Balanced arrays of strength two and nested (r,λ)-designs

Shinji Kuriki Department of Applied Mathematics Osaka Women’s University 2-1 Daisen-cho, Sakai City Osaka 590, Japan and Ryoh Fuji-Hara Institute of Socio-Economic Planning University of Tsukuba Tsukuba, Ibaraki 305, Japan

Abstract : An (r,λ)-design with mutually balanced nested subdesigns ( for brevity, (r,λ)-design with MBN ) is introduced firstly in this paper. It is shown that an (r,λ)design with MBN is equivalent to a balanced array of strength 2 with s symbols. By the use of a nested design and an orthogonal array, a construction of an (r,λ)design with MBN is given. A direct construction of such an (r,λ)-design, based on the result obtained by Wilson (1972), is also given. By these constructions, new balanced arrays with s ≥ 3 are presented.

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AMS Subject Classification : Primary 05B30; Secondary 62K15, 05B05. Key words and phrases : Balanced arrays; (r,λ)-designs with mutually balanced nested subdesigns; construction. Abbreviated title : Balanced arrays and nested (r,λ)-designs

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1. Introduction A balanced array was first introduced by Chakravarti (1956,1961) in connection with some class of statistical designs. Some constructions of balanced arrays have been studied by several authors ( e.g., Kageyama (1975), Rafter and Seiden (1974), Shirakura (1977), Srivastava (1972) and Srivastava and Chopra (1973) ). Our concern in this paper is to construct balanced arrays of strength 2 with s(≥ 3) symbols. First of all we begin with the definition of a balanced array. Let S be {0, 1, . . . , s−1}, called symbols. A balanced array of strength 2, denoted by BA(m, n, s, 2), is an m × n matrix T whose entries are from S, satisfying the following conditions: (i) in any two rowed submatrix T0 of T , a vector [i, j] occurs exactly µij times as columns in T0 for any i, j ∈ S, (ii) µij = µji for any i, j ∈ S. The µij ’s are called indices. If µij = µ for every i, j ∈ S, then the array is called an orthogonal array with the index µ, and is denoted by OA(m, n, s, 2), where n = µs2 . Example 1.1. A 6 × 20 matrix          

0 0 0 2 2 1

0 0 2 1 1 0

0 0 1 2 1 0

0 0 1 1 0 2

0 1 1 0 0 1

0 1 2 0 0 2

0 1 0 1 2 0

0 2 2 2 0 0

0 2 0 0 1 1

0 2 0 0 2 2

1 0 0 2 0 2

1 0 0 1 0 1

1 0 2 0 2 0

1 1 0 0 1 0

1 2 1 0 0 0

2 0 0 0 1 2

2 0 2 0 0 1

2 0 1 0 2 0

2 1 0 2 0 0

2 2 0 1 0 0

         

is a BA(6, 20, 3, 2) with indices µ00 = 4, µ01 = µ02 = 3 and µ11 = µ12 = µ22 = 1. In order to construct such balanced arrays, we introduce in Section 2 an (r,λ)design satisfying some conditions. And we show that it is equivalent to a balanced array of strength 2 with s symbols. In Section 3, we give some constructions of 3

such (r,λ)-designs. By these constructions, we present new balanced arrays with s symbols.

2. (r,λ)-designs with mutually balanced nested subdesigns An (r,λ)-design is a pair (V ,B), where V is a v-set ( called points ) and B is a collection of subsets of V ( called blocks ), satisfying the following conditions: (i) every point occurs in precisely r blocks, (ii) every pair of distinct points occurs in precisely λ blocks. If each block contains k points, then it is called a balanced incomplete block design, and is denoted by (v, k, λ)-BIBD. It is well-known that the existence of an (r,λ)-design (V ,B) is equivalent to the existence of a BA(v, b, 2, 2) with indices µ11 = λ, µ01 = r − λ and µ00 = b − 2r + λ, where b is the number of blocks of B. In the case of more than 2 symbols, Dey, Kulshreshtha and Saha (1972) have shown a construction of balanced arrays with 3 symbols from BIBD’s. Kageyama (1975) has improved the result for s ≥ 3. Now we introduce an (r,λ)-design with mutually balanced nested subdesigns and show that it is equivalent to a balanced array of strength 2 with s symbols. Let (V ,B) be an (r,λ)-design. Suppose that each block B ∈ B is partitioned into g subblocks B1 , B2 , . . . , Bg ( some of them may be empty ). We denote the collection of the i-th subblocks Bi ’s for each B ∈ B by Bi and Π = {B1 , B2 , . . . , Bg }. An (r,λ)-design with mutually balanced nested subdesigns ( for brevity, (r,λ)-design with MBN ) is a triple (V ,B,Π) satisfying the following conditions: (i) (V ,Bi ) is an (ri ,λi )-design for i = 1, 2, . . . , g, 4

(ii) for distinct points x and y of V , the number of blocks B of B containing x in the i-th and y in the j-th subblocks of B is λi,j which is independent of the x and y chosen. Note that λi,j = λj,i and λi = λi,i . It is easily seen that r=

g X

ri

and λ =

i=1

g X

X

λi + 2

i=1

λi,j .

1≤i
Theorem 2.1. The existence of an (r,λ)-design (V ,B,Π) with mutually balanced nested subdesigns is equivalent to the existence of a BA(v, b, s, 2) with indices  λi,j ,    

if if if if

λ, µij =  i Ps−1 r i−  u=1 λi,u ,   b − 2r + λ,

i 6= j, i, j 6= 0, i = j 6= 0, i 6= 0 and j = 0, i = j = 0,

(2.1)

where v = |V |, b is the number of blocks of B and s = |Π| + 1. Proof. Let (V ,B,Π) be an (r,λ)-design with MBN. Then we define a v × b matrix T = [tx,u ] as tx,u =

   i,  

if a point x of V occurs in the i-th subblock of the u-th block of B, otherwise.

0,

Let T0 be any two rowed submatrix of T . It is immediately seen, from the definition of an (r,λ)-design with MBN, that µij = λi,j for i 6= j, i, j 6= 0; and µii = λi for i 6= 0. Since every point of V occurs in precisely ri blocks in Bi , each of vectors [0, i] and [i, 0] for i 6= 0 occurs exactly ri −

Ps−1

u=1

λi,u times as columns in T0 . So the

vector [0, 0] occurs b−2

X

λi,j −

1≤i
=b−2

s−1 X

ri +

i=1

s−1 X

s−1 X

s−1 X

i=1

i=1

u=1

λi − 2

{ri −

s−1 X

X

i=1

1≤i
λi + 2

= b − 2r + λ 5

λi,j

λi,u }

times. Hence T is a BA(v, b, s, 2) with indices given in (2.1). Conversely, let T be a BA(v, b, s, 2) with indices given in (2.1). We give a correspondence between points of a v-set V and rows of T , and between blocks of a collection B and columns of T . Each block of B consists of points of V corresponding to nonzero entries of T . T is also a balanced array with 2 symbols, the pair (V ,B) therefore is an (r,λ)-design. For each B ∈ B, we partition B into s − 1 subblocks B1 , B2 , . . . , Bs−1 such that Bi consists of points with the entry i. Let Bi be a collection of the i-th subblocks Bi of B ∈ B and Π = {B1 , B2 , . . . , Bs−1 }. Since T is a BA(v, b, s, 2) with indices given in (2.1), the number ri of blocks of Bi containing any point is

Ps−1

u=0

µiu , and the number λi of blocks of Bi containing any pair of

distinct points is µii . For distinct points x and y of V , the number λi,j of blocks B of B containing x in the i-th and y in the j-th subblocks of B is µij which is independent of the x and y chosen. Hence the triple (V ,B,Π) is an (r,λ)-design with MBN.

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Example 2.2. Let V = {1, 2, 3, 4, 5, 6} and B be a collection of 20 blocks: {1, 2, 3}, {1, 2, 5}, {1, 3, 6}, {1, 5, 6}, {2, 3, 4}, {2, 4, 5}, {3, 4, 6}, {4, 5, 6}; {1, 2, 4}, {1, 3, 5}, {1, 4, 6}, {2, 3, 6}, {2, 5, 6}, {3, 4, 5}, twice. The pair (V ,B) is an (r,λ)-design with r = 10 and λ = 4. We partition each of these blocks into two subblocks as follows: {1, 3} ∪ {2}, {2, 4} ∪ {5}, {1} ∪ {3, 5}, {2} ∪ {3, 6},

{1, 2, 5} ∪ φ, {3, 4} ∪ {6}, {3} ∪ {1, 5}, {5, 6} ∪ {2},

{6} ∪ {1, 3}, {6} ∪ {4, 5}, {1, 4, 6} ∪ φ, φ ∪ {2, 5, 6},

{5} ∪ {1, 6}, {2} ∪ {1, 4}, {1} ∪ {4, 6}, {3, 5} ∪ {4},

φ ∪ {2, 3, 4}, {4} ∪ {1, 2}, {2, 3, 6} ∪ φ, {4, 5} ∪ {3}.

The triple (V ,B,Π) is an (r,λ)-design with MBN having parameters r1 = r2 = 5 and λ1 = λ2 = λ1,2 = 1, where Π = {B1 , B2 } and B1 and B2 are collections of the first and the second subblocks of each block of B. A 6 × 20 matrix T , as defined in the proof of Theorem 2.1 for the (r,λ)-design (V ,B,Π) with MBN, is a BA(6, 20, 3, 2) with indices µ00 = 4, µ01 = µ02 = 3 and µ11 = µ12 = µ22 = 1. After some rearrangement of columns, this balanced array is seen to be the same as the one in Example 1.1. 6

3. Constructions of (r,λ)-designs with mutually balanced nested subdesigns Let (V ,B) be an (r,λ)-design. Suppose that each block of B is partitioned into g subblocks some of which may be empty. We denote the collection of all subblocks by B 0 . A nested (r,λ)-design is a triple (V ,B,B 0 ) such that every pair of distinct points of V occurs in precisely λ0 blocks in B 0 . If each block contains k points and each subblock contains k 0 points, then it is called a nested BIBD, and is denoted by NBIBD(v; k, λ; k 0 , λ0 ). Obviously, vr = bk = b0 k 0 ,

λ(v − 1) = r(k − 1) and λ0 (v − 1) = r(k 0 − 1),

where v = |V | and b and b0 are the numbers of blocks of B and B 0 , respectively. The nested BIBD was introduced by Preece (1967) and a list of nested BIBD’s for small values of parameters was given. Note that there is a different definition for nested designs ( see, Colbourn and Colbourn (1983) and Longyear (1981) ). Using a nested (r0 ,λ0 )-design and an orthogonal array, we construct an (r,λ)design with MBN. Theorem 3.1. If there exist a nested (r0 ,λ0 )-design (V ,B 0 ,B 00 ) and an OA(g + 1, g 2 , g, 2), then there exists an (r,λ)-design (V ,B,Π) (|Π| = g) with mutually balanced nested subdesigns having parameters ri = (g − 1)r0 ,

λi = (g − 1)λ00

and λi,j = λ0 − λ00 ,

(3.1)

where g is the number of subblocks in each block of B 0 and λ00 is the number of blocks of B 00 containing any pair of distinct points of V . Proof. Without loss of generality, we may assume that an OA(g + 1, g 2 , g, 2) A = [ai,u ] is standardized as follows: ag+1,ug+u0 = u,

for any 0 ≤ u ≤ g − 1 and 1 ≤ u0 ≤ g, 7

and ai,g(g−1)+u = u − 1,

for any 1 ≤ i ≤ g

and 1 ≤ u ≤ g.

From this assumption, it is seen that, in the columns of a submatrix A˜ = [ai,u ] (1 ≤ i ≤ g, 1 ≤ u ≤ g(g − 1)) of A, entries of each column are all distinct. For each block B 0 of B 0 , we construct g(g − 1) blocks such that the i-th subblock of the u-th block is the (ai,u + 1)-th subblock of B 0 for 1 ≤ i ≤ g and 1 ≤ u ≤ g(g − 1). Let B and Bi be collections of these g(g − 1)b0 blocks and the i-th subblocks, respectively, and Π = {B1 , B2 , . . . , Bg }, where b0 is the number of blocks of B 0 . Obviously, (V ,B) is an (r,λ)-design. Since (V ,B 00 ) is an (r0 ,λ00 )-design and A is an OA(g + 1, g 2 , g, 2), the number ri of blocks of Bi containing any point is (g − 1)r0 , and the number λi of blocks of Bi containing any pair of distinct points is (g − 1)λ00 . Since (V ,B 0 ) is an (r0 ,λ0 )-design, for distinct points x and y of V , the number λi,j of blocks B of B containing x in the i-th and y in the j-th subblocks of B is λ0 − λ00 which is independent of the x and y chosen. Hence the triple (V ,B,Π) is an (r,λ)-design with MBN having parameters given in (3.1).

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Combining Theorems 2.1 and 3.1, we obtain immediately the following corollary: Corollary 3.2. If there exist a nested (r,λ)-design (V ,B,B 0 ) and an OA(s, (s − 1)2 , s − 1, 2), then there exists a BA(v, (s − 1)(s − 2)b, s, 2) with indices  λ − λ0 ,     (s − 2)λ0 , µij =  (s − 2)(r − λ),   

(s − 1)(s − 2)(b − 2r + λ),

if if if if

i 6= j, i, j 6= 0, i = j 6= 0, i 6= 0 and j = 0, i = j = 0,

where s − 1 is the number of subblocks in each block of B, v = |V |, b is the number of blocks of B and λ0 is the number of blocks of B 0 containing any pair of distinct points of V . Let (V ,B) be an (r,λ)-design. We partition each block B ∈ B into two subblocks B and φ. Then a triple (V ,B,B 0 ) is a nested (r,λ)-design. From Corollary 3.2, we 8

construct a BA(v, 2b, 3, 2) with indices µ12 = 0, µ11 = µ22 = λ, µ01 = µ02 = r − λ and µ00 = 2(b − 2r + λ). This is the case of Dey, Kulshreshtha and Saha (1972). If we partition each block B into B, φ, φ,. . ., then we have the case of Kageyama (1975). Several constructions of nested BIBD’s are shown by Homel and Robinson (1975) and Jimbo and Kuriki (1983). For example, Jimbo and Kuriki have shown: Theorem 3.3. Let q be a prime power. Then, for all positive integers d and 0

d0 such that d > d0 , there exists a NBIBD(q N ; q d , λ; q d , λ0 ), where N = d + d0 , λ = φ(N − 2, d − 2, q) and λ0 = φ(N − 2, d0 − 2, q). Here φ(N − 1, d − 1, q) =

(q N − 1)(q N −1 − 1) · · · (q N −d+1 − 1) . (q d − 1)(q d−1 − 1) · · · (q − 1)

We apply Theorem 3.3 to Corollary 3.2. Since it is well-known that there exists 0

an OA(s, (s − 1)2 , s − 1, 2) for a prime power s − 1 = q d−d , we have: Corollary 3.4. Let q be a prime power. Then, for all positive integers d and d0 0

0

such that d > d0 , there exists a BA(q N , q d (q d−d − 1)φ(N − 1, d0 − 1, q), q d−d + 1, 2) with indices

µij =

 0 0 {q d /(q d − 1)}Φ,     Φ,  q d Φ,    d−d0 N q (q − q d − 1)Φ,

if if if if

i 6= j, i, j 6= 0, i = j 6= 0, i 6= 0 and j = 0, i = j = 0,

0

where N = d + d0 and Φ = (q d−d − 1)φ(N − 2, d0 − 2, q). We give a direct construction of (r,λ)-designs with MBN. Let v be a prime power and let V = GF (v), a finite field of order v. For any subsets W and W 0 of V , let W + W 0 , W − W 0 and W ◦ W 0 be multisets {w + w0 |w ∈ W, w0 ∈ W 0 }, {w − w0 |w ∈ W, w0 ∈ W 0 } and {ww0 |w ∈ W, w0 ∈ W 0 }, respectively. And, for any nonnegative integer λ, λ × W denotes a multiset containing every element of W λ times. For 9

brevity, {w} + W 0 and {w} ◦ W 0 are denoted by w + W 0 and wW 0 , respectively. For an integer p satisfying p|v −1, we define Hup as a set {xe |e ≡ u (mod p)}, where x is a primitive element of V . Clearly H0p is a subgroup of V , which is denoted by H p . We select an element cu from each Hup and call elements of the set Cp = {c0 , c1 , . . . , cp−1 } representatives for the cosets modulo H p . Then V − {0} = H p ◦ Cp . For an `-subset L of Cp , let B = L ◦ H p , i.e., B is a union of ` cosets of p H0p , H1p , . . . , Hp−1 . And let B be a collection of blocks y + cB for y ∈ V and c ∈ Cp .

Then Wilson (1972) has shown that the pair (V ,B) is a (v, f `, `(f ` − 1))-BIBD, where v = pf + 1. Lemma 3.5. Let v = pf + 1 be a prime power. For all positive integer ` such that ` ≤ p, there exists a (v, f `, `(f ` − 1))-BIBD. By Lemma 3.5, we can show the following theorem: Theorem 3.6. Let v = pf + 1 be a prime power. For all positive integers `1 , `2 , . . . , `g such that

Pg

i=1 `i

≤ p, there exists an (r,λ)-design, r = pf ` and

λ = `(f ` − 1), with mutually balanced nested subdesigns having parameters ri = pf `i , where ` =

λi = `i (f `i − 1) and λi,j = f `i `j ,

(3.2)

Pg

i=1 `i .

Proof. For any mutually disjoint subsets L1 , L2 , . . . , Lg of Cp , `i = |Li |, let Bi = Li ◦ H p and B = B1 ∪ B2 ∪ · · · ∪ Bg . And let B and Bi be collections of blocks y + cB and the i-th subblocks y + cBi for y ∈ V and c ∈ Cp , respectively, and Π = {B1 , B2 , . . . , Bg }. Now we show that the triple (V ,B,Π) is an (r,λ)-design with MBN having parameters given in (3.2). By Wilson’s construction, (V ,B) is an (r,λ)-design and (V ,Bi ) is an (ri ,λi )-design for i = 1, 2, . . . , g, where ri = pf `i and λi = `i (f `i − 1). For any two elements xei of Li and xej of Lj , a multiset (xei H p ) − (xej H p ) consists of f cosets modulo H p , where x is a primitive element 10

of V . So a multiset (Li ◦ H p ) − (Lj ◦ H p ) consists of f `i `j cosets modulo H p . Then we have

[

{c(Li ◦ H p ) − c(Lj ◦ H p )}

c∈Cp

=

[

c{(Li ◦ H p ) − (Lj ◦ H p )}

c∈Cp

= (f `i `j ) × (V − {0}). 0

Hence, for distinct points xe and xe of V , the number λi,j of blocks B 0 of B containing 0

xe in the i-th and xe in the j-th subblocks of B 0 is f `i `j which is independent of 0

the xe and xe chosen. This completes the proof.

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Note that the design (V ,B) is a (v, f `, `(f `−1))-BIBD and the i-th subdesign (V ,Bi ) is a (v, f `i , `i (f `i − 1))-BIBD. Combining Theorems 2.1 and 3.6, we obtain: Corollary 3.7. Let v = pf + 1 be a prime power. For all positive integers `1 , `2 , . . . , `s−1 such that

Ps−1 i=1

`i ≤ p, there exists a BA(v, pv, s, 2) with indices

 f `i `j ,     ` (f ` − 1), i i µij =  `i {f (p − `) + 1},   

(p − `){f (p − `) + 1},

where ` =

Ps−1 i=1

if if if if

i 6= j, i, j 6= 0, i = j 6= 0, i 6= 0 and j = 0, i = j = 0,

`i .

Chakravarty and Dey (1976) have given, for a prime power 4t + 1, a BA(4t + 1, 8t + 2, 3, 2) with indices µ12 = 2t, µ11 = µ22 = 2t − 1, µ01 = µ02 = 1 and µ00 = 0. This construction is a special case of Corollary 3.7 with p = 2, f = 2t, s = 3 and `1 = `2 = 1.

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Shirakura, T. (1977). Contributions to balanced fractional 2m factorial designs derived from balanced arrays of strength 2`. Hiroshima Math. J. 7, 217-285. Srivastava, J.N. (1972). Some general existence conditions for balanced arrays of strength t and 2 symbols. J. Combin. Theory (A) 13, 198-206. Srivastava, J.N. and D.V. Chopra (1973). Balanced arrays and orthogonal arrays. In: J.N. Srivastava et al., Eds., A Survey of Combinatorial Theory. NorthHolland, Amsterdam, 411-428. Wilson, R.M. (1972). Cyclotomy and difference families in elementary Abelian groups. J. Number Theory 4, 17-47.

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