Construction of E(s2)-optimal supersaturated designs

Abstract

Booth and Cox proposed the E(s 2 ) criterion for constructing twolevel supersaturated designs. Nguyen [Technometrics 38 (1996) 69-73] and Tang and Wu [Canad. J. Statist 25 (1997) 191-201] independently derived a lower bound for E(s 2 ). This lower bound can be achieved only when m is a multiple of N − 1, where m is the number of factors and N is the run size. We present a method that uses difference families to construct designs that satisfy this lower bound. We also derive better lower bounds for the case where the Nguyen-Tang-Wu bound is not achievable. Our bounds cover more cases than a bound recently obtained by Butler, Mead, Eskridge and Gilmour [J. R. Stat. Soc. Ser. B Stat. Methodol. 63 (2001) 621-632]. New E(s 2 )-optimal designs are obtained by using a computer to search for designs that achieve the improved bounds.

An Overview of Two-Level Supersaturated Designs with Cyclic Structure

“…. , i + 30 of G. The generators can further be rewritten by noting the positions which hold +1, that is the positions in the first rows of the 30 cyclic subarrays that hold +1: 4,6,7,10,11 ) .…”

“…If n ≡ 2 (mod 4) with k odd, then The cyclic construction method used by Plackett and Burman (1946) for orthogonal saturated designs was generalized by Liu and Dean (2004) to obtain a class of supersaturated Liu and Dean (2004) give rise to designs that are mean-orthogonal and 24 which are either Es 2 -optimal or have at least 96.4% efficiency. The following three results summarize the links between k-circulant designs, the concatenation of k individual saturated 26 cyclic (1-circulant) designs and the designs of Nguyen (1996), Eskridge, Gilmour, Mead, Butler and Travnicek (2004), and Bulutoglu and Cheng (2004) formed from the incidence 28 matrices of cyclic incomplete block designs. These links are illustrated in Example 3.1.…”

“…Various non-cyclic solutions for Es 2 -optimal supersaturated designs for ranges of n and m have been provided 2 by Butler, Mead, Eskridge and Gilmour (2001), Bulutoglu and Cheng (2004), Ryan and Bulutoglu (2007) and Das, Dey, Chan and Chatterjee (2008). 4 …”

“…It is interesting to note very different autocorrelation function values in Table 4.1 for T i 's which comprise these two Bulutoglu and Cheng (2004) gave the complete set of solutions for 10-run supersaturated designs attaining bound (2.2) and they showed that they exist only for m = 18t factors with 2 t = 1, . .…”

“…4 (0,1,3,5,8,14,16,17,18); (2,4,5,7,9,13,14,15,17); (0,2,4,7,9,10,13,14,18); (0,1,3,5,6,7,10,13,14); (3,5,6,9,10,11,12,13,18); (0,5,6,9,13,14,16,17,18). 20 133 2964 0 , 4427 4 , 1387 8 0.…”