# General Minimum Lower-Order Confounding Designs with Multi-Block Variables

Authors: Yuna Zhao
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

Blocking the inhomogeneous units of experiments into groups is an efficient way to reduce the influence of systematic sources on the estimations of treatment effects. In practice, there are two types of blocking problems. One considers only a single block variable and the other considers multi-block variables. The present paper considers the blocking problem of multi-block variables. Theoretical results and systematical construction methods of optimal blocked2n−m designs with N/4+1≤n≤5N/16 are developed under the prevalent general minimum lower-order confounding (GMC) criterion, where N=2n−m.


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## Body

## 1. Introduction

The regular2n−m factorial experiment has played an important role in engineering, manufacturing industry, agriculture and medicine, and so on. It allows efficient and economic experimentation to estimate treatment effects. When the size of the experimental units is large, the inhomogeneity will cause unwanted variance to the estimations of treatment effects. To reduce such bad influence, a crucial way is to partition the experimental units into blocks.There are two kinds of blocking problems as pointed out in [1]. One is called the single block variable problem which considers only a single block variable, and the other is called the multi-block variable problem which considers two or more block variables. In the last decades, choosing optimal blocked 2n−m designs with a single block variable has been well investigated; for example, the authors in [2–9] studied the blocked 2n−m designs under different minimum aberration criteria; Chen et al. and Zhao et al. [10, 11] explored the blocked 2n−m designs under the clear effects criterion; Zhang and Mukerjee, Zhao et al., and Zhao and Zhao [12–14] explored the constructions of the blocked 2n−m designs under the general minimum lower-order confounding (GMC) criterion proposed in [15]; and Zhao et al. [16–18] gave construction methods of the blocked 2n−m designs under another GMC criterion proposed in [19].Compared to the large body of work on the blocking problem of single block variable, the studies on choosing optimal blocked2n−m designs with multi-block variables are relatively rare. However, it has been recognized that the blocking problem of multi-block variables can arise quite naturally in many practical situations. For example, in the agricultural context, Bisgaard [1] pointed that when designs are laid out in rectangular schemes, both row and column inhomogeneity effects probably exist in the soil. Another example of multi-block variables is from [20]. Considering the comparison of two gasoline additives by testing them on two cars with two drivers over two days, the “cars,” “drivers,” and “days” are three block variables which should be taken into account when performing experiments.Under the clear effects criterion, Zhao and Zhao [21] proposed an algorithm for finding optimal blocked 2n−m designs with multi-block variables. Under the minimum aberration criterion, Zhao and Zhao [22] developed some rules for constructing optimal blocked 2n−m designs with multi-block variables. Zhang et al. [23] extended the idea of GMC criterion to the case of multi-block variable problem and developed the blocked GMC criterion, called the B2-GMC criterion. Inheriting the advantage of the GMC criterion, the B2-GMC designs are particularly preferable when some prior information on the importance ordering of treatment effects is present. By computer search, Zhang et al. [23] tabulated some B2-GMC deigns with small n and N, where N=2n−m. When n or N is large, computer search becomes computationally expensive. Zhao et al. [24] and Zhao and Zhao [25] completed the constructions of B2-GMC designs with 5N/16+1≤n≤N−1. This paper aims at providing theories and systematical construction methods of the B2-GMC designs with N/4+1≤n≤5N/16.The rest of the paper is organized as follows. Section2 reviews doubling theory and B2-GMC criterion. Section 3 provides theoretical results and construction methods of B2-GMC designs. Section 4 gives concluding remarks. Some useful lemmas are deferred to Appendix.


## 2. Preliminaries

### 2.1. Doubling Theory

LetX=x1,…,xt be a matrix with entries 1 or −1. Denote J0=1,1′ and J1=1,−1′, where ′ denotes transpose. Define(1)DX=J0,J1⊗x1,…,J0,J1⊗xt,as a double of X, where ⊗ is the Kronecker product. Let DqX denote the design obtained by repeatedly doubling Xq times, i.e., DqX=DDq−1X. When X=1, we write Dq1=I2q,12q,22q,12q22q,…,12q22q32q,…,q2q, where the subscript 2q means the dimension of a column, I2q is a column of 1’s,(2)12q′=1,…,1,−1,…,−122q′=1,…,1,−1,…,−1,…,1,…,1,−1,…,−1⋮q2q′=1,−1,1,−1,…,1,−1,1,−1,are q independent columns, and the other columns are the component-wise products of some of these q independent columns. For example, 12q22q is the component-wise product of the columns 12q and 22q. Write Dq1=I2q,Dq⋅; then, Dq⋅ is just the regular two-level saturated design with columns arranged in Yates order. As some subdesigns in Dq⋅, denote H0q=∅, H1q=12q,Hrq=Hr−1q,r2q,r2qHr−1q for r=2,3,…,q, where ∅ denotes the empty set, the superscript q of Hrq refers to that Hrq is a subdesign of Dq⋅, and r2qHr−1q=r2qd2q:d2q∈Hr−1q, i.e., Hrq consists of the first 2r−1 columns of Dq⋅. Especially, Hqq=Dq⋅.


### 2.2.B2-GMC Criterion

Before introducing theB2-GMC criterion, we first review some principles in the multi-block variable problem. Let b1,b2,…,bs denote the s block variables which cause the inhomogeneity of the experimental units. Suppose that the block variable bj partitions the N=2n−m experimental units into 2lj blocks; then, lj independent columns are needed to carry out this blocking plan. Denote Sj as the set of the lj independent columns related to the block variable bj. The block columns should follow the following rules:(i)
Thelj block columns in Sjj=1,2,…,s are independent of each other.(ii)
A block column fromSj is not necessarily independent of the block columns from Si with j≠i.In this paper, we focus on the case where each block variable is at two levels, i.e.,lj=1.The effect hierarchy principle for blocked designs with multi-block variables is as follows (see [23]):(i)
The lower-order treatment factorial effects are more likely to be important than the higher-order ones, and the treatment factorial effects of the same order are equally likely to be important.(ii)
The lower-order block factorial effects are more likely to be important than the higher-order ones, and the block factorial effects of the same order are equally likely to be important.(iii)
All the interactions between treatment factors and block factors are negligible.Since each variable or factor is assigned to one column of the design matrix when an experiment is carried out, we do not differentiate the variable, factor, and column. Based on the effect hierarchy principle and weak assumption that the effects involving three or more factors are usually not important and negligible, Zhang et al. [23] proposed the B2-GMC criterion which pays attention to only the confounding among main treatment effects and the two-factor interactions of treatment factors (2fi’s for short). For the same reason, a common assumption in blocking problem is that only the main effects of block variables and the interactions of any two block factors are potentially significant, and if a treatment effect is confounded with a potentially significant block effect, the treatment effect cannot be estimated. Thus, the confounding between the main effects of treatment factors and any potentially significant block effect is not allowed.DenoteD=Dt:Db as a 2n−m:2s design, where Dt consists of n treatment factors corresponding to a regular 2n−m design and Db consists of s block factors each of which can partition the 2n−m runs into 2 blocks. Denote C1#2pD as the number of main treatment effects which are aliased with p2fi’s but not with any potentially significant block effects, where p=0,1,2,…,P and P=nn−1/2. Similarly, C2#2pD denotes the number of 2fi’s which are aliased with the other p2fi’s but not with any potentially significant block effects, where p=0,1,…,P. Denote(3)C1#2D=C1#20D,C1#21D,…,C1#2pD,C2#2D=C2#20D,C2#21D,…,C2#2pD,(4)C#D=C1#2D,C2#2D.A2n−m:2s blocked design D=Dt:Db is called a B2-GMC design if it sequentially maximizes (4). Let C1#2pDt be the number of main effects which are aliased with p 2fi’s of Dt and C2#2pDt be the number of 2fi’s which are aliased with the other p 2fi’s of Dt. Let(5)C1#2Dt=C1#20Dt,C1#21Dt,…,C1#2pDt,C2#2Dt=C2#20Dt,C2#21Dt,…,C2#2pDt,C#Dt=C1#2Dt,C2#2Dt.A2n−m design Dt is called a GMC design if Dt sequentially maximizes (5).Letq=n−m. Constructing a B2-GMC design is to choose Dt and Db from Dq⋅ such that (4) is sequentially maximized. In the following, without causing confusions, we omit the subscript of a column and the superscript of a design when they are taken from Dq⋅. For example, we use a, Hr, and Hq instead of a2q, Hrq, and Hqq, respectively. Denote(6)UDb=γ∈Hq:γ∈Dborγ=abwitha,b∈Db,and then UDb consists of all the potentially significant block effects. As previously stated, the confounding between main treatment effects and potentially significant block effects is not allowed. This requires Dt∩UDb=∅, and thus C1#2pD=C1#2pDt for p=0,1,2,…,P.ForDt⊂Hq and γ∈Hq, define(7)B2Dt,γ=#d1,d2:d1,d2∈Dt,d1d2=γ,where # denotes the cardinality of a set and d1d2 stands for the two-factor interaction of d1 and d2. Thus, B2Dt,γ is the number of 2fi’s of Dt appearing in the alias set that contains γ.Isomorphism introduced by Tang and Wu [26] is a useful concept which helps narrow down the search of the optimal blocked designs here. An isomorphism ϕ is a one-to-one mapping from Hq to Hq such that ϕxy=ϕxϕy for every x≠y∈Hq. The 2n−m designs Dt and Dt∗ are isomorphic if there exists an isomorphism ϕ that maps Dt onto Dt∗. The 2n−m:2s designs D=Dt:Db and D∗=Dt∗:Db∗ are isomorphic if there exists an isomorphism ϕ that maps Dt onto Dt∗ and Db onto Db∗.


## 2.1. Doubling Theory

LetX=x1,…,xt be a matrix with entries 1 or −1. Denote J0=1,1′ and J1=1,−1′, where ′ denotes transpose. Define(1)DX=J0,J1⊗x1,…,J0,J1⊗xt,as a double of X, where ⊗ is the Kronecker product. Let DqX denote the design obtained by repeatedly doubling Xq times, i.e., DqX=DDq−1X. When X=1, we write Dq1=I2q,12q,22q,12q22q,…,12q22q32q,…,q2q, where the subscript 2q means the dimension of a column, I2q is a column of 1’s,(2)12q′=1,…,1,−1,…,−122q′=1,…,1,−1,…,−1,…,1,…,1,−1,…,−1⋮q2q′=1,−1,1,−1,…,1,−1,1,−1,are q independent columns, and the other columns are the component-wise products of some of these q independent columns. For example, 12q22q is the component-wise product of the columns 12q and 22q. Write Dq1=I2q,Dq⋅; then, Dq⋅ is just the regular two-level saturated design with columns arranged in Yates order. As some subdesigns in Dq⋅, denote H0q=∅, H1q=12q,Hrq=Hr−1q,r2q,r2qHr−1q for r=2,3,…,q, where ∅ denotes the empty set, the superscript q of Hrq refers to that Hrq is a subdesign of Dq⋅, and r2qHr−1q=r2qd2q:d2q∈Hr−1q, i.e., Hrq consists of the first 2r−1 columns of Dq⋅. Especially, Hqq=Dq⋅.


## 2.2.B2-GMC Criterion

Before introducing theB2-GMC criterion, we first review some principles in the multi-block variable problem. Let b1,b2,…,bs denote the s block variables which cause the inhomogeneity of the experimental units. Suppose that the block variable bj partitions the N=2n−m experimental units into 2lj blocks; then, lj independent columns are needed to carry out this blocking plan. Denote Sj as the set of the lj independent columns related to the block variable bj. The block columns should follow the following rules:(i)
Thelj block columns in Sjj=1,2,…,s are independent of each other.(ii)
A block column fromSj is not necessarily independent of the block columns from Si with j≠i.In this paper, we focus on the case where each block variable is at two levels, i.e.,lj=1.The effect hierarchy principle for blocked designs with multi-block variables is as follows (see [23]):(i)
The lower-order treatment factorial effects are more likely to be important than the higher-order ones, and the treatment factorial effects of the same order are equally likely to be important.(ii)
The lower-order block factorial effects are more likely to be important than the higher-order ones, and the block factorial effects of the same order are equally likely to be important.(iii)
All the interactions between treatment factors and block factors are negligible.Since each variable or factor is assigned to one column of the design matrix when an experiment is carried out, we do not differentiate the variable, factor, and column. Based on the effect hierarchy principle and weak assumption that the effects involving three or more factors are usually not important and negligible, Zhang et al. [23] proposed the B2-GMC criterion which pays attention to only the confounding among main treatment effects and the two-factor interactions of treatment factors (2fi’s for short). For the same reason, a common assumption in blocking problem is that only the main effects of block variables and the interactions of any two block factors are potentially significant, and if a treatment effect is confounded with a potentially significant block effect, the treatment effect cannot be estimated. Thus, the confounding between the main effects of treatment factors and any potentially significant block effect is not allowed.DenoteD=Dt:Db as a 2n−m:2s design, where Dt consists of n treatment factors corresponding to a regular 2n−m design and Db consists of s block factors each of which can partition the 2n−m runs into 2 blocks. Denote C1#2pD as the number of main treatment effects which are aliased with p2fi’s but not with any potentially significant block effects, where p=0,1,2,…,P and P=nn−1/2. Similarly, C2#2pD denotes the number of 2fi’s which are aliased with the other p2fi’s but not with any potentially significant block effects, where p=0,1,…,P. Denote(3)C1#2D=C1#20D,C1#21D,…,C1#2pD,C2#2D=C2#20D,C2#21D,…,C2#2pD,(4)C#D=C1#2D,C2#2D.A2n−m:2s blocked design D=Dt:Db is called a B2-GMC design if it sequentially maximizes (4). Let C1#2pDt be the number of main effects which are aliased with p 2fi’s of Dt and C2#2pDt be the number of 2fi’s which are aliased with the other p 2fi’s of Dt. Let(5)C1#2Dt=C1#20Dt,C1#21Dt,…,C1#2pDt,C2#2Dt=C2#20Dt,C2#21Dt,…,C2#2pDt,C#Dt=C1#2Dt,C2#2Dt.A2n−m design Dt is called a GMC design if Dt sequentially maximizes (5).Letq=n−m. Constructing a B2-GMC design is to choose Dt and Db from Dq⋅ such that (4) is sequentially maximized. In the following, without causing confusions, we omit the subscript of a column and the superscript of a design when they are taken from Dq⋅. For example, we use a, Hr, and Hq instead of a2q, Hrq, and Hqq, respectively. Denote(6)UDb=γ∈Hq:γ∈Dborγ=abwitha,b∈Db,and then UDb consists of all the potentially significant block effects. As previously stated, the confounding between main treatment effects and potentially significant block effects is not allowed. This requires Dt∩UDb=∅, and thus C1#2pD=C1#2pDt for p=0,1,2,…,P.ForDt⊂Hq and γ∈Hq, define(7)B2Dt,γ=#d1,d2:d1,d2∈Dt,d1d2=γ,where # denotes the cardinality of a set and d1d2 stands for the two-factor interaction of d1 and d2. Thus, B2Dt,γ is the number of 2fi’s of Dt appearing in the alias set that contains γ.Isomorphism introduced by Tang and Wu [26] is a useful concept which helps narrow down the search of the optimal blocked designs here. An isomorphism ϕ is a one-to-one mapping from Hq to Hq such that ϕxy=ϕxϕy for every x≠y∈Hq. The 2n−m designs Dt and Dt∗ are isomorphic if there exists an isomorphism ϕ that maps Dt onto Dt∗. The 2n−m:2s designs D=Dt:Db and D∗=Dt∗:Db∗ are isomorphic if there exists an isomorphism ϕ that maps Dt onto Dt∗ and Db onto Db∗.


## 3. Constructions ofB2-GMC Designs

### 3.1.B2-GMC 2n−m:2s Designs with n=N/4+1

A design is of MaxC2 (see [27]) if it has resolution IV and maximum number of clear 2fi’s, where a resolutionR design has no c-factor interaction confounded with any other interaction involving less than R−c factors (see [28]), and a 2fi is called clear if it is not aliased with any main treatment effect and other 2fi’s. Cheng and Zhang [29] showed that a 2n−m design with n=N/4+1 is a MaxC2 design if and only if it is a GMC design. They also pointed out that, up to isomorphism, the GMC 2n−m design with n=N/4+1 can be uniquely expressed as SN/4+1=q−1,q,qq−1Hq−2. It is easy to obtain that(8)B2SN/4+1,γ=1,forγ∈q−1Hq−2,qHq−2,qq−1,N8−1,forγ∈Hq−2,0,otherwise.Therefore,(9)C1#2pSN/4+1=N4+1,forp=0,0,otherwise,(10)C2#2pSN/4+1=2N4−1+1,forp=0,N4−1N8−1,forp=N8−2,0,otherwise.Lemma1 is a straightforward extension of Lemma A.1, in Appendix, introduced from [24, 25].Lemma 1.
SupposeDb is any s-projection of Hr,γ,γHr with 2k≤s≤2k+1−1 for some k0≤k≤r−1, where γ is independent of the columns of Hr. We have(i)
IfDb∩γ,γHr=∅, then #UDb∩Hr≥2k+1−1 and the equality holds when Db has k+1 independent columns.(ii)
IfDb∩γ,γHr≠∅, then #UDb∩Hr≥2k−1 and the equality holds when Db has k+1 independent columns.(iii)
IfDb∩γ,γHr≠∅, then #UDb∩γ,γHr≥2k and the equality holds when Db has k+1 independent columns.(iv)
IfDb⊂Hk+1, then UDb=Hk+1.(v)
IfDb⊂Hk,γ,γHk, then UDb=Hk,γ,γHk.Lemma2 provides a necessary condition for a 2n−m:2s design D=Dt:Db with n=N/4+1 to be a B2-GMC design.Lemma 2.
SupposeD=Dt:Db is a 2n−m:2s design with n=N/4+1 and 2k≤s≤2k+1−1 for some k0≤k≤q−3; then, D=Dt:Db is a B2-GMC design only if Dt=SN/4+1.Proof.
LetD˜=SN/4+1:D˜b be a 2n−m:2s design and D˜b∈Hq−2. From (8)–(10), we can obtain(11)C1#2D˜=N4+1,0,…,0,which is sequentially maximized by D˜, and C2#20D˜=2N/4−1+1=N/2−1. Suppose that any D=SN/4+1:Db is not a B2-GMC 2n−m:2s design for n=N/4+1. Then, by the definition of B2-GMC criterion, there should be a 2n−m:2s design D∗=Dt∗:Db∗ outperforming D˜=SN/4+1:D˜b in terms of (4). This leads to(12)C1#2D∗=N4+1,0,…,0,and C2#20D∗≥N/2−1. Clearly,(13)C1#2Dt∗=C1#2D∗=N4+1,0,…,0,and C2#20Dt∗≥C2#20D∗≥N/2−1 noting that Dt∗ is the unblocked part of D∗ and Dt∗ has n=N/4+1 columns.
In fact, the inequality in the formulaC2#20Dt∗≥N/2−1 is not valid. Recall that the MaxC2 design SN/4+1 has the largest number, N/2−1, of clear 2fi’s among all the 2n−m designs with n=N/4+1. Therefore, Dt∗ has at most N/2−1 clear 2fi’s, i.e., C2#20Dt∗=N/2−1. Wu and Wu [30] showed that a 2n−m design with n=N/4+1 is a MaxC2 design if and only if this design has N/2−1 clear 2fi’s. This obtains that Dt∗=SN/4+1 up to isomorphism.With Lemma2, Theorem 1 provides the constructions of B2-GMC designs with n=N/4+1.Theorem 1.
SupposeD=Dt:Db is a 2n−m:2s design with n=N/4+1 and 2k≤s≤2k+1−1 for some k0≤k≤q−3; then, D=Dt:Db is a B2-GMC design if Dt=SN/4+1 and Db is any s-projection of Hk+1.Proof.
By Lemma2, if D=Dt:Db is a B2-GMC 2n−m:2s design with n=N/4+1, then Dt=SN/4+1 up to isomorphism. Thus, C1#2D=N/4+1,0,…,0 which is sequentially maximized by D=SN/4+1:Db. Let u1=#UDb∩q−1Hq−2,qHq−2,qq−1 and u2=#UDb∩Hq−2; then, from (10), we have(14)C2#2pD=2N4−1+1−u1,forp=0,N4−1−u2N8−1,forp=N8−2,0,otherwise.
IfD is a B2-GMC design, then D must sequentially minimize u1,u2. There are two different ways to choose Db from Hq\SN/4+1:(i)
Db∩q−1Hq−2,qHq−2,qq−1=∅.(ii)
Db∩q−1Hq−2,qHq−2,qq−1≠∅.
It is not hard to verify thatDb in case (ii) results in u1>0 while Db in case (i) gives u1=0. Therefore, if D=SN/4+1:Db is a B2-GMC design, then Db must be of case (i). Recall that Db∩SN/4+1=∅, and we have Db⊂Hq−2. By Lemma 1 (i), if Db⊂Hq−2, then u2=#UDb∩Hq−2≥2k+1−1 and the equality holds when Db⊂Hk+1 up to isomorphism. This completes the proof.The following example illustrates the construction method in Theorem1.Example 1.
Consider the construction of theB2-GMC 29−4:25 design. Here, n=9,m=4,q=5, N=32, and s=5 which leads to k=2. According to Theorem 1, let Dt=4,5,45H3 and Db be any 5-projection of H3, say Db=1,2,12,3,13. Then, D=Dt:Db is a B2-GMC 29−4:25 design.


### 3.2.B2-GMC Designs with N/4+1<n≤5N/16

To sequentially maximize (4), the first part C1#2D of (4) should be first maximized. Recall that C1#2D=C1#2Dt. If Dt has resolution IV, then C1#2D must be maximized. According to [31], when N/4+1<n≤5N/16, the Dt with resolution IV must be an n-projection of some second-order saturated (SOS) designs. In the following, we first review the concept of SOS design.A2n−m design is called an SOS design if all of its degrees of freedom can be used to estimate only the main treatment effects and 2fi’s. In terms of coding theory and projective geometry, Davydov and Tombak [32] showed that, given N, only the SOS designs of N/4+1, N2w−2+1/2w with w≥4 and N/2 factors exist. Block and Mee [31] further showed that an SOS design of N2w−2+1/2w factors can be obtained by doubling some smaller SOS 22w−2+1−2w−2+1−w designs q−w times. Zhang and Cheng [33] showed that the SOS design of N/2 factors can be uniquely represented by SN/2=q,qHq−1 up to isomorphism. Let(15)ℒw=Dq−wY:Yis an SOS22w−2+1−2w−2+1−wdesign,denote the collection of all the SOS designs obtained by doubling some SOS 22w−2+1−2w−2+1−w designs q−w times. Especially, in ℒw, we denote the SOS design obtained by doubling the MaxC2 22w−2+1−2w−2+1−w design Φw=w−12w,w2w,w2ww−12wHw−2w as SN2w−2+1/2w.With a little algebra, it is easy to verify that(16)Dq−wI2w=I2q,Hq−wq=I,Hq−wDq−w12w=q−w+12qI2q,Hq−wq=q−w+1I,Hq−wDq−w22w=q−w+22qI2q,Hq−wq=q−w+2I,Hq−w⋮Dq−ww−12w=q−12qI2q,Hq−wq=q−1I,Hq−wDq−ww2w=q2qI2q,Hq−wq=qI,Hq−w,where I2q−w⊗12w=q−w+12q,I2q−w⊗22w=q−w+22q,…,I2q−w⊗w−12w=q−12q, and I2q−w⊗w2w=q2q. Then,(17)Dq−ww−12ww2wHw−2w=qq−1Hq−2\Hq−w.Therefore,SN2w−2+1/2w can be expressed as(18)SN2w−2+1/2w=Dq−wΦw=q−1,q−1Hq−w,q,qHq−w,qq−1Hq−2\Hq−w.The following lemma provides a necessary condition for a2n−m:2s design D=Dt:Db with N/4+1<n≤5N/16 to be a B2-GMC design.Lemma 3.
SupposeD=Dt:Db is a 2n−m:2s design with N2w−1+1/2w+1<n≤N2w−2+1/2w for w≥4 and 2k≤s≤2k+1−1 for some k0≤k≤q−3; then, D=Dt:Db is a B2-GMC design only if Dt⊂SN2w−2+1/2w.Proof.
As discussed in the first paragraph of this section, ifD=Dt:Db is a B2-GMC 2n−m:2s design with N2w−1+1/2w+1<n≤N2w−2+1/2w, then Dt must be an n-projection of some SOS designs in ℒv with 4≤v≤w or SN/2.
LetP1 be an n-projection of SN/2. According to Theorem 3 in [33], we obtain(19)C2#2pP1=0,forp<n−N4−1,N4n−N4,forp=n−N4−1,0,forn−N4−1<p<N8−1.
LetXv=a1,a2,…,a2v−2+1 be any SOS 22v−2+1−2v−2+1−v design but not a MaxC2 design; then, Dq−vXv∈ℒv. Denote pi as the number of clear 2fi’s of Xv involving ai for i=1,2,…,2v−2+1 and c as the total number of clear 2fi’s of Xv. Let Pv be an n-projection of Dq−vXv and P¯v=Dq−vXv\Pv. Then, #P¯v=N2v−2+1/2v−n≤N/2v. Applying equations (23) and (24) in the proof of Theorem 3.1 of [29], among all the n-projections of Dq−vXv, Pv sequentially maximizes C2#2 in (5), only if P¯v⊂Dq−vaj, where aj is the column such that pj=maxp1,p2,…,p2v−2+1. For a Pv with P¯v⊂Dq−vaj, according to equation (24) in the proof of Theorem 3.1 of [29], it is obtained that(20)C2#2pPv=0,for0≤p<n−N4−1,pjN2vn−N/4,forp=n−N4−1,0,forn−N4−1<p<N2v−1,c−pjN2v2,forp=N2v−1.
For a MaxC222v−2+1−2v−2+1−v design, it has 2v−1−1 clear 2fi’s and pj=2v−2. Since Xv is not a MaxC2 design, we have(21)pj=2v−2,c−pj<2v−1−2v−2−1,or(22)pj<2v−2,c<2v−1−1.
LetQv be an n-projection of SN2v−2+1/2v obtained by doubling Φv, the MaxC222v−2+1−2v−2+1−v design as mentioned above. With a column permutation, rewrite Dq⋅ as(23)DRCq⋅=Hq−v,Hq−2\Hq−v,q−1Hq−2\Hq−v,qHq−2\Hq−v,qq−1I,Hq−v,SN2v−2+1/2v,in a re-changed Yates order. Write Φv as Φv=s1,s2,…,s2v−2+1, where s1=v−12v, s2=v2v and s3,…,s2v−2+1=v2vv−12vHv−2v. Applying equations (23) and (24) in the proof of Theorem 3.1 of [29], among all the n-projections of SN2v−2+1/2v, Qv sequentially maximizes C2#2 in (3), only if Q¯v⊂Dq−vsi with i=1 or 2. In the following, we investigate C2#2pQv with Q¯v⊂Dq−vs1 for which the following analysis and final conclusion are the same as that for C2#2pQv with Q¯v⊂Dq−vs2. For Qv with Q¯v⊂Dq−vs1, from equations (18) and (23), it is obtained that(24)B2Qv,γ=0,forγ∈SN2v−2+1/2v,n−N4,forγ∈qHq−2\Hq−v,qq−1I,Hq−v,N2v,forγ∈q−1Hq−2\Hq−v,N8−N2v,forγ∈Hq−2\Hq−v,B2Q¯v,γ+N8,forγ∈Hq−v,where the second equality can be easily verified by noting that(25)qHq−2\Hq−v,qq−1I,Hq−v=Dq−vs1s2,Dq−vs1s3,…,Dq−vs1s2v−2+1.
Therefore,(26)C2#2pQv=0,for0≤p<n−N4−1,N4n−N/4,forp=n−N4−1,0,forn−N4−1<p<N2v−1,N2vN/4−N/2v,forp=N2v−1,0,forN2v−1<p<N8−N2v−1,N4−N2vN8−N2v,forp=N8−N2v−1.
DenoteD∗=Dt∗:Db∗, where Dt∗⊂SN2w−2+1/2w, SN2w−2+1/2w\Dt∗⊂Dq−ws1 and Db∗⊂Hk+1. Since k≤q−3, we have UDb∗⊂Hq−2. Therefore, from (24) and (26), we obtain(27)C2#2pD∗=0,for0≤p<n−N4−1,N4n−N/4,forp=n−N4−1,0,forn−N4−1<p<N2w−1,N2wN/4−N/2w,forp=N2w−1.
Comparing (27) with (19), (20) for 4≤v≤w, and (26) for 4≤v≤w−1, it is obtained that if D=Dt:Db is a B2-GMC design, D should not be worse than D∗ in terms of (3). Therefore, if D=Dt:Db is a B2-GMC design, Dt should be an n-projection of SN2w−2+1/2w. This completes the proof.The lemma below is an extension of LemmaA.2, in Appendix, introduced from [34].Lemma 4.
SupposeDt consists of the last n, N2w−1+1/2w+1<n≤N2w−2+1/2w, columns of SN2w−2+1/2w; then, B2Dt,γ1≥B2Dt,γ2 if γ1 is ahead of γ2 in Hq−w.Proof.
SupposeO consists of the last n−N/4 columns of Dq−ww−12w. Then, Dt=O,SN2w−2+1/2w\Dq−ww−1. From (16) and (18), for any γ∈Hq−w, B2Dt,γ=B2O,γ+N/8. Straightforwardly, from Lemma A.2, it is obtained that B2O,γ1≥B2O,γ2 if γ1 is ahead of γ2 in Hq−w. This completes the proof.With Lemmas3 and 4, Theorem 2 provides the construction methods of B2-GMC 2n−m:2s designs with N/4+1<n≤5N/16.Theorem 2.
LetD=Dt:Db be a 2n−m:2s design with N2w−1+1/2w+1<n≤N2w−2+1/2w for w≥4 and 2k≤s≤2k+1−1 for some 0≤k≤q−3. Suppose 2r≤N2w−2+1/2w−n≤2r+1−1 for some 0≤r≤q−w−1; then, D is a B2-GMC design if Dt consists of the last n columns of SN2w−2+1/2w and(a)
Db is any s-projection of Hk,q−1,q−1Hk when k≤r≤q−w−1.(b)
Db is any s-projection of Hk+1 when r+1≤k≤q−w−1.(c)
Db is any s-projection of Hq−2 when q−w−1<k≤q−3.Proof.
By Lemma3 and its proof, if D∗=Dt∗:Db∗ is a B2-GMC 2n−m:2s design, there should be Dt∗⊂SN2w−2+12w and D¯t∗=SN2w−2+12w\Dt∗⊂Dq−ww−12w. Substituting v with w in (24), we obtain(28)B2Dt∗,γ=0,forγ∈SN2w−2+1/2w,n−N4,forγ∈qHq−2\Hq−w,qq−1I,Hq−w,N2w,forγ∈q−1Hq−2\Hq−w,N8−N2w,forγ∈Hq−2\Hq−w,B2D¯t∗,γ+N8,forγ∈Hq−w.
Let(29)u1=#UDb∗∩qHq−2\Hq−w,qq−1I,Hq−w,u2=#UDb∗∩q−1Hq−2\Hq−w,u3=#UDb∗∩Hq−2\Hq−w,and then(30)C2#2pD∗=n−N4N4−u1,forp=n−N4−1,N2wN4−N2w−u2,forp=N2w−1,N8−N2wN4−N2w−u3,forp=N8−N2w−1,fDt∗,p,forp≥N8−1,0,otherwise,where fDt∗,p=C2#2pDt∗−p+1#γ:γ∈Hq−w∩UDb∗,B2Dt∗,γ=p+1. Therefore, D∗ sequentially maximizes C2#2 among all the possible 2n−m:2s designs, only if D∗ sequentially minimizes(31)u1,u2,u3.
For ease of presenting, letA=qHq−2\Hq−w,qq−1I,Hq−w, B=q−1Hq−2\Hq−w, and C=Hq−2\Hq−w.
For (a), whenk≤q−w−1, there are two different ways to choose Db∗ from Hq: (i) Db∗∩A,B,C=∅ and (ii) Db∗∩A,B,C≠∅. Recall that if D∗=Dt∗:Db∗ is a B2-GMC design, then Dt∗⊂SN2w−2+12w with D¯t∗=SN2w−2+12w\Dt∗⊂Dq−ww−12w. Case (i) implies that Db∗⊂Hq−w,D¯t∗, and thus Db∗⊂Hq−w,q−1,q−1Hq−w. Therefore, choosing Db∗ according to case (i) results in u1=u2=u3=0. If Db∗ is chosen according to case (ii), then UDb∗∩A,B,C≠∅ which results in u1>0, u2>0, or u3>0. Clearly, if D∗ is a B2-GMC design, Db∗ should be chosen as (i). In the following, we consider only Db∗∩A,B,C=∅, i.e., Db∗⊂Hq−w,q−1,q−1Hq−w.
ForD=Dt:Db in (a), according to (v) in Lemma 1, if Db⊂Hk,q−1,q−1Hk with k≤q−w−1, then UDb=Hk,q−1,q−1Hk. Denote b as the last column of Hk in Yates order, and B2Dt,b=b; then, b≥N/8. According to Lemma 4, for γ∈Hq−w, if γ is ahead of b in Hq−w, then B2Dt,γ≥b by (18). For any γ∈q−1,q−1Hk, we have B2Dt,γ=0. For any γ such that B2Dt,γ<b, we have γ∈Hq\Hk. Therefore, for p≤b−2,(32)C2#2pD=p+1#γ∈Hq:γ∉UDb,B2Dt,γ=p+1=p+1#γ∈Hq:B2Dt,γ=p+1=C2#2pDt,and for p≥b,(33)C2#2pD=p+1#γ∈Hq:γ∉UDb,B2Dt,γ=p+1=p+1#γ∈Hq:B2Dt,γ=p+1−p+1#γ∈Hk:B2Dt,γ=p+1=0.
From (33), it is obtained that(34)C2#20D,…,C2#2N/8−1D,C2#2N/8D,…,C2#2b−2D=C2#20Dt,…,C2#2N/8−1Dt,C2#2N/8Dt,…,C2#2b−2Dt.
Note thatDt is a GMC design [29]; then, D maximizes(35)C2#20⋅,…,C2#2N/8−1⋅,C2#2N/8⋅,…,C2#2b−2⋅,among all the possible 2n−m:2s designs.
SupposeD is not a B2-GMC design; then, there should be a D∗=Dt∗:Db∗ which outperforms D=Dt:Db in terms of (3). This implies that there exists some p1≥b−1 such that(36)C2#20D∗,…,C2#2N/8−1D∗,C2#2N/8D∗,…,C2#2b−2D∗=C2#20D,…,C2#2N/8−1D,C2#2N/8D,…,C2#2b−2D,(37)C2#2p1−1D∗>C2#2p1−1D.
Recalling the definitions ofC2#2pD and B2Dt,γ, we have(38)∑p=0PC2#2pDp+1=∑p=0P#γ∈Hq:γ∉UDb,B2Dt,γ=p+1=∑p=0P#γ∈Hq\SN2w−2+1/2w:B2Dt,γ=p+1−∑p=0P#γ∈UDb:B2Dt,γ=p+1=#Hq\SN2w−2+1/2w−#Hk=2q−N2w−2+12w−2k,where the second and third equalities are due to B2Dt,γ=0 for any γ∈SN2w−2+1/2w. Similarly,(39)∑p=0PC2#2pD∗p+1=∑p=0P#γ∈Hq:γ∉UDb∗,B2Dt∗,γ=p+1=∑p=0P#γ∈Hq\SN2w−2+1/2w:B2Dt∗,γ=p+1−∑p=0P#γ∈UDb∗:B2Dt∗,γ=p+1,=#Hq\SN2w−2+1/2w−#UDb∗,=2q−N2w−2+12w−#UDb∗.
From (33) and (38), we have(40)∑p=0PC2#2pDp+1=∑p=0b−2C2#2pDp+1+C2#2b−1Db=2q−N2w−2+12w−2k.
By (36) and (37), it is obtained that(41)∑p=0PC2#2pD∗p+1≥∑p=0b−2C2#2pD∗p+1+C2#2b−1D∗b>∑p=0b−2C2#2pDp+1+C2#2b−1Db.
Then, according to (39)–(41), we obtain #UDb∗<2k−1 which contradicts Lemma 1 (i) and (ii).
For (b), similar to (a), whenk≤q−w−1, there are two different ways to choose Db∗ from Hq: (i) Db∗∩A,B,C=∅ and (ii) Db∗∩A,B,C≠∅. With a similar argument to (a), case (i) results in u1=u2=u3=0 while case (ii) results in u1>0,u2>0 or u3>0. Therefore, if D∗ is a B2-GMC design, Db∗ should be chosen according to case (i), i.e., Db∗⊂Hq−w,q−1,q−1Hq−w. When choosing Db∗ according to (i), there should be Db∗∩q−1,q−1Hq−w=∅, i.e., Db∗⊂Hq−w; otherwise, Dt∗∩UDb∗≠∅ noting that #UDb∗∩q−1,q−1Hq−w≥2k>2r+1−1 as shown in Lemma 1 (iii). The remainder of the proof is similar to that of (a).
For (c), whenk>q−w−1, there are two different ways to choose Db∗ from Hq: (i) Db∗∩B,C=∅ and (ii) Db∗∩B,C≠∅. With a similar argument to (a), it is not hard to verify that case (ii) results in u1>0 or u2>0 while case (i) results in u1=u2=0. Therefore, if D∗ is a B2-GMC design, Db∗ should be an s-projection of Hq−2=B,C. The remainder of the proof is similar to that of (a).Remark 1.
Theorem2 shows that when constructing the B2-GMC designs with N/4+1<n≤5N/16, we can first partition the range N/4+1,5N/16 into q−4 sequential subranges as N2w−1+1/2w+1,N2w−2+1/2w with w=4,5,…,q−1 and then obtain the B2-GMC designs according to Theorem 2.In the following, an example is provided to illustrate the construction method in Theorem2 and Remark 1.Example 2.
Consider the constructions ofB2-GMC 219−13:2s designs for s=1,3,4. Here, n=19,m=13,q=6, and N=64. The values of the parameters N and n satisfy N/4+1<n≤5N/16, i.e., n∈17,20. From Remark 1, we partition the range 17,20 into two subranges 17,18 because w=5 and 18,20 because w=4. Since 19∈18,20, according to Theorem 2, Dt should be the last 19 columns of S5N/16, where S5N/16=Dq−4Φ4=5,5H2,6,6H2,56H4\H2 from (18). Therefore, Dt=5H2,6,6H2,56H4\H2. Note that 5N/16−n=20−19=1; then, r=0. Next, we choose Db for s=1,3,4.
Fors=1, we have k=0. According to Theorem 2 (a), Db=5 as H0=∅.
Fors=3, we have k=1. According to Theorem 2 (b), Db should be a 3-projection of H2, i.e., Db=H2=1,2,12.
Fors=4, we have k=2. According to Theorem 2 (c), Db should be a 4-projection of H4. Without loss of generality, we choose Db=1,2,12,3.


## 3.1.B2-GMC 2n−m:2s Designs with n=N/4+1

A design is of MaxC2 (see [27]) if it has resolution IV and maximum number of clear 2fi’s, where a resolutionR design has no c-factor interaction confounded with any other interaction involving less than R−c factors (see [28]), and a 2fi is called clear if it is not aliased with any main treatment effect and other 2fi’s. Cheng and Zhang [29] showed that a 2n−m design with n=N/4+1 is a MaxC2 design if and only if it is a GMC design. They also pointed out that, up to isomorphism, the GMC 2n−m design with n=N/4+1 can be uniquely expressed as SN/4+1=q−1,q,qq−1Hq−2. It is easy to obtain that(8)B2SN/4+1,γ=1,forγ∈q−1Hq−2,qHq−2,qq−1,N8−1,forγ∈Hq−2,0,otherwise.Therefore,(9)C1#2pSN/4+1=N4+1,forp=0,0,otherwise,(10)C2#2pSN/4+1=2N4−1+1,forp=0,N4−1N8−1,forp=N8−2,0,otherwise.Lemma1 is a straightforward extension of Lemma A.1, in Appendix, introduced from [24, 25].Lemma 1.
SupposeDb is any s-projection of Hr,γ,γHr with 2k≤s≤2k+1−1 for some k0≤k≤r−1, where γ is independent of the columns of Hr. We have(i)
IfDb∩γ,γHr=∅, then #UDb∩Hr≥2k+1−1 and the equality holds when Db has k+1 independent columns.(ii)
IfDb∩γ,γHr≠∅, then #UDb∩Hr≥2k−1 and the equality holds when Db has k+1 independent columns.(iii)
IfDb∩γ,γHr≠∅, then #UDb∩γ,γHr≥2k and the equality holds when Db has k+1 independent columns.(iv)
IfDb⊂Hk+1, then UDb=Hk+1.(v)
IfDb⊂Hk,γ,γHk, then UDb=Hk,γ,γHk.Lemma2 provides a necessary condition for a 2n−m:2s design D=Dt:Db with n=N/4+1 to be a B2-GMC design.Lemma 2.
SupposeD=Dt:Db is a 2n−m:2s design with n=N/4+1 and 2k≤s≤2k+1−1 for some k0≤k≤q−3; then, D=Dt:Db is a B2-GMC design only if Dt=SN/4+1.Proof.
LetD˜=SN/4+1:D˜b be a 2n−m:2s design and D˜b∈Hq−2. From (8)–(10), we can obtain(11)C1#2D˜=N4+1,0,…,0,which is sequentially maximized by D˜, and C2#20D˜=2N/4−1+1=N/2−1. Suppose that any D=SN/4+1:Db is not a B2-GMC 2n−m:2s design for n=N/4+1. Then, by the definition of B2-GMC criterion, there should be a 2n−m:2s design D∗=Dt∗:Db∗ outperforming D˜=SN/4+1:D˜b in terms of (4). This leads to(12)C1#2D∗=N4+1,0,…,0,and C2#20D∗≥N/2−1. Clearly,(13)C1#2Dt∗=C1#2D∗=N4+1,0,…,0,and C2#20Dt∗≥C2#20D∗≥N/2−1 noting that Dt∗ is the unblocked part of D∗ and Dt∗ has n=N/4+1 columns.
In fact, the inequality in the formulaC2#20Dt∗≥N/2−1 is not valid. Recall that the MaxC2 design SN/4+1 has the largest number, N/2−1, of clear 2fi’s among all the 2n−m designs with n=N/4+1. Therefore, Dt∗ has at most N/2−1 clear 2fi’s, i.e., C2#20Dt∗=N/2−1. Wu and Wu [30] showed that a 2n−m design with n=N/4+1 is a MaxC2 design if and only if this design has N/2−1 clear 2fi’s. This obtains that Dt∗=SN/4+1 up to isomorphism.With Lemma2, Theorem 1 provides the constructions of B2-GMC designs with n=N/4+1.Theorem 1.
SupposeD=Dt:Db is a 2n−m:2s design with n=N/4+1 and 2k≤s≤2k+1−1 for some k0≤k≤q−3; then, D=Dt:Db is a B2-GMC design if Dt=SN/4+1 and Db is any s-projection of Hk+1.Proof.
By Lemma2, if D=Dt:Db is a B2-GMC 2n−m:2s design with n=N/4+1, then Dt=SN/4+1 up to isomorphism. Thus, C1#2D=N/4+1,0,…,0 which is sequentially maximized by D=SN/4+1:Db. Let u1=#UDb∩q−1Hq−2,qHq−2,qq−1 and u2=#UDb∩Hq−2; then, from (10), we have(14)C2#2pD=2N4−1+1−u1,forp=0,N4−1−u2N8−1,forp=N8−2,0,otherwise.
IfD is a B2-GMC design, then D must sequentially minimize u1,u2. There are two different ways to choose Db from Hq\SN/4+1:(i)
Db∩q−1Hq−2,qHq−2,qq−1=∅.(ii)
Db∩q−1Hq−2,qHq−2,qq−1≠∅.
It is not hard to verify thatDb in case (ii) results in u1>0 while Db in case (i) gives u1=0. Therefore, if D=SN/4+1:Db is a B2-GMC design, then Db must be of case (i). Recall that Db∩SN/4+1=∅, and we have Db⊂Hq−2. By Lemma 1 (i), if Db⊂Hq−2, then u2=#UDb∩Hq−2≥2k+1−1 and the equality holds when Db⊂Hk+1 up to isomorphism. This completes the proof.The following example illustrates the construction method in Theorem1.Example 1.
Consider the construction of theB2-GMC 29−4:25 design. Here, n=9,m=4,q=5, N=32, and s=5 which leads to k=2. According to Theorem 1, let Dt=4,5,45H3 and Db be any 5-projection of H3, say Db=1,2,12,3,13. Then, D=Dt:Db is a B2-GMC 29−4:25 design.


## 3.2.B2-GMC Designs with N/4+1<n≤5N/16

To sequentially maximize (4), the first part C1#2D of (4) should be first maximized. Recall that C1#2D=C1#2Dt. If Dt has resolution IV, then C1#2D must be maximized. According to [31], when N/4+1<n≤5N/16, the Dt with resolution IV must be an n-projection of some second-order saturated (SOS) designs. In the following, we first review the concept of SOS design.A2n−m design is called an SOS design if all of its degrees of freedom can be used to estimate only the main treatment effects and 2fi’s. In terms of coding theory and projective geometry, Davydov and Tombak [32] showed that, given N, only the SOS designs of N/4+1, N2w−2+1/2w with w≥4 and N/2 factors exist. Block and Mee [31] further showed that an SOS design of N2w−2+1/2w factors can be obtained by doubling some smaller SOS 22w−2+1−2w−2+1−w designs q−w times. Zhang and Cheng [33] showed that the SOS design of N/2 factors can be uniquely represented by SN/2=q,qHq−1 up to isomorphism. Let(15)ℒw=Dq−wY:Yis an SOS22w−2+1−2w−2+1−wdesign,denote the collection of all the SOS designs obtained by doubling some SOS 22w−2+1−2w−2+1−w designs q−w times. Especially, in ℒw, we denote the SOS design obtained by doubling the MaxC2 22w−2+1−2w−2+1−w design Φw=w−12w,w2w,w2ww−12wHw−2w as SN2w−2+1/2w.With a little algebra, it is easy to verify that(16)Dq−wI2w=I2q,Hq−wq=I,Hq−wDq−w12w=q−w+12qI2q,Hq−wq=q−w+1I,Hq−wDq−w22w=q−w+22qI2q,Hq−wq=q−w+2I,Hq−w⋮Dq−ww−12w=q−12qI2q,Hq−wq=q−1I,Hq−wDq−ww2w=q2qI2q,Hq−wq=qI,Hq−w,where I2q−w⊗12w=q−w+12q,I2q−w⊗22w=q−w+22q,…,I2q−w⊗w−12w=q−12q, and I2q−w⊗w2w=q2q. Then,(17)Dq−ww−12ww2wHw−2w=qq−1Hq−2\Hq−w.Therefore,SN2w−2+1/2w can be expressed as(18)SN2w−2+1/2w=Dq−wΦw=q−1,q−1Hq−w,q,qHq−w,qq−1Hq−2\Hq−w.The following lemma provides a necessary condition for a2n−m:2s design D=Dt:Db with N/4+1<n≤5N/16 to be a B2-GMC design.Lemma 3.
SupposeD=Dt:Db is a 2n−m:2s design with N2w−1+1/2w+1<n≤N2w−2+1/2w for w≥4 and 2k≤s≤2k+1−1 for some k0≤k≤q−3; then, D=Dt:Db is a B2-GMC design only if Dt⊂SN2w−2+1/2w.Proof.
As discussed in the first paragraph of this section, ifD=Dt:Db is a B2-GMC 2n−m:2s design with N2w−1+1/2w+1<n≤N2w−2+1/2w, then Dt must be an n-projection of some SOS designs in ℒv with 4≤v≤w or SN/2.
LetP1 be an n-projection of SN/2. According to Theorem 3 in [33], we obtain(19)C2#2pP1=0,forp<n−N4−1,N4n−N4,forp=n−N4−1,0,forn−N4−1<p<N8−1.
LetXv=a1,a2,…,a2v−2+1 be any SOS 22v−2+1−2v−2+1−v design but not a MaxC2 design; then, Dq−vXv∈ℒv. Denote pi as the number of clear 2fi’s of Xv involving ai for i=1,2,…,2v−2+1 and c as the total number of clear 2fi’s of Xv. Let Pv be an n-projection of Dq−vXv and P¯v=Dq−vXv\Pv. Then, #P¯v=N2v−2+1/2v−n≤N/2v. Applying equations (23) and (24) in the proof of Theorem 3.1 of [29], among all the n-projections of Dq−vXv, Pv sequentially maximizes C2#2 in (5), only if P¯v⊂Dq−vaj, where aj is the column such that pj=maxp1,p2,…,p2v−2+1. For a Pv with P¯v⊂Dq−vaj, according to equation (24) in the proof of Theorem 3.1 of [29], it is obtained that(20)C2#2pPv=0,for0≤p<n−N4−1,pjN2vn−N/4,forp=n−N4−1,0,forn−N4−1<p<N2v−1,c−pjN2v2,forp=N2v−1.
For a MaxC222v−2+1−2v−2+1−v design, it has 2v−1−1 clear 2fi’s and pj=2v−2. Since Xv is not a MaxC2 design, we have(21)pj=2v−2,c−pj<2v−1−2v−2−1,or(22)pj<2v−2,c<2v−1−1.
LetQv be an n-projection of SN2v−2+1/2v obtained by doubling Φv, the MaxC222v−2+1−2v−2+1−v design as mentioned above. With a column permutation, rewrite Dq⋅ as(23)DRCq⋅=Hq−v,Hq−2\Hq−v,q−1Hq−2\Hq−v,qHq−2\Hq−v,qq−1I,Hq−v,SN2v−2+1/2v,in a re-changed Yates order. Write Φv as Φv=s1,s2,…,s2v−2+1, where s1=v−12v, s2=v2v and s3,…,s2v−2+1=v2vv−12vHv−2v. Applying equations (23) and (24) in the proof of Theorem 3.1 of [29], among all the n-projections of SN2v−2+1/2v, Qv sequentially maximizes C2#2 in (3), only if Q¯v⊂Dq−vsi with i=1 or 2. In the following, we investigate C2#2pQv with Q¯v⊂Dq−vs1 for which the following analysis and final conclusion are the same as that for C2#2pQv with Q¯v⊂Dq−vs2. For Qv with Q¯v⊂Dq−vs1, from equations (18) and (23), it is obtained that(24)B2Qv,γ=0,forγ∈SN2v−2+1/2v,n−N4,forγ∈qHq−2\Hq−v,qq−1I,Hq−v,N2v,forγ∈q−1Hq−2\Hq−v,N8−N2v,forγ∈Hq−2\Hq−v,B2Q¯v,γ+N8,forγ∈Hq−v,where the second equality can be easily verified by noting that(25)qHq−2\Hq−v,qq−1I,Hq−v=Dq−vs1s2,Dq−vs1s3,…,Dq−vs1s2v−2+1.
Therefore,(26)C2#2pQv=0,for0≤p<n−N4−1,N4n−N/4,forp=n−N4−1,0,forn−N4−1<p<N2v−1,N2vN/4−N/2v,forp=N2v−1,0,forN2v−1<p<N8−N2v−1,N4−N2vN8−N2v,forp=N8−N2v−1.
DenoteD∗=Dt∗:Db∗, where Dt∗⊂SN2w−2+1/2w, SN2w−2+1/2w\Dt∗⊂Dq−ws1 and Db∗⊂Hk+1. Since k≤q−3, we have UDb∗⊂Hq−2. Therefore, from (24) and (26), we obtain(27)C2#2pD∗=0,for0≤p<n−N4−1,N4n−N/4,forp=n−N4−1,0,forn−N4−1<p<N2w−1,N2wN/4−N/2w,forp=N2w−1.
Comparing (27) with (19), (20) for 4≤v≤w, and (26) for 4≤v≤w−1, it is obtained that if D=Dt:Db is a B2-GMC design, D should not be worse than D∗ in terms of (3). Therefore, if D=Dt:Db is a B2-GMC design, Dt should be an n-projection of SN2w−2+1/2w. This completes the proof.The lemma below is an extension of LemmaA.2, in Appendix, introduced from [34].Lemma 4.
SupposeDt consists of the last n, N2w−1+1/2w+1<n≤N2w−2+1/2w, columns of SN2w−2+1/2w; then, B2Dt,γ1≥B2Dt,γ2 if γ1 is ahead of γ2 in Hq−w.Proof.
SupposeO consists of the last n−N/4 columns of Dq−ww−12w. Then, Dt=O,SN2w−2+1/2w\Dq−ww−1. From (16) and (18), for any γ∈Hq−w, B2Dt,γ=B2O,γ+N/8. Straightforwardly, from Lemma A.2, it is obtained that B2O,γ1≥B2O,γ2 if γ1 is ahead of γ2 in Hq−w. This completes the proof.With Lemmas3 and 4, Theorem 2 provides the construction methods of B2-GMC 2n−m:2s designs with N/4+1<n≤5N/16.Theorem 2.
LetD=Dt:Db be a 2n−m:2s design with N2w−1+1/2w+1<n≤N2w−2+1/2w for w≥4 and 2k≤s≤2k+1−1 for some 0≤k≤q−3. Suppose 2r≤N2w−2+1/2w−n≤2r+1−1 for some 0≤r≤q−w−1; then, D is a B2-GMC design if Dt consists of the last n columns of SN2w−2+1/2w and(a)
Db is any s-projection of Hk,q−1,q−1Hk when k≤r≤q−w−1.(b)
Db is any s-projection of Hk+1 when r+1≤k≤q−w−1.(c)
Db is any s-projection of Hq−2 when q−w−1<k≤q−3.Proof.
By Lemma3 and its proof, if D∗=Dt∗:Db∗ is a B2-GMC 2n−m:2s design, there should be Dt∗⊂SN2w−2+12w and D¯t∗=SN2w−2+12w\Dt∗⊂Dq−ww−12w. Substituting v with w in (24), we obtain(28)B2Dt∗,γ=0,forγ∈SN2w−2+1/2w,n−N4,forγ∈qHq−2\Hq−w,qq−1I,Hq−w,N2w,forγ∈q−1Hq−2\Hq−w,N8−N2w,forγ∈Hq−2\Hq−w,B2D¯t∗,γ+N8,forγ∈Hq−w.
Let(29)u1=#UDb∗∩qHq−2\Hq−w,qq−1I,Hq−w,u2=#UDb∗∩q−1Hq−2\Hq−w,u3=#UDb∗∩Hq−2\Hq−w,and then(30)C2#2pD∗=n−N4N4−u1,forp=n−N4−1,N2wN4−N2w−u2,forp=N2w−1,N8−N2wN4−N2w−u3,forp=N8−N2w−1,fDt∗,p,forp≥N8−1,0,otherwise,where fDt∗,p=C2#2pDt∗−p+1#γ:γ∈Hq−w∩UDb∗,B2Dt∗,γ=p+1. Therefore, D∗ sequentially maximizes C2#2 among all the possible 2n−m:2s designs, only if D∗ sequentially minimizes(31)u1,u2,u3.
For ease of presenting, letA=qHq−2\Hq−w,qq−1I,Hq−w, B=q−1Hq−2\Hq−w, and C=Hq−2\Hq−w.
For (a), whenk≤q−w−1, there are two different ways to choose Db∗ from Hq: (i) Db∗∩A,B,C=∅ and (ii) Db∗∩A,B,C≠∅. Recall that if D∗=Dt∗:Db∗ is a B2-GMC design, then Dt∗⊂SN2w−2+12w with D¯t∗=SN2w−2+12w\Dt∗⊂Dq−ww−12w. Case (i) implies that Db∗⊂Hq−w,D¯t∗, and thus Db∗⊂Hq−w,q−1,q−1Hq−w. Therefore, choosing Db∗ according to case (i) results in u1=u2=u3=0. If Db∗ is chosen according to case (ii), then UDb∗∩A,B,C≠∅ which results in u1>0, u2>0, or u3>0. Clearly, if D∗ is a B2-GMC design, Db∗ should be chosen as (i). In the following, we consider only Db∗∩A,B,C=∅, i.e., Db∗⊂Hq−w,q−1,q−1Hq−w.
ForD=Dt:Db in (a), according to (v) in Lemma 1, if Db⊂Hk,q−1,q−1Hk with k≤q−w−1, then UDb=Hk,q−1,q−1Hk. Denote b as the last column of Hk in Yates order, and B2Dt,b=b; then, b≥N/8. According to Lemma 4, for γ∈Hq−w, if γ is ahead of b in Hq−w, then B2Dt,γ≥b by (18). For any γ∈q−1,q−1Hk, we have B2Dt,γ=0. For any γ such that B2Dt,γ<b, we have γ∈Hq\Hk. Therefore, for p≤b−2,(32)C2#2pD=p+1#γ∈Hq:γ∉UDb,B2Dt,γ=p+1=p+1#γ∈Hq:B2Dt,γ=p+1=C2#2pDt,and for p≥b,(33)C2#2pD=p+1#γ∈Hq:γ∉UDb,B2Dt,γ=p+1=p+1#γ∈Hq:B2Dt,γ=p+1−p+1#γ∈Hk:B2Dt,γ=p+1=0.
From (33), it is obtained that(34)C2#20D,…,C2#2N/8−1D,C2#2N/8D,…,C2#2b−2D=C2#20Dt,…,C2#2N/8−1Dt,C2#2N/8Dt,…,C2#2b−2Dt.
Note thatDt is a GMC design [29]; then, D maximizes(35)C2#20⋅,…,C2#2N/8−1⋅,C2#2N/8⋅,…,C2#2b−2⋅,among all the possible 2n−m:2s designs.
SupposeD is not a B2-GMC design; then, there should be a D∗=Dt∗:Db∗ which outperforms D=Dt:Db in terms of (3). This implies that there exists some p1≥b−1 such that(36)C2#20D∗,…,C2#2N/8−1D∗,C2#2N/8D∗,…,C2#2b−2D∗=C2#20D,…,C2#2N/8−1D,C2#2N/8D,…,C2#2b−2D,(37)C2#2p1−1D∗>C2#2p1−1D.
Recalling the definitions ofC2#2pD and B2Dt,γ, we have(38)∑p=0PC2#2pDp+1=∑p=0P#γ∈Hq:γ∉UDb,B2Dt,γ=p+1=∑p=0P#γ∈Hq\SN2w−2+1/2w:B2Dt,γ=p+1−∑p=0P#γ∈UDb:B2Dt,γ=p+1=#Hq\SN2w−2+1/2w−#Hk=2q−N2w−2+12w−2k,where the second and third equalities are due to B2Dt,γ=0 for any γ∈SN2w−2+1/2w. Similarly,(39)∑p=0PC2#2pD∗p+1=∑p=0P#γ∈Hq:γ∉UDb∗,B2Dt∗,γ=p+1=∑p=0P#γ∈Hq\SN2w−2+1/2w:B2Dt∗,γ=p+1−∑p=0P#γ∈UDb∗:B2Dt∗,γ=p+1,=#Hq\SN2w−2+1/2w−#UDb∗,=2q−N2w−2+12w−#UDb∗.
From (33) and (38), we have(40)∑p=0PC2#2pDp+1=∑p=0b−2C2#2pDp+1+C2#2b−1Db=2q−N2w−2+12w−2k.
By (36) and (37), it is obtained that(41)∑p=0PC2#2pD∗p+1≥∑p=0b−2C2#2pD∗p+1+C2#2b−1D∗b>∑p=0b−2C2#2pDp+1+C2#2b−1Db.
Then, according to (39)–(41), we obtain #UDb∗<2k−1 which contradicts Lemma 1 (i) and (ii).
For (b), similar to (a), whenk≤q−w−1, there are two different ways to choose Db∗ from Hq: (i) Db∗∩A,B,C=∅ and (ii) Db∗∩A,B,C≠∅. With a similar argument to (a), case (i) results in u1=u2=u3=0 while case (ii) results in u1>0,u2>0 or u3>0. Therefore, if D∗ is a B2-GMC design, Db∗ should be chosen according to case (i), i.e., Db∗⊂Hq−w,q−1,q−1Hq−w. When choosing Db∗ according to (i), there should be Db∗∩q−1,q−1Hq−w=∅, i.e., Db∗⊂Hq−w; otherwise, Dt∗∩UDb∗≠∅ noting that #UDb∗∩q−1,q−1Hq−w≥2k>2r+1−1 as shown in Lemma 1 (iii). The remainder of the proof is similar to that of (a).
For (c), whenk>q−w−1, there are two different ways to choose Db∗ from Hq: (i) Db∗∩B,C=∅ and (ii) Db∗∩B,C≠∅. With a similar argument to (a), it is not hard to verify that case (ii) results in u1>0 or u2>0 while case (i) results in u1=u2=0. Therefore, if D∗ is a B2-GMC design, Db∗ should be an s-projection of Hq−2=B,C. The remainder of the proof is similar to that of (a).Remark 1.
Theorem2 shows that when constructing the B2-GMC designs with N/4+1<n≤5N/16, we can first partition the range N/4+1,5N/16 into q−4 sequential subranges as N2w−1+1/2w+1,N2w−2+1/2w with w=4,5,…,q−1 and then obtain the B2-GMC designs according to Theorem 2.In the following, an example is provided to illustrate the construction method in Theorem2 and Remark 1.Example 2.
Consider the constructions ofB2-GMC 219−13:2s designs for s=1,3,4. Here, n=19,m=13,q=6, and N=64. The values of the parameters N and n satisfy N/4+1<n≤5N/16, i.e., n∈17,20. From Remark 1, we partition the range 17,20 into two subranges 17,18 because w=5 and 18,20 because w=4. Since 19∈18,20, according to Theorem 2, Dt should be the last 19 columns of S5N/16, where S5N/16=Dq−4Φ4=5,5H2,6,6H2,56H4\H2 from (18). Therefore, Dt=5H2,6,6H2,56H4\H2. Note that 5N/16−n=20−19=1; then, r=0. Next, we choose Db for s=1,3,4.
Fors=1, we have k=0. According to Theorem 2 (a), Db=5 as H0=∅.
Fors=3, we have k=1. According to Theorem 2 (b), Db should be a 3-projection of H2, i.e., Db=H2=1,2,12.
Fors=4, we have k=2. According to Theorem 2 (c), Db should be a 4-projection of H4. Without loss of generality, we choose Db=1,2,12,3.


## 4. Concluding Remarks

The regular2n−m designs have wide applications in engineering, manufacturing industry, agriculture and medicine, and so on. When the size of experimental units is large, the inhomogeneity of experimental units results in unwanted variances of estimations of treatment effects. An essential way to solve this problem is to partition the experimental units into blocks.There are two kinds of blocking problems as pointed out in [1]. One is called the single block variable problem which considers only a single block variable, and the other is called multi-block variable problem which considers two or more block variables. As stated in Section 1, experiments which involve multi-block variables are more widely concerned in practice than those which involve only a single block variable. However, due to the complexity of multi-block variable problem, the studies on constructing optimal designs with multi-block variables are relatively rare.In this paper, we aim at exploring the theories and constructions of optimal blocked2n−m designs with multi-block variables. The prevalent B2-GMC criterion is adopted. This criterion is preferable when there is some prior knowledge on the importance ordering of the treatment effects. The systematical construction methods of the B2-GMC 2n−m:2s designs with N/4+1≤n≤5N/16 are developed. The construction methods are concise and easy to implement as indicated by the examples provided.



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