Metrika (2012) 75:33–53
DOI 10.1007/s00184-010-0313-9
Construction of equidistant and weak equidistant
supersaturated designs
Yan Liu · Min-Qian Liu
Received: 27 December 2009 / Published online: 4 May 2010
© Springer-Verlag 2010
Abstract Supersaturated designs (SSDs) have been highly valued in recent years
for their ability of screening out important factors in the early stages of experiments.
Recently, Liu and Lin (in Statist Sinica 19:197–211, 2009) proposed a method to
construct optimal mixed-level SSDs from smaller multi-level SSDs and transposed
orthogonal arrays (OAs). This paper extends their method to construct more equidistant optimal SSDs by replacing the multi-level SSDs and transposed OAs with
mixed-level SSDs and general transposed difference matrices, respectively, and then
proposes two practical methods for constructing weak equidistant SSDs based on this
extended method. A large number of new optimal SSDs can be constructed from
these three methods. Some examples are provided and more new designs are listed in
“Appendix” for practical use.
Keywords Balanced design · Coincidence number · Difference matrix · Equidistant
design · Kronecker sum
1 Introduction
A factorial design is called a supersaturated design (SSD) if its run size is insufficient
for estimating all its main effects. In the early stages of industrial or scientific experiments involving a large number of factors, SSDs are usually adopted to screen out
important factors. The analysis of the data collected by SSDs depends on the assumption of effect sparsity, which states that the response is actually affected by a few
important factors. SSDs were proposed by Satterthwaite (1959) firstly and studied
Y. Liu · M.-Q. Liu (B)
Department of Statistics, School of Mathematical Sciences and LPMC, Nankai University,
Tianjin 300071, China
e-mail: [email protected]
123
34
Y. Liu, M.-Q. Liu
further by Booth and Cox (1962), Lin (1993), and Wu (1993). After then, SSDs
have received much interest because they can identify the important factors effectively. Some most recent studies focused on multi-level and mixed-level SSDs include
Yamada and Lin (1999, 2002), Yamada and Matsui (2002), Fang et al. (2003a,b), Xu
(2003), Fang et al. (2004a,b,c), Li et al. (2004), Xu and Wu (2005), Koukouvinos and
Mantas (2005), Georgiou and Koukouvinos (2006), Georgiou et al. (2006), Liu et al.
(2006, 2007), Yamada et al. (2006), Ai et al. (2007), Zhang et al. (2007), Tang et al.
(2007), Chen and Liu (2008a,b), Liu and Lin (2009), Liu and Cai (2009) and Liu and
Zhang (2009).
The E( f N O D ) criterion, proposed by Fang et al. (2003a), is one of the most commonly used criteria for evaluating SSDs. This paper considers the construction of
E( f N O D ) optimal SSDs. According to this criterion, E( f N O D ) optimal designs whose
E( f N O D ) values reach their lower bounds (Fang et al. 2004b) are divided into two
kinds: equidistant and weak equidistant designs. While many papers have addressed
the construction methods for equidistant designs, weak equidistant designs still need
more attention.
For constructing E( f N O D ) optimal SSDs, this paper proposes three extensions of
Liu and Lin’s (2009) method, one of which is concerned with equidistant designs and
the other two are about weak equidistant designs. Section 2 reviews some definitions,
notations and the E( f N O D ) criterion. Section 3 introduces the method for constructing
equidistant SSDs and Sect. 4 proposes the two methods for constructing weak equidistant SSDs. Using these three methods, we can construct a large number of new
E( f N O D ) optimal SSDs. Some examples are given in Sects. 3 and 4 for illustration.
Section 5 discusses some good properties and the application of these three methods.
More new designs are listed in Tables 19, 20, 21, 22, 23, 24 in “Appendix”.
2 Preliminaries
2.1 Some definitions and notations
Let D(n; p1 , . . . , pm ) represent a mixed-level (asymmetrical) design of n runs and
m factors with levels p1 , . . . , pm , where the tth factor takes values from a set of
pt symbols {0, . . . , pt − 1}. This design can also be written as an n × m matrix
D = (dit ), where the ith row and the tth column are denoted by ri and ct , respectively.
A D(n; p1 , . . . , pm ) is called an orthogonal array (OA) of strength two, denoted by
L n ( p1 , . . . , p
m ), if all possible level-combinations for any two columns appear equally
m
often. When
m t=1 ( pt − 1) = n − 1, the D(n; p1 , . . . , pm ) design is called saturated;
and when t=1 ( pt − 1) > n − 1, the design is called a supersaturated design (SSD).
When some pt ’s are equal, we use notations D(n; p1s1 . . . plsl ) and L n ( p1s1 . . . plsl ),
respectively, where lk=1 sk = m; when all the pt ’s are equal, we call them symmetrical designs, and use notations D(n; p m ) and L n ( p m ), respectively. Two columns
are called fully aliased if one can be obtained from the other by permuting levels.
Fully aliased columns should be avoided in a design as they cannot accommodate two
different factors at the same time. A design is called balanced if each of its columns
123
Construction of equidistant and weak equidistant SSDs
35
has the equal occurrence property of the levels. Throughout this paper, we consider
balanced designs only.
Let G, +̇ denote an abelian group, where G = {0, . . . , q − 1} and +̇ is a binary
operation on G. The notation −̇ denotes the inverse operation of +̇. For any two
matrices A = (ait ) of order n × m and B = (buv ) of order w × z, both with
entries from G, +̇, define their Kronecker sum to be A ⊕q B = [B ait ]1≤i≤n,1≤t≤m ,
where B ait = (B +̇ait J ) = (buv +̇ait )1≤u≤w,1≤v≤z , and J denotes the w × z matrix
of 1’s.
A δq × r matrix is called a difference matrix (DM) on G, +̇ and denoted by
Mδq,r ;q if (i) all the elements in Mδq,r ;q come from G; (ii) for any two columns of
Mδq,r ;q , for example, ci = (c1,i , . . . , cδq,i ) and c j = (c1, j , . . . , cδq, j ) , where i = j,
each element of G occurs exactly δ times in ci −̇c j = (c1,i −̇c1, j , . . . , cδq,i −̇cδq, j ) .
Generally, the operation +̇ is defined as x +̇y = (x + y) mod q, and thus the −̇ is
defined as x −̇y = (x − y) mod q. But sometimes they are differently defined, see
e.g. the case in Example 2. Except this case, the definitions of x +̇y = (x + y) mod q
and x −̇y = (x − y) mod q are used in all other places in this paper.
Obviously, OAs are a special kind of DMs, and for an Mδq,r ;q , any m columns of
it form an Mδq,m;q , where m = 2, . . . , r − 1.
According to the definition of DM, we can easily get the following properties
presented in Lemma 1.
Lemma 1 Let ci = (c1,i , . . . , cδq,i ) and c j = (c1, j , . . . , cδq, j ) denote any two distinct
columns of an Mδq,r ;q , where 1 ≤ i = j ≤ r , then we have
(i) there are exactly δ coincidence positions between ci and k ⊕q c j , for any k =
0, . . . , q − 1;
(ii) there is no coincidence position between ci and k ⊕q ci , for any k = 1, . . . , q −1.
2.2 The E( f N O D ) Criterion
For a D(n; p1s1 . . . plsl ) design with lk=1 sk = m, we suppose the ith and jth columns
ci and c j have pi and p j levels, respectively. The E( f N O D ) criterion is defined as
E( f N O D ) =
f N O D (ci , c j )
1≤i< j≤m
f N O D (ci , c j ) =
p
j −1 i −1 p
(i j)
n uv
u=0 v=0
n
−
pi p j
m
2
, where
2
,
(i j)
and n uv is the number of (u, v)-pairs in (ci , c j ). For this criterion, some properties
obtained by Fang et al. (2003a, 2004b) are summarized in the following lemma.
123
36
Y. Liu, M.-Q. Liu
Lemma 2 For any D(n; p1s1 . . . plsl ) design D with
n
i, j=1,i= j
l
k=1 sk
= m,
λi2j
+Cf
m(m − 1)
n(n − 1)
[(λ
+ 1 − λ)(λ − λ
) + λ2 ] + C f , where
≥
(1)
m(m − 1)
⎞
⎛
l
l
l
n2
sk
sk (sk − 1)
s h sk ⎠
nm
⎝
−
+
+
,
Cf =
2
m − 1 m(m − 1)
p
p p
pk
k=1 k
k=1
k,h=1,h=k h k
E( f N O D ) =
λi j denotes
the coincidence
number between the ith and jth rows,
l
λ = n k=1 sk / pk − m /(n − 1), and λ
denotes the integer part of λ. The lower
bound in (1) can be achieved if and only if all the values of λi j (i = j) take at most
two different values λ
and λ
+ 1.
Remark 1 When λ is an integer, the design whose E( f N O D ) value reaches its lower
bound in (1) is called an equidistant design, for all the values of λi j (i = j) take only
one value λ, and when λ is not an integer, the design whose E( f N O D ) value reaches
its lower bound in (1) is called a weak equidistant design (Zhang et al. 2005), for all
the values of λi j (i = j) take two different values λ
and λ
+ 1.
Remark 2 From Mukerjee and Wu (1995), a saturated OA L n ( p m ) with n − 1 =
m( p − 1) is also an equidistant design with integer λi j = λ = (m − 1)/ p, for i = j.
3 Construction of equidistant SSDs
Recently, Liu and Lin (2009) proposed a method to construct mixed-level SSDs from
smaller multi-level SSDs and transposed OAs. Now, let us extend their method to
construct equidistant SSDs. The construction is given in the following theorem.
Theorem 1 If there exist two designs D0 and D1 such that (i) D0 is an equidistant
D(n 0 ; p1s1 . . . plsl ) design with a constant coincidence number λ between any two
distinct rows, where lk=1 sk = m 0 ; (ii) D1 is the transpose of an Mδq,n 0 ;q ; (iii)
λ + δ = m 0 , then
D = [0q ⊕ p D0 , L q ⊕q D1 ]
(2)
is an equidistant D(qn 0 ; p1s1 . . . plsl q δq ) design with a constant coincidence number
m 0 between any two distinct rows, where p = max pi , 0q denotes a q × 1 vector
of 0’s and L q = (0, . . . , q − 1) .
123
i∈{1,...,l}
Construction of equidistant and weak equidistant SSDs
Table 1 L 4 (23 )
Table 2 M6,6;3
37
0
0
0
0
1
1
1
0
1
1
1
0
0
0
0
0
0
0
0
1
2
0
2
1
0
2
1
2
0
1
0
0
1
1
2
2
0
1
0
2
1
2
0
2
2
1
1
0
Proof Apparently, the resulting design D in (2) has the form:
⎡
D0
⎢ D0
⎢
⎢ ..
⎣ .
D1
1 ⊕q D1
..
.
⎤
⎥
⎥
⎥.
⎦
D0 (q − 1) ⊕q D1
According to the properties of a DM presented in Lemma 1, we can get that, for any
two rows of D, say the ith and jth rows, where 1 ≤ i < j ≤ qn 0 , if j − i = 0
mod n 0 , they have m 0 coincidence positions at the 0q ⊕ p D0 part and no coincidence
position at the L q ⊕q D1 part; otherwise, they have λ coincidence positions at the
0q ⊕ p D0 part and δ coincidence positions at the L q ⊕q D1 part. Since λ + δ = m 0 ,
the design D has a constant coincidence number m 0 between any two distinct rows,
and thus is an equidistant design.
Theorem 1 intimates that though neither 0q ⊕ p D0 nor L q ⊕q D1 has a constant
coincidence number between any two distinct rows, when we juxtapose them, they
can mutually complement each other, and thus the resulting design D has a constant
coincidence number between any two distinct rows. Three examples are shown below
to illustrate the construction method given in Theorem 1. More new designs are listed
in Tables 19 and 20.
Example 1 The saturated OA L 4 (23 ) shown in Table 1 is an equidistant design with
constant coincidence numbers 1. The transpose of an M6,6;3 is shown in Table 2. Let
D0 denote the L 4 (23 ) and D1 denote the design consisting of the first four rows of the
transpose of this M6,6;3 . Then the D(12; 23 36 ) design constructed from D0 and D1
via (2), as shown in Table 3, is a new equidistant design with a constant coincidence
number 3 between any two distinct rows.
Example 2 From the FSOA method of Fang et al. (2003a), we can construct an equidistant D(6; 21 33 ) with constant coincidence numbers 1 as shown in Table 4. Table 6
shows the transpose of an M12,6;4 , where the binary operation +̇ is defined as in
123
38
Table 3 D(12; 23 36 )
Table 4 D(6; 21 33 )
Table 5 definition of +̇
Y. Liu, M.-Q. Liu
0
0
0
0
0
0
0
0
0
0
1
1
0
1
2
0
2
1
1
0
1
0
2
1
2
0
1
1
1
0
0
0
1
1
2
2
0
0
0
1
1
1
1
1
1
0
1
1
1
2
0
1
0
2
1
0
1
1
0
2
0
1
2
1
1
0
1
1
2
2
0
0
0
0
0
2
2
2
2
2
2
0
1
1
2
0
1
2
1
0
1
0
1
2
1
0
1
2
0
1
1
0
2
2
0
0
1
1
0
0
0
0
0
1
1
1
0
2
2
2
1
0
1
2
1
1
2
0
1
2
0
1
+̇
0
1
2
3
0
0
1
2
3
1
1
0
3
2
2
2
3
0
1
3
3
2
1
0
Table 5. Then the D(24; 21 33 412 ) design constructed from these two designs via (2)
is a new equidistant design with a constant coincidence number 4 between any two
distinct rows. This design is shown in Table 7.
Example 3 Adding (0, . . . , 5) to the D(6; 21 33 ) shown in Table 4, then we get an
equidistant D(6; 61 21 33 ) design with a constant coincidence number 1 between any
two distinct rows as shown in Table 8. Table 9 shows the transpose of an OA L 8 (27 ).
Let D0 denote the D(6; 61 21 33 ) and D1 denote the design consisting of the first six
rows of the transpose of this L 8 (27 ), then the equidistant D(12; 61 29 33 ) design constructed from D0 and D1 via (2) with a constant coincidence number 5 between any
two distinct rows is presented in Table 10.
This extended method can construct a large number of new equidistant designs,
while the next section will present two convenient methods for constructing weak
equidistant designs based on this extension.
123
Construction of equidistant and weak equidistant SSDs
Table 6 M12,6;4
Table 7 D(24; 21 33 412 )
Table 8 D(6; 61 21 33 )
39
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
1
1
2
2
2
3
3
3
0
0
0
3
3
3
1
1
1
2
2
2
0
2
3
0
1
2
0
1
3
1
2
3
0
2
3
2
0
1
1
3
0
2
3
1
0
2
3
1
2
0
3
0
1
3
1
2
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
1
1
0
0
0
1
1
1
2
2
2
3
3
3
0
2
2
2
0
0
0
3
3
3
1
1
1
2
2
2
1
0
1
2
0
2
3
0
1
2
0
1
3
1
2
3
1
1
2
0
0
2
3
2
0
1
1
3
0
2
3
1
1
2
0
1
0
2
3
1
2
0
3
0
1
3
1
2
0
0
0
0
1
1
1
1
1
1
1
1
1
1
1
1
0
1
1
1
1
1
1
0
0
0
3
3
3
2
2
2
0
2
2
2
1
1
1
2
2
2
0
0
0
3
3
3
1
0
1
2
1
3
2
1
0
3
1
0
2
0
3
2
1
1
2
0
1
3
2
3
1
0
0
2
1
3
2
0
1
2
0
1
1
3
2
0
3
1
2
1
0
2
0
3
0
0
0
0
2
2
2
2
2
2
2
2
2
2
2
2
0
1
1
1
2
2
2
3
3
3
0
0
0
1
1
1
0
2
2
2
2
2
2
1
1
1
3
3
3
0
0
0
1
0
1
2
2
0
1
2
3
0
2
3
1
3
0
1
1
1
2
0
2
0
1
0
2
3
3
1
2
0
1
3
1
2
0
1
2
0
1
3
0
2
1
2
3
1
3
0
0
0
0
0
3
3
3
3
3
3
3
3
3
3
3
3
0
1
1
1
3
3
3
2
2
2
1
1
1
0
0
0
0
2
2
2
3
3
3
0
0
0
2
2
2
1
1
1
1
0
1
2
3
1
0
3
2
1
3
2
0
2
1
0
1
1
2
0
3
1
0
1
3
2
2
0
3
1
0
2
1
2
0
1
3
1
0
2
1
3
0
3
2
0
2
1
0
0
0
0
0
1
0
1
1
1
2
0
2
2
2
3
1
0
1
2
4
1
1
2
0
5
1
2
0
1
4 Constructions of weak equidistant SSDs
Replacing the equidistant design D0 in Theorem 1 with a weak equidistant one, we
can prove that the resulting design D will be a weak equidistant design when the
parameters satisfy some conditions.
123
40
Table 9 L 8 (27 )
Table 10 D(12; 61 29 33 )
Table 11 D(4; 41 22 )
Y. Liu, M.-Q. Liu
0
0
0
0
1
1
1
1
0
0
1
1
0
0
1
1
0
1
0
1
0
1
0
1
0
1
1
0
0
1
1
0
0
0
1
1
1
1
0
0
0
1
0
1
1
0
1
0
0
1
1
0
1
0
0
1
0
0
0
0
0
0
0
0
0
1
1
1
1
1
0
1
1
1
0
0
1
1
0
0
1
1
2
0
2
2
2
0
1
0
1
0
1
0
1
3
1
0
1
2
0
1
1
0
0
1
1
0
4
1
1
2
0
0
0
1
1
1
1
0
0
5
1
2
0
1
0
1
0
1
1
0
1
0
0
0
0
0
0
1
1
1
1
0
0
0
0
1
0
1
1
1
1
1
0
0
1
1
0
0
2
0
2
2
2
1
0
1
0
1
0
1
0
3
1
0
1
2
1
0
0
1
1
0
0
1
4
1
1
2
0
1
1
0
0
0
0
1
1
5
1
2
0
1
1
0
1
0
0
1
0
1
0
0
0
1
0
1
2
1
0
3
1
1
Corollary 1 If there exist two designs D0 and D1 such that (i) D0 is a weak equidistant D(n 0 ; p1s1 . . . plsl ) design with coincidence numbers λ
and λ
+ 1, where
l
k=1 sk = m 0 ; (ii) D1 is the transpose of an Mδq,n 0 ;q ; (iii) λ
+ δ = m 0 or
λ
+ δ + 1 = m 0 , then
D = [0q ⊕ p D0 , L q ⊕q D1 ]
(3)
is a weak equidistant D(qn 0 ; p1s1 . . . plsl q δq ) design with coincidence numbers m 0 and
m 0 + 1 or m 0 − 1 and m 0 , respectively, where p = max pi .
i∈{1,...,l}
The proof is similar to that of Theorem 1, and thus omitted here. Now, let us see
some illustrative examples. More new designs are listed in Tables 21 and 22.
Example 4 It is easy to see that the D(4; 41 22 ) design shown in Table 11 is a weak
equidistant design with coincidence numbers 0 and 1. Table 12 shows the transpose
of an OA L 9 (34 ). A new D(12; 41 22 39 ) with coincidence numbers 3 and 4 can be
constructed from these two designs via (3), see Table 13 for this new design.
123
Construction of equidistant and weak equidistant SSDs
Table 12 L 9 (34 )
Table 13 D(12; 41 22 39 )
Table 14 D(12; 41 22 36 )
41
0
0
0
1
1
1
2
2
2
0
1
2
0
1
2
0
1
2
0
1
2
1
2
0
2
0
1
0
2
1
1
0
2
2
1
0
0
0
0
0
0
0
1
1
1
2
2
2
1
0
1
0
1
2
0
1
2
0
1
2
2
1
0
0
1
2
1
2
0
2
0
1
3
1
1
0
2
1
1
0
2
2
1
0
0
0
0
1
1
1
2
2
2
0
0
0
1
0
1
1
2
0
1
2
0
1
2
0
2
1
0
1
2
0
2
0
1
0
1
2
3
1
1
1
0
2
2
1
0
0
2
1
0
0
0
2
2
2
0
0
0
1
1
1
1
0
1
2
0
1
2
0
1
2
0
1
2
1
0
2
0
1
0
1
2
1
2
0
3
1
1
2
1
0
0
2
1
1
0
2
0
0
0
0
0
0
0
0
0
1
0
1
0
1
2
0
2
1
2
1
0
0
2
1
2
0
1
3
1
1
0
0
1
1
2
2
0
0
0
1
1
1
1
1
1
1
0
1
1
2
0
1
0
2
2
1
0
1
0
2
0
1
2
3
1
1
1
1
2
2
0
0
0
0
0
2
2
2
2
2
2
1
0
1
2
0
1
2
1
0
2
1
0
2
1
0
1
2
0
3
1
1
2
2
0
0
1
1
Example 5 Let D0 be the D(4; 41 22 ) design shown in Table 11, and D1 be the design
consisting the first four rows of the transpose of the M6,6;3 which is shown in Table 2,
then the D(12; 41 22 36 ) design obtained from D0 and D1 via (3), as shown in Table 14,
is a new weak equidistant design with coincidence numbers 2 and 3.
Suppose D0 is still an equidistant design, and D1 a transposed DM as in Theorem 1,
when we juxtapose 0q ⊕ p D0 and L q ⊕q D1 , if the coincidence numbers of D are
not equal to a constant but take two different values, with difference 1, the resulting
design D is a weak equidistant design. Thus we have the following corollary, whose
proof is similar to that of Theorem 1, and thus omitted.
123
42
Table 15 D(12; 29 33 )
Table 16 D(18; 21 39 )
Table 17 M5,4;5
Y. Liu, M.-Q. Liu
0
0
0
0
0
0
1
1
0
0
1
1
0
1
1
1
0
1
0
1
0
1
0
1
0
2
2
2
0
1
1
0
0
1
1
0
1
0
1
2
0
0
1
1
1
1
0
0
1
1
2
0
0
1
0
1
1
0
1
0
1
2
0
1
0
1
1
0
1
0
0
1
0
0
0
0
1
1
0
0
1
1
0
0
0
1
1
1
1
0
1
0
1
0
1
0
0
2
2
2
1
0
0
1
1
0
0
1
1
0
1
2
1
1
0
0
0
0
1
1
1
1
2
0
1
0
1
0
0
1
0
1
1
2
0
1
1
0
0
1
0
1
1
0
0
0
0
0
0
0
0
0
0
0
0
1
1
1
0
1
2
0
2
1
0
2
2
2
0
2
1
2
0
1
1
0
1
2
0
0
1
1
2
2
1
1
2
0
0
1
0
2
1
2
1
2
0
1
0
2
2
1
1
0
0
0
0
0
1
1
1
1
1
1
0
1
1
1
1
2
0
1
0
2
0
2
2
2
1
0
2
0
1
2
1
0
1
2
1
1
2
2
0
0
1
1
2
0
1
2
1
0
2
0
1
2
0
1
1
0
0
2
2
1
0
0
0
0
2
2
2
2
2
2
0
1
1
1
2
0
1
2
1
0
0
2
2
2
2
1
0
1
2
0
1
0
1
2
2
2
0
0
1
1
1
1
2
0
2
0
2
1
0
1
1
2
0
1
2
1
1
0
0
2
0
0
0
0
0
0
1
2
3
4
1
3
0
2
4
3
1
4
2
0
Corollary 2 Suppose D0 and D1 are two designs satisfying conditions (i) and (ii) in
Theorem 1, and (iii) λ + δ = m 0 + 1 or λ + δ = m 0 − 1, then
D = [0q ⊕ p D0 , L q ⊕q D1 ]
123
(4)
Construction of equidistant and weak equidistant SSDs
Table 18 D(20; 23 55 )
43
0
0
0
0
0
0
0
0
0
1
1
0
1
2
3
4
1
0
1
1
3
0
2
4
1
1
0
3
1
4
2
0
0
0
0
1
1
1
1
1
0
1
1
1
2
3
4
0
1
0
1
2
4
1
3
0
1
1
0
4
2
0
3
1
0
0
0
2
2
2
2
2
0
1
1
2
3
4
0
1
1
0
1
3
0
2
4
1
1
1
0
0
3
1
4
2
0
0
0
3
3
3
3
3
0
1
1
3
4
0
1
2
1
0
1
4
1
3
0
2
1
1
0
1
4
2
0
3
0
0
0
4
4
4
4
4
0
1
1
4
0
1
2
3
1
0
1
0
2
4
1
3
1
1
0
2
0
3
1
4
is a weak equidistant D(qn 0 ; p1s1 . . . plsl q δq ) design with coincidence numbers m 0 and
m 0 + 1 or m 0 − 1 and m 0 , respectively, where p = max pi .
i∈{1,...,l}
Now, let us see two examples for illustrating the construction method. More new
weak equidistant designs are listed in Tables 23 and 24.
Example 6 Let D0 denote the D(6; 21 33 ) shown in Table 4, and D1 denote the design
consisting of the last six rows of the transpose of the L 8 (27 ) shown in Table 9, then the
D(12; 29 33 ) design obtained from D0 and D1 via (4) is a new weak equidistant design
with coincidence numbers 4 and 5. Further, if D1 denotes the transpose of the M6,6;3
shown in Table 2, then the D(18; 21 39 ) design obtained from D0 and D1 via (4) is a
new weak equidistant design with coincidence numbers 3 and 4. The two designs are
given in Tables 15 and 16, respectively.
Example 7 The D(20; 23 55 ) design constructed via (4) from the L 4 (23 ) shown in
Table 1 and the transpose of the M5,4;5 shown in Table 17 is a new weak equidistant
design with coincidence numbers 2 and 3, which is shown in Table 18.
5 Properties of the constructed designs
Some properties of the designs constructed from the above methods are presented in
the following theorem, which can be easily proved.
123
44
Y. Liu, M.-Q. Liu
Theorem 2 Suppose D is a D(qn 0 ; p1s1 . . . plsl q δq ) design constructed through (2),
(3) or (4), then
(i) D has no fully aliased columns if both D0 and D1 have no fully aliased columns;
(ii) for any two runs of D, either they take the same level symbol on each of the
first m 0 factors and totally different level symbols on each of the last δq factors,
or they take the same level symbol on each of some λ∗ positions of the first m 0
factors and take the same level symbol on each of some δ positions of the last δq
factors, where λ∗ = λ for the design D constructed via (2) or (4), and λ∗ = λ
or λ
+ 1 for the D constructed via (3).
Previous papers have provided a lot of equidistant designs, such as Liu and Zhang
(2000), Fang et al. (2002a,b, 2003a,b, 2004a,b,c), Lu et al. (2003), Georgiou and
Koukouvinos (2006), Georgiou et al. (2006), Koukouvinos and Mantas (2005), Chen
and Liu (2008a) and Liu and Cai (2009). For an equidistant design with n runs, adding
an n-level column, say (0, . . . , n − 1) , to it also results in an equidistant design, and
a weak equidistant design can be obtained by adding or deleting a balanced column to
or from an equidistant design. Hedayat et al. (1999) also provided many methods for
constructing OAs. OAs are a special kind of DMs and saturated OAs are a special kind
of equidistant designs. Many existing DMs can be found at the website http://support.
sas.com/techsup/technote/ts723_DifferenceSchemes.txt. Using these existing (weak)
equidistant designs, OAs and DMs, we can construct a large amount of equidistant
and weak equidistant designs by the new methods. Some new designs with n 0 ≤ 12
and q ≤ 7 are tabulated in Tables 19, 20, 21, 22, 23, 24 for practical use. Note that
in these tables, all the DMs come from the above website, and we use abbreviation
FGL04 to represent the reference Fang et al. (2004a), GK06 to represent Georgiou
and Koukouvinos (2006), and so on.
Acknowledgments This work was supported by the Program for New Century Excellent Talents in
University (NCET-07-0454) of China and the National Natural Science Foundation of China (Grant Nos.
10671099, 10971107). The authors thank the editor and two anonymous referees for their valuable comments.
Appendix
See Tables 19, 20, 21, 22, 23, 24.
Table 19 Equidistant SSDs constructed from Theorem 1
No.
D(n 0 ; p m 0 )
[Source]
Mδq,n 0 ;q
Resulting design
1
D(6; 35 )
[FGL04]
M4q,6;q
D(6q; 35 q 4q ), q = 3, . . . , 7
2
D(6; 310 )
[GK06]
M8q,6;q
D(6q; 310 q 8q ), q = 3, . . . , 7
3
D(6; 315 )
[GK06]
M12q,6;q
D(6q; 315 q 12q ), q = 3, . . . , 6
4
L 8 (27 )
OA
M4q,8;q
D(8q; 27 q 4q ), q = 3, 4, 5, 7
5
D(8; 228 )
[LZ00]
M16q,8;q
D(8q; 228 q 16q ), q = 3, 4, 5
6
D(8; 47 )
[FGL02a]
M6q,8;q
D(8q; 47 q 6q ), q = 3, 4, 5, 7
7
D(8; 414 )
[FGL02a]
M12q,8;q
D(8q; 414 q 12q ), q = 3, 4∗ , 5, 6
123
Construction of equidistant and weak equidistant SSDs
45
Table 19 continued
No.
D(n 0 ; p m 0 )
[Source]
Mδq,n 0 ;q
Resulting design
8
D(8; 421 )
[GK06]
M18q,8;q
D(8q; 421 q 18q ), q = 3, 4, 5
9
D(8; 428 )
[GK06]
M24q,8;q
D(8q; 428 q 24q ), q = 3, 4∗
10
D(8; 435 )
[GK06]
M30q,8;q
D(8q; 435 q 30q ), q = 3, 4
11
D(8; 442 )
[GK06]
M36q,8;q
D(8q; 442 q 36q ), q = 3
12
OA
M3q,9;q
D(9q; 34 q 3q ), q = 3, 4, 7
13
L 9 (34 )
D(9; 38 )
[FGL04]
M6q,9;q
D(9q; 38 q 6q ), q = 3, 4, 5, 7
14
D(9; 312 )
[FGL04]
M9q,9;q
D(9q; 312 q 9q ), q = 3∗ , 4, 5, 7
15
D(9; 316 )
[FGL04]
M12q,9;q
D(9q; 316 q 12q ), q = 3∗ , 4, 5, 6
16
D(9; 320 )
[FGL04]
M15q,9;q
D(9q; 320 q 15q ), q = 3∗ , 4, 5
17
D(9; 324 )
[FGL04]
M18q,9;q
D(9q; 324 q 18q ), q = 3∗ , 4, 5
18
D(9; 328 )
[FGL04]
M21q,9;q
D(9q; 328 q 21q ), q = 3∗ , 4
19
D(9; 332 )
[GKM06]
M24q,9;q
D(9q; 332 q 24q ), q = 3∗ , 4
20
D(9; 336 )
[GKM06]
M27q,9;q
D(9q; 336 q 27q ), q = 3∗ , 4
21
D(9; 340 )
[GK06]
M30q,9;q
D(9q; 340 q 30q ), q = 3∗ , 4
22
D(9; 348 )
[GK06]
M36q,9;q
D(9q; 348 q 36q ), q = 3∗
23
D(10; 59 )
[FGL02b]
M8q,10;q
D(10q; 59 q 8q ), q = 3, . . . , 7
24
D(10; 518 )
[GK06]
M16q,10;q
D(10q; 518 q 16q ), q = 3, 4, 5
25
D(10; 527 )
[GK06]
M24q,10;q
D(10q; 527 q 24q ), q = 3, 4
26
L 12 (211 )
OA
M6q,12;q
D(12q; 211 q 6q ), q = 3, 4, 7
27
D(12; 233 )
[LZ00]
M18q,12;q
D(12q; 233 q 18q ), q = 3, 4, 5
28
D(12; 244 )
[LZ00]
M24q,12;q
D(12q; 244 q 24q ), q = 3, 4
29
D(12; 266 )
[LZ00]
M36q,12;q
D(12q; 266 q 36q ), q = 3
30
D(12; 288 )
[LZ00]
M48q,12;q
D(12q; 288 q 48q ), q = 3
31
D(12; 299 )
[LZ00]
M54q,12;q
D(12q; 299 q 54q ), q = 3
32
D(12; 311 )
[LHZ03]
M8q,12;q
D(12q; 311 q 8q ), q = 3, 4, 5, 7
33
D(12; 322 )
[GK06]
M16q,12;q
D(12q; 322 q 16q ), q = 3, 4, 5
34
D(12; 333 )
[GK06]
M24q,12;q
D(12q; 333 q 24q ), q = 3∗ , 4
35
D(12; 344 )
[GK06]
M32q,12;q
D(12q; 344 q 32q ), q = 3, 4
36
D(12; 355 )
[GK06]
M40q,12;q
D(12q; 355 q 40q ), q = 3∗
37
D(12; 411 )
[FGLQ03]
M9q,12;q
D(12q; 411 q 9q ), q = 3, 4, 5, 7
38
D(12; 422 )
[GK06]
M18q,12;q
D(12q; 422 q 18q ), q = 3, 4, 5
39
D(12; 433 )
[GK06]
M27q,12;q
D(12q; 433 q 27q ), q = 3, 4
40
D(12; 611 )
[LHZ03]
M10q,12;q
D(12q; 611 q 10q ), q = 3, 4, 5, 7
41
D(12; 622 )
[GK06]
M20q,12;q
D(12q; 622 q 20q ), q = 3, 4, 5
42
D(6; 21 33 )
[FLL03]
M3q,6;q
D(6q; 21 33 q 3q ), q = 3, 4, 5, 7
43
D(8; 21 44 )
[FLL03]
M4q,8;q
D(8q; 21 44 q 4q ), q = 3, 4, 5, 7
44
D(8; 28 44 )
[KM05]
M8q,8;q
D(8q; 28 44 q 8q ), q = 3, . . . , 7
∗ The designs can also be constructed by other existing methods
123
46
Y. Liu, M.-Q. Liu
Table 20 More equidistant SSDs constructed from Theorem 1
No.
D(n 0 ; n 10 p m 0 −1 )
[Source† ]
Mδq,n 0 ;q
Resulting design
1
D(6; 61 35 )
[FGL04]
M5q,6;q
D(6q; 61 35 q 5q ), q = 3, 4, 5, 7
2
D(6; 61 310 )
[GK06]
M9q,6;q
D(6q; 61 310 q 9q ), q = 3, 4, 5, 7
3
D(6; 61 315 )
[GK06]
M13q,6;q
D(6q; 61 315 q 13q ), q = 3, 4, 5
4
D(8; 81 27 )
OA
M5q,8;q
D(8q; 81 27 q 5q ), q = 3, 4, 5, 7
5
D(8; 81 228 )
[LZ00]
M17q,8;q
D(8q; 81 228 q 17q ), q = 3, 4, 5
6
D(8; 81 47 )
[FGL02a]
M7q,8;q
D(8q; 81 47 q 7q ), q = 3, 4, 5, 7
7
D(8; 81 414 )
[FGL02a]
M13q,8;q
D(8q; 81 414 q 13q ), q = 3, 4, 5
8
D(8; 81 421 )
[GK06]
M19q,8;q
D(8q; 81 421 q 19q ), q = 3, 4, 5
9
D(8; 81 428 )
[GK06]
M25q,8;q
D(8q; 81 428 q 25q ), q = 3, 4
10
D(8; 81 435 )
[GK06]
M31q,8;q
D(8q; 81 435 q 31q ), q = 3, 4
11
D(8; 81 442 )
[GK06]
M37q,8;q
D(8q; 81 442 q 37q ), q = 3
12
D(9; 91 34 )
OA
M4q,9;q
D(9q; 91 34 q 4q ), q = 3, 4, 5, 7
13
D(9; 91 38 )
[FGL04]
M7q,9;q
D(9q; 91 33 q 7q ), q = 3, 4, 5, 7
14
D(9; 91 312 )
[FGL04]
M10q,9;q
D(9q; 91 312 q 10q ), q = 3, 4, 5, 7
15
D(9; 91 316 )
[FGL04]
M13q,9;q
D(9q; 91 316 q 13q ), q = 3, 4, 5
16
D(9; 91 320 )
[FGL04]
M16q,9;q
D(9q; 91 320 q 16q ), q = 3, 4, 5
17
D(9; 91 324 )
[FGL04]
M19q,9;q
D(9q; 91 324 q 19q ), q = 3, 4, 5
18
D(9; 91 328 )
[FGL04]
M22q,9;q
D(9q; 91 328 q 22q ), q = 3, 4
19
D(9; 91 332 )
[GKM06]
M25q,9;q
D(9q; 91 332 q 25q ), q = 3, 4
20
D(9; 91 336 )
[GKM06]
M28q,9;q
D(9q; 91 336 q 28q ), q = 3, 4
21
D(9; 91 340 )
[GK06]
M31q,9;q
D(9q; 91 340 q 31q ), q = 3, 4
22
D(9; 91 348 )
[GK06]
M37q,9;q
D(9q; 91 348 q 37q ), q = 3
23
D(10; 101 59 )
[FGL02b]
M9q,10;q
D(10q; 101 59 q 9q ), q = 3, 4, 5, 7
24
D(10; 101 518 )
[GK06]
M17q,10;q
D(10q; 101 518 q 17q ), q = 3, 4, 5
25
D(10; 101 527 )
[GK06]
M25q,10;q
D(10q; 101 527 q 25q ), q = 3, 4
26
D(12; 121 211 )
OA
M7q,12;q
D(12q; 121 211 q 7q ), q = 3, 4, 7
27
D(12; 121 233 )
[LZ00]
M19q,12;q
D(12q; 121 233 q 19q ), q = 3, 4, 5
28
D(12; 121 244 )
[LZ00]
M25q,12;q
D(12q; 121 244 q 25q ), q = 3, 4
29
D(12; 121 266 )
[LZ00]
M37q,12;q
D(12q; 121 266 q 37q ), q = 3
30
D(12; 121 288 )
[LZ00]
M49q,12;q
D(12q; 121 288 q 49q ), q = 3
31
D(12; 121 299 )
[LZ00]
M55q,12;q
D(12q; 121 299 q 55q ), q = 3
32
D(12; 121 311 )
[LHZ03]
M9q,12;q
D(12q; 121 311 q 9q ), q = 3, 4, 5, 7
33
D(12; 121 322 )
[GK06]
M17q,12;q
D(12q; 121 322 q 17q ), q = 3, 4, 5
34
D(12; 121 333 )
[GK06]
M25q,12;q
D(12q; 121 333 q 25q ), q = 3, 4
35
D(12; 121 344 )
[GK06]
M33q,12;q
D(12q; 121 344 q 33q ), q = 3
36
D(12; 121 355 )
[GK06]
M41q,12;q
D(12q; 121 355 q 41q ), q = 3
37
D(12; 121 411 )
[FGLQ03]
M10q,12;q
D(12q; 121 411 q 10q ), q = 3, 4, 5, 7
123
Construction of equidistant and weak equidistant SSDs
47
Table 20 continued
No.
D(n 0 ; n 10 p m 0 −1 )
[Source† ]
Mδq,n 0 ;q
Resulting design
38
D(12; 121 422 )
[GK06]
M19q,12;q
D(12q; 121 422 q 19q ), q = 3, 4, 5
39
D(12; 121 433 )
[GK06]
M28q,12;q
D(12q; 121 433 q 28q ), q = 3, 4
40
D(12; 121 611 )
[LHZ03]
M11q,12;q
D(12q; 121 611 q 11q ), q = 3, 4
41
D(12; 121 622 )
[GK06]
M21q,12;q
D(12q; 121 622 q 21q ), q = 3, 4
42
D(6; 61 21 33 )
[FLL03]
M4q,6;q
D(6q; 61 21 33 q 4q ), q = 3, . . . , 7
43
D(8; 81 21 44 )
[FLL03]
M5q,8;q
D(8q; 81 21 44 q 5q ), q = 3, 4, 5, 7
44
D(8; 81 28 44 )
[KM05]
M9q,8;q
D(8q; 81 28 44 q 9q ), q = 3, 4, 5, 7
† Each design D(n ; n 1 p m 0 −1 ) is obtained by adding an n -level column to the
0 0
0
source design D(n 0 ; p m 0 −1 )
Table 21 Weak equidistant SSDs constructed from Corollary 1
No.
D(n 0 ; n 10 p m 0 −1 )
[Source† ]
‡
Mδq,n
0 ;q
Resulting design‡
1
D(6; 61 34 )
[FGL04]
M(4.5±0.5)q,6;q
D(6q; 61 34 q (4.5±0.5)q ), q = 3, 4, 5, 7 or q = 3, . . . , 7
2
D(6; 61 39 )
[GK06]
M(8.5±0.5)q,6;q
D(6q; 61 39 q (8.5±0.5)q ), q = 3, 4, 5, 7 or q = 3, . . . , 7
3
D(6; 61 314 )
[GK06]
M(12.5±0.5)q,6;q
D(6q; 61 314 q (12.5±0.5)q ), q = 3, 4, 5 or q = 3, 4, 5, 6
4
D(8; 81 26 )
OA
M(4.5±0.5)q,8;q
D(8q; 81 26 q (4.5±0.5)q ), q = 3, 4, 5, 7
5
D(8; 81 227 )
[LZ00]
M(16.5±0.5)q,8;q
D(8q; 81 227 q (16.5±0.5)q ), q = 3, 4, 5
6
D(8; 81 46 )
[FGL02a]
M(6.5±0.5)q,8;q
D(8q; 81 46 q (6.5±0.5)q ), q = 3, 4, 5, 7
7
D(8; 81 413 )
[FGL02a]
M(12.5±0.5)q,8;q
D(8q; 81 413 q (12.5±0.5)q ), q = 3, 4, 5 or q = 3, 4, 5, 6
8
D(8; 81 420 )
[GK06]
M(18.5±0.5)q,8;q
D(8q; 420 q (18.5±0.5)q ), q = 3, 4, 5
9
D(8; 81 427 )
[GK06]
M(24.5±0.5)q,8;q
D(8q; 427 q (24.5±0.5)q ), q = 3, 4
10
D(8; 81 434 )
[GK06]
M(30.5±0.5)q,8;q
D(8q; 434 q (30.5±0.5)q ), q = 3, 4
11
D(8; 81 441 )
[GK06]
M(36.5±0.5)q,8;q
D(8q; 441 q (36.5±0.5)q ), q = 3
12
D(9; 91 33 )
OA
M(3.5±0.5)q,9;q
D(9q; 91 33 q (3.5±0.5)q ), q = 3, 4, 5, 7 or q = 3, 4, 7
13
D(9; 91 37 )
[FGL04]
M(6.5±0.5)q,9;q
D(9q; 91 37 q (6.5±0.5)q ), q = 3, 4, 5, 7
14
D(9; 91 311 )
[FGL04]
M(9.5±0.5)q,9;q
D(9q; 91 311 q (9.5±0.5)q ), q = 3, 4, 5, 7
15
D(9; 91 315 )
[FGL04]
M(12.5±0.5)q,9;q
D(9q; 91 315 q (12.5±0.5)q ), q = 3, 4, 5 or q = 3, 4, 5, 6
16
D(9; 91 319 )
[FGL04]
M(15.5±0.5)q,9;q
D(9q; 91 319 q (15.5±0.5)q ), q = 3, 4, 5
17
D(9; 91 323 )
[FGL04]
M(18.5±0.5)q,9;q
D(9q; 91 323 q (18.5±0.5)q ), q = 3, 4, 5
18
D(9; 91 327 )
[FGL04]
M(21.5±0.5)q,9;q
D(9q; 91 327 q (21.5±0.5)q ), q = 3, 4
19
D(9; 91 331 )
[GKM06]
M(24.5±0.5)q,9;q
D(9q; 91 331 q (24.5±0.5)q ), q = 3, 4
20
D(9; 91 335 )
[GKM06]
M(27.5±0.5)q,9;q
D(9q; 91 335 q (27.5±0.5)q ), q = 3, 4
21
D(9; 91 339 )
[GK06]
M(30.5±0.5)q,9;q
D(9q; 91 339 q (30.5±0.5)q ), q = 3, 4
22
D(9; 91 347 )
[GK06]
M(36.5±0.5)q,9;q
D(9q; 91 347 q (36.5±0.5)q ), q = 3
23
D(10; 101 58 )
[FGL02b]
M(8.5±0.5)q,10;q
D(10q; 101 58 q (8.5±0.5)q ), q = 3, 4, 5, 7 or q = 3, . . . , 7
24
D(10; 101 517 )
[GK06]
M(16.5±0.5)q,10;q
D(10q; 101 517 q (16.5±0.5)q ), q = 3, 4, 5
25
D(10; 101 526 )
[GK06]
M(24.5±0.5)q,10;q
D(10q; 101 526 q (24.5±0.5)q ), q = 3, 4
26
D(12; 121 210 )
OA
M(6.5±0.5)q,12;q
D(12q; 121 210 q (6.5±0.5)q ), q = 3, 4, 7
27
D(12; 121 232 )
[LZ00]
M(18.5±0.5)q,12;q
D(12q; 121 232 q (18.5±0.5)q ), q = 3, 4, 5
123
48
Y. Liu, M.-Q. Liu
Table 21 continued
No.
D(n 0 ; n 10 p m 0 −1 ) [Source† ]
‡
Mδq,n
0 ;q
Resulting design‡
28
D(12; 121 243 )
[LZ00]
M(24.5±0.5)q,12;q
D(12q; 121 243 q (24.5±0.5)q ), q = 3, 4
29
D(12; 121 265 )
[LZ00]
M(36.5±0.5)q,12;q
D(12q; 121 265 q (36.5±0.5)q ), q = 3
30
D(12; 121 287 )
[LZ00]
M(48.5±0.5)q,12;q
D(12q; 121 287 q (48.5±0.5)q ), q = 3
31
D(12; 121 298 )
[LZ00]
M(54.5±0.5)q,12;q
D(12q; 121 298 q (54.5±0.5)q ), q = 3
32
33
D(12; 121 310 )
D(12; 121 321 )
[LHZ03]
[GK06]
M(8.5±0.5)q,12;q
M(16.5±0.5)q,12;q
D(12q; 121 310 q (8.5±0.5)q ), q = 3, 4, 5, 7
D(12q; 121 321 q (16.5±0.5)q ), q = 3, 4, 5
34
D(12; 121 332 )
[GK06]
M(24.5±0.5)q,12;q
D(12q; 121 332 q (24.5±0.5)q ), q = 3, 4
35
D(12; 121 343 )
[GK06]
M(32.5±0.5)q,12;q
D(12q; 121 343 q (32.5±0.5)q ), q = 3 or q = 3, 4
36
D(12; 121 354 )
[GK06]
M(40.5±0.5)q,12;q
D(12q; 121 354 q (40.5±0.5)q ), q = 3
37
D(12; 121 410 )
[FGLQ03] M(9.5±0.5)q,12;q
D(12q; 121 410 q (9.5±0.5)q ), q = 3, 4, 5, 7
38
D(12; 121 421 )
[GK06]
M(18.5±0.5)q,12;q
D(12q; 121 421 q (18.5±0.5)q ), q = 3, 4, 5
39
D(12; 121 432 )
[GK06]
M(27.5±0.5)q,12;q
D(12q; 121 432 q (27.5±0.5)q ), q = 3, 4
40
D(12; 121 610 )
[LHZ03]
M(10.5±0.5)q,12;q
D(12q; 121 610 q (10.5±0.5)q ), q = 3, 4 or q = 3, 4, 5, 7
41
D(12; 121 621 )
[GK06]
M(20.5±0.5)q,12;q
D(12q; 121 621 q (20.5±0.5)q ), q = 3, 4 or q = 3, 4, 5
42
D(6; 61 21 32 )
[FLL03]
M(3.5±0.5)q,6;q
D(6q; 61 21 32 q (3.5±0.5)q ), q = 3, . . . , 7 or q = 3, 4, 5, 7
43
D(8; 81 21 43 )
[FLL03]
M(4.5±0.5)q,8;q
D(8q; 81 21 43 q (4.5±0.5)q ), q = 3, 4, 5, 7
44
D(8; 81 28 43 )
[KM05]
M(8.5±0.5)q,8;q
D(8q; 81 28 43 q (8.5±0.5)q ), q = 3, 4, 5, 7 or q = 3, . . . , 7
† Each design D(n ; n 1 p m 0 −1 ) is obtained by deleting a balanced p-level column from and adding
0 0
an n 0 -level column to the source design D(n 0 ; p m 0 )
‡ δ = 4.5 + 0.5 = 5 for q = 3, 4, 5, 7, and δ = 4.5 − 0.5 = 4 for q = 3, . . . , 7, etc
Table 22 More weak equidistant SSDs constructed from Corollary 1
No.
D(n 0 ; n 10 p m 0 −1 )
[Source† ]
‡
Mδq,n
0 ;q
Resulting design‡
1
D(6; 61 36 )
[FGL04]
M(5.5±0.5)q,6;q
D(6q; 61 36 q (5.5±0.5)q ), q = 3, . . . , 7 or q = 3, 4, 5, 7
2
D(6; 61 310 21 )
[GK06]
M(9.5±0.5)q,6;q
D(6q; 61 310 21 q (9.5±0.5)q ), q = 3, . . . , 7 or q = 3, 4, 5, 7
3
D(6; 61 315 21 )
[GK06]
M(13.5±0.5)q,6;q
D(6q; 61 315 21 q (13.5±0.5)q ), q = 3, 4, 5, 6 or q = 3, 4, 5
4
D(8; 81 28 )
OA
M(5.5±0.5)q,8;q
D(8q; 81 27 q (5.5±0.5)q ), q = 3, 4, 5, 7
5
D(8; 81 229 )
[LZ00]
M(17.5±0.5)q,8;q
D(8q; 81 229 q (17.5±0.5)q ), q = 3, 4, 5
6
D(8; 81 48 )
[FGL02a]
M(7.5±0.5)q,8;q
D(8q; 81 48 q (7.5±0.5)q ), q = 3, . . . , 7 or q = 3, 4, 5, 7
7
D(8; 81 415 )
[FGL02a]
M(13.5±0.5)q,8;q
D(8q; 81 415 q (13.5±0.5)q ), q = 3, 4, 5, 6 or q = 3, 4, 5
8
D(8; 81 422 )
[GK06]
M(19.5±0.5)q,8;q
D(8q; 81 422 q (19.5±0.5)q ), q = 3, 4, 5
9
D(8; 81 429 )
[GK06]
M(25.5±0.5)q,8;q
D(8q; 81 429 q (25.5±0.5)q ), q = 3, 4
10
D(8; 81 436 )
[GK06]
M(31.5±0.5)q,8;q
D(8q; 81 436 q (31.5±0.5)q ), q = 3, 4
11
D(8; 81 442 )
[GK06]
M(37.5±0.5)q,8;q
D(8q; 81 443 q (37.5±0.5)q ), q = 3
12
D(9; 91 35 )
OA
M(4.5±0.5)q,9;q
D(9q; 91 35 q (4.5±0.5)q ), q = 3, 4, 5, 7
13
D(9; 91 39 )
[FGL04]
M(7.5±0.5)q,9;q
D(9q; 91 39 q (7.5±0.5)q ), q = 3, . . . , 7 or q = 3, 4, 5, 7
14
D(9; 91 313 )
[FGL04]
M(10.5±0.5)q,9;q
D(9q; 91 313 q (10.5±0.5)q ), q = 3, 4, 5 or q = 3, 4, 5, 7
15
D(9; 91 317 )
[FGL04]
M(13.5±0.5)q,9;q
D(9q; 91 317 q (13.5±0.5)q ), q = 3, 4, 5
16
D(9; 91 321 )
[FGL04]
M(16.5±0.5)q,9;q
D(9q; 91 321 q (16.5±0.5)q ), q = 3, 4, 5
17
D(9; 91 325 )
[FGL04]
M(19.5±0.5)q,9;q
D(9q; 91 325 q (19.5±0.5)q ), q = 3, 4, 5
123
Construction of equidistant and weak equidistant SSDs
49
Table 22 continued
No.
D(n 0 ; n 10 p m 0 −1 ) [Source† ]
‡
Mδq,n
0 ;q
Resulting design‡
18
D(9; 91 329 )
[FGL04]
M(22.5±0.5)q,9;q
D(9q; 91 329 q (22.5±0.5)q ), q = 3, 4
19
D(9; 91 333 )
[GKM06]
M(25.5±0.5)q,9;q
D(9q; 91 333 q (25.5±0.5)q ), q = 3, 4
20
D(9; 91 337 )
[GKM06]
M(28.5±0.5)q,9;q
D(9q; 91 337 q (28.5±0.5)q ), q = 3, 4
21
D(9; 91 341 )
[GK06]
M(31.5±0.5)q,9;q
D(9q; 91 341 q (31.5±0.5)q ), q = 3, 4
22
D(9; 91 349 )
[GK06]
M(37.5±0.5)q,9;q
D(9q; 91 349 q (37.5±0.5)q ), q = 3
23
D(10; 101 510 )
[FGL02b]
M(9.5±0.5)q,10;q
D(10q; 101 510 q (9.5±0.5)q ), q = 3, 4, 5, 7
24
D(10; 101 519 )
[GK06]
M(17.5±0.5)q,10;q
D(10q; 101 519 q (17.5±0.5)q ), q = 3, 4, 5
25
D(10; 101 528 )
[GK06]
M(25.5±0.5)q,10;q
D(10q; 101 528 q (25.5±0.5)q ), q = 3, 4
26
D(12; 121 212 )
OA
M(7.5±0.5)q,12;q
D(12q; 121 212 q (7.5±0.5)q ), q = 3, 4, 5, 7 or q = 3, 4, 7
27
D(12; 121 234 )
[LZ00]
M(19.5±0.5)q,12;q
D(12q; 121 234 q (19.5±0.5)q ), q = 3, 4, 5
28
D(12; 121 245 )
[LZ00]
M(25.5±0.5)q,12;q
D(12q; 121 245 q (25.5±0.5)q ), q = 3, 4
29
D(12; 121 267 )
[LZ00]
M(37.5±0.5)q,12;q
D(12q; 121 267 q (37.5±0.5)q ), q = 3
30
D(12; 121 289 )
[LZ00]
M(49.5±0.5)q,12;q
D(12q; 121 289 q (49.5±0.5)q ), q = 3
31
D(12; 121 2100 )
[LZ00]
M(55.5±0.5)q,12;q
D(12q; 121 2100 q (55.5±0.5)q ), q = 3
32
D(12; 121 312 )
[LHZ03]
M(9.5±0.5)q,12;q
D(12q; 121 312 q (9.5±0.5)q ), q = 3, 4, 5, 7
33
D(12; 121 323 )
[GK06]
M(17.5±0.5)q,12;q
D(12q; 121 323 q (17.5±0.5)q ), q = 3, 4, 5
34
D(12; 121 334 )
[GK06]
M(25.5±0.5)q,12;q
D(12q; 121 334 q (25.5±0.5)q ), q = 3, 4
35
D(12; 121 345 )
[GK06]
M(33.5±0.5)q,12;q
D(12q; 121 345 q (33.5±0.5)q ), q = 3
36
37
D(12; 121 356 )
D(12; 121 412 )
[GK06]
[FGLQ03]
M(41.5±0.5)q,12;q
M(10.5±0.5)q,12;q
D(12q; 121 356 q (41.5±0.5)q ), q = 3
D(12q; 121 412 q (10.5±0.5)q ), q = 3, 4 or q = 3, 4, 5, 7
38
D(12; 121 423 )
[GK06]
M(19.5±0.5)q,12;q
D(12q; 121 423 q (19.5±0.5)q ), q = 3, 4, 5
39
D(12; 121 434 )
[GK06]
M(28.5±0.5)q,12;q
D(12q; 121 434 q (28.5±0.5)q ), q = 3, 4
40
D(12; 121 612 )
[LHZ03]
M(11.5±0.5)q,12;q
D(12q; 121 612 q (11.5±0.5)q ), q = 3, 4, 5, 6 or q = 3, 4
41
D(12; 121 623 )
[GK06]
M(21.5±0.5)q,12;q
D(12q; 121 623 q (21.5±0.5)q ), q = 3, 4
42
D(6; 61 21 34 )
[FLL03]
M(4.5±0.5)q,6;q
D(6q; 61 21 34 q (4.5±0.5)q ), q = 3, 4, 5, 7 or q = 3, . . . , 7
43
D(8; 81 21 45 )
[FLL03]
M(5.5±0.5)q,8;q
D(8q; 81 21 45 q (5.5±0.5)q ), q = 3, 4, 5, 7
44
D(8; 81 28 45 )
[KM05]
M(9.5±0.5)q,8;q
D(8q; 81 28 45 q (9.5±0.5)q ), q = 3, . . . , 7 or q = 3, 4, 5, 7
D(n 0 ; n 10 p m 0 −1 )
† Each design
is obtained by adding a balanced p-level column and an n 0 -level
column to the source design D(n 0 ; p m 0 −2 )
‡ δ = 5.5 + 0.5 = 6 for q = 3, . . . , 7, and δ = 5.5 − 0.5 = 5 for q = 3, 4, 5, 7, etc
Table 23 Weak equidistant SSDs constructed from Corollary 2
No.
D(n 0 ; p m 0 )
[Source]
‡
Mδq,n
0 ;q
Resulting design‡
1
D(6; 35 )
[FGL04]
M(4±1)q,6;q
D(6q; 35 q (4±1)q ), q = 3, 4, 5, 7
2
D(6; 310 )
[GK06]
M(8±1)q,6;q
3
D(6; 315 )
[GK06]
M(12±1)q,6;q
4
OA
M(4±1)q,8;q
5
L 8 (27 )
D(8; 228 )
[LZ00]
M(16±1)q,8;q
6
D(8; 47 )
[FGL02a]
M(6±1)q,8;q
7
D(8; 414 )
[FGL02a]
M(12±1)q,8;q
D(6q; 310 q (8±1)q ), q = 3, 4, 5, 7
D(6q; 315 q (12±1)q ), q = 3, 4, 5
D(8q; 27 q (4±1)q ), q = 3, 4, 5, 7
D(8q; 228 q (16±1)q ), q = 3, 4, 5
D(8q; 47 q (6±1)q ), q = 3, 4, 5, 7
D(8q; 414 q (12±1)q ), q = 3, 4, 5
123
50
Y. Liu, M.-Q. Liu
Table 23 continued
No.
D(n 0 ; p m 0 )
[Source]
‡
Mδq,n
0 ;q
Resulting design‡
8
D(8; 421 )
[GK06]
M(18±1)q,8;q
D(8q; 421 q (18±1)q ), q = 3, 4, 5
9
D(8; 428 )
[GK06]
M(24±1)q,8;q
10
D(8; 435 )
[GK06]
M(30±1)q,8;q
11
D(8; 442 )
[GK06]
M(36±1)q,8;q
12
OA
M(3±1)q,9;q
13
L 9 (34 )
D(9; 38 )
[FGL04]
M(6±1)q,9;q
14
D(9; 312 )
[FGL04]
M(9±1)q,9;q
15
D(9; 316 )
[FGL04]
M(12±1)q,9;q
16
D(9; 320 )
[FGL04]
M(15±1)q,9;q
17
D(9; 324 )
[FGL04]
M(18±1)q,9;q
18
D(9; 328 )
[FGL04]
M(21±1)q,9;q
19
D(9; 332 )
[GKM06]
M(24±1)q,9;q
20
D(9; 336 )
[GKM06]
M(27±1)q,9;q
21
D(9; 340 )
[GK06]
M(30±1)q,9;q
22
D(9; 348 )
[GK06]
M(36±1)q,9;q
23
D(10; 59 )
[FGL02b]
M(8±1)q,10;q
24
D(10; 518 )
[GK06]
M(16±1)q,10;q
25
D(10; 527 )
[GK06]
M(24±1)q,10;q
OA
M(6±1)q,12;q
(211 )
26
L 12
27
D(12; 233 )
[LZ00]
M(18±1)q,12;q
28
D(12; 244 )
[LZ00]
M(24±1)q,12;q
29
D(12; 266 )
[LZ00]
M(36±1)q,12;q
30
D(12; 288 )
[LZ00]
M(48±1)q,12;q
31
D(12; 299 )
[LZ00]
M(54±1)q,12;q
32
D(12; 311 )
[LHZ03]
M(8±1)q,12;q
33
D(12; 322 )
[GK06]
M(16±1)q,12;q
34
D(12; 333 )
[GK06]
M(24±1)q,12;q
35
D(12; 344 )
[GK06]
M(32±1)q,12;q
36
D(12; 355 )
[GK06]
M(40±1)q,12;q
37
D(12; 411 )
[FGLQ03]
M(9±1)q,12;q
38
D(12; 422 )
[GK06]
M(18±1)q,12;q
39
D(12; 433 )
[GK06]
M(27±1)q,12;q
40
D(12; 611 )
[LHZ03]
M(10±1)q,12;q
41
D(12; 622 )
[GK06]
M(20±1)q,12;q
42
D(6; 21 33 )
[FLL03]
M(3±1)q,6;q
123
D(8q; 428 q (24±1)q ), q = 3, 4
D(8q; 435 q (30±1)q ), q = 3, 4
D(8q; 442 q (36±1)q ), q = 3
D(9q; 34 q (3±1)q ), q = 3, 4, 5, 7 or q = 5, 7
D(9q; 38 q (6±1)q ), q = 3, 4, 5, 7
D(9q; 312 q (9±1)q ), q = 3, 4, 5, 7 or q = 3, . . . , 7
D(9q; 316 q (12±1)q ), q = 3, 4, 5
D(9q; 320 q (15±1)q ), q = 3, 4, 5
D(9q; 324 q (18±1)q ), q = 3, 4, 5
D(9q; 328 q (21±1)q ), q = 3, 4 or q = 3, 4, 5
D(9q; 332 q (24±1)q ), q = 3, 4
D(9q; 336 q (27±1)q ), q = 3, 4
D(9q; 340 q (30±1)q ), q = 3, 4
D(9q; 348 q (36±1)q ), q = 3
D(10q; 59 q (8±1)q ), q = 3, 4, 5, 7
D(10q; 518 q (16±1)q ), q = 3, 4, 5
D(10q; 527 q (24±1)q ), q = 3, 4
D(12q; 211 q (6±1)q ), q = 3, 4, 7 or q = 5
D(12q; 233 q (18±1)q ), q = 3, 4, 5
D(12q; 244 q (24±1)q ), q = 3, 4
D(12q; 266 q (36±1)q ), q = 3
D(12q; 288 q (48±1)q ), q = 3
D(12q; 299 q (54±1)q ), q = 3
D(12q; 311 q (8±1)q ), q = 3, 4, 5, 7 or q = 3, 4, 7
D(12q; 322 q (16±1)q ), q = 3, 4, 5
D(12q; 333 q (24±1)q ), q = 3, 4
D(12q; 344 q (32±1)q ), q = 3 or q = 3, 4
D(12q; 355 q (40±1)q ), q = 3
D(12q; 411 q (9±1)q ), q = 3, 4, 5, 7
D(12q; 422 q (18±1)q ), q = 3, 4, 5
D(12q; 433 q (27±1)q ), q = 3, 4
D(12q; 611 q (10±1)q ), q = 3, 4 or q = 3, 4, 5, 7
D(12q; 622 q (20±1)q ), q = 3, 4 or q = 3, 4, 5
D(6q; 21 33 q (3±1)q ), q = 3, . . . , 7
Construction of equidistant and weak equidistant SSDs
51
Table 23 continued
No.
D(n 0 ; p m 0 )
[Source]
‡
Mδq,n
0 ;q
Resulting design‡
43
D(8; 21 44 )
[FLL03]
M(4±1)q,8;q
D(8q; 21 44 q (4±1)q ), q = 3, 4, 5, 7
44
D(8; 28 44 )
[KM05]
M(8±1)q,8;q
D(8q; 28 44 q (8±1)q ), q = 3, 4, 5, 7
‡ δ = 4 + 1 = 5 and δ = 4 − 1 = 3 for q = 3, 4, 5, 7, etc
Table 24 More weak equidistant SSDs constructed from Corollary 2
No.
D(n 0 ; n 10 p m 0 −1 )
[Source† ]
‡
Mδq,n
0 ;q
Resulting design‡
1
D(6; 61 35 )
[FGL04]
M(5±1)q,6;q
D(6q; 61 35 q (5±1)q ), q = 3, . . . , 7
2
D(6; 61 310 )
[GK06]
M(9±1)q,6;q
D(6q; 61 310 q (9±1)q ), q = 3, . . . , 7
3
D(6; 61 315 )
[GK06]
M(13±1)q,6;q
D(6q; 61 315 q (13±1)q ), q = 3, 4, 5, 6
4
D(8; 81 27 )
OA
M(5±1)q,8;q
D(8q; 81 27 q (5±1)q ), q = 3, 4, 5, 7
5
D(8; 81 228 )
[LZ00]
M(17±1)q,8;q
D(8q; 81 228 q (17±1)q ), q = 3, 4, 5
6
D(8; 81 47 )
[FGL02a]
M(7±1)q,8;q
D(8q; 81 47 q (7±1)q ), q = 3, . . . , 7 or q = 3, 4, 5, 7
7
D(8; 81 414 )
[FGL02a]
M(13±1)q,8;q
D(8q; 81 414 q (13±1)q ), q = 3, 4, 5, 6
8
D(8; 81 421 )
[GK06]
M(19±1)q,8;q
D(8q; 81 421 q (19±1)q ), q = 3, 4, 5
9
D(8; 81 428 )
[GK06]
M(25±1)q,8;q
D(8q; 81 428 q (25±1)q ), q = 3, 4
10
D(8; 81 435 )
[GK06]
M(31±1)q,8;q
D(8q; 81 435 q (31±1)q ), q = 3, 4
11
D(8; 81 442 )
[GK06]
M(37±1)q,8;q
D(8q; 81 442 q (36±1)q ), q = 3
12
D(9; 91 34 )
OA
M(4±1)q,9;q
D(9q; 91 34 q (4±1)q ), q = 3, 4, 5, 7 or q = 3, 4, 7
13
D(9; 91 38 )
[FGL04]
M(7±1)q,9;q
D(9q; 91 38 q (7±1)q ), q = 3, . . . , 7 or q = 3, 4, 5, 7
14
D(9; 91 312 )
[FGL04]
M(10±1)q,9;q
D(9q; 91 312 q (10±1)q ), q = 3, 4, 5 or q = 3, 4.5, 7
15
D(9; 91 316 )
[FGL04]
M(13±1)q,9;q
D(9q; 91 316 q (13±1)q ), q = 3, 4, 5 or q = 3, 4, 5, 6
16
D(9; 91 320 )
[FGL04]
M(16±1)q,9;q
D(9q; 91 320 q (16±1)q ), q = 3, 4, 5
17
D(9; 91 324 )
[FGL04]
M(19±1)q,9;q
D(9q; 91 324 q (19±1)q ), q = 3, 4, 5
18
D(9; 91 328 )
[FGL04]
M(22±1)q,9;q
D(9q; 91 328 q (22±1)q ), q = 3, 4
19
D(9; 91 332 )
[GKM06]
M(25±1)q,9;q
D(9q; 91 332 q (25±1)q ), q = 3, 4
20
D(9; 91 336 )
[GKM06]
M(28±1)q,9;q
D(9q; 91 336 q (28±1)q ), q = 3, 4
21
D(9; 91 340 )
[GK06]
M(31±1)q,9;q
D(9q; 91 340 q (31±1)q ), q = 3, 4
22
D(9; 91 348 )
[GK06]
M(37±1)q,9;q
D(9q; 91 348 q (37±1)q ), q = 3
23
D(10; 101 59 )
[FGL02b]
M(9±1)q,10;q
D(10q; 101 59 q (9±1)q ), q = 3, 4, 5, 7 or q = 3, . . . , 7
24
D(10; 101 518 )
[GK06]
M(17±1)q,10;q
D(10q; 101 518 q (17±1)q ), q = 3, 4, 5
25
D(10; 101 527 )
[GK06]
M(25±1)q,10;q
D(10q; 101 527 q (25±1)q ), q = 3, 4
26
D(12; 121 211 )
OA
M(7±1)q,12;q
D(12q; 121 211 q (7±1)q ), q = 3, 4, 5, 7 or q = 3, 4, 7
27
D(12; 121 233 )
[LZ00]
M(19±1)q,12;q
D(12q; 121 233 q (19±1)q ), q = 3, 4, 5
28
D(12; 121 244 )
[LZ00]
M(25±1)q,12;q
D(12q; 121 244 q (25±1)q ), q = 3, 4
29
D(12; 121 266 )
[LZ00]
M(37±1)q,12;q
D(12q; 121 266 q (37±1)q ), q = 3
30
D(12; 121 288 )
[LZ00]
M(49±1)q,12;q
D(12q; 121 288 q (49±1)q ), q = 3
31
D(12; 121 299 )
[LZ00]
M(55±1)q,12;q
D(12q; 121 299 q (55±1)q ), q = 3
32
D(12; 121 311 )
[LHZ03]
M(9±1)q,12;q
D(12q; 121 311k q (9±1)q ), q = 3, 4, 5, 7
33
D(12; 121 322 )
[GK06]
M(17±1)q,12;q
D(12q; 121 322 q (17±1)q ), q = 3, 4, 5
123
52
Y. Liu, M.-Q. Liu
Table 24 continued
No.
D(n 0 ; n 10 p m 0 −1 )
[Source† ]
‡
Mδq,n
0 ;q
Resulting design‡
34
35
D(12; 121 333 )
D(12; 121 344 )
[GK06]
[GK06]
M(25±1)q,12;q
M(33±1)q,12;q
D(12q; 121 333 q (25±1)q ), q = 3, 4
D(12q; 121 344 q (33±1)q ), q = 3 or q = 3, 4
36
D(12; 121 355 )
[GK06]
M(41±1)q,12;q
D(12q; 121 355 q (41±1)q ), q = 3
37
D(12; 121 411 )
[FGLQ03]
M(10±1)q,12;q
D(12q; 121 411 q (10±1)q ), q = 3, 4 or q = 3, 4, 5, 7
38
D(12; 121 422 )
[GK06]
M(19±1)q,12;q
D(12q; 121 422 q (19±1)q ), q = 3, 4, 5
39
D(12; 121 433 )
[GK06]
M(28±1)q,12;q
D(12q; 121 433 q (28±1)q ), q = 3, 4
40
D(12; 121 611 )
[LHZ03]
M(11±1)q,12;q
D(12q; 121 611 q (11±1)q ), q = 3, 4, 5, 6 or q = 3, 4, 5, 7
41
D(12; 121 622 )
[GK06]
M(21±1)q,12;q
D(12q; 121 622 q (21±1)q ), q = 3, 4 or q = 3, 4, 5
42
D(6; 61 21 33 )
[FLL03]
M(4±1)q,6;q
D(6q; 61 21 33 q (4±1)q ), q = 3, 4, 5, 7
43
D(8; 81 21 44 )
[FLL03]
M(5±1)q,8;q
D(8q; 81 21 44 q (5±1)q ), q = 3, 4, 5, 7
44
D(8; 81 28 44 )
[KM05]
M(9±1)q,8;q
D(8q; 81 28 44 q (9±1)q ), q = 3, . . . , 7
†
‡
D(n 0 ; n 10 p m 0 −1 )
Each design
is obtained by adding an n 0 -level column to the source design D(n 0 ; p m 0 −1 )
δ = 5 + 1 = 6 and δ = 5 − 1 = 4 for q = 3, . . . , 7, etc
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