Orthogonal arrays obtained by repeating-column ... - Semantic Scholar
Discrete Mathematics 307 (2007) 246 – 261 www.elsevier.com/locate/disc
Orthogonal arrays obtained by repeating-column difference matrices Yingshan Zhang Department of Statistics, East China Normal University, Shanghai, 200062, People’s Republic of China Received 2 June 2003; received in revised form 11 April 2006; accepted 9 June 2006 Available online 26 September 2006
Abstract In this paper, by using the repeating-column difference matrices and orthogonal decompositions of projection matrices, we propose a new general approach to construct asymmetrical orthogonal arrays. As an application of the method, some new orthogonal arrays with run sizes 72 and 96 are constructed. © 2006 Elsevier B.V. All rights reserved. MSC: primary 62K15; secondary 05B1 Keywords: Asymmetrical orthogonal arrays; Generalized hadamard products; Generalized Kronecker products; Repeating-column difference matrices; Projection matrices; Permutation matrices
1. Introduction An n × m matrix A, having ki columns with pi ( 2) levels, i = 1, . . . , t, m = ti=1 ki , pi = pj , for i = j , is called an orthogonal array (OA) of strength d and size n if each n × d submatrix of A contains all possible 1 × d row vectors with the same frequency. Unless stated otherwise, we consider an orthogonal array of strength 2, using the notation Ln (p1k1 , . . . , ptkt ) for such an array. An orthogonal array is said to be mixed-level (or asymmetrical) if t 2. Difference matrices are essential for the construction of many asymmetrical orthogonal arrays [2]. Using the notation for additive groups, a difference matrix having level p is an p × m matrix with the entries from a finite Abelian group G of cardinality p such that the vector differences of any two columns of the array, say di − dj if i = j, contains every element of G exactly times. We will denote such an array by D(p, m; p), although this notation suppresses the relevance of the group G. In most of our examples, G will correspond to the additive group associated with a Galois field GF(p). The difference matrix D(p, m; p) is called a generalized Hadamard matrix if p = m. In particular, D(2, 2; 2) is the usual Hadamard matrix. If a D(p, m; p) exists, it can always be constructed so that one of its rows and one of its columns contain only the zero element of G. Deleting this column from D(p, m; p), we obtain a difference matrix, denoted by D 0 (p, m − 1; p) called an atom of difference matrix D(p, m; p) (or an atomic difference matrix). Without loss of generality,
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the D(p, m; p) can be written as 0 0 D(p, m; p) = = (0 D 0 (p, m − 1; p)). 0 A The property is important for the following discussions. For two matrices A = (aij )n×m and B = (bij )s×t both with the entries from group G, define their Kronecker sum [4] to be A B = (aij B)1 i n,1 j m , where each submatrix aij B of A B is obtained by adding aij to each entry of B. Shrikhande [4] showed that A B is a difference matrix if both A and B are difference matrices. In the contrast, Zhang [7] showed that A is a difference matrix if both A B and B are difference matrices. It is known that the Kronecker sum L = Lp (p s ) D(p, m; p) (or L = D(p, m; p) Lp (p s )) is an orthogonal array if Lp (p s ) is an orthogonal array and D(p, r; p) is a difference matrix [1]. By setting = s = 1, the Kronecker sum method reduces to the well-known construction of Bose and Bush [2], i.e., L = (p) D(p, m; p) (or L = D(p, m; p) (p)) is an orthogonal array if D(p, m; p) is a difference matrix. In the contrast, Zhang [7] has found that the difference matrix D(p, m; p) can be also constructed by using the orthogonal array L = (p) D(p, m; p), i.e., D(p, m; p) is a difference matrix if L = (p) D(p, m; p) is an orthogonal array. Let D(r, m; p) = (dij ), we have L = D(r, m; p) (p) = [S1 (0r (p)), . . . , Sm (0r (p))],
(1)
where Sj = diag((d1j ), . . . , (drj )) and (dij ) is a permutation matrix such that (dij )(p) = dij (p),
(2)
for any i, j , the dij (p) can be obtained by adding dij to each entry of (p). The idea of Kronecker sum and difference matrices can be generalized as follows [7]. Let n=pq. If A is an orthogonal array Lp (p1k1 , . . . , ptkt ) with the partition A = [Lp (p1k1 ), . . . , Lp (ptkt )], and if there exist the atoms: D 0 (1 p1 , m1 − 1; p1 ), . . . , D 0 (t pt , mt − 1; pt ), of difference matrices D(1 p1 , m1 ; p1 ), . . . , D(t pt , mt ; pt ), respectively, where q = i pi and p and q are both multiples the pi ’s, then the following array: [Lp 0q , 0p Lq , Lp (p1k1 ) D 0 (1 p1 , m1 − 1; p1 ), . . . , Lp (ptkt ) D 0 (t pt , mt − 1; pt )] is also an orthogonal array for any orthogonal arrays Lp and Lq . In this paper, we will prove that the following array: [Lp 0q , 0p Lq , (p) D 0 (0 p, m0 − 1; p), Lp (p11 ) D 0 (1 p1 , m1 − 1; p1 ), . . . , Lp (pt1 ) D 0 (t pt , mt − 1; pt )], is also an orthogonal array for any orthogonal arrays Lp and Lq if A = [Lp (p11 ), . . . , Lp (pt1 )] is a normal orthogonal array and D 0 = [D 0 (0 p, m0 − 1; p), D 0 (1 p1 , m1 − 1; p1 ), . . . , D 0 (t pt , mt − 1; pt )] is an atomic repeating-column difference matrix (Section 2).
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Section 2 contains the basic concepts and main theorems of repeating-column difference matrices while in Section 3 we describe the method of construction. Some new orthogonal arrays with run sizes 72 and 96 are constructed in Section 4. 2. Repeating-column difference matrices In order to define the repeating-column difference matrices, we must define a generalized Hadamard product [11]. Definition 2.1. Let h(x, y) be a mapping from 1 × 2 to V, where 1 × 2 = {(x, y) : x ∈ 1 , y ∈ 2 } and 1 , 2 , V are some sets. For two matrices A = (aij )n×m with entries from 1 and B = (bij )n×m with entries from 2 , h
define their generalized Hadamard product, denoted by ◦, as follows: h
A ◦ B = (h(aij , bij ))n×m = (h(aij , bij ))1 i n,1 j m , h
where each of the entries h(aij , bij ) of A ◦ B may be a scalar or vector or matrix under the mapping h(x, y). Unless stated otherwise, we consider the sets 1 and 2 to be finite, using the notations 1 = {0, 1, . . . , p − 1} and 2 = {0, 1, . . . , q − 1} for two example sets. When V is a row-vector space of m-dimensions, the mapping h(i, j ) can be represented by a pq × m matrix D, i.e., h
h : [(p) 0q ] ◦[0p (q)] = D = (d(1) , . . . , d(pq) )T , T with h(i, j ) = d(iq+j +1) (or h(i, j ) is the (iq + j + 1)th row of D). For this case in the following discussions, the h
h
h
generalized Hadamard product ◦ will only be defined by [(p) 0q ] ◦[0p (q)] = D. Note that [(p) 0q ] ◦[0p (q)] = h
h
h
(p) ⊗(q), the generalized Hadamard product ◦ will be also defined by (p) ⊗(q) = D. This is a form of generalized Kronecker product. h
Let 1 , 2 , V be multiplicative groups. When h(i, j )=ij , the generalized Hadamard ◦ is the usual Hadamard product in matrix theory, denoted by ◦. Let 1 , 2 , V be additive (Abelian) groups. When h(i, j ) = i + j, the generalized h
Hadamard ◦ is the usual addition of matrices in matrix theory, denoted by +. Let 1 = 2 = V = {0, 1, . . . , p − 1} and h
h(i, j ) = i + j, mod p, Then, the generalized Hadamard product ◦ is really the usual modulus addition of matrices in +
+
+
the theory of matrices, denoted by ◦, In particular, denote A ◦ A = A + A, mod p by A ◦ 2 , which is very useful for the construction of orthogonal arrays. Let h(i, j ) = (i, j ), where 1 = {0, 1, . . . , p − 1}, 2 = {0, 1, . . . , q − 1}, V = {(i, j ); i ∈ 1 , j ∈ 2 }. The h
generalized Hadamard product ◦ is also called a repeating operation, denoted by ♦, which can be used for the construction of repeating-column difference matrices. Note that we often write the elements (i, j ) of V in a form ij instead of (i, j ). Similarly let h(i, j ) = iq + j, where 1 = {0, 1, . . . , p − 1}, 2 = {0, 1, . . . , q − 1}, V = {0, 1, . . . , pq − 1}. The h
generalized Hadamard product ◦ is also called a joining operation, denoted by , which can be used for the construction of asymmetrical orthogonal arrays with large levels from those with small levels. Theorem 2.2. Let A, B, C, D be matrices and T a permutation matrix. Then, (A♦B) (C♦D) = (A C)♦(B D), and T (A♦B♦C♦D) = T A♦T B♦T C♦T D. Let m(A) be the matrix image of array A ([10,11] or Section 3), we have
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Theorem 2.3. Suppose that a and b are two orthogonal arrays which have only one column with run size n, i.e., a = Ln (p) = (a1 , . . . , an )T , b = Ln (q) = (b1 , . . . , bn )T . Then the matrix image of a♦b (or a b) is the following orthogonal decomposition: m(a♦b) = m(a) + m(b) + nm(a) ◦ m(b) = m(b♦a) = m(a b) = m(b a), if m(a)m(b) = 0, where a♦b = (a1 b1 , . . . , an bn )T (or a b = (a1 q + b1 , . . . , an q + bn )T ) is the repeating (or joining) operation of a and b in Definition 2.1, and m(a) ◦ m(b) is the usual Hadamard product in matrix theory. Corollary 2.4. Let K 1 =Ln (p1 , . . . , pm )=(Ln (p1 ), . . . , Ln (pm )) and K 2 =Ln (q1 , . . . , qm )=(Ln (q1 ), . . . , Ln (qm )) be two orthogonal arrays of run size n. Denote (Ln (p1 )♦Ln (q1 ), . . . , Ln (pm )♦Ln (qm )) =: K 1 ♦K 2 , then the matrix image of m(K 1 ♦K 2 ) satisfies m(K 1 ♦K 2 )m(K 1 ) + m(K 2 ) + nm(K 1 ) ◦ m(K 2 ), if m(K 1 )m(K 2 ) = 0. Corollary 2.5. Suppose that Ln1 = Ln1 (p1 , . . . , pm ) = (Ln1 (p1 ), . . . , Ln1 (pm )) and Ln2 = Ln2 (q1 , . . . , qm ) = (Ln2 (q1 ), . . . , Ln2 (qm )) are two orthogonal arrays. Then (Ln1 0n2 )♦(0n1 Ln2 ) is also an orthogonal array. In this case, its matrix image satisfies m((Ln1 0n2 )♦(0n1 Ln2 ))m(Ln1 ) ⊗ Pn2 + Pn1 ⊗ m(Ln2 ) + m(Ln1 ) ⊗ m(Ln2 ). Corollary 2.6. Suppose that p is a prime and a and b are OA’s which have only one column with run size n and p levels, i.e., a =Ln (p)=(a1 , . . . , an )T , b =Ln (p)=(b1 , . . . , bn )T . Then, Ln (p p−1 )=(a +b, . . . , [p −1]a +b), mod p, is also +
an OA whose MI is nm(a)◦m(b) if m(a)m(b)=0. In particular, Lp2 (p p−1 )=((p) (p), . . . , (p) ◦[p−1] (p)), mod p, is also an OA whose MI is p ⊗ p . These theorems and corollaries can be proved easily and are also found in Zhang [5–7]. In order to find a generalized result of repeating-column difference matrices, we must study the structure of associated orthogonal arrays. Definition 2.7. Let Lp = Lp (p1 , . . . , pm ) = (C1 , . . . , Cm ) be an orthogonal array where Cl is a vector with entries from a additive group Gl of order pl for any l. The array Lp is called normal over G0 if the set consisted of all entries of vector C0 = C1 ♦ · · · ♦Cm is a additive group G0 of order p, where G0 ⊂ G1 × · · · × Gm := {(x1 , . . . , xm ); xl ∈ Gl , l = 1, 2, . . . , m} with the usual addition: (x1 , x2 , . . . , xm ) + (y1 , y2 , . . . , ym ) = (x1 + y1 , x2 + y2 , . . . , xm + ym ) ∀(x1 , x2 , . . . , xm ), (y1 , y2 , . . . , ym ) ∈ G0 . Example 2.8. The following arrays are normal: a. ⎛ ⎞ ⎛ 0 0 0 0 ⎜0 1⎟ ⎜0 1 2 3 L4 (2 ) = ⎝ ⎠ , L4 (2 ) = ⎝ 1 0 1 0 1 1 1 1
⎞ 0 1⎟ ⎠, 1 0
over G40 = {(0, 0), (0, 1), (1, 0), (1, 1)} and G40 = {(0, 0, 0), (0, 1, 1), (1, 0, 1), (1, 1, 0)}, respectively.
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b. ⎛
0 ⎜0 ⎜ ⎜1 L6 (31 21 ) = ⎜ ⎜1 ⎝ 2 2
⎞
0 1⎟ ⎟ 0⎟ ⎟, 1⎟ ⎠ 0 1
⎛
0 ⎜0 ⎜ ⎜1 ⎜ ⎜1 7 L8 (2 ) = ⎜ ⎜0 ⎜ ⎜0 ⎝ 1 1
0 1 0 1 0 1 0 1
0 1 1 0 0 1 1 0
0 0 0 0 1 1 1 1
0 0 1 1 1 1 0 0
0 1 0 1 1 0 1 0
⎞ 0 1⎟ ⎟ 1⎟ ⎟ 0⎟ ⎟, 1⎟ ⎟ 0⎟ ⎠ 0 1
over G60 = {(0, 0), (0, 1), . . . , (2, 1)} and G80 = {(0, 0, 0, 0, 0, 0, 0), . . . , (1, 1, 0, 1, 0, 0, 1)}, respectively. c. ⎛ ⎛ ⎛ ⎞ ⎞ ⎞ 0 0 0 0 0 0 0 0 ⎜0 1⎟ ⎜0 1⎟ ⎜0 1 1 0⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜0 2⎟ ⎜0 2⎟ ⎜1 0 1 0⎟ ⎜ ⎜ ⎜ ⎟ ⎟ ⎟ ⎜0 3⎟ ⎜1 0⎟ ⎜1 1 0 0⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜0 4⎟ ⎜1 1⎟ ⎜0 0 0 1⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜0 5⎟ ⎜1 2⎟ ⎜0 1 1 1⎟ L12 (21 61 ) = ⎜ ⎟ , L12 (41 31 ) = ⎜ ⎟ , L12 (23 31 ) = ⎜ ⎟, ⎜1 0⎟ ⎜2 0⎟ ⎜1 0 1 1⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜1 1⎟ ⎜2 1⎟ ⎜1 1 0 1⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜1 2⎟ ⎜2 2⎟ ⎜0 0 0 2⎟ ⎜ ⎜ ⎜ ⎟ ⎟ ⎟ ⎜1 3⎟ ⎜3 0⎟ ⎜0 1 1 2⎟ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ 1 4 3 1 1 0 1 2 1 5 3 2 1 1 0 2 12 over G12 0 = {(0, 0), (0, 1), . . . , (1, 5)}, G0 = {(0, 0), (0, 1), . . . , (3, 2)} and
G12 0 = {(0, 0, 0, 0), (0, 1, 1, 0), . . . , (1, 1, 0, 2)}, respectively. Definition 2.9. Let Lp = Lp (p1 , . . . , pm ) = (C1 , . . . , Cm ) be a normal orthogonal array over G0 and denote
be a difference matrix over G having p levels.And suppose C0 =C1 ♦ · · · ♦Cm and let D0 =D(q, k0 ; p)=D1 ♦ · · · ♦Dm 0
that [Di , Di ] is an p × (k0 + ki ) difference matrix having pi levels for i = 1, 2, . . . , m. Thus, C = [C0 , C1 , . . . , Cm ] is a partitioned matrix in which the m+1 columns C0 , C1 , . . . , Cm are orthogonal arrays of strength 1 having p, p1 , . . . , pm levels, respectively. We call D = [D0 , D1 , . . . , Dm ] a repeating-column difference matrix about C = [C0 , C1 , . . . , Cm ]. The repeating-column difference matrix D is called an atomic repeating-column difference matrix if Dj is atomic for any j. Theorem 2.10. Let Lp = Lp (p1 , . . . , pm ) = (C1 , . . . , Cm ) be a normal orthogonal array and denote C0 = C1 ♦ · · · ♦Cm . Then, there exist p permutation matrices 0 (x), ∀x ∈ G0 and pl permutation matrices l (xl ), ∀xl ∈ Gl , l = 1, 2, . . . , m, such that 0 (x) · C0 = x C0 , 0 (x) · Cl = l (xl ) · Cl = xl Cl , ∀x = (x1 , x2 , . . . , xm ) ∈ G0 ,
l = 1, 2, . . . , m
where xl Cl stands for the vector obtained by xl to each entry of Cl . In other words, we have 0 (x)Lp = (1 (x1 )C1 , . . . , m (xm )Cm ). In this case, the matrix images of C0 , C1 , C2 , . . . , Cm satisfy the following equations: m(Cl ) = m(0 (x)Cl )
∀x = (x1 , x2 , . . . , xm ) ∈ G0 , l = 0, 1, 2, . . . , m.
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In particular, if let = p − m(C1 ) − · · · − m(Cm ), then, we have 0 (x) 0 (x)T = . Proof. Since the set of entries of vector C0 = C1 ♦ · · · ♦Cm is an additive group G0 , for any given x ∈ G0 , there exists a permutation matrix 0 (x) such that 0 (x) · C0 = x C0 , where x C0 stands for the vector obtained by x to each entry of C0 . Furthermore, since the order of group G0 is p, i.e., x C0 = C0 iff x = (0, 0, . . . , 0), we have 0 (x) = 0 (y) if x = y. Similarly, because Cl is a vector with entries from an group Gl and having a form Cl = Tl (0l (pl )) where l pl = p and Tl is a permutation matrix for any l, for any given xl ∈ Gl there exists a permutation matrix l (xl ) such that l (xl ) · Cl = xl Cl = Tl (0l [xl (pl )]), since there exists a permutation matrix l (xl ) such that l (xl )(pl ) = xl (pl ), where l pl = p and Tl is a permutation matrix for any l. From above results and Theorem 2.2, we have (0 (x)C1 )♦ · · · ♦(0 (x)Cm ) = 0 (x)C0 = x C0 = (x1 C1 )♦ · · · ♦(xm Cm ) = (1 (x1 )C1 )♦ · · · ♦(m (xm )Cm ), i.e., 0 (x)Cl = l (xl )Cl , l = 1, 2, . . . , m. Now, we prove that m(Cl ) = m(0 (x)Cl ), ∀x ∈ G0 , l = 0, 1, 2, . . . , m. In fact, that m(C0 )=m(0 (x)C0 ) is trivial since m(C0 )=p =0 (x)p 0 (x)T =m(0 (x)C0 ). Since Gl is an additive group of order sl , there exists a permutation matrix l (xl ) such that l (xl )(pl ) = xl (pl ). By the form of Cl and the definition of matrix image (Section 2), for any xl ∈ Gl , m(0 (x)Cl ) = m(l (xl )Cl ) = m(xl Cl ) = m(Tl (0l [xl (pl )])) = m(Tl (0l [l (xl )(pl )])) = Tl (Pl ⊗ [l (xl )pl l (xl )T ])TlT = Tl (Pl ⊗ pl )TlT = m(Cl ), for any l. This completes the proof.
Corollary 2.11. Let Lq = (C1 , . . . , Cm ) be normal and D = (D0 , D1 , . . . , Dm ) a repeating-column difference matrix about C0 , C1 , . . . , Cm with entries from G0 , G1 , . . . , Gm where C0 = C1 ♦ · · · ♦Cm . Then, for any vector a0 = a1 ♦ · · · ♦am with entries from G0 , the following array: [a0 + D0 , a1 + D1 , . . . , am + Dm ] is also a repeating-column difference matrix, where aj + Dj means that aj is added to each column of Dj . Definition 2.12. Let Lq = (C1 , . . . , Cm ) be normal and D = (D0 , D1 , . . . , Dm ) a repeating-column difference matrix about C0 , C1 , . . . , Cm with entries from G0 , G1 , . . . , Gm where C0 =C1 ♦ · · · ♦Cm . Then, the following every operation is called a transformation of repeating-column difference matrices. (a) Exchange any two runs of D or any two columns of Dl for a given l = 0, 1, 2, . . . , m. (b) Add an element xl ∈ Gl to some column of Dl for a given l = 0, 1, 2, . . . , m. (c) Add an element x = (x1 , . . . , xm ) ∈ G0 to some row of D0 while add xl ∈ Gl to the same row of Dl for all l = 1, 2, . . . , m. By Theorem 2.10 and Corollary 2.11, it is easy to see that D is still a repeating-column difference matrix if do a transformation of repeating-column difference matrices on D.
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3. Orthogonal arrays and repeating-column difference matrices Suppose that an experiment is performed according to the array A=(aij )n×m =(a1 , . . . , am ) and Y =(y1 , y2 , . . . , yn )T is the experimental data vector. In the analysis of variance Sj2 , the sum of squares of the jth factor, is defined as Sj2
=
pj i=1
⎛ 1 ⎝ |Iij |
s∈Iij
⎞2
2 n 1 Ys , Ys ⎠ − n s=1
where Iij = {s : asj = i} and |Iij | is the number of elements in Iij . From the definition, Sj2 is a quadratic form in Y and there exists a unique symmetric matrix Aj such that Sj2 = Y T Aj Y. The matrix Aj is called the matrix image (MI) of the jth column aj of A, denoted by m(aj ) = Aj . The MI of a subarray of A is defined as the sum of the MIs of all its columns. In particular, we denote the MI of A by m(A) and the MIs of 1r , (r) are Pr , n , respectively. If a design is an orthogonal array, then the MIs of its columns has some interesting properties, which can be used to construct orthogonal arrays. Theorem 3.1. For any permutation matrix S and any array L, m(S(L ⊗ 1r )) = S(m(L) ⊗ Pr )S T
and m(S(1r ⊗ L)) = S(Pr ⊗ m(L))S T .
Theorem 3.2. Let the array A be an orthogonal array of strength 1, i.e., A = (a1 , . . . , am ) = (S1 (0r1 (p1 )), . . . , Sm (0rm (pm ))), where ri pi = n, Si is a permutation matrix, for i = 1, . . . , m. Then, the following statements are equivalent. (1) (2) (3) (4)
A is an orthogonal array of strength 2. The MI of A is a projection matrix. The MIs of any two columns of A are orthogonal, i.e., m(ai )m(aj ) = 0 (i = j ). The projection matrix n can be decomposed as n = m(a1 ) + · · · + m(am ) + , where rk( ) = n − 1 − m j =1 (pj − 1) is the rank of the matrix .
Definition 3.3. An orthogonal array A is said to be saturated if
m
j =1 (pj
− 1) = n − 1 (or, equivalently, m(A) = n ).
Corollary 3.4. Let (L, H ) and K be orthogonal arrays of size n. Then, (K, H ) is an orthogonal array if m(K)m(L), where m(K)m(L) means that the difference m(L) − m(K) is nonnegative definite. Corollary 3.5. Suppose that L and H are orthogonal arrays. Then, K = (L, H ) is also an orthogonal array if m(L) and m(H ) are orthogonal, i.e., m(L)m(H ) = 0. In this case, m(K) = m(L) + m(H ). Theorem 3.6. Suppose that D 0 (q, m; p) is an atomic difference matrix. Then, (p) D 0 (q, m; p) is an orthogonal array whose matrix image satisfying m((p) D 0 (rq, m; p))p ⊗ q . These theorems and corollaries can be found in Zhang [5–7] and Zhang et al. [8]. Our procedure of constructing mixed-level orthogonal arrays consists of the following three steps [10]: Step 1: Orthogonally decompose the projection matrix n : n = A1 + · · · + Ak
where Ai Aj = 0 (i = j ).
Step 2: Find an orthogonal array Li such that m(Li ) Ai .
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Step 3: Lay out the new orthogonal array L by Corollaries 3.4 and 3.5: L = (L1 , . . . , Lk1 ) (k1 k). Let Lp =Lp (p1 , . . . , pm )=(C1 , . . . , Cm ) be a normal orthogonal array over G0 and denote C0 =C1 ♦ · · · ♦Cm and let
be an atomic difference matrix over G having p levels. And suppose that [D , D ] is D0 = D(q, k0 ; p) = D1 ♦ · · · ♦Dm 0 i i an atomic q ×(k0 +ki ) difference matrix having pi levels for i =1, 2, . . . , m. Thus, C =[C0 , C1 , . . . , Cm ] is a partitioned matrix in which the m + 1 columns C0 , C1 , . . . , Cm are orthogonal arrays of strength 1 having p, p1 , . . . , pm levels, respectively. Then, D=[D0 , D1 , . . . , Dm ] is an atomic repeating-column difference matrix about C =[C0 , C1 , . . . , Cm ] if D is a repeating-column difference matrix about C. Theorem 3.7. The matrix D = [D0 , D1 , . . . , Dm ] is an atomic repeating-column difference matrix about C = [C0 , C1 , . . . , Cm ] if and only if L = [C0 D0 , C1 D1 , . . . , Cm Dm ] is an orthogonal array whose matrix image satisfying m(L) p ⊗q . In particular, m(C0 D0 ) ⊗ q where = p − (m(C1 ) + · · · + m(Cm )).
m
i=1 m(Ci Di )+
Proof. ⇒ Consider the following orthogonal decomposition of projection matrix p ⊗ q : p ⊗ q = m(C1 ) ⊗ q + · · · + m(Cm ) ⊗ q + ⊗ q , where p = m(Lp ) + = m(C1 ) + · · · + m(Cm ) + . By Theorems 3.1, 3.2 and 3.6, we have m(Cj [Dj , Dj ])m(Cj ) ⊗ q ,
j = 1, 2, . . . , m,
, D ]] is an orthogonal array. By Corollary 3.5, we have that m(C D ) + i.e., [C1 [D1 , D1 ], . . . , Cm [Dm m i i
m(Ci Di ) = m(Ci [Di , Di ]), i = 1, 2, . . . , m. Thus,
m((C1 D1 ) + · · · + m(Cm Dm )) + ⊗ q p ⊗ q ,
m((C1 D1 ) + · · · + m(Cm Dm ))p ⊗ q , and
[m((C1 D1 ) + · · · + m(Cm Dm )) + ⊗ q ][m((C1 D1 ) + · · · + m(Cm Dm ))] = 0.
By their orthogonality and Theorem 3.2, L = [C0 D0 , C1 D1 , . . . , Cm Dm ] is an orthogonal array whose matrix image satisfies m(L)p ⊗ q if
m(C0 D0 )(m(C1 D1 ) + · · · + m(Cm Dm )) + ⊗ q .
In fact, let Dl = (dijl )q×k0 = (d1l , . . . , dkl 0 ) be an atomic difference matrix for any l. And denote C0 = (p). From Eqs. (1) and (2), we have [0, D0 ] (p) = (S00 (0q (p)), S10 (0q (p)), . . . , Sk00 (0q (p))), [0, Dl ] Cl = (S0l (0q Cl ), S1l (0n Cl ), . . . , Skl 0 (0q Cl )), l ), . . . , (d l )), j = 1, 2, . . . , m; l = 1, . . . , m. By Theorems 3.1 and 3.2, we have where S0l = Ipq and Sjl = diag(l (d1j l qj
m([0, D0 ] C0 ) = m([0, D0 ] (p)) =
k0 j =0
=
k0 m i=1 j =0
Sj0 (Pq ⊗ p )(Sj0 )T
Sj0 (Pq ⊗ m(Ci ))(Sj0 )T +
k0 j =0
Sj0 (Pq ⊗ )(Sj0 )T ,
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Y. Zhang / Discrete Mathematics 307 (2007) 246 – 261
since p =m(Lp )+ =m(C1 )+· · ·+m(Cm )+ . The above decompositions are orthogonal because of the orthogonality in each step. Thus, all items Sj0 (Pq ⊗m(Ci ))(Sj0 )T , Ss0 (Pq ⊗ )(Ss0 )T , i =1, 2, . . . , m; j, s =0, 1, . . . , k0 , are orthogonal to each other. By Theorem 2.10, we have that 0 T 0 0 0 T ) ) ) , . . . , 0 (dqj ) 0 (dqj ) 0 (d1j Sj0 (Pq ⊗ )(Sj0 )T Sj0 (Iq ⊗ )(Sj0 )T = diag(0 (d1j
= diag( , . . . , ) = Iq ⊗ , for j = 0, 1, . . . , m; and that Sj0 (Pq ⊗ m(Ci ))(Sj0 )T = m(Sj0 (0q Ci )) = m(Sji (0q Ci )) = m(dji Ci ). Thus, we obtain m([0, D0 ] C0 ) Pq ⊗ p +
m
m(Di Ci ) + q ⊗ ,
i=1
i.e., m(C0 D0 ) = K(p, q)m(D0 C0 )K(p, q)T = K(p, q)(m([0, D0 ] C0 ) − Pq ⊗ p )K(p, q)T m m
T m(Ci Di ) + ⊗ q . K(p, q) m(Di Ci ) + q ⊗ K(p, q) = i=1
i=1
This completes the proof of ⇒ . ⇐ Let L=[C0 D0 , C1 D1 , . . . , Cm Dm ] be an orthogonal array whose matrix image satisfies m(L) p ⊗q . Then, C0 [0, D0 ] and Ci [0, Di , Di ] are orthogonal arrays for all i. Thus, [0, D0 ] and [0, Di , Di ] are difference matrices for all i, i.e., [0, D] is a repeating-column difference matrix. It means that the matrix D is an atomic repeatingcolumn difference matrix about C. This completes the proof. Corollary 3.8. The matrix D = [D0 , D1 , …, Dm ] be an atomic repeating-column difference matrix about C if and only if L = [Lp 0q , 0p Lq , C0 D0 , C1 D1 , . . . , Cm Dm ] is an orthogonal array for any orthogonal arrays Lp and Lq . 4. Examples 4.1. Constructions of orthogonal arrays of run size 72 Zhang et al. [11] has constructed an orthogonal array L72 (28 38 64 121 ) whose structure is (=)
(=)
L72 (28 38 64 121 ) = [03 (12) 02 , 03 L24 (28 ), L36 (38 ) 02 , (−)
(−)
(M1 ⊗ Q1 )(L36 (62 ) 02 ), (M2 ⊗ Q2 )(L36 (62 ) 02 )], where Q1 = K(2, 2), Q2 = K(2, 2) diag(I2 , N2 )K(2, 2)T and M1 = K(3, 6) diag(N3 , N32 , Q1 ⊗ I3 )K(3, 6)T ,
M2 = K(3, 6) diag(N32 , N3 , Q2 ⊗ I3 )K(3, 6)T ;
and where the orthogonal arrays satisfy (=)
L24 (28 ) = D(12, 8; 2) (2),
8 L= 36 (3 ) = (3) D(12, 8; 3) +
L36 (62 ) = [[(3) (3) 04 ] (018 (2)], [(3) (3) ◦ 2 04 ] [09 (2) (2)]], (−)
in which D(12, 8; 2) and D(12, 8; 3) are some difference matrices.
Y. Zhang / Discrete Mathematics 307 (2007) 246 – 261
255
It is easy to prove that there exists a difference matrix D(12, 4; 6) = D(12, 4; 3)♦D(12, 4; 2) such that [(3) D(12, 4; 3) 02 ]♦[03 D(12, 4; 2) (2)] (−)
(−)
= [(M1 ⊗ Q1 )(L36 (62 ) 02 ), (M2 ⊗ Q2 )(L36 (62 ) 02 )]. Hence the following array, [D(12, 4; 6) [((3) 02 )♦(03 (2))], D(12, 8; 3) ((3) 02 ), D(12, 8; 2) (03 (2))] is also an orthogonal array. By Theorem 3.7, we have D = [D(12, 4; 6), D(12, 8; 3), D(12, 8; 2)] is a repeating-column difference matrix about [(6), (3) 02 , 02 (2)]. Let D(12, 4; 6) =(a0 , D(12, 3; 6)) where a0 = a1 ♦a2 . By the transformation of repeating-column difference matrix, we have [0, D 0 ] := [D(12, 4; 6) − a0 , D(12, 8; 3) − a1 , D(12, 8; 2) − a2 ] is also a repeating-column difference matrix. Thus, D 0 is an atomic repeating-column difference matrix. From above D 0 , we can construct some atomic repeating-column difference matrices of run size 12 having a large number of columns such as ⎛ ⎞ 00 00 00 00 0000000 0000000 ⎜ 01 21 20 11 2021011 0101100 ⎟ ⎜ ⎟ ⎜ 11 21 10 20 1002201 1110010 ⎟ ⎜ ⎟ ⎜ 01 20 10 00 0212112 1011101 ⎟ ⎜ ⎟ ⎜ 00 11 21 20 0110221 0011110 ⎟ ⎜ ⎟ ⎜ 20 10 00 21 1020112 0110111 ⎟ D 0 = [D 0 (12, 4; 6), D 0 (12, 7; 3), D 0 (12, 7; 2)] = ⎜ ⎟, ⎜ 10 11 11 10 2200022 1100101 ⎟ ⎜ ⎟ ⎜ 11 00 01 21 2211210 1000110 ⎟ ⎜ ⎟ ⎜ 21 01 11 11 0121202 0010001 ⎟ ⎜ ⎟ ⎜ 21 20 01 10 1112020 0101011 ⎟ ⎝ ⎠ 10 10 21 01 2122100 1111000 20 01 20 01 1201121 1001011 ⎛
00 ⎜ 20 ⎜ ⎜ 01 ⎜ ⎜ 00 ⎜ ⎜ 11 ⎜ ⎜ 10 0 0 0 0 D = [D (12, 5; 6), D (12, 3; 3), D (12, 6; 2)] = ⎜ ⎜ 11 ⎜ ⎜ 10 ⎜ ⎜ 21 ⎜ ⎜ 21 ⎝ 20 01
00 00 11 20 01 01 20 11 21 10 21 10
00 11 21 01 20 21 10 00 01 10 20 11
00 21 10 20 10 20 00 01 11 21 01 11
00 20 20 01 00 11 21 21 11 01 10 10
000 011 000 212 221 102 102 221 120 120 011 212
⎞ 000000 011011 ⎟ ⎟ 110001 ⎟ ⎟ 101101 ⎟ ⎟ 001111 ⎟ ⎟ 010110 ⎟ ⎟, 111010 ⎟ ⎟ 100011 ⎟ ⎟ 001000 ⎟ ⎟ 010101 ⎟ ⎠ 111100 100110
over G60 . Define 0 = 00, 1 = 11, 2 = 20, 3 = 01, 4 = 10, 5 = 21. Then, the group is Z6 . By using Corollary 3.8 and above atomic repeating-column difference matrices D 0 s, we can construct a lot of new orthogonal arrays of run size 72, which are exhibited in Table 1 or in Kuhfeld [3].
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Y. Zhang / Discrete Mathematics 307 (2007) 246 – 261
Table 1 Orthogonal arrays L72 (· · ·) obtained in Section 4.1 No.
f1 − f10
c1 − c10
b1 − b7
b8 − b18
lf dc
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48
0000000000 1111111111 2222222222 3333333333 4444444444 5555555555 0220512405 1331023510 2442134021 3553245132 4004350243 5115401354 0231454114 1342505225 2453010330 3504121441 4015232552 5120343003 0302233240 1413344351 2524455402 3035500513 4140011024 5 2 5 1 1 22 1 3 5 0013420152 1124531203 2235042314 3340153425 4451204530 5502315041 0354543521 1405054032 2510105143 3021210254 4132321305 5243432410 0512041542 1023152053 2134203104 3245314215 4350425320 5401530431 0541301035 1052412140 2103523251 3214034302 4325145413 5430250524
0000000000 1111111111 2222222222 0000000000 1111111111 2222222222 0111020112 1222101220 2000212001 0111020112 1222101220 2000212001 0002200022 1110011100 2221122211 0002200022 1110011100 2221122211 2120212112 0201020220 1012101001 2120212112 0201020220 1012101001 2210110221 0021221002 1102002110 2210110221 0021221002 1102002110 1202021011 2010102122 0121210200 1202021011 2010102122 0121210200 1021002201 2102110012 0210221120 1021002201 2102110012 0210221120 2212211210 0020022021 1101100102 2212211210 0020022021 1101100102
0000000 1111111 0000000 1111111 0000000 1111111 0110111 1001000 0110111 1001000 0110111 1001000 1100101 0011010 1100101 0011010 1100101 0011010 1011101 0100010 1011101 0100010 1011101 0100010 0011110 1100001 0011110 1100001 0011110 1100001 0101100 1010011 0101100 1010011 0101100 1010011 1110010 0001101 1110010 0001101 1110010 0001101 1000110 0111001 1000110 0111001 1000110 0111001
00000000000 00000000000 00000000000 00000000000 00000000000 00000000000 11100010101 11100010101 11100010101 11100010101 11100010101 11100010101 01011110001 01011110001 01011110001 01011110001 01011110001 01011110001 10111010010 10111010010 10111010010 10111010010 10111010010 10111010010 00010011111 00010011111 00010011111 00010011111 00010011111 00010011111 11001001011 11001001011 11001001011 11001001011 11001001011 11001001011 01100111010 01100111010 01100111010 01100111010 01100111010 01100111010 10110101001 10110101001 10110101001 10110101001 10110101001 10110101001
0000 0030 0000 0030 0000 0030 1010 1020 1010 1020 1010 1020 2130 2100 2130 2100 2130 2100 3120 3110 3120 3110 3120 3110 4201 4231 4201 4231 4201 4231 5211 5221 5211 5221 5211 5221 6331 6301 6331 6301 6331 6301 7321 7311 7321 7311 7321 7311
49 50 51 52 53 54 55 56 57
0155135311 1200240422 2311351533 3422402044 4533513155 5044024200 0143255234 1254300345 2305411450
1200121202 2011202010 0122010121 1200121202 2011202010 0122010121 1021112020 2102220101 0210001212
0010001 1101110 0010001 1101110 0010001 1101110 0101011 1010100 0101011
00101100111 00101100111 00101100111 00101100111 00101100111 00101100111 11010100110 11010100110 11010100110
8402 8432 8402 8432 8402 8432 9412 9422 9412
Y. Zhang / Discrete Mathematics 307 (2007) 246 – 261
257
Table 1 (continued) No.
f1 − f10
c1 − c10
b1 − b7
b8 − b18
lf dc
58 59 60 61 62 63 64 65 66 67 68 69 70 71 72
3410522501 4521033012 5032144123 0425324453 1530435504 2041540015 3152051120 4203102231 5314213342 0434112323 1545223434 2050334545 3101445050 4212550101 5323001212
1021112020 2102220101 0210001212 0112122100 1220200211 2001011022 0112122100 1220200211 2001011022 2121201121 0202012202 1010120010 2121201121 0202012202 1010120010
1010100 0101011 1010100 1111000 0000111 1111000 0000111 1111000 0000111 1001011 0110100 1001011 0110100 1001011 0110100
11010100110 11010100110 11010100110 01111001100 01111001100 01111001100 01111001100 01111001100 01111001100 10001111100 10001111100 10001111100 10001111100 10001111100 10001111100
9422 9412 9422 10 5 3 2 10 5 0 2 10 5 3 2 10 5 0 2 10 5 3 2 10 5 0 2 11 5 2 2 11 5 1 2 11 5 2 2 11 5 1 2 11 5 2 2 11 5 1 2
L72 (121 65 37 27 ) = (l f1 f7 − f10 c4 − c10 b1 − b7 )
(old)
L72 (41 65 37 215 ) = (d f1 f7 − f10 c4 − c10 b3 − b7 b9 − b18 ) L72 (41 65 38 28 ) = (d f1 f7 − f10 c4 − c10 c b3 − b7 b9 − b11 ) L72 (41 66 37 26 ) = (d f1 f7 − f10 f c4 − c10 b3 − b7 b9 ) L72 (121 66 33 26 ) = (l f1 − f6 c1 − c3 b1 − b6 )
(old)
L72 (41 66 33 214 ) = (d f1 − f6 c1 − c3 b3 − b6 b9 − b18 ) L72 (41 66 34 27 ) = (d f1 − f6 c1 − c3 c b3 − b6 b9 − b11 ) L72 (41 67 33 25 ) = (d f1 − f6 f c1 − c3 b3 − b6 b9 ) where d = b1 ♦b2 ♦b8 ,
l = b8 ♦ · · · ♦b18 ,
m(c) m(b12 − b18 ),
m(f ) m(b10 − b18 )
4.2. Construction of orthogonal arrays of run size 96 Zhang et al. [9] has constructed an orthogonal array L96 (212 420 241 ) whose structure is L96 (212 420 241 ) = [D 1 (12, 4; 2) 02 (2) 02 , D 2 (12, 4; 2) 04 (2), D 3 (12, 4; 2) (2) (2) (2), D(24, 20; 4) (4), (24) 04 ], where D(24, 20; 4), D 1 (12, 4; 2) 02 , D 2 (12, 4; 2) 02 and D 3 (12, 4; 2) (2) are some difference matrices.
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Y. Zhang / Discrete Mathematics 307 (2007) 246 – 261
It is easy to prove that L4 (23 )=((2) 02 , 02 (2), (2) (2)) is normal (Example 2.8). By Theorem 3.7, the following array: D = [D(24, 20; 4), D 1 (12, 4; 2) 02 , D 2 (12, 4; 2) 02 , D 3 (12, 4; 2) (2)] is a repeating-column difference matrix about [(4), (2) 02 , 02 (2), (2) (2)]. Let D(24, 20; 4) = (a0 , D(24, 19; 4)), where a0 = a1 ♦a2 ♦a3 . By the transformation of repeating-column difference matrix, we have [0, D 0 ] : =[a0 + D(24, 4; 4), a1 + D 1 (12, 4; 2) 02 , a2 + D 2 (12, 4; 2) 02 , a3 + D 3 (12, 4; 2) (2)] is also a repeating-column difference matrix. Thus, D 0 is an atomic repeating-column difference matrix ⎛ ⎞ 0000111112222233333 0011 0101 0110 ⎜ ⎟ ⎜ 1332102232311032001 0011 0101 1001 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 0000222223333311111 0110 0011 0101 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 1332231103200110223 0110 0011 1010 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 0000333331111122222 0101 0110 0011 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 1332320011022323110 0101 0110 1100 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 2230022300223002230 1111 1111 0000 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 3102031020310203102 1111 1111 1111 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 2230330223302233022 1000 1000 0000 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 3102323103231032310 1000 1000 1111 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 2230203032030320303 1111 1000 0111 ⎟ ⎜ ⎟ ⎜ ⎟ 3102210312103121031 1111 1000 1000 ⎜ ⎟ ⎟, D0 = ⎜ ⎜ ⎟ ⎜ 0311311030031113030 1010 1011 0001 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 1023302310102312302 1010 1011 1110 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 0311003111303031103 1100 1110 0010 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 1023010231230230231 1100 1110 1101 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 0311130303110300311 1001 1101 0100 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 1023123023023101023 1001 1101 1011 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 2121200120212111200 0001 0100 0101 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 3213213200321310132 0001 0100 1010 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 2121112002001202121 0100 0010 0110 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 3213101322132003213 0100 0010 1001 ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ 2121021211120020012 0010 0001 0011 ⎠ 3213032131013221320
0010
0001
1100
where 0 = 000, 1 = 011, 2 = 101, 3 = 110 and x + y = y + x, x + x = 0, 0 + x = x, 2 + 3 = 1, 3 + 1 = 2, 2 + 1 = 3. By using Corollary 3.8 and above atomic repeating-column difference matrix D 0 , we can construct a lot of new orthogonal arrays of run size 96, which are exhibited in Table 2 or in Kuhfeld [3].
Y. Zhang / Discrete Mathematics 307 (2007) 246 – 261
259
Table 2 Orthogonal arrays L96 (· · ·) obtained in Section 4.2 L96 (420 235 )
reL96
d1 − d20
b1 − b35
f d 21 − d23 clx
00000111112222233333 11111000003333322222 22222333330000011111 33333222221111100000 01332102232311032001 10223013323200123110 23110320010133210223 32001231101022301332 00000222223333311111 11111333332222200000 22222000001111133333 33333111110000022222 01332231103200110223 10223320012311001332 23110013321022332001 32001102230133223110 00000333331111122222 11111222220000033333 22222111113333300000 33333000002222211111 01332320011022323110 10223231100133232001 23110102233200101332 32001013322311010223 02230022300223002230 13321133211332113321 20012200122001220012 31103311033110331103 03102031020310203102 12013120131201312013 21320213202132021320 30231302313023130231 02230330223302233022 13321221332213322133 20012112001120011200 31103003110031100311 03102323103231032310 12013232012320123201 21320101321013210132 30231010230102301023 02230203032030320303 13321312123121231212 20012021210212102121 31103130301303013030 03102210312103121031 1 20 1 3 3 0 1 2 0 3 0 1 2 0 3 0 1 2 0 21320032130321303213 30231123021230212302 00311311030031113030 11200200121120002121 22133133212213331212 33022022303302220303 01023302310102312302 10132213201013203213 23201120132320130120 32310031023231021031
00110101011000010111101111110000010 00111010100100010111101111110000010 11000101100100010111101111110000010 11001010011000010111101111110000010 00110101100111100110100110000010111 00111010011011100110100110000010111 11000101011011100110100110000010111 11001010100111100110100110000010111 01100011010101111101000010110110100 01101100101001111101000010110110100 10010011101001111101000010110110100 10011100010101111101000010110110100 01100011101010001101110011000101110 01101100010110001101110011000101110 10010011010110001101110011000101110 10011100101010001101110011000101110 01010110001101111101000101001001011 0 1 0 1 1 0 0 1 1 10 0 0 1 1 1 1 1 0 1 0 0 0 1 0 1 0 0 1 0 0 1 0 1 1 10100110110001111101000101001001011 10101001001101111101000101001001011 01010110110010001101110100111010001 01011001001110001101110100111010001 10100110001110001101110100111010001 10101001110010001101110100111010001 11111111000000010111101000001111101 11110000111100010111101000001111101 00001111111100010111101000001111101 00000000000000010111101000001111101 11111111111111100110100001111101000 11110000000011100110100001111101000 00001111000011100110100001111101000 00000000111111100110100001111101000 10001000000000000000000000000000000 10000111111100000000000000000000000 01111000111100000000000000000000000 01110111000000000000000000000000000 10001000111101100001111000110001111 10000111000001100001111000110001111 01111000000001100001111000110001111 01110111111101100001111000110001111 11111000011111011000101001011010110 11110111100011011000101001011010110 00001000100011011000101001011010110 00000111011111011000101001011010110 11111000100001001110011001100110011 11110111011101001110011001100110011 00001000011101001110011001100110011 00000111100001001110011001100110011 10101011000100000000000111111111111 10100100111000000000000111111111111 01011011111000000000000111111111111 01010100000100000000000111111111111 10101011111001100001111111001110000 10100100000101100001111111001110000 01011011000101100001111111001110000 01010100111001100001111111001110000
0012000 0011000 0022000 0021000 0112011 0111011 0122011 0121011 1331022 1332022 1301022 1302022 1231033 1232033 1201033 1202033 1023024 1020024 1013024 1010024 1123035 1120035 1113035 1110035 0333006 0330006 0303006 0300006 0233017 0230017 0203017 0200017 2000148 2003148 2030148 2033148 3000159 3003159 3030159 3033159 4 1 3 0 2 6 10 4 1 3 3 2 6 10 4 1 0 0 2 6 10 4 1 0 3 2 6 10 5 3 3 0 2 7 11 5 3 3 3 2 7 11 5 3 0 0 2 7 11 5 3 0 3 2 7 11 2 3 1 1 1 4 12 2 3 1 2 1 4 12 2 3 2 1 1 4 12 2 3 2 2 1 4 12 3 3 1 1 1 5 13 3 3 1 2 1 5 13 3 3 2 1 1 5 13 3 3 2 2 1 5 13
260
Y. Zhang / Discrete Mathematics 307 (2007) 246 – 261
Table 2 (continued) L96 (420 235 )
reL96
d1 − d20
b1 − b35
f d 21 − d23 clx
00311003111303031103 11200112000212120012 22133221333121213321 33022330222030302230 01023010231230230231 10132101320321321320 23201232013012012013 32310323102103103102 00311130303110300311 11200021212001211200 22133312121332122133 33022203030223033022 01023123023023101023 10132032132132010132 23201301201201323201 32310210310310232310 02121200120212111200 13030311031303000311 20303022302030333022 31212133213121222133 03213213200321310132 12302302311230201023 21031031022103132310 30120120133012023201 02121112002001202121 13030003113110313030 20303330220223020303 31212221331332131212
11001110001011010011010101010100101 11000001110111010011010101010100101 00111110110111010011010101010100101 00110001001011010011010101010100101 11001110110110110100011101100011100 11000001001010110100011101100011100 00111110001010110100011101100011100 00110001110110110100011101100011100 10011101010011010011010010101011010 10010010101111010011010010101011010 01101101101111010011010010101011010 01100010010011010011010010101011010 10011101101110110100011010011100011 10010010010010110100011010011100011 01101101010010110100011010011100011 01100010101110110100011010011100011 0 0 0 1 0 1 0 0 0 1 01 0 0 1 1 1 0 1 0 1 1 0 0 1 1 1 0 1 0 0 0 1 0 1 00011011101000111010110011101000101 11100100101000111010110011101000101 11101011010100111010110011101000101 00010100101010101011001011010011001 00011011010110101011001011010011001 11100100010110101011001011010011001 11101011101010101011001011010011001 01000010011011011000101110100101001 01001101100111011000101110100101001 10110010100111011000101110100101001 10111101011011011000101110100101001
2 2 2 3 1 8 14 2 2 2 0 1 8 14 2 2 1 3 1 8 14 2 2 1 0 1 8 14 3 1 2 3 1 9 15 3 1 2 0 1 9 15 3 1 1 3 1 9 15 3 1 1 0 1 9 15 2 1 0 2 1 8 16 2 1 0 1 1 8 16 2 1 3 2 1 8 16 2 1 3 1 1 8 16 3 2 0 2 1 9 17 3 2 0 1 1 9 17 3 2 3 2 1 9 17 3 2 3 1 1 9 17 4 0 0 2 2 10 18 4 0 0 1 2 10 18 4 0 3 2 2 10 18 4 0 3 1 2 10 18 5 1 0 2 2 11 19 5 1 0 1 2 11 19 5 1 3 2 2 11 19 5 1 3 1 2 11 19 4 2 2 1 2 6 20 4 2 2 2 2 6 20 4 2 1 1 2 6 20 4 2 1 2 2 6 20
03213101322132003213 12302010233023112302 21031323100310221031 30120232011201330120 02121021211120020012 13030130300031131103 20303203033302202230 31212312122213313321 03213032131013221320 12302123020102330231 21031210313231003102 30120301202320112013
01000010100101001110011110011001100 01001101011001001110011110011001100 10110010011001001110011110011001100 10111101100101001110011110011001100 00100001001100111010110100010111010 00101110110000111010110100010111010 11010001110000111010110100010111010 11011110001100111010110100010111010 00100001110010101011001100101100110 00101110001110101011001100101100110 11010001001110101011001100101100110 11011110110010101011001100101100110
5 0 2 1 2 7 21 5 0 2 2 2 7 21 5 0 1 1 2 7 21 5 0 1 2 2 7 21 4 3 1 0 2 10 22 4 3 1 3 2 10 22 4 3 2 0 2 10 22 4 3 2 3 2 10 22 5 2 1 0 2 11 23 5 2 1 3 2 11 23 5 2 2 0 2 11 23 5 2 2 3 2 11 23
L96 (241 420 212 ) = (x d1 − d20 b1 − b12 )
L96 (121 422 218 ) = (l d1 − d20 d22 d23 b1 b4 b5 b8 − b12 b26 − b35 )
L96 (423 226 ) = (d1 − d23 b1 b4 b5 b8 − b12 b14 − b23 b26 − b29 b32 − b35 )
L96 (31 423 219 ) = (c d1 − d23 b1 b4 b5 b8 − b12 b14 − b16 b26 − b29 b32 − b35 )
L96 (61 423 217 ) = (f d1 − d23 b1 b4 b5 b8 − b12 b14 b26 − b29 b32 − b35 ).
Y. Zhang / Discrete Mathematics 307 (2007) 246 – 261
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Table 2 (continued) L96 (420 235 ) d1 − d20
reL96 f d 21 − d23 clx
b1 − b35
where d21 = b13 ♦b30 ♦b31 , x = b13 ♦ · · · ♦b35 , m(c) m(b13 − b23 ),
d22 = b2 ♦b3 ♦b24 ,
d23 = b6 ♦b7 ♦b25 ,
l = b13 ♦ · · · ♦b23 , m(f ) m(b15 − b23 ).
Acknowledgements The author would like to thank the referee for his many valuable suggestions and comments. The work was supported by National Social Science Foundations (No. 10571045, No. 0224370051 (Henan) and No. 0211063100 (Henan)) in China. References [1] T. Beth, D. Jungnickel, H. Lenz, 1985. Design Theory, Bibliographishes Institut, Mannheinu-Wien-Zürich, 1985, and Cambridge University Press, Cambridge, 1986. [2] R.C. Bose, K.A. Bush, Orthogonal arrays of strength two and three, Ann. Math. Statist. 23 (1952) 508–524. [3] W.F. Kuhfeld, Orthogonal arrays, http://support.sas.com/technote/ts723.html, 2006. [4] S. Shrikhande, Generalized Hadamard matrices and orthogonal arrays strength two, Canad. J. Math. 16 (1964) 736–740. [5] Y.S. Zhang, Asymmetrical orthogonal design by multi-matrix methods, J. Chinese Statist. Assoc. 29 (1991) 197–218. [6] Y.S. Zhang, Orthogonal array and matrices, J. Math. Res. Exposition 12 (3) (1992) 438–440. [7] Y.S. Zhang, Theory of Multilateral Matrix, Chinese Statistic Press, 1993. [8] Y.S. Zhang, W.G. Li, S.S. Mao, Z.Q. Zheng, A simple method for constructing orthogonal arrays by the Kronecker sum, J. Syst. Sci. Complexity 19 (2006) 266–273. [9] Y.S. Zhang, Orthogonal arrays obtained by generalized Kronecker product, J. Statist. Plann. Inference (2000), in review. [10] Y.S. Zhang, Y.Q. Lu, S.Q. Pang, Orthogonal arrays obtained by orthogonal decomposition of projection matrices, Statist. Sinica 9 (1999) 595–604. [11] Y.S. Zhang, S.Q. Pang, Y.P. Wang, Orthogonal arrays obtained by generalized Hadamard product, Discrete Math. 238 (2001) 151–170.
Orthogonal arrays obtained by repeating-column difference matrices Yingshan Zhang Department of Statistics, East China Normal University, Shanghai, 200062, People’s Republic of China Received 2 June 2003; received in revised form 11 April 2006; accepted 9 June 2006 Available online 26 September 2006
Abstract In this paper, by using the repeating-column difference matrices and orthogonal decompositions of projection matrices, we propose a new general approach to construct asymmetrical orthogonal arrays. As an application of the method, some new orthogonal arrays with run sizes 72 and 96 are constructed. © 2006 Elsevier B.V. All rights reserved. MSC: primary 62K15; secondary 05B1 Keywords: Asymmetrical orthogonal arrays; Generalized hadamard products; Generalized Kronecker products; Repeating-column difference matrices; Projection matrices; Permutation matrices
1. Introduction An n × m matrix A, having ki columns with pi ( 2) levels, i = 1, . . . , t, m = ti=1 ki , pi = pj , for i = j , is called an orthogonal array (OA) of strength d and size n if each n × d submatrix of A contains all possible 1 × d row vectors with the same frequency. Unless stated otherwise, we consider an orthogonal array of strength 2, using the notation Ln (p1k1 , . . . , ptkt ) for such an array. An orthogonal array is said to be mixed-level (or asymmetrical) if t 2. Difference matrices are essential for the construction of many asymmetrical orthogonal arrays [2]. Using the notation for additive groups, a difference matrix having level p is an p × m matrix with the entries from a finite Abelian group G of cardinality p such that the vector differences of any two columns of the array, say di − dj if i = j, contains every element of G exactly times. We will denote such an array by D(p, m; p), although this notation suppresses the relevance of the group G. In most of our examples, G will correspond to the additive group associated with a Galois field GF(p). The difference matrix D(p, m; p) is called a generalized Hadamard matrix if p = m. In particular, D(2, 2; 2) is the usual Hadamard matrix. If a D(p, m; p) exists, it can always be constructed so that one of its rows and one of its columns contain only the zero element of G. Deleting this column from D(p, m; p), we obtain a difference matrix, denoted by D 0 (p, m − 1; p) called an atom of difference matrix D(p, m; p) (or an atomic difference matrix). Without loss of generality,
E-mail address: [email protected]. 0012-365X/$ - see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.disc.2006.06.029
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the D(p, m; p) can be written as 0 0 D(p, m; p) = = (0 D 0 (p, m − 1; p)). 0 A The property is important for the following discussions. For two matrices A = (aij )n×m and B = (bij )s×t both with the entries from group G, define their Kronecker sum [4] to be A B = (aij B)1 i n,1 j m , where each submatrix aij B of A B is obtained by adding aij to each entry of B. Shrikhande [4] showed that A B is a difference matrix if both A and B are difference matrices. In the contrast, Zhang [7] showed that A is a difference matrix if both A B and B are difference matrices. It is known that the Kronecker sum L = Lp (p s ) D(p, m; p) (or L = D(p, m; p) Lp (p s )) is an orthogonal array if Lp (p s ) is an orthogonal array and D(p, r; p) is a difference matrix [1]. By setting = s = 1, the Kronecker sum method reduces to the well-known construction of Bose and Bush [2], i.e., L = (p) D(p, m; p) (or L = D(p, m; p) (p)) is an orthogonal array if D(p, m; p) is a difference matrix. In the contrast, Zhang [7] has found that the difference matrix D(p, m; p) can be also constructed by using the orthogonal array L = (p) D(p, m; p), i.e., D(p, m; p) is a difference matrix if L = (p) D(p, m; p) is an orthogonal array. Let D(r, m; p) = (dij ), we have L = D(r, m; p) (p) = [S1 (0r (p)), . . . , Sm (0r (p))],
(1)
where Sj = diag((d1j ), . . . , (drj )) and (dij ) is a permutation matrix such that (dij )(p) = dij (p),
(2)
for any i, j , the dij (p) can be obtained by adding dij to each entry of (p). The idea of Kronecker sum and difference matrices can be generalized as follows [7]. Let n=pq. If A is an orthogonal array Lp (p1k1 , . . . , ptkt ) with the partition A = [Lp (p1k1 ), . . . , Lp (ptkt )], and if there exist the atoms: D 0 (1 p1 , m1 − 1; p1 ), . . . , D 0 (t pt , mt − 1; pt ), of difference matrices D(1 p1 , m1 ; p1 ), . . . , D(t pt , mt ; pt ), respectively, where q = i pi and p and q are both multiples the pi ’s, then the following array: [Lp 0q , 0p Lq , Lp (p1k1 ) D 0 (1 p1 , m1 − 1; p1 ), . . . , Lp (ptkt ) D 0 (t pt , mt − 1; pt )] is also an orthogonal array for any orthogonal arrays Lp and Lq . In this paper, we will prove that the following array: [Lp 0q , 0p Lq , (p) D 0 (0 p, m0 − 1; p), Lp (p11 ) D 0 (1 p1 , m1 − 1; p1 ), . . . , Lp (pt1 ) D 0 (t pt , mt − 1; pt )], is also an orthogonal array for any orthogonal arrays Lp and Lq if A = [Lp (p11 ), . . . , Lp (pt1 )] is a normal orthogonal array and D 0 = [D 0 (0 p, m0 − 1; p), D 0 (1 p1 , m1 − 1; p1 ), . . . , D 0 (t pt , mt − 1; pt )] is an atomic repeating-column difference matrix (Section 2).
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Section 2 contains the basic concepts and main theorems of repeating-column difference matrices while in Section 3 we describe the method of construction. Some new orthogonal arrays with run sizes 72 and 96 are constructed in Section 4. 2. Repeating-column difference matrices In order to define the repeating-column difference matrices, we must define a generalized Hadamard product [11]. Definition 2.1. Let h(x, y) be a mapping from 1 × 2 to V, where 1 × 2 = {(x, y) : x ∈ 1 , y ∈ 2 } and 1 , 2 , V are some sets. For two matrices A = (aij )n×m with entries from 1 and B = (bij )n×m with entries from 2 , h
define their generalized Hadamard product, denoted by ◦, as follows: h
A ◦ B = (h(aij , bij ))n×m = (h(aij , bij ))1 i n,1 j m , h
where each of the entries h(aij , bij ) of A ◦ B may be a scalar or vector or matrix under the mapping h(x, y). Unless stated otherwise, we consider the sets 1 and 2 to be finite, using the notations 1 = {0, 1, . . . , p − 1} and 2 = {0, 1, . . . , q − 1} for two example sets. When V is a row-vector space of m-dimensions, the mapping h(i, j ) can be represented by a pq × m matrix D, i.e., h
h : [(p) 0q ] ◦[0p (q)] = D = (d(1) , . . . , d(pq) )T , T with h(i, j ) = d(iq+j +1) (or h(i, j ) is the (iq + j + 1)th row of D). For this case in the following discussions, the h
h
h
generalized Hadamard product ◦ will only be defined by [(p) 0q ] ◦[0p (q)] = D. Note that [(p) 0q ] ◦[0p (q)] = h
h
h
(p) ⊗(q), the generalized Hadamard product ◦ will be also defined by (p) ⊗(q) = D. This is a form of generalized Kronecker product. h
Let 1 , 2 , V be multiplicative groups. When h(i, j )=ij , the generalized Hadamard ◦ is the usual Hadamard product in matrix theory, denoted by ◦. Let 1 , 2 , V be additive (Abelian) groups. When h(i, j ) = i + j, the generalized h
Hadamard ◦ is the usual addition of matrices in matrix theory, denoted by +. Let 1 = 2 = V = {0, 1, . . . , p − 1} and h
h(i, j ) = i + j, mod p, Then, the generalized Hadamard product ◦ is really the usual modulus addition of matrices in +
+
+
the theory of matrices, denoted by ◦, In particular, denote A ◦ A = A + A, mod p by A ◦ 2 , which is very useful for the construction of orthogonal arrays. Let h(i, j ) = (i, j ), where 1 = {0, 1, . . . , p − 1}, 2 = {0, 1, . . . , q − 1}, V = {(i, j ); i ∈ 1 , j ∈ 2 }. The h
generalized Hadamard product ◦ is also called a repeating operation, denoted by ♦, which can be used for the construction of repeating-column difference matrices. Note that we often write the elements (i, j ) of V in a form ij instead of (i, j ). Similarly let h(i, j ) = iq + j, where 1 = {0, 1, . . . , p − 1}, 2 = {0, 1, . . . , q − 1}, V = {0, 1, . . . , pq − 1}. The h
generalized Hadamard product ◦ is also called a joining operation, denoted by , which can be used for the construction of asymmetrical orthogonal arrays with large levels from those with small levels. Theorem 2.2. Let A, B, C, D be matrices and T a permutation matrix. Then, (A♦B) (C♦D) = (A C)♦(B D), and T (A♦B♦C♦D) = T A♦T B♦T C♦T D. Let m(A) be the matrix image of array A ([10,11] or Section 3), we have
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Theorem 2.3. Suppose that a and b are two orthogonal arrays which have only one column with run size n, i.e., a = Ln (p) = (a1 , . . . , an )T , b = Ln (q) = (b1 , . . . , bn )T . Then the matrix image of a♦b (or a b) is the following orthogonal decomposition: m(a♦b) = m(a) + m(b) + nm(a) ◦ m(b) = m(b♦a) = m(a b) = m(b a), if m(a)m(b) = 0, where a♦b = (a1 b1 , . . . , an bn )T (or a b = (a1 q + b1 , . . . , an q + bn )T ) is the repeating (or joining) operation of a and b in Definition 2.1, and m(a) ◦ m(b) is the usual Hadamard product in matrix theory. Corollary 2.4. Let K 1 =Ln (p1 , . . . , pm )=(Ln (p1 ), . . . , Ln (pm )) and K 2 =Ln (q1 , . . . , qm )=(Ln (q1 ), . . . , Ln (qm )) be two orthogonal arrays of run size n. Denote (Ln (p1 )♦Ln (q1 ), . . . , Ln (pm )♦Ln (qm )) =: K 1 ♦K 2 , then the matrix image of m(K 1 ♦K 2 ) satisfies m(K 1 ♦K 2 )m(K 1 ) + m(K 2 ) + nm(K 1 ) ◦ m(K 2 ), if m(K 1 )m(K 2 ) = 0. Corollary 2.5. Suppose that Ln1 = Ln1 (p1 , . . . , pm ) = (Ln1 (p1 ), . . . , Ln1 (pm )) and Ln2 = Ln2 (q1 , . . . , qm ) = (Ln2 (q1 ), . . . , Ln2 (qm )) are two orthogonal arrays. Then (Ln1 0n2 )♦(0n1 Ln2 ) is also an orthogonal array. In this case, its matrix image satisfies m((Ln1 0n2 )♦(0n1 Ln2 ))m(Ln1 ) ⊗ Pn2 + Pn1 ⊗ m(Ln2 ) + m(Ln1 ) ⊗ m(Ln2 ). Corollary 2.6. Suppose that p is a prime and a and b are OA’s which have only one column with run size n and p levels, i.e., a =Ln (p)=(a1 , . . . , an )T , b =Ln (p)=(b1 , . . . , bn )T . Then, Ln (p p−1 )=(a +b, . . . , [p −1]a +b), mod p, is also +
an OA whose MI is nm(a)◦m(b) if m(a)m(b)=0. In particular, Lp2 (p p−1 )=((p) (p), . . . , (p) ◦[p−1] (p)), mod p, is also an OA whose MI is p ⊗ p . These theorems and corollaries can be proved easily and are also found in Zhang [5–7]. In order to find a generalized result of repeating-column difference matrices, we must study the structure of associated orthogonal arrays. Definition 2.7. Let Lp = Lp (p1 , . . . , pm ) = (C1 , . . . , Cm ) be an orthogonal array where Cl is a vector with entries from a additive group Gl of order pl for any l. The array Lp is called normal over G0 if the set consisted of all entries of vector C0 = C1 ♦ · · · ♦Cm is a additive group G0 of order p, where G0 ⊂ G1 × · · · × Gm := {(x1 , . . . , xm ); xl ∈ Gl , l = 1, 2, . . . , m} with the usual addition: (x1 , x2 , . . . , xm ) + (y1 , y2 , . . . , ym ) = (x1 + y1 , x2 + y2 , . . . , xm + ym ) ∀(x1 , x2 , . . . , xm ), (y1 , y2 , . . . , ym ) ∈ G0 . Example 2.8. The following arrays are normal: a. ⎛ ⎞ ⎛ 0 0 0 0 ⎜0 1⎟ ⎜0 1 2 3 L4 (2 ) = ⎝ ⎠ , L4 (2 ) = ⎝ 1 0 1 0 1 1 1 1
⎞ 0 1⎟ ⎠, 1 0
over G40 = {(0, 0), (0, 1), (1, 0), (1, 1)} and G40 = {(0, 0, 0), (0, 1, 1), (1, 0, 1), (1, 1, 0)}, respectively.
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b. ⎛
0 ⎜0 ⎜ ⎜1 L6 (31 21 ) = ⎜ ⎜1 ⎝ 2 2
⎞
0 1⎟ ⎟ 0⎟ ⎟, 1⎟ ⎠ 0 1
⎛
0 ⎜0 ⎜ ⎜1 ⎜ ⎜1 7 L8 (2 ) = ⎜ ⎜0 ⎜ ⎜0 ⎝ 1 1
0 1 0 1 0 1 0 1
0 1 1 0 0 1 1 0
0 0 0 0 1 1 1 1
0 0 1 1 1 1 0 0
0 1 0 1 1 0 1 0
⎞ 0 1⎟ ⎟ 1⎟ ⎟ 0⎟ ⎟, 1⎟ ⎟ 0⎟ ⎠ 0 1
over G60 = {(0, 0), (0, 1), . . . , (2, 1)} and G80 = {(0, 0, 0, 0, 0, 0, 0), . . . , (1, 1, 0, 1, 0, 0, 1)}, respectively. c. ⎛ ⎛ ⎛ ⎞ ⎞ ⎞ 0 0 0 0 0 0 0 0 ⎜0 1⎟ ⎜0 1⎟ ⎜0 1 1 0⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜0 2⎟ ⎜0 2⎟ ⎜1 0 1 0⎟ ⎜ ⎜ ⎜ ⎟ ⎟ ⎟ ⎜0 3⎟ ⎜1 0⎟ ⎜1 1 0 0⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜0 4⎟ ⎜1 1⎟ ⎜0 0 0 1⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜0 5⎟ ⎜1 2⎟ ⎜0 1 1 1⎟ L12 (21 61 ) = ⎜ ⎟ , L12 (41 31 ) = ⎜ ⎟ , L12 (23 31 ) = ⎜ ⎟, ⎜1 0⎟ ⎜2 0⎟ ⎜1 0 1 1⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜1 1⎟ ⎜2 1⎟ ⎜1 1 0 1⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜1 2⎟ ⎜2 2⎟ ⎜0 0 0 2⎟ ⎜ ⎜ ⎜ ⎟ ⎟ ⎟ ⎜1 3⎟ ⎜3 0⎟ ⎜0 1 1 2⎟ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ 1 4 3 1 1 0 1 2 1 5 3 2 1 1 0 2 12 over G12 0 = {(0, 0), (0, 1), . . . , (1, 5)}, G0 = {(0, 0), (0, 1), . . . , (3, 2)} and
G12 0 = {(0, 0, 0, 0), (0, 1, 1, 0), . . . , (1, 1, 0, 2)}, respectively. Definition 2.9. Let Lp = Lp (p1 , . . . , pm ) = (C1 , . . . , Cm ) be a normal orthogonal array over G0 and denote
be a difference matrix over G having p levels.And suppose C0 =C1 ♦ · · · ♦Cm and let D0 =D(q, k0 ; p)=D1 ♦ · · · ♦Dm 0
that [Di , Di ] is an p × (k0 + ki ) difference matrix having pi levels for i = 1, 2, . . . , m. Thus, C = [C0 , C1 , . . . , Cm ] is a partitioned matrix in which the m+1 columns C0 , C1 , . . . , Cm are orthogonal arrays of strength 1 having p, p1 , . . . , pm levels, respectively. We call D = [D0 , D1 , . . . , Dm ] a repeating-column difference matrix about C = [C0 , C1 , . . . , Cm ]. The repeating-column difference matrix D is called an atomic repeating-column difference matrix if Dj is atomic for any j. Theorem 2.10. Let Lp = Lp (p1 , . . . , pm ) = (C1 , . . . , Cm ) be a normal orthogonal array and denote C0 = C1 ♦ · · · ♦Cm . Then, there exist p permutation matrices 0 (x), ∀x ∈ G0 and pl permutation matrices l (xl ), ∀xl ∈ Gl , l = 1, 2, . . . , m, such that 0 (x) · C0 = x C0 , 0 (x) · Cl = l (xl ) · Cl = xl Cl , ∀x = (x1 , x2 , . . . , xm ) ∈ G0 ,
l = 1, 2, . . . , m
where xl Cl stands for the vector obtained by xl to each entry of Cl . In other words, we have 0 (x)Lp = (1 (x1 )C1 , . . . , m (xm )Cm ). In this case, the matrix images of C0 , C1 , C2 , . . . , Cm satisfy the following equations: m(Cl ) = m(0 (x)Cl )
∀x = (x1 , x2 , . . . , xm ) ∈ G0 , l = 0, 1, 2, . . . , m.
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In particular, if let = p − m(C1 ) − · · · − m(Cm ), then, we have 0 (x) 0 (x)T = . Proof. Since the set of entries of vector C0 = C1 ♦ · · · ♦Cm is an additive group G0 , for any given x ∈ G0 , there exists a permutation matrix 0 (x) such that 0 (x) · C0 = x C0 , where x C0 stands for the vector obtained by x to each entry of C0 . Furthermore, since the order of group G0 is p, i.e., x C0 = C0 iff x = (0, 0, . . . , 0), we have 0 (x) = 0 (y) if x = y. Similarly, because Cl is a vector with entries from an group Gl and having a form Cl = Tl (0l (pl )) where l pl = p and Tl is a permutation matrix for any l, for any given xl ∈ Gl there exists a permutation matrix l (xl ) such that l (xl ) · Cl = xl Cl = Tl (0l [xl (pl )]), since there exists a permutation matrix l (xl ) such that l (xl )(pl ) = xl (pl ), where l pl = p and Tl is a permutation matrix for any l. From above results and Theorem 2.2, we have (0 (x)C1 )♦ · · · ♦(0 (x)Cm ) = 0 (x)C0 = x C0 = (x1 C1 )♦ · · · ♦(xm Cm ) = (1 (x1 )C1 )♦ · · · ♦(m (xm )Cm ), i.e., 0 (x)Cl = l (xl )Cl , l = 1, 2, . . . , m. Now, we prove that m(Cl ) = m(0 (x)Cl ), ∀x ∈ G0 , l = 0, 1, 2, . . . , m. In fact, that m(C0 )=m(0 (x)C0 ) is trivial since m(C0 )=p =0 (x)p 0 (x)T =m(0 (x)C0 ). Since Gl is an additive group of order sl , there exists a permutation matrix l (xl ) such that l (xl )(pl ) = xl (pl ). By the form of Cl and the definition of matrix image (Section 2), for any xl ∈ Gl , m(0 (x)Cl ) = m(l (xl )Cl ) = m(xl Cl ) = m(Tl (0l [xl (pl )])) = m(Tl (0l [l (xl )(pl )])) = Tl (Pl ⊗ [l (xl )pl l (xl )T ])TlT = Tl (Pl ⊗ pl )TlT = m(Cl ), for any l. This completes the proof.
Corollary 2.11. Let Lq = (C1 , . . . , Cm ) be normal and D = (D0 , D1 , . . . , Dm ) a repeating-column difference matrix about C0 , C1 , . . . , Cm with entries from G0 , G1 , . . . , Gm where C0 = C1 ♦ · · · ♦Cm . Then, for any vector a0 = a1 ♦ · · · ♦am with entries from G0 , the following array: [a0 + D0 , a1 + D1 , . . . , am + Dm ] is also a repeating-column difference matrix, where aj + Dj means that aj is added to each column of Dj . Definition 2.12. Let Lq = (C1 , . . . , Cm ) be normal and D = (D0 , D1 , . . . , Dm ) a repeating-column difference matrix about C0 , C1 , . . . , Cm with entries from G0 , G1 , . . . , Gm where C0 =C1 ♦ · · · ♦Cm . Then, the following every operation is called a transformation of repeating-column difference matrices. (a) Exchange any two runs of D or any two columns of Dl for a given l = 0, 1, 2, . . . , m. (b) Add an element xl ∈ Gl to some column of Dl for a given l = 0, 1, 2, . . . , m. (c) Add an element x = (x1 , . . . , xm ) ∈ G0 to some row of D0 while add xl ∈ Gl to the same row of Dl for all l = 1, 2, . . . , m. By Theorem 2.10 and Corollary 2.11, it is easy to see that D is still a repeating-column difference matrix if do a transformation of repeating-column difference matrices on D.
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3. Orthogonal arrays and repeating-column difference matrices Suppose that an experiment is performed according to the array A=(aij )n×m =(a1 , . . . , am ) and Y =(y1 , y2 , . . . , yn )T is the experimental data vector. In the analysis of variance Sj2 , the sum of squares of the jth factor, is defined as Sj2
=
pj i=1
⎛ 1 ⎝ |Iij |
s∈Iij
⎞2
2 n 1 Ys , Ys ⎠ − n s=1
where Iij = {s : asj = i} and |Iij | is the number of elements in Iij . From the definition, Sj2 is a quadratic form in Y and there exists a unique symmetric matrix Aj such that Sj2 = Y T Aj Y. The matrix Aj is called the matrix image (MI) of the jth column aj of A, denoted by m(aj ) = Aj . The MI of a subarray of A is defined as the sum of the MIs of all its columns. In particular, we denote the MI of A by m(A) and the MIs of 1r , (r) are Pr , n , respectively. If a design is an orthogonal array, then the MIs of its columns has some interesting properties, which can be used to construct orthogonal arrays. Theorem 3.1. For any permutation matrix S and any array L, m(S(L ⊗ 1r )) = S(m(L) ⊗ Pr )S T
and m(S(1r ⊗ L)) = S(Pr ⊗ m(L))S T .
Theorem 3.2. Let the array A be an orthogonal array of strength 1, i.e., A = (a1 , . . . , am ) = (S1 (0r1 (p1 )), . . . , Sm (0rm (pm ))), where ri pi = n, Si is a permutation matrix, for i = 1, . . . , m. Then, the following statements are equivalent. (1) (2) (3) (4)
A is an orthogonal array of strength 2. The MI of A is a projection matrix. The MIs of any two columns of A are orthogonal, i.e., m(ai )m(aj ) = 0 (i = j ). The projection matrix n can be decomposed as n = m(a1 ) + · · · + m(am ) + , where rk( ) = n − 1 − m j =1 (pj − 1) is the rank of the matrix .
Definition 3.3. An orthogonal array A is said to be saturated if
m
j =1 (pj
− 1) = n − 1 (or, equivalently, m(A) = n ).
Corollary 3.4. Let (L, H ) and K be orthogonal arrays of size n. Then, (K, H ) is an orthogonal array if m(K)m(L), where m(K)m(L) means that the difference m(L) − m(K) is nonnegative definite. Corollary 3.5. Suppose that L and H are orthogonal arrays. Then, K = (L, H ) is also an orthogonal array if m(L) and m(H ) are orthogonal, i.e., m(L)m(H ) = 0. In this case, m(K) = m(L) + m(H ). Theorem 3.6. Suppose that D 0 (q, m; p) is an atomic difference matrix. Then, (p) D 0 (q, m; p) is an orthogonal array whose matrix image satisfying m((p) D 0 (rq, m; p))p ⊗ q . These theorems and corollaries can be found in Zhang [5–7] and Zhang et al. [8]. Our procedure of constructing mixed-level orthogonal arrays consists of the following three steps [10]: Step 1: Orthogonally decompose the projection matrix n : n = A1 + · · · + Ak
where Ai Aj = 0 (i = j ).
Step 2: Find an orthogonal array Li such that m(Li ) Ai .
Y. Zhang / Discrete Mathematics 307 (2007) 246 – 261
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Step 3: Lay out the new orthogonal array L by Corollaries 3.4 and 3.5: L = (L1 , . . . , Lk1 ) (k1 k). Let Lp =Lp (p1 , . . . , pm )=(C1 , . . . , Cm ) be a normal orthogonal array over G0 and denote C0 =C1 ♦ · · · ♦Cm and let
be an atomic difference matrix over G having p levels. And suppose that [D , D ] is D0 = D(q, k0 ; p) = D1 ♦ · · · ♦Dm 0 i i an atomic q ×(k0 +ki ) difference matrix having pi levels for i =1, 2, . . . , m. Thus, C =[C0 , C1 , . . . , Cm ] is a partitioned matrix in which the m + 1 columns C0 , C1 , . . . , Cm are orthogonal arrays of strength 1 having p, p1 , . . . , pm levels, respectively. Then, D=[D0 , D1 , . . . , Dm ] is an atomic repeating-column difference matrix about C =[C0 , C1 , . . . , Cm ] if D is a repeating-column difference matrix about C. Theorem 3.7. The matrix D = [D0 , D1 , . . . , Dm ] is an atomic repeating-column difference matrix about C = [C0 , C1 , . . . , Cm ] if and only if L = [C0 D0 , C1 D1 , . . . , Cm Dm ] is an orthogonal array whose matrix image satisfying m(L) p ⊗q . In particular, m(C0 D0 ) ⊗ q where = p − (m(C1 ) + · · · + m(Cm )).
m
i=1 m(Ci Di )+
Proof. ⇒ Consider the following orthogonal decomposition of projection matrix p ⊗ q : p ⊗ q = m(C1 ) ⊗ q + · · · + m(Cm ) ⊗ q + ⊗ q , where p = m(Lp ) + = m(C1 ) + · · · + m(Cm ) + . By Theorems 3.1, 3.2 and 3.6, we have m(Cj [Dj , Dj ])m(Cj ) ⊗ q ,
j = 1, 2, . . . , m,
, D ]] is an orthogonal array. By Corollary 3.5, we have that m(C D ) + i.e., [C1 [D1 , D1 ], . . . , Cm [Dm m i i
m(Ci Di ) = m(Ci [Di , Di ]), i = 1, 2, . . . , m. Thus,
m((C1 D1 ) + · · · + m(Cm Dm )) + ⊗ q p ⊗ q ,
m((C1 D1 ) + · · · + m(Cm Dm ))p ⊗ q , and
[m((C1 D1 ) + · · · + m(Cm Dm )) + ⊗ q ][m((C1 D1 ) + · · · + m(Cm Dm ))] = 0.
By their orthogonality and Theorem 3.2, L = [C0 D0 , C1 D1 , . . . , Cm Dm ] is an orthogonal array whose matrix image satisfies m(L)p ⊗ q if
m(C0 D0 )(m(C1 D1 ) + · · · + m(Cm Dm )) + ⊗ q .
In fact, let Dl = (dijl )q×k0 = (d1l , . . . , dkl 0 ) be an atomic difference matrix for any l. And denote C0 = (p). From Eqs. (1) and (2), we have [0, D0 ] (p) = (S00 (0q (p)), S10 (0q (p)), . . . , Sk00 (0q (p))), [0, Dl ] Cl = (S0l (0q Cl ), S1l (0n Cl ), . . . , Skl 0 (0q Cl )), l ), . . . , (d l )), j = 1, 2, . . . , m; l = 1, . . . , m. By Theorems 3.1 and 3.2, we have where S0l = Ipq and Sjl = diag(l (d1j l qj
m([0, D0 ] C0 ) = m([0, D0 ] (p)) =
k0 j =0
=
k0 m i=1 j =0
Sj0 (Pq ⊗ p )(Sj0 )T
Sj0 (Pq ⊗ m(Ci ))(Sj0 )T +
k0 j =0
Sj0 (Pq ⊗ )(Sj0 )T ,
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Y. Zhang / Discrete Mathematics 307 (2007) 246 – 261
since p =m(Lp )+ =m(C1 )+· · ·+m(Cm )+ . The above decompositions are orthogonal because of the orthogonality in each step. Thus, all items Sj0 (Pq ⊗m(Ci ))(Sj0 )T , Ss0 (Pq ⊗ )(Ss0 )T , i =1, 2, . . . , m; j, s =0, 1, . . . , k0 , are orthogonal to each other. By Theorem 2.10, we have that 0 T 0 0 0 T ) ) ) , . . . , 0 (dqj ) 0 (dqj ) 0 (d1j Sj0 (Pq ⊗ )(Sj0 )T Sj0 (Iq ⊗ )(Sj0 )T = diag(0 (d1j
= diag( , . . . , ) = Iq ⊗ , for j = 0, 1, . . . , m; and that Sj0 (Pq ⊗ m(Ci ))(Sj0 )T = m(Sj0 (0q Ci )) = m(Sji (0q Ci )) = m(dji Ci ). Thus, we obtain m([0, D0 ] C0 ) Pq ⊗ p +
m
m(Di Ci ) + q ⊗ ,
i=1
i.e., m(C0 D0 ) = K(p, q)m(D0 C0 )K(p, q)T = K(p, q)(m([0, D0 ] C0 ) − Pq ⊗ p )K(p, q)T m m
T m(Ci Di ) + ⊗ q . K(p, q) m(Di Ci ) + q ⊗ K(p, q) = i=1
i=1
This completes the proof of ⇒ . ⇐ Let L=[C0 D0 , C1 D1 , . . . , Cm Dm ] be an orthogonal array whose matrix image satisfies m(L) p ⊗q . Then, C0 [0, D0 ] and Ci [0, Di , Di ] are orthogonal arrays for all i. Thus, [0, D0 ] and [0, Di , Di ] are difference matrices for all i, i.e., [0, D] is a repeating-column difference matrix. It means that the matrix D is an atomic repeatingcolumn difference matrix about C. This completes the proof. Corollary 3.8. The matrix D = [D0 , D1 , …, Dm ] be an atomic repeating-column difference matrix about C if and only if L = [Lp 0q , 0p Lq , C0 D0 , C1 D1 , . . . , Cm Dm ] is an orthogonal array for any orthogonal arrays Lp and Lq . 4. Examples 4.1. Constructions of orthogonal arrays of run size 72 Zhang et al. [11] has constructed an orthogonal array L72 (28 38 64 121 ) whose structure is (=)
(=)
L72 (28 38 64 121 ) = [03 (12) 02 , 03 L24 (28 ), L36 (38 ) 02 , (−)
(−)
(M1 ⊗ Q1 )(L36 (62 ) 02 ), (M2 ⊗ Q2 )(L36 (62 ) 02 )], where Q1 = K(2, 2), Q2 = K(2, 2) diag(I2 , N2 )K(2, 2)T and M1 = K(3, 6) diag(N3 , N32 , Q1 ⊗ I3 )K(3, 6)T ,
M2 = K(3, 6) diag(N32 , N3 , Q2 ⊗ I3 )K(3, 6)T ;
and where the orthogonal arrays satisfy (=)
L24 (28 ) = D(12, 8; 2) (2),
8 L= 36 (3 ) = (3) D(12, 8; 3) +
L36 (62 ) = [[(3) (3) 04 ] (018 (2)], [(3) (3) ◦ 2 04 ] [09 (2) (2)]], (−)
in which D(12, 8; 2) and D(12, 8; 3) are some difference matrices.
Y. Zhang / Discrete Mathematics 307 (2007) 246 – 261
255
It is easy to prove that there exists a difference matrix D(12, 4; 6) = D(12, 4; 3)♦D(12, 4; 2) such that [(3) D(12, 4; 3) 02 ]♦[03 D(12, 4; 2) (2)] (−)
(−)
= [(M1 ⊗ Q1 )(L36 (62 ) 02 ), (M2 ⊗ Q2 )(L36 (62 ) 02 )]. Hence the following array, [D(12, 4; 6) [((3) 02 )♦(03 (2))], D(12, 8; 3) ((3) 02 ), D(12, 8; 2) (03 (2))] is also an orthogonal array. By Theorem 3.7, we have D = [D(12, 4; 6), D(12, 8; 3), D(12, 8; 2)] is a repeating-column difference matrix about [(6), (3) 02 , 02 (2)]. Let D(12, 4; 6) =(a0 , D(12, 3; 6)) where a0 = a1 ♦a2 . By the transformation of repeating-column difference matrix, we have [0, D 0 ] := [D(12, 4; 6) − a0 , D(12, 8; 3) − a1 , D(12, 8; 2) − a2 ] is also a repeating-column difference matrix. Thus, D 0 is an atomic repeating-column difference matrix. From above D 0 , we can construct some atomic repeating-column difference matrices of run size 12 having a large number of columns such as ⎛ ⎞ 00 00 00 00 0000000 0000000 ⎜ 01 21 20 11 2021011 0101100 ⎟ ⎜ ⎟ ⎜ 11 21 10 20 1002201 1110010 ⎟ ⎜ ⎟ ⎜ 01 20 10 00 0212112 1011101 ⎟ ⎜ ⎟ ⎜ 00 11 21 20 0110221 0011110 ⎟ ⎜ ⎟ ⎜ 20 10 00 21 1020112 0110111 ⎟ D 0 = [D 0 (12, 4; 6), D 0 (12, 7; 3), D 0 (12, 7; 2)] = ⎜ ⎟, ⎜ 10 11 11 10 2200022 1100101 ⎟ ⎜ ⎟ ⎜ 11 00 01 21 2211210 1000110 ⎟ ⎜ ⎟ ⎜ 21 01 11 11 0121202 0010001 ⎟ ⎜ ⎟ ⎜ 21 20 01 10 1112020 0101011 ⎟ ⎝ ⎠ 10 10 21 01 2122100 1111000 20 01 20 01 1201121 1001011 ⎛
00 ⎜ 20 ⎜ ⎜ 01 ⎜ ⎜ 00 ⎜ ⎜ 11 ⎜ ⎜ 10 0 0 0 0 D = [D (12, 5; 6), D (12, 3; 3), D (12, 6; 2)] = ⎜ ⎜ 11 ⎜ ⎜ 10 ⎜ ⎜ 21 ⎜ ⎜ 21 ⎝ 20 01
00 00 11 20 01 01 20 11 21 10 21 10
00 11 21 01 20 21 10 00 01 10 20 11
00 21 10 20 10 20 00 01 11 21 01 11
00 20 20 01 00 11 21 21 11 01 10 10
000 011 000 212 221 102 102 221 120 120 011 212
⎞ 000000 011011 ⎟ ⎟ 110001 ⎟ ⎟ 101101 ⎟ ⎟ 001111 ⎟ ⎟ 010110 ⎟ ⎟, 111010 ⎟ ⎟ 100011 ⎟ ⎟ 001000 ⎟ ⎟ 010101 ⎟ ⎠ 111100 100110
over G60 . Define 0 = 00, 1 = 11, 2 = 20, 3 = 01, 4 = 10, 5 = 21. Then, the group is Z6 . By using Corollary 3.8 and above atomic repeating-column difference matrices D 0 s, we can construct a lot of new orthogonal arrays of run size 72, which are exhibited in Table 1 or in Kuhfeld [3].
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Y. Zhang / Discrete Mathematics 307 (2007) 246 – 261
Table 1 Orthogonal arrays L72 (· · ·) obtained in Section 4.1 No.
f1 − f10
c1 − c10
b1 − b7
b8 − b18
lf dc
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48
0000000000 1111111111 2222222222 3333333333 4444444444 5555555555 0220512405 1331023510 2442134021 3553245132 4004350243 5115401354 0231454114 1342505225 2453010330 3504121441 4015232552 5120343003 0302233240 1413344351 2524455402 3035500513 4140011024 5 2 5 1 1 22 1 3 5 0013420152 1124531203 2235042314 3340153425 4451204530 5502315041 0354543521 1405054032 2510105143 3021210254 4132321305 5243432410 0512041542 1023152053 2134203104 3245314215 4350425320 5401530431 0541301035 1052412140 2103523251 3214034302 4325145413 5430250524
0000000000 1111111111 2222222222 0000000000 1111111111 2222222222 0111020112 1222101220 2000212001 0111020112 1222101220 2000212001 0002200022 1110011100 2221122211 0002200022 1110011100 2221122211 2120212112 0201020220 1012101001 2120212112 0201020220 1012101001 2210110221 0021221002 1102002110 2210110221 0021221002 1102002110 1202021011 2010102122 0121210200 1202021011 2010102122 0121210200 1021002201 2102110012 0210221120 1021002201 2102110012 0210221120 2212211210 0020022021 1101100102 2212211210 0020022021 1101100102
0000000 1111111 0000000 1111111 0000000 1111111 0110111 1001000 0110111 1001000 0110111 1001000 1100101 0011010 1100101 0011010 1100101 0011010 1011101 0100010 1011101 0100010 1011101 0100010 0011110 1100001 0011110 1100001 0011110 1100001 0101100 1010011 0101100 1010011 0101100 1010011 1110010 0001101 1110010 0001101 1110010 0001101 1000110 0111001 1000110 0111001 1000110 0111001
00000000000 00000000000 00000000000 00000000000 00000000000 00000000000 11100010101 11100010101 11100010101 11100010101 11100010101 11100010101 01011110001 01011110001 01011110001 01011110001 01011110001 01011110001 10111010010 10111010010 10111010010 10111010010 10111010010 10111010010 00010011111 00010011111 00010011111 00010011111 00010011111 00010011111 11001001011 11001001011 11001001011 11001001011 11001001011 11001001011 01100111010 01100111010 01100111010 01100111010 01100111010 01100111010 10110101001 10110101001 10110101001 10110101001 10110101001 10110101001
0000 0030 0000 0030 0000 0030 1010 1020 1010 1020 1010 1020 2130 2100 2130 2100 2130 2100 3120 3110 3120 3110 3120 3110 4201 4231 4201 4231 4201 4231 5211 5221 5211 5221 5211 5221 6331 6301 6331 6301 6331 6301 7321 7311 7321 7311 7321 7311
49 50 51 52 53 54 55 56 57
0155135311 1200240422 2311351533 3422402044 4533513155 5044024200 0143255234 1254300345 2305411450
1200121202 2011202010 0122010121 1200121202 2011202010 0122010121 1021112020 2102220101 0210001212
0010001 1101110 0010001 1101110 0010001 1101110 0101011 1010100 0101011
00101100111 00101100111 00101100111 00101100111 00101100111 00101100111 11010100110 11010100110 11010100110
8402 8432 8402 8432 8402 8432 9412 9422 9412
Y. Zhang / Discrete Mathematics 307 (2007) 246 – 261
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Table 1 (continued) No.
f1 − f10
c1 − c10
b1 − b7
b8 − b18
lf dc
58 59 60 61 62 63 64 65 66 67 68 69 70 71 72
3410522501 4521033012 5032144123 0425324453 1530435504 2041540015 3152051120 4203102231 5314213342 0434112323 1545223434 2050334545 3101445050 4212550101 5323001212
1021112020 2102220101 0210001212 0112122100 1220200211 2001011022 0112122100 1220200211 2001011022 2121201121 0202012202 1010120010 2121201121 0202012202 1010120010
1010100 0101011 1010100 1111000 0000111 1111000 0000111 1111000 0000111 1001011 0110100 1001011 0110100 1001011 0110100
11010100110 11010100110 11010100110 01111001100 01111001100 01111001100 01111001100 01111001100 01111001100 10001111100 10001111100 10001111100 10001111100 10001111100 10001111100
9422 9412 9422 10 5 3 2 10 5 0 2 10 5 3 2 10 5 0 2 10 5 3 2 10 5 0 2 11 5 2 2 11 5 1 2 11 5 2 2 11 5 1 2 11 5 2 2 11 5 1 2
L72 (121 65 37 27 ) = (l f1 f7 − f10 c4 − c10 b1 − b7 )
(old)
L72 (41 65 37 215 ) = (d f1 f7 − f10 c4 − c10 b3 − b7 b9 − b18 ) L72 (41 65 38 28 ) = (d f1 f7 − f10 c4 − c10 c b3 − b7 b9 − b11 ) L72 (41 66 37 26 ) = (d f1 f7 − f10 f c4 − c10 b3 − b7 b9 ) L72 (121 66 33 26 ) = (l f1 − f6 c1 − c3 b1 − b6 )
(old)
L72 (41 66 33 214 ) = (d f1 − f6 c1 − c3 b3 − b6 b9 − b18 ) L72 (41 66 34 27 ) = (d f1 − f6 c1 − c3 c b3 − b6 b9 − b11 ) L72 (41 67 33 25 ) = (d f1 − f6 f c1 − c3 b3 − b6 b9 ) where d = b1 ♦b2 ♦b8 ,
l = b8 ♦ · · · ♦b18 ,
m(c) m(b12 − b18 ),
m(f ) m(b10 − b18 )
4.2. Construction of orthogonal arrays of run size 96 Zhang et al. [9] has constructed an orthogonal array L96 (212 420 241 ) whose structure is L96 (212 420 241 ) = [D 1 (12, 4; 2) 02 (2) 02 , D 2 (12, 4; 2) 04 (2), D 3 (12, 4; 2) (2) (2) (2), D(24, 20; 4) (4), (24) 04 ], where D(24, 20; 4), D 1 (12, 4; 2) 02 , D 2 (12, 4; 2) 02 and D 3 (12, 4; 2) (2) are some difference matrices.
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Y. Zhang / Discrete Mathematics 307 (2007) 246 – 261
It is easy to prove that L4 (23 )=((2) 02 , 02 (2), (2) (2)) is normal (Example 2.8). By Theorem 3.7, the following array: D = [D(24, 20; 4), D 1 (12, 4; 2) 02 , D 2 (12, 4; 2) 02 , D 3 (12, 4; 2) (2)] is a repeating-column difference matrix about [(4), (2) 02 , 02 (2), (2) (2)]. Let D(24, 20; 4) = (a0 , D(24, 19; 4)), where a0 = a1 ♦a2 ♦a3 . By the transformation of repeating-column difference matrix, we have [0, D 0 ] : =[a0 + D(24, 4; 4), a1 + D 1 (12, 4; 2) 02 , a2 + D 2 (12, 4; 2) 02 , a3 + D 3 (12, 4; 2) (2)] is also a repeating-column difference matrix. Thus, D 0 is an atomic repeating-column difference matrix ⎛ ⎞ 0000111112222233333 0011 0101 0110 ⎜ ⎟ ⎜ 1332102232311032001 0011 0101 1001 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 0000222223333311111 0110 0011 0101 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 1332231103200110223 0110 0011 1010 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 0000333331111122222 0101 0110 0011 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 1332320011022323110 0101 0110 1100 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 2230022300223002230 1111 1111 0000 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 3102031020310203102 1111 1111 1111 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 2230330223302233022 1000 1000 0000 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 3102323103231032310 1000 1000 1111 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 2230203032030320303 1111 1000 0111 ⎟ ⎜ ⎟ ⎜ ⎟ 3102210312103121031 1111 1000 1000 ⎜ ⎟ ⎟, D0 = ⎜ ⎜ ⎟ ⎜ 0311311030031113030 1010 1011 0001 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 1023302310102312302 1010 1011 1110 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 0311003111303031103 1100 1110 0010 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 1023010231230230231 1100 1110 1101 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 0311130303110300311 1001 1101 0100 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 1023123023023101023 1001 1101 1011 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 2121200120212111200 0001 0100 0101 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 3213213200321310132 0001 0100 1010 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 2121112002001202121 0100 0010 0110 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 3213101322132003213 0100 0010 1001 ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ 2121021211120020012 0010 0001 0011 ⎠ 3213032131013221320
0010
0001
1100
where 0 = 000, 1 = 011, 2 = 101, 3 = 110 and x + y = y + x, x + x = 0, 0 + x = x, 2 + 3 = 1, 3 + 1 = 2, 2 + 1 = 3. By using Corollary 3.8 and above atomic repeating-column difference matrix D 0 , we can construct a lot of new orthogonal arrays of run size 96, which are exhibited in Table 2 or in Kuhfeld [3].
Y. Zhang / Discrete Mathematics 307 (2007) 246 – 261
259
Table 2 Orthogonal arrays L96 (· · ·) obtained in Section 4.2 L96 (420 235 )
reL96
d1 − d20
b1 − b35
f d 21 − d23 clx
00000111112222233333 11111000003333322222 22222333330000011111 33333222221111100000 01332102232311032001 10223013323200123110 23110320010133210223 32001231101022301332 00000222223333311111 11111333332222200000 22222000001111133333 33333111110000022222 01332231103200110223 10223320012311001332 23110013321022332001 32001102230133223110 00000333331111122222 11111222220000033333 22222111113333300000 33333000002222211111 01332320011022323110 10223231100133232001 23110102233200101332 32001013322311010223 02230022300223002230 13321133211332113321 20012200122001220012 31103311033110331103 03102031020310203102 12013120131201312013 21320213202132021320 30231302313023130231 02230330223302233022 13321221332213322133 20012112001120011200 31103003110031100311 03102323103231032310 12013232012320123201 21320101321013210132 30231010230102301023 02230203032030320303 13321312123121231212 20012021210212102121 31103130301303013030 03102210312103121031 1 20 1 3 3 0 1 2 0 3 0 1 2 0 3 0 1 2 0 21320032130321303213 30231123021230212302 00311311030031113030 11200200121120002121 22133133212213331212 33022022303302220303 01023302310102312302 10132213201013203213 23201120132320130120 32310031023231021031
00110101011000010111101111110000010 00111010100100010111101111110000010 11000101100100010111101111110000010 11001010011000010111101111110000010 00110101100111100110100110000010111 00111010011011100110100110000010111 11000101011011100110100110000010111 11001010100111100110100110000010111 01100011010101111101000010110110100 01101100101001111101000010110110100 10010011101001111101000010110110100 10011100010101111101000010110110100 01100011101010001101110011000101110 01101100010110001101110011000101110 10010011010110001101110011000101110 10011100101010001101110011000101110 01010110001101111101000101001001011 0 1 0 1 1 0 0 1 1 10 0 0 1 1 1 1 1 0 1 0 0 0 1 0 1 0 0 1 0 0 1 0 1 1 10100110110001111101000101001001011 10101001001101111101000101001001011 01010110110010001101110100111010001 01011001001110001101110100111010001 10100110001110001101110100111010001 10101001110010001101110100111010001 11111111000000010111101000001111101 11110000111100010111101000001111101 00001111111100010111101000001111101 00000000000000010111101000001111101 11111111111111100110100001111101000 11110000000011100110100001111101000 00001111000011100110100001111101000 00000000111111100110100001111101000 10001000000000000000000000000000000 10000111111100000000000000000000000 01111000111100000000000000000000000 01110111000000000000000000000000000 10001000111101100001111000110001111 10000111000001100001111000110001111 01111000000001100001111000110001111 01110111111101100001111000110001111 11111000011111011000101001011010110 11110111100011011000101001011010110 00001000100011011000101001011010110 00000111011111011000101001011010110 11111000100001001110011001100110011 11110111011101001110011001100110011 00001000011101001110011001100110011 00000111100001001110011001100110011 10101011000100000000000111111111111 10100100111000000000000111111111111 01011011111000000000000111111111111 01010100000100000000000111111111111 10101011111001100001111111001110000 10100100000101100001111111001110000 01011011000101100001111111001110000 01010100111001100001111111001110000
0012000 0011000 0022000 0021000 0112011 0111011 0122011 0121011 1331022 1332022 1301022 1302022 1231033 1232033 1201033 1202033 1023024 1020024 1013024 1010024 1123035 1120035 1113035 1110035 0333006 0330006 0303006 0300006 0233017 0230017 0203017 0200017 2000148 2003148 2030148 2033148 3000159 3003159 3030159 3033159 4 1 3 0 2 6 10 4 1 3 3 2 6 10 4 1 0 0 2 6 10 4 1 0 3 2 6 10 5 3 3 0 2 7 11 5 3 3 3 2 7 11 5 3 0 0 2 7 11 5 3 0 3 2 7 11 2 3 1 1 1 4 12 2 3 1 2 1 4 12 2 3 2 1 1 4 12 2 3 2 2 1 4 12 3 3 1 1 1 5 13 3 3 1 2 1 5 13 3 3 2 1 1 5 13 3 3 2 2 1 5 13
260
Y. Zhang / Discrete Mathematics 307 (2007) 246 – 261
Table 2 (continued) L96 (420 235 )
reL96
d1 − d20
b1 − b35
f d 21 − d23 clx
00311003111303031103 11200112000212120012 22133221333121213321 33022330222030302230 01023010231230230231 10132101320321321320 23201232013012012013 32310323102103103102 00311130303110300311 11200021212001211200 22133312121332122133 33022203030223033022 01023123023023101023 10132032132132010132 23201301201201323201 32310210310310232310 02121200120212111200 13030311031303000311 20303022302030333022 31212133213121222133 03213213200321310132 12302302311230201023 21031031022103132310 30120120133012023201 02121112002001202121 13030003113110313030 20303330220223020303 31212221331332131212
11001110001011010011010101010100101 11000001110111010011010101010100101 00111110110111010011010101010100101 00110001001011010011010101010100101 11001110110110110100011101100011100 11000001001010110100011101100011100 00111110001010110100011101100011100 00110001110110110100011101100011100 10011101010011010011010010101011010 10010010101111010011010010101011010 01101101101111010011010010101011010 01100010010011010011010010101011010 10011101101110110100011010011100011 10010010010010110100011010011100011 01101101010010110100011010011100011 01100010101110110100011010011100011 0 0 0 1 0 1 0 0 0 1 01 0 0 1 1 1 0 1 0 1 1 0 0 1 1 1 0 1 0 0 0 1 0 1 00011011101000111010110011101000101 11100100101000111010110011101000101 11101011010100111010110011101000101 00010100101010101011001011010011001 00011011010110101011001011010011001 11100100010110101011001011010011001 11101011101010101011001011010011001 01000010011011011000101110100101001 01001101100111011000101110100101001 10110010100111011000101110100101001 10111101011011011000101110100101001
2 2 2 3 1 8 14 2 2 2 0 1 8 14 2 2 1 3 1 8 14 2 2 1 0 1 8 14 3 1 2 3 1 9 15 3 1 2 0 1 9 15 3 1 1 3 1 9 15 3 1 1 0 1 9 15 2 1 0 2 1 8 16 2 1 0 1 1 8 16 2 1 3 2 1 8 16 2 1 3 1 1 8 16 3 2 0 2 1 9 17 3 2 0 1 1 9 17 3 2 3 2 1 9 17 3 2 3 1 1 9 17 4 0 0 2 2 10 18 4 0 0 1 2 10 18 4 0 3 2 2 10 18 4 0 3 1 2 10 18 5 1 0 2 2 11 19 5 1 0 1 2 11 19 5 1 3 2 2 11 19 5 1 3 1 2 11 19 4 2 2 1 2 6 20 4 2 2 2 2 6 20 4 2 1 1 2 6 20 4 2 1 2 2 6 20
03213101322132003213 12302010233023112302 21031323100310221031 30120232011201330120 02121021211120020012 13030130300031131103 20303203033302202230 31212312122213313321 03213032131013221320 12302123020102330231 21031210313231003102 30120301202320112013
01000010100101001110011110011001100 01001101011001001110011110011001100 10110010011001001110011110011001100 10111101100101001110011110011001100 00100001001100111010110100010111010 00101110110000111010110100010111010 11010001110000111010110100010111010 11011110001100111010110100010111010 00100001110010101011001100101100110 00101110001110101011001100101100110 11010001001110101011001100101100110 11011110110010101011001100101100110
5 0 2 1 2 7 21 5 0 2 2 2 7 21 5 0 1 1 2 7 21 5 0 1 2 2 7 21 4 3 1 0 2 10 22 4 3 1 3 2 10 22 4 3 2 0 2 10 22 4 3 2 3 2 10 22 5 2 1 0 2 11 23 5 2 1 3 2 11 23 5 2 2 0 2 11 23 5 2 2 3 2 11 23
L96 (241 420 212 ) = (x d1 − d20 b1 − b12 )
L96 (121 422 218 ) = (l d1 − d20 d22 d23 b1 b4 b5 b8 − b12 b26 − b35 )
L96 (423 226 ) = (d1 − d23 b1 b4 b5 b8 − b12 b14 − b23 b26 − b29 b32 − b35 )
L96 (31 423 219 ) = (c d1 − d23 b1 b4 b5 b8 − b12 b14 − b16 b26 − b29 b32 − b35 )
L96 (61 423 217 ) = (f d1 − d23 b1 b4 b5 b8 − b12 b14 b26 − b29 b32 − b35 ).
Y. Zhang / Discrete Mathematics 307 (2007) 246 – 261
261
Table 2 (continued) L96 (420 235 ) d1 − d20
reL96 f d 21 − d23 clx
b1 − b35
where d21 = b13 ♦b30 ♦b31 , x = b13 ♦ · · · ♦b35 , m(c) m(b13 − b23 ),
d22 = b2 ♦b3 ♦b24 ,
d23 = b6 ♦b7 ♦b25 ,
l = b13 ♦ · · · ♦b23 , m(f ) m(b15 − b23 ).
Acknowledgements The author would like to thank the referee for his many valuable suggestions and comments. The work was supported by National Social Science Foundations (No. 10571045, No. 0224370051 (Henan) and No. 0211063100 (Henan)) in China. References [1] T. Beth, D. Jungnickel, H. Lenz, 1985. Design Theory, Bibliographishes Institut, Mannheinu-Wien-Zürich, 1985, and Cambridge University Press, Cambridge, 1986. [2] R.C. Bose, K.A. Bush, Orthogonal arrays of strength two and three, Ann. Math. Statist. 23 (1952) 508–524. [3] W.F. Kuhfeld, Orthogonal arrays, http://support.sas.com/technote/ts723.html, 2006. [4] S. Shrikhande, Generalized Hadamard matrices and orthogonal arrays strength two, Canad. J. Math. 16 (1964) 736–740. [5] Y.S. Zhang, Asymmetrical orthogonal design by multi-matrix methods, J. Chinese Statist. Assoc. 29 (1991) 197–218. [6] Y.S. Zhang, Orthogonal array and matrices, J. Math. Res. Exposition 12 (3) (1992) 438–440. [7] Y.S. Zhang, Theory of Multilateral Matrix, Chinese Statistic Press, 1993. [8] Y.S. Zhang, W.G. Li, S.S. Mao, Z.Q. Zheng, A simple method for constructing orthogonal arrays by the Kronecker sum, J. Syst. Sci. Complexity 19 (2006) 266–273. [9] Y.S. Zhang, Orthogonal arrays obtained by generalized Kronecker product, J. Statist. Plann. Inference (2000), in review. [10] Y.S. Zhang, Y.Q. Lu, S.Q. Pang, Orthogonal arrays obtained by orthogonal decomposition of projection matrices, Statist. Sinica 9 (1999) 595–604. [11] Y.S. Zhang, S.Q. Pang, Y.P. Wang, Orthogonal arrays obtained by generalized Hadamard product, Discrete Math. 238 (2001) 151–170.
