Some results on constructing general minimum lower order confounding $2^{n-m}$ designs for $n\le 2^{n-m-2}$

Abstract

Zhang et al. (Stat Sinica 18:1689–1705, 2008) introduced an aliased effect-number pattern for two-level regular designs and proposed a general minimum lower-order confounding (GMC) criterion for choosing optimal designs. All the GMC $2^{n-m}$ designs with $N/4+1\le n\le N-1$ were constructed by Li et al. (Stat Sinica 21:1571–1589, 2011), Zhang and Cheng (J Stat Plan Inference 140:1719–1730, 2010) and Cheng and Zhang (J Stat Plan Inference 140:2384–2394, 2010), where $N=2^{n-m}$ is run number and $n$ is factor number. In this paper, we first study some further properties of GMC design, then we construct all the GMC $2^{n-m}$ designs respectively with the three parameter cases of $n\le N-1$: (i) $m\le 4$, (ii) $m\ge 5$ and $n=(2^m-1)u+r$ for $u>0$ and $r=0,1,2$, and (iii) $m\ge 5$ and $n=(2^m-1)u+r$ for $u\ge 0$ and $r=2^m-3,2^m-2$.