Orthogonal arrays obtained by repeating-column difference matrices

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1 Discrete Mathematics 307 (2007) Orthogonal arrays obtained by repeating-column difference matrices Yingshan Zhang Department of Statistics, East China Normal University, Shanghai, , 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 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 k i columns with p i ( 2) levels, i = 1,...,t,m= t i=1 k i,p i p j, 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 L n (p k 1 1,...,pk t t ) 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 λp 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 d i d j if i j, contains every element of G exactly λ times. We will denote such an array by D(λp, 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(λp, m; p) is called a generalized Hadamard matrix if λp = m. In particular, D(λ2, λ2; 2) is the usual Hadamard matrix. If a D(λp, 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(λp, m; p), we obtain a difference matrix, denoted by D 0 (λp, m 1; p) called an atom of difference matrix D(λp, m; p) (or an atomic difference matrix). Without loss of generality, address: ysh_zhang@163.com X/$ - see front matter 2006 Elsevier B.V. All rights reserved. doi: /j.disc

2 Y. Zhang / Discrete Mathematics 307 (2007) the D(λp, m; p) can be written as ( ) 0 0 D(λp, m; p) = = (0 D 0 (λp, m 1; p)). 0 A The property is important for the following discussions. For two matrices A = (a ij ) n m and B = (b ij ) s t both with the entries from group G, define their Kronecker sum [4] to be A B = (a ij B) 1 i n,1 j m, where each submatrix a ij B of A B is obtained by adding a ij 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 = L μp (p s ) D(λp, m; p) (or L = D(λp, m; p) L μp (p s )) is an orthogonal array if L μp (p s ) is an orthogonal array and D(λp, r; p) is a difference matrix [1]. By setting μ=s =1, the Kronecker sum method reduces to the well-known construction of Bose and Bush [2], i.e., L = (p) D(λp, m; p) (or L = D(λp, m; p) (p)) is an orthogonal array if D(λp, m; p) is a difference matrix. In the contrast, Zhang [7] has found that the difference matrix D(λp, m; p) can be also constructed by using the orthogonal array L = (p) D(λp, m; p), i.e., D(λp, m; p) is a difference matrix if L = (p) D(λp, m; p) is an orthogonal array. Let D(r, m; p) = (d ij ),wehave L = D(r, m; p) (p) =[S 1 (0 r (p)),...,s m (0 r (p))], (1) where S j = diag(σ(d 1j ),...,σ(d rj )) and σ(d ij ) is a permutation matrix such that σ(d ij )(p) = d ij (p), (2) for any i, j, the d ij (p) can be obtained by adding d ij to each entry of (p). The idea of Kronecker sum and difference matrices can be generalized as follows [7]. Let n=pq. IfA is an orthogonal array L p (p k 1 1,...,pk t t ) with the partition A =[L p (p k 1 1 ),...,L p(p k t t )], and if there exist the atoms: D 0 (λ 1 p 1,m 1 1; p 1 ),...,D 0 (λ t p t,m t 1; p t ), of difference matrices D(λ 1 p 1,m 1 ; p 1 ),...,D(λ t p t,m t ; p t ), respectively, where q = λ i p i and p and q are both multiples the p i s, then the following array: [L p 0 q, 0 p L q,l p (p k 1 1 ) D0 (λ 1 p 1,m 1 1; p 1 ),...,L p (p k t t ) D 0 (λ t p t,m t 1; p t )] is also an orthogonal array for any orthogonal arrays L p and L q. In this paper, we will prove that the following array: [L p 0 q, 0 p L q,(p) D 0 (λ 0 p, m 0 1; p), L p (p 1 1 ) D0 (λ 1 p 1,m 1 1; p 1 ),...,L p (p 1 t ) D0 (λ t p t,m t 1; p t )], is also an orthogonal array for any orthogonal arrays L p and L q if A =[L p (p1 1),...,L p(pt 1 )] is a normal orthogonal array and D 0 =[D 0 (λ 0 p, m 0 1; p), D 0 (λ 1 p 1,m 1 1; p 1 ),...,D 0 (λ t p t,m t 1; p t )] is an atomic repeating-column difference matrix (Section 2).

3 248 Y. Zhang / Discrete Mathematics 307 (2007) 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 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 Ω 1 Ω 2 to V, where Ω 1 Ω 2 ={(x, y) : x Ω 1,y Ω 2 } and Ω 1, Ω 2,V are some sets. For two matrices A = (a ij ) n m with entries from Ω 1 and B = (b ij ) n m with entries from Ω 2, define their generalized Hadamard product, denoted by, h as follows: A h B = (h(a ij,b ij )) n m = (h(a ij,b ij )) 1 i n,1 j m, where each of the entries h(a ij,b ij ) of A h B may be a scalar or vector or matrix under the mapping h(x, y). Unless stated otherwise, we consider the sets Ω 1 and Ω 2 to be finite, using the notations Ω 1 ={0, 1,...,p 1} and Ω 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 :[(p) 0 q ] h [0 p (q)]=d = (d (1),...,d (pq) ) T, with h(i, j) = d(iq+j+1) T (or h(i, j) is the (iq + j + 1)th row of D). For this case in the following discussions, the generalized Hadamard product h will only be defined by [(p) 0 q ] [0 h p (q)]=d. Note that [(p) 0 q ] [0 h p (q)]= (p) (q), h the generalized Hadamard product h will be also defined by (p) (q) h = D. This is a form of generalized Kronecker product. Let Ω 1, Ω 2,V be multiplicative groups. When h(i, j)=ij, the generalized Hadamard h is the usual Hadamard product in matrix theory, denoted by. Let Ω 1, Ω 2,V be additive (Abelian) groups. When h(i, j) = i + j, the generalized Hadamard h is the usual addition of matrices in matrix theory, denoted by +. Let Ω 1 = Ω 2 = V ={0, 1,...,p 1} and h(i, j) = i + j, mod p, Then, the generalized Hadamard product h 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 Ω 1 ={0, 1,...,p 1}, Ω 2 ={0, 1,...,q 1},V ={(i, j); i Ω 1,j Ω 2 }. The generalized Hadamard product h 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 Ω 1 ={0, 1,...,p 1}, Ω 2 ={0, 1,...,q 1},V ={0, 1,...,pq 1}. The generalized Hadamard product h 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, and (A B) (C D) = (A C) (B D), T(A B C D) = TA TB TC TD. Let m(a) be the matrix image of array A ([10,11] or Section 3), we have

4 Y. Zhang / Discrete Mathematics 307 (2007) Theorem 2.3. Suppose that a and b are two orthogonal arrays which have only one column with run size n, i.e., a = L n (p) = (a 1,...,a n ) T,b= L n (q) = (b 1,...,b n ) 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 = (a 1 b 1,...,a n b n ) T (or a b = (a 1 q + b 1,...,a n q + b n ) 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 =L n (p 1,...,p m )=(L n (p 1 ),...,L n (p m )) and K 2 =L n (q 1,...,q m )=(L n (q 1 ),...,L n (q m )) be two orthogonal arrays of run size n. Denote (L n (p 1 ) L n (q 1 ),...,L n (p m ) L n (q m )) =: 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 L n1 = L n1 (p 1,...,p m ) = (L n1 (p 1 ),...,L n1 (p m )) and L n2 = L n2 (q 1,...,q m ) = (L n2 (q 1 ),...,L n2 (q m )) are two orthogonal arrays. Then (L n1 0 n2 ) (0 n1 L n2 ) is also an orthogonal array. In this case, its matrix image satisfies m((l n1 0 n2 ) (0 n1 L n2 )) m(l n1 ) P n2 + P n1 m(l n2 ) + m(l n1 ) m(l n2 ). 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 =L n (p)=(a 1,...,a n ) T,b=L n (p)=(b 1,...,b n ) T. Then, L n (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, L p 2(p p 1 )=((p) (p),...,(p) + [p 1] (p)), mod p, is also an OA whose MI is τ p τ 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 L p = L p (p 1,...,p m ) = (C 1,...,C m ) be an orthogonal array where C l is a vector with entries from a additive group G l of order p l for any l. The array L p is called normal over G 0 if the set consisted of all entries of vector C 0 = C 1 C m is a additive group G 0 of order p, where G 0 G 1 G m := {(x 1,...,x m ); x l G l,l= 1, 2,...,m} with the usual addition: (x 1,x 2,...,x m ) + (y 1,y 2,...,y m ) = (x 1 + y 1,x 2 + y 2,...,x m + y m ) (x 1,x 2,...,x m ), (y 1,y 2,...,y m ) G 0. Example 2.8. The following arrays are normal: a L 4 ( ) =, L ( ) =, over G 4 0 ={(0, 0), (0, 1), (1, 0), (1, 1)} and G4 0 ={(0, 0, 0), (0, 1, 1), (1, 0, 1), (1, 1, 0)}, respectively.

5 250 Y. Zhang / Discrete Mathematics 307 (2007) b L 6 ( ) =, L ( ) =, over G 6 0 ={(0, 0), (0, 1),...,(2, 1)} and G8 0 ={(0, 0, 0, 0, 0, 0, 0),...,(1, 1, 0, 1, 0, 0, 1)}, respectively. c L 12 ( ) =, L ( ) =, L ( ) =, over G 12 0 G 12 0 respectively. ={(0, 0), (0, 1),...,(1, 5)}, G12 0 ={(0, 0, 0, 0), (0, 1, 1, 0),...,(1, 1, 0, 2)}, ={(0, 0), (0, 1),...,(3, 2)} and Definition 2.9. Let L p = L p (p 1,...,p m ) = (C 1,...,C m ) be a normal orthogonal array over G 0 and denote C 0 =C 1 C m and let D 0 =D(q, k 0 ; p)=d 1 D m be a difference matrix over G 0 having p levels.and suppose that [D i,d i] is an p (k 0 + k i ) difference matrix having p i levels for i = 1, 2,...,m.Thus, C =[C 0,C 1,...,C m ] is a partitioned matrix in which the m+1 columns C 0,C 1,...,C m are orthogonal arrays of strength 1 having p, p 1,...,p m levels, respectively. We call D =[D 0,D 1,...,D m ] a repeating-column difference matrix about C =[C 0,C 1,...,C m ]. The repeating-column difference matrix D is called an atomic repeating-column difference matrix if D j is atomic for any j. Theorem Let L p = L p (p 1,...,p m ) = (C 1,...,C m ) be a normal orthogonal array and denote C 0 = C 1 C m. Then, there exist p permutation matrices σ 0 (x), x G 0 and p l permutation matrices σ l (x l ), x l G l, l = 1, 2,...,m,such that σ 0 (x) C 0 = x C 0, σ 0 (x) C l = σ l (x l ) C l = x l C l, x = (x 1,x 2,...,x m ) G 0, l = 1, 2,...,m where x l C l stands for the vector obtained by x l to each entry of C l. In other words, we have σ 0 (x)l p = (σ 1 (x 1 )C 1,...,σ m (x m )C m ). In this case, the matrix images of C 0,C 1,C 2,...,C m satisfy the following equations: m(c l ) = m(σ 0 (x)c l ) x = (x 1,x 2,...,x m ) G 0, l = 0, 1, 2,...,m.

6 Y. Zhang / Discrete Mathematics 307 (2007) In particular, if let =τ p m(c 1 ) m(c m ), then, we have σ 0 (x) σ 0 (x) T =. Proof. Since the set of entries of vector C 0 = C 1 C m is an additive group G 0, for any given x G 0, there exists a permutation matrix σ 0 (x) such that σ 0 (x) C 0 = x C 0, where x C 0 stands for the vector obtained by x to each entry of C 0. Furthermore, since the order of group G 0 is p, i.e., x C 0 = C 0 iff x = (0, 0,...,0), we have σ 0 (x) σ 0 (y) if x y. Similarly, because C l is a vector with entries from an group G l and having a form C l = T l (0 λl (p l )) where λ l p l = p and T l is a permutation matrix for any l, for any given x l G l there exists a permutation matrix σ l (x l ) such that σ l (x l ) C l = x l C l = T l (0 λl [x l (p l )]), since there exists a permutation matrix π l (x l ) such that π l (x l )(p l ) = x l (p l ), where λ l p l = p and T l is a permutation matrix for any l. From above results and Theorem 2.2, we have (σ 0 (x)c 1 ) (σ 0 (x)c m ) = σ 0 (x)c 0 = x C 0 = (x 1 C 1 ) (x m C m ) = (σ 1 (x 1 )C 1 ) (σ m (x m )C m ), i.e., σ 0 (x)c l = σ l (x l )C l, l = 1, 2,...,m. Now, we prove that m(c l ) = m(σ 0 (x)c l ), x G 0,l= 0, 1, 2,...,m. In fact, that m(c 0 )=m(σ 0 (x)c 0 ) is trivial since m(c 0 )=τ p =σ 0 (x)τ p σ 0 (x) T =m(σ 0 (x)c 0 ). Since G l is an additive group of order s l, there exists a permutation matrix π l (x l ) such that π l (x l )(p l ) = x l (p l ). By the form of C l and the definition of matrix image (Section 2), for any x l G l, m(σ 0 (x)c l ) = m(σ l (x l )C l ) = m(x l C l ) = m(t l (0 λl [x l (p l )])) = m(t l (0 λl [π l (x l )(p l )])) = T l (P λl [π l (x l )τ pl π l (x l ) T ])Tl T = T l (P λl τ pl )Tl T = m(c l ), for any l. This completes the proof. Corollary Let L q = (C 1,...,C m ) be normal and D = (D 0,D 1,...,D m ) a repeating-column difference matrix about C 0,C 1,...,C m with entries from G 0,G 1,...,G m where C 0 = C 1 C m. Then, for any vector a 0 = a 1 a m with entries from G 0, the following array: [a 0 + D 0,a 1 + D 1,...,a m + D m ] is also a repeating-column difference matrix, where a j + D j means that a j is added to each column of D j. Definition Let L q = (C 1,...,C m ) be normal and D = (D 0,D 1,...,D m ) a repeating-column difference matrix about C 0,C 1,...,C m with entries from G 0,G 1,...,G m where C 0 =C 1 C m. 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 D l for a given l = 0, 1, 2,...,m. (b) Add an element x l G l to some column of D l for a given l = 0, 1, 2,...,m. (c) Add an element x = (x 1,...,x m ) G 0 to some row of D 0 while add x l G l to the same row of D l 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.

7 252 Y. Zhang / Discrete Mathematics 307 (2007) Orthogonal arrays and repeating-column difference matrices Suppose that an experiment is performed according to the array A=(a ij ) n m =(a 1,...,a m ) and Y =(y 1,y 2,...,y n ) T is the experimental data vector. In the analysis of variance Sj 2, the sum of squares of the jth factor, is defined as p j Sj 2 = i=1 1 I ij s Iij Y s 2 ( n ) 2 1 Y s, n s=1 where I ij ={s : a sj = i} and I ij is the number of elements in I ij. From the definition, Sj 2 is a quadratic form in Y and there exists a unique symmetric matrix A j such that Sj 2 = Y T A j Y. The matrix A j is called the matrix image (MI) of the jth column a j of A, denoted by m(a j ) = A j. 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 1 r,(r)are P r, τ 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 1 r )) = S(m(L) P r )S T and m(s(1 r L)) = S(P r m(l))s T. Theorem 3.2. Let the array A be an orthogonal array of strength 1, i.e., A = (a 1,...,a m ) = (S 1 (0 r1 (p 1 )),...,S m (0 rm (p m ))), where r i p i = n, S i is a permutation matrix, for i = 1,...,m. Then, the following statements are equivalent. (1) A is an orthogonal array of strength 2. (2) The MI of A is a projection matrix. (3) The MIs of any two columns of A are orthogonal, i.e., m(a i )m(a j ) = 0 (i j). (4) The projection matrix τ n can be decomposed as τ n = m(a 1 ) + +m(a m ) +, where rk( ) = n 1 m j=1 (p j 1) is the rank of the matrix. Definition 3.3. An orthogonal array A is said to be saturated if m j=1 (p j 1) = n 1 (or, equivalently, m(a) = τ 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)) τ p τ 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 τ n : τ n = A 1 + +A k where A i A j = 0 (i j). Step 2: Find an orthogonal array L i such that m(l i ) A i.

8 Y. Zhang / Discrete Mathematics 307 (2007) Step 3: Lay out the new orthogonal array L by Corollaries 3.4 and 3.5: L = (L 1,...,L k1 ) (k 1 k). Let L p =L p (p 1,...,p m )=(C 1,...,C m ) be a normal orthogonal array over G 0 and denote C 0 =C 1 C m and let D 0 =D(q, k 0 ; p)=d 1 D m be an atomic difference matrix over G 0 having p levels. And suppose that [D i,d i] is an atomic q (k 0 +k i ) difference matrix having p i levels for i=1, 2,...,m.Thus, C=[C 0,C 1,...,C m ] is a partitioned matrix in which the m + 1 columns C 0,C 1,...,C m are orthogonal arrays of strength 1 having p, p 1,...,p m levels, respectively. Then, D=[D 0,D 1,...,D m ] is an atomic repeating-column difference matrix about C=[C 0,C 1,...,C m ] if D is a repeating-column difference matrix about C. Theorem 3.7. The matrix D =[D 0,D 1,...,D m ] is an atomic repeating-column difference matrix about C =[C 0,C 1,...,C m ] if and only if L =[C 0 D 0,C 1 D 1,...,C m D m ] is an orthogonal array whose matrix image satisfying m(l) τ p τ q. In particular, m(c 0 D 0 ) m i=1 m(ci D i )+ τ q where =τ p (m(c 1 ) + +m(c m )). Proof. Consider the following orthogonal decomposition of projection matrix τ p τ q : τ p τ q = m(c 1 ) τ q + +m(c m ) τ q + τ q, where τ p = m(l p ) + =m(c 1 ) + +m(c m ) +. By Theorems 3.1, 3.2 and 3.6, we have m(c j [D j,d j ]) m(c j ) τ q, j = 1, 2,...,m, i.e., [C 1 [D 1,D 1],...,C m [D m,d m]] is an orthogonal array. By Corollary 3.5, we have that m(c i D i ) + m(c i D i ) = m(c i [D i,d i]), i = 1, 2,...,m.Thus, and m((c 1 D 1 ) + +m(c m D m )) + τ q τ p τ q, m((c 1 D 1 ) + +m(c m D m )) τ p τ q, [m((c 1 D 1 ) + +m(c m D m )) + τ q][m((c 1 D 1 ) + +m(c m D m ))]=0. By their orthogonality and Theorem 3.2, L =[C 0 D 0,C 1 D 1,...,C m D m ] is an orthogonal array whose matrix image satisfies m(l) τ p τ q if m(c 0 D 0 ) (m(c 1 D 1 ) + +m(c m D m )) + τ q. In fact, let D l = (dl ij ) q k 0 = (d l 1,...,dl k 0 ) be an atomic difference matrix for any l.and denote C 0 = (p). From Eqs. (1) and (2), we have [0,D 0 ] (p) = (S 0 0 (0 q (p)), S 0 1 (0 q (p)),...,s 0 k 0 (0 q (p))), [0,D l ] C l = (S l 0 (0 q C l ), S l 1 (0 n C l ),...,S l k 0 (0 q C l )), where S0 l = I pq and Sj l = diag(σ l(d1j l ),...,σ l(dqj l )), j = 1, 2,...,m; l = 1,...,m.By Theorems 3.1 and 3.2, we have m([0,d 0 ] C 0 ) = m([0,d 0 ] (p)) = = m k 0 i=1 j=0 k 0 j=0 S 0 j (P q τ p )(S 0 j )T S 0 j (P q m(c i ))(S 0 j )T + k 0 j=0 S 0 j (P q )(S 0 j )T,

9 254 Y. Zhang / Discrete Mathematics 307 (2007) since τ p =m(l p )+ =m(c 1 )+ +m(c m )+. The above decompositions are orthogonal because of the orthogonality in each step. Thus, all items S 0 j (P q m(c i ))(S 0 j )T,S 0 s (P q )(S 0 s )T,i=1, 2,...,m; j,s=0, 1,...,k 0, are orthogonal to each other. By Theorem 2.10, we have that Sj 0 (P q )(Sj 0 )T Sj 0 (I q )(Sj 0 )T = diag(σ 0 (d1j 0 ) σ 0(d1j 0 )T,...,σ 0 (dqj 0 ) σ 0(dqj 0 )T ) = diag(,..., ) = I q, for j = 0, 1,...,m; and that S 0 j (P q m(c i ))(S 0 j )T = m(s 0 j (0 q C i )) = m(s i j (0 q C i )) = m(d i j C i). Thus, we obtain i.e., m([0,d 0 ] C 0 ) P q τ p + m m(d i C i) + τ q, i=1 m(c 0 D 0 ) = K(p, q)m(d 0 C 0 )K(p, q) T = K(p, q)(m([0,d 0 ] C 0 ) P q τ p )K(p, q) T [ m ] m K(p,q) m(d i C i) + τ q K(p,q) T = m(c i D i ) + τ q. i=1 This completes the proof of. Let L=[C 0 D 0,C 1 D 1,...,C m D m ] be an orthogonal array whose matrix image satisfies m(l) τ p τ q. Then, C 0 [0,D 0 ] and C i [0,D i,d i] are orthogonal arrays for all i. Thus, [0,D 0 ] and [0,D i,d i] 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 = [D 0, D 1,,D m ] be an atomic repeating-column difference matrix about C if and only if L =[L p 0 q, 0 p L q,c 0 D 0,C 1 D 1,...,C m D m ] is an orthogonal array for any orthogonal arrays L p and L q. 4. Examples 4.1. Constructions of orthogonal arrays of run size 72 Zhang et al. [11] has constructed an orthogonal array L 72 ( ) whose structure is L 72 ( ) =[0 3 (12) 0 2, 0 3 L (=) 24 (28 ), L (=) 36 (38 ) 0 2, (M 1 Q 1 )(L ( ) 36 (62 ) 0 2 ), (M 2 Q 2 )(L ( ) 36 (62 ) 0 2 )], where Q 1 = K(2, 2), Q 2 = K(2, 2) diag(i 2,N 2 )K(2, 2) T and M 1 = K(3, 6) diag(n 3,N 2 3,Q 1 I 3 )K(3, 6) T, M 2 = K(3, 6) diag(n 2 3,N 3,Q 2 I 3 )K(3, 6) T ; and where the orthogonal arrays satisfy L (=) 24 (28 ) = D(12, 8; 2) (2), L = 36 (38 ) = (3) D(12, 8; 3) L ( ) 36 (62 ) =[[(3) (3) 0 4 ] (0 18 (2)], [(3) (3) ] [0 9 (2) (2)]], in which D(12, 8; 2) and D(12, 8; 3) are some difference matrices. i=1

10 Y. Zhang / Discrete Mathematics 307 (2007) 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) 0 2 ] [0 3 D(12, 4; 2) (2)] =[(M 1 Q 1 )(L ( ) 36 (62 ) 0 2 ), (M 2 Q 2 )(L ( ) 36 (62 ) 0 2 )]. Hence the following array, [D(12, 4; 6) [((3) 0 2 ) (0 3 (2))],D(12, 8; 3) ((3) 0 2 ), D(12, 8; 2) (0 3 (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) 0 2, 0 2 (2)]. Let D(12, 4; 6)=(a 0,D(12, 3; 6)) where a 0 =a 1 a 2. By the transformation of repeating-column difference matrix, we have [0,D 0 ]:=[D(12, 4; 6) a 0,D(12, 8; 3) a 1,D(12, 8; 2) a 2 ] 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 D 0 =[D 0 (12, 4; 6), D 0 (12, 7; 3), D (12, 7; 2)]=, D 0 =[D 0 (12, 5; 6), D 0 (12, 3; 3), D (12, 6; 2)]=, over G 6 0. Define 0 = 00, 1 = 11, 2 = 20, 3 = 01, 4 = 10, 5 = 21. Then, the group is Z 6. 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].

11 256 Y. Zhang / Discrete Mathematics 307 (2007) Table 1 Orthogonal arrays L 72 ( ) obtained in Section 4.1 No. f 1 f 10 c 1 c 10 b 1 b 7 b 8 b 18 lf dc

12 Y. Zhang / Discrete Mathematics 307 (2007) Table 1 (continued) No. f 1 f 10 c 1 c 10 b 1 b 7 b 8 b 18 lf dc L 72 ( ) = (l f 1 f 7 f 10 c 4 c 10 b 1 b 7 ) (old) L 72 ( ) = (d f 1 f 7 f 10 c 4 c 10 b 3 b 7 b 9 b 18 ) L 72 ( ) = (d f 1 f 7 f 10 c 4 c 10 c b 3 b 7 b 9 b 11 ) L 72 ( ) = (d f 1 f 7 f 10 f c 4 c 10 b 3 b 7 b 9 ) L 72 ( ) = (l f 1 f 6 c 1 c 3 b 1 b 6 ) (old) L 72 ( ) = (d f 1 f 6 c 1 c 3 b 3 b 6 b 9 b 18 ) L 72 ( ) = (d f 1 f 6 c 1 c 3 c b 3 b 6 b 9 b 11 ) L 72 ( ) = (d f 1 f 6 f c 1 c 3 b 3 b 6 b 9 ) where d = b 1 b 2 b 8, l = b 8 b 18, m(c) m(b 12 b 18 ), m(f ) m(b 10 b 18 ) 4.2. Construction of orthogonal arrays of run size 96 Zhang et al. [9] has constructed an orthogonal array L 96 ( ) whose structure is L 96 ( ) =[D 1 (12, 4; 2) 0 2 (2) 0 2,D 2 (12, 4; 2) 0 4 (2), D 3 (12, 4; 2) (2) (2) (2), D(24, 20; 4) (4), (24) 0 4 ], where D(24, 20; 4), D 1 (12, 4; 2) 0 2, D 2 (12, 4; 2) 0 2 and D 3 (12, 4; 2) (2) are some difference matrices.

13 258 Y. Zhang / Discrete Mathematics 307 (2007) It is easy to prove that L 4 (2 3 )=((2) 0 2, 0 2 (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) 0 2,D 2 (12, 4; 2) 0 2,D 3 (12, 4; 2) (2)] is a repeating-column difference matrix about [(4), (2) 0 2, 0 2 (2), (2) (2)]. Let D(24, 20; 4)=(a 0,D(24, 19; 4)), where a 0 =a 1 a 2 a 3. By the transformation of repeating-column difference matrix, we have [0,D 0 ]:=[a 0 + D(24, 4; 4), a 1 + D 1 (12, 4; 2) 0 2,a 2 + D 2 (12, 4; 2) 0 2,a 3 + D 3 (12, 4; 2) (2)] is also a repeating-column difference matrix. Thus, D 0 is an atomic repeating-column difference matrix D =, where 0 = 000, 1 = 011, 2 = 101, 3 = 110 and x + y = y + x,x + x = 0, 0 + x = x,2 + 3 = 1, = 2, = 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].

14 Y. Zhang / Discrete Mathematics 307 (2007) Table 2 Orthogonal arrays L 96 ( ) obtained in Section 4.2 L 96 ( ) rel 96 d 1 d 20 b 1 b 35 fd 21 d 23 clx

15 260 Y. Zhang / Discrete Mathematics 307 (2007) Table 2 (continued) L 96 ( ) rel 96 d 1 d 20 b 1 b 35 fd 21 d 23 clx L 96 ( ) = (x d 1 d 20 b 1 b 12 ) L 96 ( ) = (l d 1 d 20 d 22 d 23 b 1 b 4 b 5 b 8 b 12 b 26 b 35 ) L 96 ( ) = (d 1 d 23 b 1 b 4 b 5 b 8 b 12 b 14 b 23 b 26 b 29 b 32 b 35 ) L 96 ( ) = (c d 1 d 23 b 1 b 4 b 5 b 8 b 12 b 14 b 16 b 26 b 29 b 32 b 35 ) L 96 ( ) = (f d 1 d 23 b 1 b 4 b 5 b 8 b 12 b 14 b 26 b 29 b 32 b 35 ).

16 Y. Zhang / Discrete Mathematics 307 (2007) Table 2 (continued) L 96 ( ) rel 96 d 1 d 20 b 1 b 35 fd 21 d 23 clx where d 21 = b 13 b 30 b 31, d 22 = b 2 b 3 b 24, d 23 = b 6 b 7 b 25, x = b 13 b 35, l = b 13 b 23, m(c) m(b 13 b 23 ), m(f ) m(b 15 b 23 ). 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 , No (Henan) and No (Henan)) in China. References [1] T. Beth, D. Jungnickel, H. Lenz, Design Theory, Bibliographishes Institut, Mannheinu-Wien-Zürich, 1985, and Cambridge University Press, Cambridge, [2] R.C. Bose, K.A. Bush, Orthogonal arrays of strength two and three, Ann. Math. Statist. 23 (1952) [3] W.F. Kuhfeld, Orthogonal arrays, [4] S. Shrikhande, Generalized Hadamard matrices and orthogonal arrays strength two, Canad. J. Math. 16 (1964) [5] Y.S. Zhang, Asymmetrical orthogonal design by multi-matrix methods, J. Chinese Statist. Assoc. 29 (1991) [6] Y.S. Zhang, Orthogonal array and matrices, J. Math. Res. Exposition 12 (3) (1992) [7] Y.S. Zhang, Theory of Multilateral Matrix, Chinese Statistic Press, [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) [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) [11] Y.S. Zhang, S.Q. Pang, Y.P. Wang, Orthogonal arrays obtained by generalized Hadamard product, Discrete Math. 238 (2001)