Neighbour Designs

“…The neighbour concurrence matrix is set out below. Many computer search procedures for the construction of efficient incomplete block designs routinely use the idea that high E 1 is associated with (i) binary designs and (ii) minimizing the range of the off-diagonal elements of NN' (John and Williams, 1982;Paterson, 1983). These ideas can also be applied to (16).…”