Discrete Mathematics 259 (2002) 91 – 119

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Balanced nested designs and balanced arrays R. Fuji-Haraa , S. Kageyamab , S. Kurikic , Y. Miaoa; ∗ , S. Shinoharad a Institute

of Policy and Planning Sciences, University of Tsukuba, 1-1-1 Tennodai, Tsukuba 305-8573, Japan b Department of Mathematics, Faculty of Education, Hiroshima University, Higashi-Hiroshima 739-8524, Japan c Department of Mathematical Sciences, College of Engineering, Osaka Prefecture University, Sakai, Osaka 593-8231, Japan d Department of Management Information, Faculty of Informatics, Meisei University, Oume-shi, Tokyo 198-8655, Japan Received 4 May 1999; received in revised form 2 January 2002; accepted 22 January 2002

Abstract Balanced nested designs are closely related to other combinatorial structures such as balanced arrays and balanced n-ary designs. In particular, the existence of symmetric balanced nested designs is equivalent to the existence of some balanced arrays. In this paper, various constructions for symmetric balanced nested designs are provided. They are used to determine the spectrum of symmetric balanced nested balanced incomplete block designs with block size 3 and 4. c 2002 Elsevier Science B.V. All rights reserved.
Keywords: Balanced array; Frame; Latin square; Ordered design; Room frame; Symmetric balanced nested design.

1. Introduction The notion of a balanced array was ;rst introduced by Chakravarti [7] in connection with some class of statistical designs. Later on, many people contributed to the theory and construction of balanced arrays; see, for example, [8,12,18,22]. Let S be a set {0; 1; : : : ; s−1} of s elements, and let X be the set of all t-dimensional column vectors with elements from S. A balanced array of strength t, denoted by ∗

Corresponding author. Tel.: +81-298-53-5009; fax: +81-298-53-5070. E-mail addresses: [email protected] (R. Fuji-Hara), [email protected] (S. Kageyama), [email protected] (S. Kuriki), [email protected] (Y. Miao), [email protected] (S. Shinohara). c 2002 Elsevier Science B.V. All rights reserved. 0012-365X/02/$ - see front matter  PII: S 0 0 1 2 - 3 6 5 X ( 0 2 ) 0 0 3 5 6 - 4

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BA(m; n; s; t), over S is an m × n matrix A with entries from S which satis;es the following two conditions: (A1) in any t-rowed submatrix A
of A, any t-vector x ∈ X occurs exactly (x) times as columns in A
, and (A2) for any permutation of order t and for any x ∈ X; (x) = ( (x)). The (x)’s are the indices of the balanced array. If (x) = for every x ∈ X, then the balanced array coincides with the well-known combinatorial structure called orthogonal array of strength t, which is usually denoted by OA(m; n; s; t). For convenience, we denote the index (( ji )) of a balanced array of strength 2 by (i; j). Example 1.1. The following matrix is a BA(6; 30; 4; 2) de;ned over S = Z4 with (0; 0)= (1; 1) = (0; 2) = (2; 0) = (0; 3) = (3; 0) = (1; 2) = (2; 1) = (1; 3) = (3; 1) = 2; (0; 1) = (1; 0) = 4; (2; 2) = (3; 3) = 0, and (2; 3) = (3; 2) = 1: 

3 0  0  1  1 2

1 3 0 0 1 2

1 1 3 0 0 2

0 1 1 3 0 2

0 0 1 1 3 2

2 0 1 1 0 3

0 2 0 1 1 3

1 0 2 0 1 3

1 1 0 2 0 3

0 1 1 0 2 3

0 2 3 1 0 1

0 0 2 3 1 1

1 0 0 2 3 1

3 1 0 0 2 1

2 3 1 0 0 1

0 1 2 0 3 1

3 0 1 2 0 1

0 3 0 1 2 1

2 0 3 0 1 1

1 2 0 3 0 1

0 3 1 2 1 0

1 0 3 1 2 0

2 1 0 3 1 0

1 2 1 0 3 0

3 1 2 1 0 0

1 0 1 3 2 0

2 1 0 1 3 0

3 2 1 0 1 0

1 3 2 1 0 0

 0 1  3 : 2  1 0

The following matrix is a BA(4; 9; 3; 2) de;ned over S = Z3 with (x) = 1 for every x ∈ X, where X is the set of all two-dimensional column vectors with elements from S = Z3 . In fact, it is an OA(4; 9; 3; 2) over S = Z3 .   0 0 0 1 1 1 2 2 2 0 1 2 0 1 2 0 1 2   0 1 2 2 0 1 1 2 0: 0 1 2 1 2 0 2 0 1 Let V be a set of v elements and B a collection of subsets of V. The elements of V and B are called points and blocks, respectively, and the pair (V; B) is called a design. There are many types of designs, including the well-known pairwise balanced designs, (r; )-designs, group divisible designs, and balanced incomplete block designs. An (r; )-design is a design which satis;es the following two conditions: (B1) every point occurs in precisely r blocks of B, and (B2) every pair of distinct points occurs in precisely  blocks. In particular, when each block contains k points, it is a balanced incomplete block design, denoted by B(v; k; ). Let (V; B) be a design where each block B ∈ B is partitioned into n subblocks B1 ; : : : ; Bn (some of them may be empty). We denote by Bi the collection of the ith subblock Bi for each B ∈ B and let  = {B1 ; : : : ; Bn }. The triple (V; B; ) is called a nested design. Note that the sizes of subblocks are not required to be the same in each block, and some of them may even take the value 0. A nested design having

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some properties of “balance” among its subdesigns is called a balanced nested design. Its exact meaning will be described later. We consider a class of balanced nested designs. Let  ij (x; y) denote the number of blocks B ∈ B containing x in the ith subblock and y in the jth subblock of B. A symmetric balanced nested design is a triple (V; B; ) such that for any distinct points x and y of V;  ij (x; y) is independent of the points x and y chosen, say  ij (x; y) =  ij . In this case we can easily see that  ij = ji . This is the reason why we call this balanced nested design symmetric. Example 1.2. Consider the following blocks de;ned over V = Z 5 ∪ {∞}: {3; 4; ∞; 0};

{2; 3; 0; ∞};

{1; ∞; 2; 4};

{2; 4; 3; 1};

{∞; 3; 1; 2}; {0; 2; 4; 3}:

If we develop each of the above 6 blocks modulo 5, then we can obtain a collection B of 30 blocks. It can be easily veri;ed that (V; B) forms a balanced incomplete block design over V = Z 5 ∪ {∞}, with block size k = 4, and  = 12. We partition each block of B into 3 subblocks of size 2; 1 and 1, respectively, to obtain  = {B1 (mod 5); B2 (mod 5); B3 (mod 5)}, where B1 = {{3; 4}; {2; 3}; {∞; 3}; {1; ∞}; {2; 4}; {0; 2}}, B2 = {{∞}; {0}; {1}; {2}; {3}; {4}}, and B3 = {{0}; {∞}; {2}; {4}; {1}; {3}}. Then we can also easily check that (V; B; ) forms a symmetric balanced nested design with 11 = 12 =  21 = 13 =  31 = 2, and  23 =  32 = 1. Clearly, (V; B1 ) forms a B(6; 2; 2), while for i = 2; 3; (V; Bi ) forms a 1-(6; 1; 5) design. Here a 1-(|V|; k; |B
|=|V|) design means a pair (V; B
) such that each point of V occurs in exactly |B
|=|V| blocks of B
, where each block in B
is of size k. Kuriki and Fuji-Hara [18] de;ned a special class of symmetric balanced nested designs. In each of such designs, the base design (V; B) and the subdesigns (V; Bi ) are an (r; )-design and (ri ;  i )-designs, respectively, by recalling that  i =  ii . This design is called in [18] an (r; )-design with mutually balanced nested subdesigns. In this paper we will call it a symmetric balanced nested (r; )-design. In [18], Kuriki and Fuji-Hara not only considered the constructions for symmetric balanced nested (r; )designs, but also established an equivalence between such a design and a balanced array of strength 2, as Theorem 1.3 shows. Theorem 1.3 (Kuriki and Fuji-Hara [18]). The existence of a symmetric balanced nested (r; )-design (V; B; ) is equivalent to the existence of a balanced array BA(v; b; n + 1; 2) with indices   ij if i = j and i; j = 0;     i if i = j = 0; n (i; j) = (1.1)  r −  if i = 0 and j = 0; i iu   u=1  b − 2r +  if i = j = 0; where v = |V|; b is the number of blocks of B, and n = ||.

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A generalization of Theorem 1.3 to balanced arrays of strength t is given by FujiHara and Kuriki [12]. Recently, Fuji-Hara et al. [13] de;ned another type of balanced nested designs, in which  ij (x; y) + ji (x; y) is independent of x and y chosen for any distinct points belonging to V. Such a balanced nested design is called a pairsum balanced nested design in [13]. It is clear that if a balanced nested design is of symmetric type, then the design is also of pair-sum type. It is shown [13] that the existence of a pair-sum balanced nested design satisfying some additional properties is equivalent to the existence of a balanced n-ary design. The notion of a balanced n-ary design was ;rst introduced by Tocher [24]. Some good surveys on balanced n-ary designs can be found in [4,5]. From now on, we will mainly investigate a symmetric balanced nested balanced incomplete block design B(|V|; k; ); (V; B; ), in which each ith subblock Bi of B ∈ B contains k i points for 16i6n, denoted by symmetric (|V|; k; )-BND of form (k 1 ; : : : ; k n ). Clearly each (V; Bi ) forms a B(|V|; k i ;  i ) if k i ¿2, and forms a 1-(|V|; k i ; |B|=|V|) design if k i = 1. In this case,  ij = k i k j =(k(k − 1)) for all i; j ∈ {1; : : : ; n}; i = j, and  i = (k i (k i − 1))=(k(k − 1)) for all i ∈ {1; : : : ; n} with k i ¿2. The symmetric balanced nested design described in Example 1.2 is in fact a symmetric (6; 4; 12)-BND of form (2; 1; 1). In this paper, we will provide various constructions for symmetric balanced nested designs. As a consequence, the spectrum of symmetric balanced nested balanced incomplete block designs with block size 3 and 4 will be completely determined. 2. Group divisible balanced arrays and symmetric balanced nested group divisible designs Let S = {0; 1; : : : ; s − 1} be a set of s elements, and T be an m × n matrix with entries from S satisfying the following three conditions:    (C1) T can be partitioned into si -rowed submatrices Ti ; 06i6u−1, i.e., T = 

T0 T1 .. . Tu−1

 , 

(C2) in any two-rowed submatrix T
of T , a vector ( ji ) occurs exactly (i; j) or (i; j) times, respectively, as columns in T
for any i; j ∈ S when these two rows belong to distinct submatrices, or the same submatrix of T0 ; T1 ; : : : ; Tu−1 , (C3) (i; j) = ( j; i) and (i; j) = ( j; i) for any i; j ∈ S. Then T is called a group divisible balanced array, and denoted by GDBA(m; n; s; T0 ; : : : ; Tu−1 ). A group divisible balanced array is a special case of a partially balanced array of strength 2 discussed in, for example, [22]. The (i; j)’s and (i; j)’s are the indices of the group divisible balanced array. When (i; j) = (i; j) for every i; j ∈ S, the group divisible balanced array becomes a balanced array of strength 2.

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95

In a GDBA(m; n; s; T0 ; T1 ; : : : ; Tu−1 ); T , if the submatrices T0 ; T1 ; : : : ; Tu−1 are all (m=u) × n matrices, then T is said to be uniform, and is denoted simply by GDBA (m; n; s; u). For each i ∈ S = {0; 1; : : : ; s − 1}, let k i be the number of times the element i appears in a column of a GDBA(m; n; s; u). When k i is independent of the column chosen for every i ∈ S = {0; 1; : : : ; s − 1}, we say that the GDBA(m; n; s; u) is of form (k 0 ; k 1 ; : : : ; k s−1 ). Group divisible balanced arrays are useful in the construction of balanced arrays. We now introduce the concept of a symmetric balanced nested group divisible design, and show that such a nested design coincides with a group divisible balanced array. Some recursive constructions for symmetric balanced nested group divisible designs will be presented. They are analogues of Wilson’s constructions for group divisible designs. Symmetric balanced nested group divisible designs can be used to construct symmetric balanced nested balanced incomplete block designs, or in other words, group divisible balanced arrays can be used to construct balanced arrays. First we give some de;nitions. Let K be a set of some positive integers. A group divisible design (K; )-GDD is a triple (V; G; B) where V is a set of points, G is a partition of V into groups, and B is a collection of subsets of V, called blocks, such that (D1) |B| ∈ K for every B ∈ B, (D2) |G ∩ B|61 for every G ∈ G and every B ∈ B, and (D3) every pair of points {x; y}, where x and y belong to distinct groups, is contained in exactly  blocks of B. The type of a GDD (V; G; B) is the multiset {|G|: G ∈ G}. An exponential notation is used to describe types: a type 1i 2j · · · denotes i occurrences of 1; j occurrences of 2, etc. A GDD of type g|V|=g , that is, the groups all have the same size g, is said to be uniform. A symmetric balanced nested group divisible design, denoted by symmetric (k; )BNGDD, (V; G; B) of type T and of form (k 1 ; : : : ; k n ) is a uniform ({k}; )-GDD of type T where each block B ∈ B is partitioned into n mutually disjoint subblocks Bi of size k i such that (V; G; Bi ) forms a ({k i };  i )-GDD of type T for each integer i ∈ {1; : : : ; n} with k i ¿2 and a 1-(|V|; k i ; |B|=|V|) design for each integer i ∈ {1; : : : ; n} with k i = 1, where Bi denotes the collection of the ith subblocks Bi for each B ∈ B, and for any two distinct points x and y, the number of blocks B of B containing x in the ith and y in the jth subblocks of B is  ij which is independent of the x and y chosen. From the de;nition of the type of a GDD, each ({k i };  i )-GDD is of the same type T as the base ({k}; )-GDD. They all have the same groups. Note that  ij = k i k j =(k(k − 1)) for all i = j ∈ {1; : : : ; n} and  ii =  i = (k i (k i − 1))=(k(k − 1)) for all i ∈ {1; : : : ; n} such that k i ¿2. Clearly a symmetric (v; k; )-BND of form F is a symmetric (k; )-BNGDD of type 1v and of form F. Theorem 2.1. The existence of a symmetric (k; )-BNGDD of type g u and of form (k 1 ; : : : ; k n ) is equivalent to the existence of a GDBA(v; b; n + 1; u) of form

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(v − k; k 1 ; : : : ; k n ) with indices   ij if i = j and i; j = 0;     i if i = j = 0; (i; j) = (g(u−1)−k+1)  ki if i = 0 and j = 0; or i = 0 and j = 0;  k(k−1)   b − 2r +  if i = j = 0;  0 if i = j and i; j = 0;    0 if i = j = 0; (i; j) =  if i = 0 and j = 0; or i = 0 and j = 0; ri    b − 2r if i = j = 0; where v = gu; b = (g2 u(u − 1))=(k(k − 1)), r = (g(u − 1))=(k − 1) and  g(u−1) i if ki ¿2; ki −1 ri = g(u−1) if ki = 1: k(k−1)

(2.2)

(2.3)

(2.4)

Proof. Let (V; G; B) be a symmetric (k; )-BNGDD of type g u and of form (k 1 ; : : : ; k n ), where G = {G0 ; : : : ; Gu−1 }. Then a v × b matrix T = (txw ) is de;ned as    i if a point x of V occurs in the ith subblock of the wth block of B; txw =   0 otherwise: Let T
be any two-rowed submatrix of T corresponding to two points, say, x and y. It is immediately seen, from the de;nition of a symmetric (k; )-BNGDD of type g u and of form (k 1 ; : : : ; k n ), that if the two points x and y belong to distinct groups among G0 ; G1 ; : : : ; Gu−1 , then (i; j) =  ij for i = j; i; j = 0; (i; i) =  i for i = 0. Since every point of V occurs in precisely ri blocks in Bi , where ri is shown 0 0 in (2.4),

n each of the vectors ( i ) and ( i ) for i = 0 occurs

n exactly (0; i) = (i; 0) = ri − j = 1 (i; j) = [(g(u−1)−k +1)=(k(k −1))]k i = ri − j=1  ij times as columns in T
with (i; i) =  i = 0 if k i = 1. So the vector ( 00 ) occurs   n n  n      ri − b−2  ij − i − 2  ij   16i¡j6n

=b − 2

n 

i=1

ri +

i=1

n  i=1

i=1

i + 2



j=1

 ij

16i¡j6n

= b − 2r +  times. If the two points x and y belong to the same group among G0 ; G1 ; : : : ; Gu−1 , then (i; j) = 0 for i = j; i; j = 0, and (i; i) = 0 for i = 0. Since every point of V occurs

R. Fuji-Hara et al. / Discrete Mathematics 259 (2002) 91 – 119

in precisely ri blocks in Bi , each of the vectors

  0 i

and

(0; i) = (i; 0) = ri times as columns in T
. So the vector b−2

 16i¡j6n

(i; j) −

n  i=1

(i; i) −

n 

i

0  0 0

97

for i = 0 occurs exactly occurs

((0; i) + (i; 0)) = b − 2r

i=1

times. Hence T is a GDBA(v; b; n + 1; u) of form (v − k; k 1 ; : : : ; k n ) with indices in (2.2) and (2.3). Conversely, let T be a GDBA(v; b; n + 1; u) of form (v − k; k 1 ; : : : ; k n ) and with indices given in (2.2) and (2.3). We give a correspondence between points of a v-set V and rows of T , between groups of G and the partitioning submatrices 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 . Since T is also a group divisible balanced array with B) is a ({k}; )-GDD

n 2 elements, the

triple (V; G; of type (v=u)u , where k = i=1 k i and  = 16i6n  i + 2 16i¡j6n  ij . For each B ∈ B, we partition B into n subblocks B1 ; B2 ; : : : ; Bn such that Bi consists of points with entry i. Let Bi be the collection of the ith subblocks Bi of B ∈ B. Since T is a GDBA(v; b; n + 1; u) of form (v − k; k 1 ; : : : ; k n ) and with indices given in (2.2) and (2.3), (V; G; Bi ) is a ({k i };  i )-GDD of type (v=u)u for each i ∈ {1; : : : ; n} with k i ¿2 and a 1-(v; k i ; ri ) design for each i ∈ {1; : : : ; n} with k i = 1. For distinct points x and y of V, the number  ij of blocks B of B containing x in the ith and y in the jth subblocks of B is (i; j) if they belong to distinct submatrices of T , or (i; j) if they belong to the same submatrix of T , which is independent of the x and y chosen. Hence (V; G; B) is a symmetric (k; )-BNGDD of type gn and of form (k 1 ; : : : ; k n ). Note that Theorem 2.1 is a generalization of Theorem 1.3. Also note that the BA(6; 30; 4; 2) in Example 1.1 can be constructed from the symmetric (6; 4; 12)-BND of form (2; 1; 1) in Example 1.2, and vice versa, in the ways described in the proof of Theorem 2.1. In order to describe our recursive constructions for symmetric balanced nested group divisible designs, we need the well-known recursive constructions for group divisible designs due to Wilson [25]. Theorem 2.2 (Wilson [25]). Let (V; G; B) be a GDD with index . Further let w : V→ N ∪ {0} be a weight function, where N is the set of positive integers.
For  each B ∈ B, suppose that there exists a (K;  )-GDD of type {w(x): x ∈ B}; ( x∈B S(x); {S(x): x ∈ B}; B(B)), where S(x) = {x1 ; x2 ; : : : ; xw(x) } for every x ∈ V and
B(B) is the collection of blocks of theingredient GDD. Then there exists

  a (K;  )GDD of type { x∈G w(x): G ∈ G}, ( x∈V S(x); { x∈G S(x): G ∈ G}; B∈B B(B)). Theorem 2.3 (Wilson [25]). Let (V; G; B) be a (K; )-GDD. Further let G0 be a set of new points, that is, G0 ∩ V = ∅, and suppose that for each group G ∈ G, there exists a (K; )-GDD (G ∪ G0 ; HG ∪ {G0 }; BG ), where HG is the set of groups except

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G0 and BG is the collection of blocks of the ingredient GDD. Then there exists a  (K; )-GDD (V ∪ G0 ; {G0 } ∪ {HG : G ∈ G}; B ∪ ( G∈G BG )). Now we can state our recursive constructions for symmetric balanced nested group divisible designs. Theorem 2.4. Let (V; G; B) be a uniform GDD with index . Further let w : V → N∪ {0} be a weight function. For each B ∈ B, suppose that there  exists a symmetric (k; 
)-BNGDD of type {w(x): x ∈ B} and of form (k 1 ; : : : ; k n); ( x∈B S(x); {S(x): x ∈B}; B(B)), where S(x) = {x1 ; x2 ; : : : ; xw(x) } for every x ∈ V and w(x)

= w(y) for any x; y ∈ V. Then there existsa symmetric(k; 
)-BNGDD of  type { x∈G w(x): G ∈ G} and of form (k 1 ; : : : ; k n ); ( x∈V S(x); { x∈G S(x): G ∈ G}; B∈B B(B)). Theorem 2.5. Let (V; G; B) be a symmetric (k; )-BNGDD of form F. Further let G0 be a set of new points, that is, G0 ∩ V = ∅, and suppose that for each group G ∈ G, there exists a symmetric (k; )-BNGDD of form F; (G ∪ G0 ; {G0 } ∪ HG ; BG ). Then thereexists a symmetric (k; )-BNGDD of form F; (V ∪ G0 ; {G0 } ∪ {HG : G ∈ G}; B ∪( G∈G BG )). Theorem 2.5 leads to the following corollary which is very useful in the construction of symmetric balanced nested balanced incomplete block designs. Corollary 2.6. Let (V; G; B) be a symmetric (k; )-BNGDD of form F. If for each group G ∈ G, there exists a symmetric (|G|+1; k; )-BND of form F, then there exists a symmetric (|V| + 1; k; )-BND of form F. Meanwhile, as an immediate consequence of Theorem 2.4, a structural property of symmetric balanced nested balanced incomplete block designs can be obtained. A pairwise balanced design B(v; K; ) is a (K; )-GDD of type 1v . A set K of positive integers is said to be PBD-closed if K = B(K), where B(K) = {v: a B(v; K; 1) exists}. Corollary 2.7. Let BND(k; ; F) = {v: a symmetric(v; k; )-BND of form F exists}. Then BND(k; ; F) is a PBD-closed set. Proof. For convenience of notation, let BND denote BND(k; ; F). Obviously, BND ⊆ B(BND). We need only to show B(BND) ⊆ BND. Let v ∈ B(BND), i.e., a B(v; BND; 1) exists. Then Theorem 2.4 with w(x) = 1 for all points x shows that v ∈ BND, which completes the proof. 3. Frames and symmetric balanced nested designs To apply the recursive constructions described in Section 2, we must ;rst have some constructions to produce ingredient symmetric balanced nested designs. The method of diOerences is the most widely used direct construction for many types of designs.

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Symmetric balanced nested designs can also be constructed from this method. They will be described in Section 6 of this paper. In this section, we use another combinatorial structure, called frames, to construct symmetric balanced nested designs. Let (V; G; B) be a ({k}; )-GDD of type T . If the collection B of blocks can be partitioned into partial resolution classes each of which partitions V\G for some G ∈ G, then the ({k}; )-GDD is called a (k; )-frame of type T . Frames were formally introduced by Stinson [23] to construct Kirkman triple systems with subsystems. Together with a so-called “;lling in holes” construction, frames provide a powerful mechanism for the construction of resolvable designs and their related structures. Existence and construction of frames have been investigated by many people; see, for example, [15] and the relevant references therein. Here we provide a construction for symmetric balanced nested GDDs by means of frames. Theorem 3.1. The existence of a (k; )-frame of type g u implies the existence of a symmetric (k + 1; 
)-BNGDD of type g u and of form (k; 1), where k +1 if  is a factor of k − 1;
 = 2  + k−1 if k − 1 is a factor of : Proof. For each group of a (k; )-frame of type g u , (V; G; B), it is known (see, for example, [15]) that there are exactly g=(k − 1) partial resolution classes associated with it. When |(k − 1), let k − 1 = m; m ∈ N. Then there are g=m partial resolution classes, say, PG; i ; 16i6g=m, associated with the group G ∈ G. For each block B of a partial resolution class PG; i , where G = {x1 ; : : : ; xg }, we form m new blocks B1 = B ∪ {x(i−1)m+1 }; : : : ; Bm = B ∪ {xim }. For any two given points x and y from distinct groups, they appear m = k − 1 times in the new blocks of size k + 1 constructed from the blocks of partial resolution classes not associated with any group containing x or y, and twice in the new blocks constructed from the blocks of partial resolution classes associated with the group containing x and y respectively. Then (V; G; B
) with B
= {Bi : i = 1; : : : ; m; B ∈ B} is a symmetric (k + 1; k + 1)-BNGDD of type g u and of form (k; 1). When (k − 1)|, let  = n(k − 1); n ∈ N. Then there are ng partial resolution classes, say, QG; i ; 16i6ng, associated with the group G = {x1 ; : : : ; xg } ∈ G. For each block B of the n partial resolution classes QG; (t−1)n+1 ; : : : ; QG; tn ; t = 1; : : : ; g, we form a new block B ∪ {xt }. For any two given points x and y from distinct groups, they appear  times in the new blocks constructed from the blocks of partial resolution classes not associated with any group containing x or y, and 2n times in the new blocks constructed from the blocks of partial resolution classes associated with the group con



taining  n x and y respectively. Then (V; G; B ) with B = {B ∪ {xt }: t = 1; :u: : ; g; B ∈ j=1 QG; (t−1)n+j } is a symmetric (k + 1;  + 2=(k − 1))-BNGDD of type g and of form (k; 1). We provide an example to illustrate the ;rst part of this construction.

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Example 3.2. A (3; 1)-frame of type 24 over V = {1; 2; 3; 4; 5; 6; 7; 8} is listed below. The set of groups is G = {G1 ; G2 ; G3 ; G4 }, where G1 = {1; 2}; G2 = {3; 6}; G3 = {4; 8}, and G4 = {5; 7}. The partial resolution class PGi ;1 associated with Gi ; 16i64, is PG1 ;1 = {{3; 4; 5}; {6; 7; 8}}; PG2 ;1 = {{1; 4; 7}; {2; 5; 8}}, PG3 ;1 = {{1; 5; 6}; {2; 3; 7}}, and PG4 ;1 = {{1; 3; 8}; {2; 4; 6}}, respectively. Then we de;ne the following set B
of 16 new blocks: {3; 4; 5; 1};

{3; 4; 5; 2};

{6; 7; 8; 1};

{6; 7; 8; 2};

{1; 4; 7; 3};

{1; 4; 7; 6};

{2; 5; 8; 3}; {2; 5; 8; 6}; {1; 5; 6; 4};

{1; 5; 6; 8};

{2; 3; 7; 4};

{2; 3; 7; 8};

{1; 3; 8; 5}; {1; 3; 8; 7}; {2; 4; 6; 5};

{2; 4; 6; 7}:

We then obtain a symmetric (4; 4)-BNGDD of type 24 and of form (3; 1). As an immediate consequence of Theorem 3.1, we have the following corollary. Corollary 3.3. Let u; g and  be positive integers such that u¿4; g(u−1) ≡ 0 (mod 3) and g ≡ 0 (mod 2). Then there exists a symmetric (4; )-BNGDD of type g u and of form (3; 1), where


 =

4 2

if  = 1 or 2; if  is even:

Proof. Stinson [23] and Assaf and Hartman [2] proved that a (3; )-frame of type g u exists if and only if u¿4; g(u − 1) ≡ 0 (mod 3) and g ≡ 0 (mod 2). Then apply Theorem 3.1. One particular class of frames has drawn a lot of interest in the literature. This is a (k; k − 1)-frame of type 1tk+1 where t is any positive integer, commonly referred to as an almost resolvable design, denoted by (tk + 1; k; k − 1)-ARD. Corollary 3.4. The existence of a (tk + 1; k; k − 1)-ARD implies the existence of a symmetric (tk + 1; k + 1; k + 1)-BND of form (k; 1). The existence of a (tk + 1; k; k − 1)-ARD has been established for those k which are not too large (see [15]). For example, if k = 3; 4 or 5, then a (tk + 1; k; k − 1)-ARD exists for any positive integer t.

R. Fuji-Hara et al. / Discrete Mathematics 259 (2002) 91 – 119

101

Corollary 3.5. There exists a symmetric (tk + 1; k + 1; k + 1)-BND of form (k; 1) for k = 3; 4; 5 and for any positive integer t. 4. Room frames and symmetric balanced nested designs Room frames can also be used to construct symmetric balanced nested designs. They have been studied extensively, and a large number of their applications are known (see [10]). Let S be a set of elements, and {S1 ; : : : ; Sn } be a partition of S. An {S1 ; : : : ; Sn }Room frame is an |S| × |S| array, F, indexed by S, which satis;es the following four properties: (E1) every cell of F either is empty or contains an unordered pair of elements of S, (E2) the subarrays Si × Si are empty for 16i6n (these subarrays are referred to as holes), (E3) each element x ∈= Si occurs once in row (or column)  s for any s ∈ Si , and (E4) the pairs in F are those {s; t}, with (s; t) ∈ (S × S)\ 16i6n (Si × Si ). The type of a Room frame F is de;ned to be the multiset {|Si |: 16i6n}. As usually done in the literature, we use an “exponential” notation to describe types, i.e., a type t1u1 : : : tmum denotes ui occurrences of ti ; 16i6m. A skew Room frame is a Room frame in which a cell oO the main diagonal is occupied if and only if its symmetric cell is empty. Example 4.1. The following 7 × 7 array is a skew Room frame of type 17 . 45 72 63

26 56 13 74

67 24 15

71 35

37

12 46

57 41

34 61 52

23

Theorem 4.2. The existence of a skew Room frame of type g u implies the existence of a symmetric (4; 6)-BNGDD of type g u and form (2; 2). Proof. Let F be a skew Room frame of type g u over V = {1; : : : ; gu} with holes G = {G1 ; : : : ; Gu } of size g each. Let B = {{x; y; r; c} : x and y belong to diOerent holes of G and {x; y} occurs in the cell (r; c) of F}. Then we can see that (V; B) is a symmetric (4; 6)-BNGDD of type g u and of form (2; 2). Example 4.3. We apply Theorem 4.2 to Example 4.1, and we obtain the following collection of blocks of a symmetric (4; 6)-BNGDD of type 17 and of form (2; 2),

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where V = {1; : : : ; 7} with holes G = {{1}; : : : ; {u}}: {1; 2; 6; 5}; {2; 3; 7; 6}; {3; 4; 1; 7}; {4; 5; 2; 1}; {5; 6; 3; 2}; {6; 7; 4; 3}; {7; 1; 5; 4}; {2; 4; 5; 3}; {4; 6; 7; 5}; {6; 1; 2; 7}; {4; 1; 3; 6}; {6; 3; 5; 1};

{1; 3; 4; 2}; {3; 5; 6; 4}; {5; 7; 1; 6}; {7; 2; 3; 1}; {5; 2; 4; 7}; {7; 4; 6; 2};

{1; 5; 7; 3}; {2; 6; 1; 4}; {3; 7; 2; 5}: Note that in this case, g = 1, so every pair of distinct elements occurs in the ;rst subblocks. The next construction is another one which makes use of skew Room frames. Theorem 4.4. The existence of a skew Room frame of type g u implies the existence of a symmetric (4; 4)-BNGDD of type (3g)u and of form (3; 1). Proof. Let F be a skew Room frame with holes H = {{1; : : : ; g}; {g + 1; : : : ; 2g}; : : : ; {(u − 1)g + 1; : : : ; ug}} over X = {1; : : : ; ug}. Let V = X × Z3 ; G = {H × Z3 : H ∈ H}, and B = {{(x; 0); (y; 0); (r; 1); (c; 1)} (mod (−; 3)); {(x; 0); (y; 0); (c; 2); (r; 2)}(mod (−; 3)): {x; y} ∈= H; H ∈ H; {x; y} in the cell (r; c)}. Then we can also check that (V; G; B) is a symmetric (4; 4)-BNGDD of type (3g)u and of form (3; 1). Example 4.5. This time we apply Theorem 4.4 to Example 4.1, and we obtain the collection of blocks of a symmetric (4; 4)-BNGDD of type 17 and of form (3; 1) by developing the following blocks modulo (−; 3), where V = {1; : : : ; 7} × Z3 ; G = {{i} × Z3 : i ∈ V}: {(1; 0); (2; 0); (6; 1); (5; 1)}; {(1; 0); (2; 0); (5; 2); (6; 2)}; {(2; 0); (3; 0); (7; 1); (6; 1)}; {(2; 0); (3; 0); (6; 2); (7; 2)}; {(3; 0); (4; 0); (1; 1); (7; 1)}; {(3; 0); (4; 0); (7; 2); (1; 2)}; {(4; 0); (5; 0); (2; 1); (1; 1)}; {(5; 0); (6; 0); (3; 1); (2; 1)}; {(6; 0); (7; 0); (4; 1); (3; 1)}; {(7; 0); (1; 0); (5; 1); (4; 1)};

{(4; 0); (5; 0); (1; 2); (2; 2)}; {(5; 0); (6; 0); (2; 2); (3; 2)}; {(6; 0); (7; 0); (3; 2); (4; 2)}; {(7; 0); (1; 0); (4; 2); (5; 2)};

R. Fuji-Hara et al. / Discrete Mathematics 259 (2002) 91 – 119

{(1; 0); (3; 0); (4; 1); (2; 1)}; {(2; 0); (4; 0); (5; 1); (3; 1)}; {(3; 0); (5; 0); (6; 1); (4; 1)}; {(4; 0); (6; 0); (7; 1); (5; 1)};

{(1; 0); (3; 0); (2; 2); (4; 2)}; {(2; 0); (4; 0); (3; 2); (5; 2)}; {(3; 0); (5; 0); (4; 2); (6; 2)}; {(4; 0); (6; 0); (5; 2); (7; 2)};

{(5; 0); (7; 0); (1; 1); (6; 1)}; {(6; 0); (1; 0); (2; 1); (7; 1)}; {(7; 0); (2; 0); (3; 1); (1; 1)}; {(4; 0); (1; 0); (3; 1); (6; 1)}; {(5; 0); (2; 0); (4; 1); (7; 1)};

{(5; 0); (7; 0); (6; 2); (1; 2)}; {(6; 0); (1; 0); (7; 2); (2; 2)}; {(7; 0); (2; 0); (1; 2); (3; 2)}; {(4; 0); (1; 0); (6; 2); (3; 2)}; {(5; 0); (2; 0); (7; 2); (4; 2)};

103

{(6; 0); (3; 0); (5; 1); (1; 1)}; {(6; 0); (3; 0); (1; 2); (5; 2)}; {(7; 0); (4; 0); (6; 1); (2; 1)}; {(7; 0); (4; 0); (2; 2); (6; 2)}; {(1; 0); (5; 0); (7; 1); (3; 1)}; {(1; 0); (5; 0); (3; 2); (7; 2)}; {(2; 0); (6; 0); (1; 1); (4; 1)}; {(2; 0); (6; 0); (4; 2); (1; 2)}; {(3; 0); (7; 0); (2; 1); (5; 1)}; {(3; 0); (7; 0); (5; 2); (2; 2)}: By applying Theorems 4.2 and 4.4 with the existence results on skew Room frames, we can easily obtain a few existence results for symmetric balanced nested designs. Here we will only state one of these immediate consequences. Corollary 4.6. There exist both a symmetric (4; 6)-BNGDD of type t u and form (2; 2) and a symmetric (4; 4)-BNGDD of type (3t)u and of form (3; 1) for t = 1 and u an odd integer greater than or equal to 7, and for t = 2 and u a positive integer not belonging to the set {6; 22; 23; 24; 26; 27; 28; 30; 34; 38}. Proof. It is known (see, for example, [10]) that in these cases, there exists a skew Room frame of type t u . Then apply Theorems 4.2 and 4.4. In the proofs of Theorems 4.2 and 4.4, we used x; y; r and c. If all the quadruples (x; y; r; c) can be partitioned into some sets such that each set forms a partition of S − Si for some i, and each Si corresponds to 2|Si | of the sets, then the skew Room frame is said to be partitionable. Theorem 4.7. The existence of a partitionable skew Room frame of type g u implies the existence of a symmetric (5; 5)-BNGDD of type (6g)u and of form (4; 1). Proof. Colbourn et al. [9] showed that the existence of a partitionable skew Room frame of type g u implies the existence of a (4; 1)-frame of type (6g)u . Then apply Theorem 3.1. Theorem 4.8. The existence of a partitionable skew Room frame of type g u implies the existence of a symmetric (5; 5)-BNGDD of type (2g)u and of form (4; 1).

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Proof. Furino et al. [14] showed that the existence of a partitionable skew Room frame of type g u implies the existence of a (4; 3)-frame of type (2g)u . Then apply Theorem 3.1. With the aid of a computer, Colbourn et al. [9] found some partitionable skew Room frames by using the starter–adder technique. Their partitionable skew Room frames can be used to produce some symmetric balanced nested designs with block size 5 by means of Theorems 4.7 and 4.8. Corollary 4.9. There exist both a symmetric (5; 5)-BNGDD of type (6g)u and of form (4; 1) and a symmetric (5; 5)-BNGDD of type (2g)u and of form (4; 1) for (g; u) ∈ {(1; 13); (1; 17); (1; 29); (3; 5); (3; 9); (4; 8)}. Proof. A partitionable skew frame of type g u was found in [9] for (g; u) = (1; 13); (1; 17); (1; 29); (3; 5); (3; 9); (4; 8). Then apply Theorems 4.7 and 4.8. 5. Nested BIBDs, ordered designs, Latin squares and symmetric balanced nested designs In [13], Fuji-Hara et al. described a construction for pair–sum balanced nested designs from perpendicular arrays and nested designs. In this section we ;rst propose a variation of this construction. Instead of using perpendicular arrays as ingredients, we use ordered designs. Their related structure, namely, mutually orthogonal Latin squares, will be also employed in the construction of symmetric balanced nested designs. An ordered design OD(g; s; ) is a s(s − 1) × g array with s elements such that each row has g distinct elements, and that each tuple of two columns contains each ordered row tuple of two distinct elements precisely  times. It can be easily seen, from the de;nition of an ordered design OD(g; s; ), that each element appears in any column the same number of times, (s−1), and that a symmetric (s; g; g(g − 1))-BND of form (1; : : : ; 1) is equivalent to a regular OD(g; s; ). We also need the de;nition of a nested BIBD due to Preece [21], which was later generalized by Kageyama and Miao [17]. Let (V; B) be a (v; k; )-BIBD. Suppose that each block of B is partitioned into k=k
subblocks with k
points each. We denote the collection of all subblocks by B
. If (V; B
) is a (v; k
; 
)-BIBD, then the triple (V; B; B
) is called a nested BIBD, denoted by (v; k; ; k
; 
)-NBIBD. Obviously, vr = bk = b
k
;

(v − 1) = r(k − 1)

and


(v − 1) = r(k
− 1);

where r is the number of blocks of B containing each point of V; b and b
are the numbers of blocks and subblocks of B and B
, respectively. Now we can describe our ;rst construction for symmetric balanced nested designs from ordered designs and nested BIBDs. It is in fact a type of recursion employing a symmetric balanced nested design of form (1; : : : ; 1) and a nested BIBD to get another symmetric balanced nested design with larger block size.

R. Fuji-Hara et al. / Discrete Mathematics 259 (2002) 91 – 119

105

Theorem 5.1. If there exist an OD(g; s; 

) and a (v; k; ; k
; 
)-NBIBD such that k=k
= s, then there exists a symmetric (v; gk
; ∗ )-BND of form (k
; : : : ; k
), where ∗ = 


[g(gk
− 1)(k − k
)]=[k
(k
− 1)]. Proof. Let (V; B
; B

) be a (v; k; ; k
; 
)-NBIBD and let A = (aij ) be an OD(g; s; 

) such that s = k=k
. For each block B
of B
, we construct 

s(s − 1) blocks of size gk
such that the ith subblock of the uth block is the aui th subblock of B
for i = 1; : : : ; g and u = 1; : : : ; 

s(s − 1). Let B and Bi be the collections of these b


s(s − 1) blocks and their ith subblocks, respectively, and  = {B1 ; : : : ; Bg }, where b
is the number of blocks of B
. We show that the triple (V; B; ) is the desired symmetric BND. Since (V; B

) is a BIBD B(v; k
; 
) and the ordered design A contains each element the same number of times in each column, (V; Bi ) is a BIBD B(v; k
; 


(s − 1)) for i = 1; : : : ; g. Also since (V; B
; B

) is a (v; k; ; k
; 
)-NBIBD and A is an OD(g; s; 

), the number  ij (x; y) is equal to ( − 
)

, which is independent of the distinct points x and y chosen. We illustrate Theorem 5.1 with an example. Example 5.2. Let  1 1 2  A=2 3 1 3 2 3

2

3

3 1

1 2

3



 2: 1

Then AT , the transpose of A, is an OD(3; 3; 1). Meanwhile, it can be checked that {{1; 6; 2; 5; 4; 3} (mod 7}) is the collection of blocks of a (7; 6; 5; 2; 1)-NBIBD over Z 7 . For the block {1; 6; 2; 5; 4; 3}, we construct 6 blocks of size 6 as follows: {1; 6; 2; 5; 4; 3}; {1; 6; 4; 3; 2; 5}; {2; 5; 1; 6; 4; 3}; {2; 5; 4; 3; 1; 6}; {4; 3; 1; 6; 2; 5}; {4; 3; 2; 5; 1; 6}: In this way, we obtain the following collection B of new blocks of size 6: {1; 6; 2; 5; 4; 3} (mod 7);

{1; 6; 4; 3; 2; 5} (mod 7);

{2; 5; 1; 6; 4; 3} (mod 7);

{2; 5; 4; 3; 1; 6} (mod 7);

{4; 3; 1; 6; 2; 5} (mod 7); {4; 3; 2; 5; 1; 6} (mod 7); the following collection B1 of subblocks of size 2: {1; 6} (mod 7);

{1; 6} (mod 7);

{2; 5} (mod 7);

{2; 5} (mod 7);

{4; 3} (mod 7);

{4; 3} (mod 7);

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the following collection B2 of subblocks of size 2: {2; 5} (mod 7); {1; 6} (mod 7); {1; 6} (mod 7);

{4; 3} (mod 7); {4; 3} (mod 7); {2; 5} (mod 7);

and the following collection B3 of subblocks of size 2: {4; 3} (mod 7); {2; 5} (mod 7); {4; 3} (mod 7); {1; 6} (mod 7); {2; 5} (mod 7); {1; 6} (mod 7): Let  = {B1 ; B2 ; B3 }. Then it is easy to verify that (Z 7 ; B; ) is a symmetric (7; 6; 30)BND of form (2; 2; 2). Our second construction for symmetric balanced nested designs by means of ordered designs is the following, which is quite simple but useful. Theorem 5.3. If there exists an OD(g; s; ), then there exists a symmetric g (s; g; g(g − 1))-BND of form (k 1 ; : : : ; k g ) where k i ¿0 for 16i6g and i=1 k i = g. Proof. Let V = {1; : : : ; s}, and B be the collection of row vectors of the OD(g; s; ) over V. Then (V; B) is a BIBD B(s; g; g(g − 1)). Divide each block (row vector) B g of B into subblocks B1 ; : : : ; Bg of sizes k 1 ; : : : ; k g , respectively, such that i=1 k i = g and k i ¿0 for each 16i6g. Denote by Bi the collection of the ith subblocks Bi for all B ∈ B. Then it follows that (V; Bi ) is a BIBD B(s; k i ; k i (k i − 1)) for any integer i ∈ {1; : : : ; g} with k i ¿2 and a 1-(s; k i ; (s − 1)) design for any integer i ∈ {1; : : : ; g} with k i = 1, and that for any distinct points x and y of V, the number of blocks B of B containing x in the ith subblock and y in the jth subblock of B is k i k j which is independent of the x and y chosen. A large number of constructions for nested BIBDs can be found, for example, in [19,20]. For the result on ordered designs, the interested reader is referred to [3]. Ordered designs are closely related to mutually orthogonal Latin squares. In what follows, we will show that mutually orthogonal Latin squares with self-orthogonality may yield symmetric balanced nested designs with smaller indices. Let L be a v × v array in which each cell contains a single element from a v-set S, say S = {1; 2; : : : ; v}. If each element occurs exactly once in each row and exactly once in each column, then L is called a Latin square of order v. The transpose of a Latin square L, denoted by LT , is the Latin square which results from L when the role of rows and columns are exchanged, i.e., LT (i; j) = L( j; i) for all 16i; j6v, where L( j; i) represents the element in the cell of the jth row and ith column of L. A Latin square L of order v is said to be symmetric if L(i; j) = L( j; i) for all 16i; j6v, and is called idempotent if L(i; i) = i for all 16i6v in the case S = {1; 2; : : : ; v}.

R. Fuji-Hara et al. / Discrete Mathematics 259 (2002) 91 – 119

107

A transversal in a Latin square of order v is a set of v cells, one from each row and column, which contains each of the v elements exactly once. Two Latin squares L and L
of the same order are said to be orthogonal if L(a; b) = L(c; d) and L
(a; b) = L
(c; d) imply a = c and b = d. Or equivalently, two Latin squares of order v; L = (aij ) (over element set S) and L
= (bij ) (over element set S
) are orthogonal if every element in S × S
occurs exactly once among the v2 pairs (aij ; bij ); 16i; j6v. A set of Latin squares of order v; L1 ; : : : ; Lm , is mutually orthogonal, or forms a set of m MOLS(v), if for every 16i ¡ j6m; Li and Lj are orthogonal. It is well known (see [3]) that the existence of a set of k idempotent MOLS(v) is equivalent to the existence of an OD(k + 2; v; 1). Generally, the existence of an OD(k + 2; v; ) is equivalent to the existence of an idempotent orthogonal array OA(k + 2; v(v − 1); v; 2), where the term “idempotent” means that this orthogonal array contains exactly  times the column vector (i; : : : ; i)T for each element i. The next construction makes use of mutually orthogonal Latin squares with selforthogonality to produce symmetric balanced nested designs with smaller indices. This is a variation of Theorem 5.3. What we are to do is to take half of the blocks from the collection of blocks so that their indices become half of the former ones. A Latin square L is said to be self-orthogonal if L is orthogonal to its transpose LT . A set of t Latin squares L1 ; : : : ; Lt of order v is said to be a set of t self-orthogonal Latin squares of order v, denoted by t SOLS(v), if {Li ; LTi : 16i6t} is a set of 2t mutually orthogonal Latin squares. A self-orthogonal Latin square has at least one transversal, its main diagonal. Hence, any self-orthogonal Latin square can be changed into an idempotent one by renaming the elements. Theorem 5.4. The existence of a set of t SOLS(v) implies the existence of a symmetric (v; 2t + 2; (t + 1)(2t + 1))-BND of form (k 1 ; : : : ; k 2t+2 ) where each k i is even or 2t+2 zero and i = 1 k i = 2t + 2. Proof. Let V = {1; 2; : : : ; v}, and let A1 = (a1ij ); : : : ; At = (aijt ) be t SOLS(v) over V, where 16i; j6v; 16a‘ij 6v; 16‘6t. Let B
= {{i; j; a1ij ; : : : ; aijt ; a1ji ; : : : ; ajit }: 16i; j 6v; i = j}. Observe that for each cell (i; j), the (2t +2)-tuple {i; j; a1ij ; : : : ; aijt ; a1ji ; : : : ; ajit } equals the (2t + 2)-tuple { j; i; a1ji ; : : : ; ajit ; a1ij ; : : : ; aijt }, which corresponds to another cell ( j; i). Then eliminate one of these two (2t +2)-tuples from B
to get B. Clearly (V; B) is a BIBD B(v; 2t + 2; (2t + 1)(t + 1)). Divide each block B of B into B1 ; : : : ; B2t+2 2t+2 of sizes k 1 ; : : : ; k 2t+2 , respectively, such that i=1 k i = 2t + 2 and k i ¿0 for each 16i62t + 2. Denote by Bi the collection of the ith subblocks Bi for each B ∈ B. Then it follows that (V; Bi ) is a BIBD B(v; k i ; k i (k i −1)=2) for any integer i ∈ {1; : : : ; 2t +2} with k i ¿0, and that for any distinct points x and y of V, the number of blocks B of B containing x in the ith and y in the jth subblocks of B is k i k j =2 which is independent of the x and y chosen.

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Example 5.5. It  1  4   5   A1 =  6  7   2  3

is easy to check that  5 6 7 2 3 4  2 5 1 3 7 6   7 3 6 1 4 2   3 2 4 7 1 5   6 4 3 5 2 1   1 7 5 4 6 3  4 1 2 6 5 7

and



1  7  2   A2 =  3  4   5 6

3 2 6

4 1 3

5 6 1

6 4 7

7 3 5

5 7 4

7 6 2

4 2 7

1 5 3

2 1 6

1

5

3

2

4

 2  5  4   6  3   1 7

are 2 idempotent SOLS(7) over S = {1; : : : ; 7}. Then we obtain the following collection B of 42=2 blocks of size 6: {1; 2; 5; 3; 4; 7}; {1; 3; 6; 4; 5; 2}; {1; 4; 7; 5; 6; 3}; {1; 5; 2; 6; 7; 4}; {1; 6; 3; 7; 2; 5}; {1; 7; 4; 2; 3; 6}; {2; 3; 5; 1; 7; 6}; {2; 4; 1; 6; 3; 5}; {2; 5; 3; 4; 6; 7}; {2; 6; 7; 3; 1; 4}; {2; 7; 6; 5; 4; 1}; {3; 4; 6; 1; 2; 7}; {3; 5; 1; 7; 4; 6}; {3; 6; 4; 5; 7; 2}; {3; 7; 2; 4; 1; 5}; {4; 5; 7; 1; 3; 2}; {4; 6; 1; 2; 5; 7}; {4; 7; 5; 6; 2; 3}; {5; 6; 2; 1; 4; 3}; {5; 7; 1; 3; 6; 2}; {6; 7; 3; 1; 5; 4}: We can divide each of the above blocks into subblocks of even sizes, say, for example, 4 and 2, as follows, to obtain two collections B1 and B2 of subblocks of size 4 and

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109

2, respectively: {1; 2; 5; 3};

{1; 3; 6; 4};

{1; 4; 7; 5};

{1; 5; 2; 6};

{1; 6; 3; 7};

{1; 7; 4; 2};

{2; 3; 5; 1};

{2; 4; 1; 6};

{2; 5; 3; 4};

{2; 6; 7; 3};

{2; 7; 6; 5};

{3; 4; 6; 1};

{3; 5; 1; 7};

{3; 6; 4; 5};

{3; 7; 2; 4};

{4; 5; 7; 1};

{4; 6; 1; 2};

{4; 7; 5; 6};

{5; 6; 2; 1};

{5; 7; 1; 3};

{6; 7; 3; 1}; {4; 7};

{5; 2};

{6; 3};

{7; 4};

{2; 5};

{3; 6};

{7; 6};

{3; 5};

{6; 7};

{1; 4};

{4; 1};

{2; 7};

{4; 6};

{7; 2};

{1; 5};

{3; 2};

{5; 7};

{2; 3};

{4; 3};

{6; 2};

{5; 4}: Then it is easily checked that (S; B; {B1 ; B2 }) forms a symmetric (7; 6; 15)-BND of form (4; 2). Note that the index in this Example is 15, quite smaller than the index in Example 5.2, which is 30. Theorem 5.4 gives a construction for symmetric balanced nested designs with even block size. The next theorem, which utilizes another special kind of mutually orthogonal Latin square, presents a construction for symmetric balanced nested designs with odd block size. Let S1 ; : : : ; Sk and M be (k + 1) Latin squares of order v. If {S1 ; : : : ; Sk } is a set of k self-orthogonal Latin squares, and M is a symmetric Latin square of order v, for which {Si ; SiT : 16i6k} ∪ {M } is a set of (2k + 1) mutually orthogonal Latin squares

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of order v, then {S1 ; : : : ; Sk ; M } is called a set of k self-orthogonal Latin squares with a symmetric orthogonal mate, denoted briePy by k SOLSSOM(v). Theorem 5.6. Let v be an odd integer. Then the existence of a set of t SOLSSOM(v) implies the existence of a symmetric (v; 2t + 3; (2t + 3)(t + 1))-BND of form 2t+3 (k 1 ; : : : ; k 2t+3 ) where all but one k i are even or zero such that i=1 k i = 2t + 3. Proof. Let V = {1; 2; : : : ; v}, where v is odd, and let A1 = (a1ij ); : : : ; At = (aijt ); B1 = (b1ij ) = (a1ji ) = AT1 ; : : : ; Bt = (btij ) = (ajit ) = ATt and C = (cij ) = (cji ) = C T be the t SOLSSOM (v) over V, where 16i; j6v; 16a‘ij ; b‘ij ; cij 6v; 16‘6t. Since v is odd, we can assume, w.l.o.g., that aii = bii = cii = i for all i; 16i6v. Let B

= {{i; j; a1ij ; : : : ; aijt ; b1ij ; : : : ; btij ; cij }: 16i; j6v; i = j}. Observe that for each cell (i; j), the (2t + 3)-tuple {i; j; a1ij ; : : : ; aijt ; b1ij ; : : : ; btij ; cij } equals the (2t+3)-tuple { j; i; a1ji ; : : : ; ajit ; b1ji ; : : : ; btji ; cji }, since a‘ij = b‘ji , and cij = cji for all i; j; 16i; j6v, and for all ‘; 16‘6t, which corresponds to another cell (j; i). Then eliminate one of these two (2t + 3)-tuples from B

to obtain B. Clearly (V; B) is a BIBD B(v; 2t + 3; (2t + 3)(t + 1)). Divide each block B of B 2t+3 into B1 ; : : : ; B2t+3 of sizes k 1 ; : : : ; k 2t+3 , respectively, such that i=1 k i = 2t + 3 and k i ¿0 for each 16i62t + 3, where all but one k i are even or zero. Denote by Bi the collection of the ith subblocks Bi for each B ∈ B. Then it follows that (V; Bi ) is a BIBD B(v; k i ; k i (k i − 1)=2) for any integer i ∈ {1; : : : ; 2t + 3} with k i ¿2, and a 1-(v; k i ; (v − 1)=2) design for at most one i ∈ {1; : : : ; 2t + 3} with k i = 1, and that for any distinct points x and y of V, the number of blocks B of B containing x in the ith and y in the jth subblocks of B is k i k j =2, which is independent of the x and y chosen. Example 5.7. It  1  6  7   M =2  3   4 5

is also easy to check that  6 7 2 3 4 5  2 4 7 1 5 3  4 3 5 2 1 6   7 5 4 6 3 1  1 2 6 5 7 4   5 1 3 7 6 2 3 6 1 4 2 7

and A1 and A2 in Example 5.5 together form a set of 2 SOLSSOM(7) over S ={1; : : : ; 7}. Then we obtain the following collection B of 42=2 blocks of size 7: {1; 2; 5; 3; 4; 7; 6}; {1; 4; 7; 5; 6; 3; 2}; {1; 6; 3; 7; 2; 5; 4}; {2; 3; 5; 1; 7; 6; 4};

{1; 3; 6; 4; 5; 2; 7}; {1; 5; 2; 6; 7; 4; 3}; {1; 7; 4; 2; 3; 6; 5}; {2; 4; 1; 6; 3; 5; 7};

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{2; 5; 3; 4; 6; 7; 1}; {2; 6; 7; 3; 1; 4; 5}; {2; 7; 6; 5; 4; 1; 3}; {3; 4; 6; 1; 2; 7; 5}; {3; 5; 1; 7; 4; 6; 2}; {3; 6; 4; 5; 7; 2; 1}; {3; 7; 2; 4; 1; 5; 6}; {4; 5; 7; 1; 3; 2; 6}; {4; 6; 1; 2; 5; 7; 3}; {4; 7; 5; 6; 2; 3; 1}; {5; 6; 2; 1; 4; 3; 7}; {5; 7; 1; 3; 6; 2; 4}; {6; 7; 3; 1; 5; 4; 2}: We can divide each of the above blocks into subblocks of even sizes and one of odd size, say, for example, 4 and 3, as follows, to obtain two collections B1 and B2 of subblocks of size 4 and 3, respectively: {1; 2; 5; 3}; {1; 4; 7; 5};

{1; 3; 6; 4}; {1; 5; 2; 6};

{1; 6; 3; 7}; {2; 3; 5; 1}; {2; 5; 3; 4};

{1; 7; 4; 2}; {2; 4; 1; 6}; {2; 6; 7; 3};

{2; 7; 6; 5}; {3; 5; 1; 7}; {3; 7; 2; 4}; {4; 6; 1; 2};

{3; 4; 6; 1}; {3; 6; 4; 5}; {4; 5; 7; 1}; {4; 7; 5; 6};

{5; 6; 2; 1}; {5; 7; 1; 3}; {6; 7; 3; 1}; {4; 7; 6}; {6; 3; 2}; {2; 5; 4}; {7; 6; 4}; {6; 7; 1}; {4; 1; 3};

{5; 2; 7}; {7; 4; 3}; {3; 6; 5}; {3; 5; 7}; {1; 4; 5}; {2; 7; 5};

{4; 6; 2}; {7; 2; 1}; {1; 5; 6}; {3; 2; 6}; {5; 7; 3}; {2; 3; 1}; {4; 3; 7}; {6; 2; 4}; {5; 4; 2}: Then it is easily checked that (S; B; {B1 ; B2 }) forms a symmetric (7; 7; 21)-BND of form (4; 3).

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Idempotent mutually orthogonal Latin squares, self-orthogonal Latin squares, selforthogonal Latin squares with a symmetric orthogonal mate and their related structures have been studied extensively. The interested reader is referred to [1,11,26] for their recent surveys.

6. Symmetric balanced nested designs over 1nite 1elds For the recursive constructions described in Section 2 to work, we must ;rst have some methods of construction to produce symmetric balanced nested designs directly. The most commonly used direct construction technique is the method of diOerences developed by Bose [6] in 1939. This method will be used here to provide several in;nite series of symmetric balanced nested designs over ;nite ;elds. Similar results on pair-sum balanced nested designs have been reported in [13]. Let q be a prime power and let V = GF(q), a ;nite ;eld of order q. For any subsets W and W
of V, let W + W
; W − W
and W◦W
be multisets {w + w
: w ∈ W; w
∈ W
}, {w−w
: w ∈ W; w
∈ W
} and {ww
: w ∈ W; w
∈ W
}, respectively. For any nonnegative integer ;  × W denotes the multiset containing every element of W  times. For brevity, {w} + W
and {w}◦W
are denoted by w + W
and wW
, respectively. For an integer h satisfying h|(q − 1), we de;ne Huh as the set {/e : e ≡ u (mod h}), where / is a primitive element of V. Clearly H0h is a subgroup of the multiplicative group GF∗ (q), which is denoted by H h . We select an element cu from each Huh and call the set Ch = {c0 ; c1 ; : : : ; ch−1 } the system of distinct representatives for the cosets modulo H h , and denote it by SDRC(Hh ). Obviously GF(q)\{0} = H h ◦Ch . For an ‘-subset L of Ch , let B = L◦H h , i.e., B is a union of ‘ cosets of h h H0 ; H1h ; : : : ; Hh−1 . Further let B be a collection of blocks y + cB for y ∈ V and c ∈ Ch . Then Wilson [25] and Hanani [16] showed that the pair (V; B) forms a BIBD B(q; f‘; ‘(f‘ − 1)), where q = hf + 1. Lemma 6.1 (Hanani [16] and Wilson [25]). Let q = hf + 1 be a prime power. For any positive integer ‘ such that ‘6h, there exists a BIBD B(q; f‘; ‘(f‘ − 1)). By applying Lemma 6.1, Kuriki and Fuji-Hara [18] showed the following. Theorem 6.2 (Kuriki and Fuji-Hara [18]). Let q = hf + 1 be a prime power. For n all positive integers ‘1 ; ‘2 ; : : : ; ‘n such that i=1 ‘i 6h, there exists a symmetric (q; f‘; ‘(f‘ − 1))-BND of form (f‘ ; f‘ ; : : : ; f‘ 1 2 n ), in which  i = ‘i (f‘i − 1) and

n  ij = f‘i ‘j , where ‘ = i=1 ‘i . In fact, this approach can lead to two more series of symmetric balanced nested designs. For an ‘-subset L of Ch , let B = L◦H h , and B
= {0} ∪ B, i.e., B
is a union h of the singleton subset {0} and ‘ cosets of H0h ; H1h ; : : : ; Hh−1 . Let B
be a collection
of blocks y + cB for y ∈ V and c ∈ Ch . Then Wilson [25] and Hanani [16] showed that the pair (V; B
) forms a B(q; f‘ + 1; ‘(f‘ + 1)), where q = hf + 1.

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113

Lemma 6.3 (Hanani [16] and Wilson [25]). Let q = hf + 1 be a prime power. For any positive integer ‘ such that ‘6h, there exists a B(q; f‘ + 1; ‘(f‘ + 1)). By Lemma 6.3, the following two theorems can be shown. Theorem 6.4.

nLet q = hf + 1 be a prime power. For all positive integers ‘1 ; ‘2 ; : : : ; ‘n such that i=1 ‘i 6h, there exists a symmetric (q; f‘ + 1; ‘(f‘ + 1))-BND of + 1);  i = ‘i (f‘i − 1); 1j = j1 = form (f‘1 + 1; f‘2 ; : : : ; f‘n ) in which 1 = ‘1 (f‘1 n ‘j (f‘1 + 1);  ij = f‘i ‘j for 26i; j6n, where ‘ = i=1 ‘i . Proof. For any mutually disjoint subsets L1 ; L2 ; : : : ; Ln of Ch ; ‘i = |Li |, let B1
= {0} ∪ (L1 ◦H h); Bi
=Li ◦H h ; 26i6n, and B
=B1
∪ B2
∪ · · · ∪ Bn
. Let B
; B1
and Bi
; 26i6n, be the collections of blocks y + cB
, the ;rst subblocks y + cB1
, and the ith subblocks, 26i6n; y + cBi
for y ∈ V and c ∈ Ch , respectively, and 
= {B1
; B2
; : : : ; Bn
}. Now we show that the triple (V; B
; 
) is the desired

n symmetric (q; f‘+1; ‘(f‘+1))-BND of form (f‘1 + 1; f‘2 ; : : : ; f‘n ) where ‘ = i=1 ‘i . By Lemmas 6.1 and 6.3, (V; B
) is a B(q; f‘ + 1; ‘(f‘ + 1)), (V; B1
) is a B(q; f‘1 + 1; ‘1 (f‘1 + 1)), and (V; Bi
) is a B(q; f‘i ; ‘i (f‘i − 1)) for i = 2; : : : ; n. For any two elements / ei of Li and / ej of Lj , the multiset / ei H h \/ ej H h consists of f cosets modulo H h , where / is a primitive element of V. Hence the multiset (Li ◦H h )\(Lj ◦H h ) consists of f‘i ‘j cosets modulo H h , while the multiset {0} ∪ (L1 ◦H h )\(Lj ◦H h ) consists of (f‘1 + 1)‘j cosets modulo H h . Then it holds that  {c(Li ◦ H h ) − c(Lj ◦ H h )} c∈Ch

=



c{(Li ◦ H h ) − (Lj ◦ H h )}

c∈Ch

= f‘i ‘j × (V − {0}); and



{c({0} ∪ (L1 ◦ H h )) − c(Lj ◦ H h )}

c∈Ch

=



c{({0} ∪ L1 ◦ H h ) − (Lj ◦ H h )}

c∈Ch

= (f‘1 + 1)‘j × (V − {0}):


Therefore, for distinct points / e and / e of V, the number 1j of blocks B
of B

containing / e in the ;rst and / e in the jth subblocks of B
is (f‘1 + 1)‘j , where the
number  ij of blocks B
of B
containing / e in the ith and / e in the jth subblocks
of B
is f‘i ‘j , which are independent of the / e and / e chosen for 26i; j6n. This completes the proof. Theorem 6.5.

n Let q = hf + 1 be a prime power. For all positive integers ‘1 ; ‘2 ; : : : ; ‘n such that i=1 ‘i 6h, there exists a symmetric (q; f‘ + 1; ‘(f‘ + 1))-BND of form

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(‘0 ; ‘1 ; : : : ; ‘n ) in which  i = ‘i (f‘i − 1); 0j = j0 = ‘j ;  ij = f‘i ‘j for 16i; j6n, where n ‘0 = 1 and ‘ = i=1 ‘i . Proof. For any mutually disjoint subsets L1 ; L2 ; : : : ; Ln of Ch ; ‘i = |Li |, let B0

= {0}; Bi

= Li ◦H h and B

= B0

∪ B1

∪ B2

∪ · · · ∪ Bn

. Let B

; B0

and Bi

; 16i6n, be the collections of blocks y + cB

, the subblocks y + cB0

and the ith subblocks, 16i6n; y + cBi

for y ∈ V and c ∈ Ch , respectively, and 

= {B0

; B1

; B2

; : : : ; Bn

}. Then, in a manner similar to the proof of Theorem 6.4, we can show that the triple (V; B

; 

) is the desired symmetric (q; f‘ + 1; ‘(f‘ + 1))-BND of form (‘0 ; ‘1 ; ‘2 ; : : : ; ‘n ) where

n ‘0 = 1 and ‘ = i=1 ‘i . Example 6.6. Consider the case q = 11, where h = 5 and f = 2. It is easy to know that 2 is a primitive element of GF(11), and H 5 = {1; 10}; H15 = {2; 9}; H25 = {4; 7}, H35 = {8; 3} and H45 = {5; 6}. The set C5 = {1; 2; 3; 4; 6} is a system of distinct representatives for the cosets modulo H 5 . For L1 = {1} ⊂ C5 and L2 = {2; 3} ⊂ C5 ; ‘1 = 1 and ‘2 = 2. Let B1
= {0; 1; 10}; B2
= {2; 3; 8; 9}, and B
= {0; 1; 10; 2; 3; 8; 9}. Let B
be the collection of blocks y +{0; 1; 10; 2; 3; 8; 9}; y +{0; 2; 9; 4; 6; 5; 7}, y +{0; 4; 7; 8; 1; 10; 3}; y + {0; 3; 8; 6; 9; 2; 5} and y + {0; 6; 5; 1; 7; 4; 10} for y ∈ GF(11). Let B1
be the collection of blocks y + {0; 1; 10}; y + {0; 2; 9}, y + {0; 4; 7}; y + {0; 3; 8} and y + {0; 6; 5} for y ∈ GF(11). Let B2
be the collection of blocks y + {2; 3; 8; 9}; y + {4; 6; 5; 7}; y+{8; 1; 10; 3}; y+{6; 9; 2; 5} and y+{1; 7; 4; 10} for y ∈ GF(11). De;ne 
= {B1
; B2
}. Then (GF(11); B
; 
) forms a symmetric (11; 7; 21)-BND of form (3; 4). If we divide the block B1
= {0; 1; 10} into B0

= {0} and B1

= {1; 10}, and let B0

be the collection of blocks y + {0}; y + {0}; y + {0}; y + {0} and y + {0} for y ∈ GF(11); B1

the collection of blocks y + {1; 10}; y + {2; 9}; y + {4; 7}; y + {3; 8} and y + {6; 5} for y ∈ GF(11), and 

= {B0

; B1

; B2
}, then (GF(11); B
; 

) forms a symmetric (11; 7; 21)-BND of form (1; 2; 4). 7. Existence results Balanced nested designs are of interest in their own right, as well as having many applications in the construction of other types of combinatorial structures, such as balanced arrays and balanced n-ary designs. In the previous sections, we described various methods of construction for symmetric balanced nested designs. In the remainder of this paper, we try to establish the existence results of symmetric balanced nested balanced incomplete block designs with small block size k. These would also imply some existence results of balanced arrays and balanced n-ary designs. Let (V; B) be a symmetric (v; k; )-BND of form (k 1 ; : : : ; k n ), where k i ¿1 for all 16i6n. Denote by Bi the collection of the ith subblocks Bi of size k i for all B ∈ B; 16i6n. Let b be the number of blocks in B, and thus in B1 ; : : : ; Bn ; r; r1 ; : : : ; rn be the replication numbers of each point x ∈ V in B; B1 ; : : : ; Bn , 1 ; : : : ; n the indices of the subdesigns (V; B1 ); : : : ; (V; Bn ), and  ij the numbers of blocks B ∈ B containing a point x ∈ V in the ith and another point y ∈ V in the jth subblocks of B; 16i; j6n.

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Then by some simple counting arguments, we obtain the following equalities: (v − 1) = r(k − 1); v(v − 1) = bk(k − 1);  i (v − 1) = ri (ki − 1)

for 16i6n with 2 6 k i 6k − 1;

 i v(v − 1) = bki (ki − 1) b = ri v

for 16i6n with 2 6 k i 6k − 1;

for 16i 6 n with k i = 1;

 ij v(v − 1) = bk i k j

for 16i; j6n:

Therefore, we can easily obtain the following statement. Theorem 7.1. The necessary conditions for the existence of a symmetric (v; k; )-BND of form (k 1 ; : : : ; k n ) are as follows. (i) If k i ¿2 for all 16i6n, then for all 16i; j6n, =

k(k − 1) k(k − 1)  ij ; i = ki (ki − 1) ki k j

 i (v − 1) ≡ 0 (mod (ki − 1));  i v(v − 1) ≡ 0 (mod ki (ki − 1)); (v − 1) ≡ 0 (mod (k − 1)): (ii) Otherwise, =

k(k − 1)  ij k i kj

for all 16i; j6n;

=

k(k − 1) i ki (ki − 1)

for 16i; j6n with ki ¿2;

 i (v − 1) ≡ 0 (mod ki (ki − 1))

for 16i; j6n with ki ¿2:

These necessary conditions are in fact also suQcient in most cases for the existence of a symmetric (v; k; )-BND. We ;rst consider the case k = 3. Theorem 7.2. The necessary and su=cient conditions for the existence of a symmetric (v; 3; )-BND of form (1; 1; 1) are that v¿3 and  is a multiple of 6. Proof. The necessity follows from Theorem 7.1. For the suQciency, we apply Theorem 5.3. It is known (see [3], for example) that an OD(3; v; 1) exists for any integer v¿3. Then we obtain a symmetric (v; 3; 6)-BND of form (1; 1; 1). Repeating such a design =6 times, we obtain a symmetric (v; 3; )-BND of form (1; 1; 1).

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Theorem 7.3. The necessary and su=cient conditions for the existence of a symmetric (v; 3; )-BND of form (2; 1) are that v¿3; 1 (v − 1) ≡ 0 (mod 2) and  = 31 = 312 =3 21 . Proof. The necessity follows from Theorem 7.1. Now we consider the suQciency. If 1 ≡ 0 (mod 2), then we apply Theorem 5.3. Similarly to the proof of Theorem 7.2, we can get a symmetric (v; 3; 6)-BND of form (2; 1) for any integer v¿3. Repeating such a design =6 times, we obtain a symmetric (v; 3; )-BND of form (2; 1) for any integer v¿3. If 1 ≡ 1 (mod 2), then v must be odd. Clearly an idempotent symmetric Latin square of odd order v¿3 exists. For example, if we let the point set be Zv , and L(i; j) ≡ (i + j)=2 (mod v), then L = (L(i; j)) is an idempotent symmetric Latin square of odd order v. Therefore, (Zv ; {{i; j; L(i; j)}: i; j ∈ {1; : : : ; v}; i¡j}) is a symmetric (v; 3; 3)-BND of form (2; 1). Repeating such a design =3 times, we obtain a symmetric (v; 3; )-BND of form (2; 1) for any odd integer v¿3. Noting that there are only two possible forms (1; 1; 1) and (2; 1) in any symmetric (v; 3; )-BND, we can obtain the following existence result by combining Theorems 7.2 and 7.3. Theorem 7.4. The necessary conditions described in Theorem 7.1 are also su=cient for the existence of a symmetric (v; 3; )-BND. Next we investigate the existence of a symmetric (v; 4; )-BND. There are only four possible forms, i.e., (1; 1; 1; 1); (2; 1; 1); (2; 2) and (3; 1), need to be considered. Theorem 7.5. The necessary and su=cient conditions for the existence of a symmetric (v; 4; )-BND of form (1; 1; 1; 1) are that v¿4, and  is a multiple of 12, except for a nonexisting symmetric (6; 4; 12)-BND of form (1; 1; 1; 1). Proof. The necessity follows from Theorem 7.1. For the suQciency, we apply Theorem 5.3. It is easy to see that the existence of a symmetric (v; 4; 12)-BND of form (1; 1; 1; 1) is equivalent to that of a pair of 2 idempotent MOLS(v). Since there does not exist a pair of 2 MOLS(6), we know that there does not exist a symmetric (6; 4; 12)BND of form (1; 1; 1; 1), too. It is known (see [1,3], for example) that an OD(4; v; ) exists for any integer v¿4 except for v = 6 when  = 1. This implies that a symmetric (v; 4; 12)-BND of form (1; 1; 1; 1) exists for any integer v¿4 except for v = 6 when  = 1. Theorem 7.6. The necessary and su=cient conditions for the existence of a symmetric (v; 4; )-BND of form (2; 1; 1) are that v¿4, and  = 61 = 612 = 613 = 12 23 . Proof. The necessity follows from Theorem 7.1. Now we consider the suQciency. Similarly to the proof of Theorem 7.5, we know that there exists a symmetric (v; 4; 12)BND of form (2; 1; 1) for any integer v¿4; v = 6. For the case v = 6, we can let the

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117

point set V be Z 5 ∪ {∞}, and the collection of base blocks B be {3; 4; ∞; 0};

{2; 3; 0; ∞};

{∞; 3; 1; 2};

{1; ∞; 2; 4};

{2; 4; 3; 1};

{0; 2; 4; 3}:

Then we repeat each block of the symmetric (v; 4; 12)-BND constructed above =12 times. Theorem 7.7. The necessary and su=cient conditions for the existence of a symmetric (v; 4; )-BND of form (2; 2) are that v¿4, and  = 61 = 6 2 = 312 . Proof. The necessity follows from Theorem 7.1. For the suQciency, we apply Theorem 5.4. It is well known (see, for example, [26]) that there exists an idempotent SOLS(v) for any integer v¿4 except for v = 6, which can imply that there exists a symmetric (v; 4; 6)-BND of form (2; 2) for any integer v¿4 except for v = 6. For the case v = 6, we can let the point set V be Z 5 ∪ {∞}, and the collection of base blocks B be {{∞; 1; 0; 2}; {0; 1; ∞; 3}; {2; 4; 0; 1}}. Then we repeat each block of the symmetric (v; 4; 6)-BND constructed above =6 times. Theorem 7.8. The necessary and su=cient conditions for the existence of a symmetric (v; 4; )-BND of form (3; 1) are that v¿4; 12 (v − 1) ≡ 0 (mod 3) and  = 21 = 412 . Proof. The necessity follows from Theorem 7.1. For the suQciency, we ;rst consider the cases 12 ≡ 1; 2 (mod 3). In these cases, v ≡ 1 (mod 3). We apply Corollary 3.5 to obtain a symmetric (v; 4; 4)-BND of form (3; 1) for any integer v ≡ 1 (mod 3). Repeat each block =4 times we get the desired design. Next we consider the case 12 ≡ 0 (mod 3). In this case, the necessary conditions are simpli;ed to v¿4. Similarly to the proof of Theorem 7.5, we can know that there exists a symmetric (v; 4; 12)-BND of form (3; 1) for any integer v¿4; v = 6. For the case v = 6, let the point set V be Z 5 ∪ {∞}, and the collection of base blocks B be {∞; 0; 2; 1};

{∞; 0; 1; 2};

{∞; 0; 3; 1};

{0; 1; 3; ∞};

{0; 1; 2; 4};

{0; 1; 2; 4}:

Then we repeat each block of the symmetric (v; 4; 12)-BND constructed above =12 times. This completes the proof. Combining Theorems 7.5– 7.8, we have the following existence result. Theorem 7.9. The necessary conditions described in Theorem 7.1 are also su=cient for the existence of a symmetric (v; 4; )-BND, except for the nonexisting symmetric (6; 4; 12)-BND of form (1; 1; 1; 1).

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As a summary, we obtain the main existence result of this paper. Theorem 7.10. The necessary conditions described in Theorem 7.1 are also su=cient for the existence of a symmetric (v; k; )-BND for k = 3; 4, except for the nonexisting symmetric (6; 4; 12)-BND of form (1; 1; 1; 1). Acknowledgements The authors wish to thank the anonymous referees for their useful comments which lead to substantial improvements in the exposition of the results of this paper. The ;rst and the fourth author would like to acknowledge the ;nancial supports from JSPS by Grant-in-Aid for Scienti;c Research (C) and Grant-in-Aid for Encouragement of Young Scientists under Contract Numbers 11640099 and 12740054, respectively. References [1] R.J.R. Abel, A.E. Brouwer, C.J. Colbourn, J.H. Dinitz, Mutually orthogonal latin squares (MOLS), in: C.J. Colbourn, J.H. Dinitz (Eds.), The CRC Handbook of Combinatorial Designs, CRC Press, Boca Raton, FL, 1996, pp. 111–142. [2] A.M. Assaf, A. Hartman, Resolvable group divisible designs with block size 3, Discrete Math. 77 (1989) 5–20. [3] J. Bierbrauer, Ordered designs and perpendicular arrays, in: C.J. Colbourn, J.H. Dinitz (Eds.), The CRC Handbook of Combinatorial Designs, CRC Press, Boca Raton, FL, 1996, pp. 396–399. [4] E.J. Billington, Balanced n-ary designs: combinatorial survey and some new results, Ars Combin. 17A (1984) 37–72. [5] E.J. Billington, Designs with repeated elements in blocks: a survey and some recent results, Congr. Numer. 68 (1989) 123–146. [6] R.C. Bose, On the construction of balanced incomplete block designs, Ann. Eugenics 91 (1939) 353–399. [7] I.M. Chakravarti, Fractional replication in asymmetrical factorial designs and partially balanced arrays, SankhyRa 17 (1956) 143–164. [8] I.M. Chakravarti, On some methods of construction of partially balanced arrays, Ann. Math. Statist. 32 (1961) 1181–1185. [9] C.J. Colbourn, D.R. Stinson, L. Zhu, More frames with block size four, J. Combin. Math. Combin. Comput. 23 (1997) 3–19. [10] J.H. Dinitz, D.R. Stinson, Room squares and related designs, in: J.H. Dinitz, D.R. Stinson (Eds.), Contemporary Design Theory: A Collection of Surveys, Wiley, New York, 1992, pp. 137–204. [11] N.J. Finizio, SOLS with a symmetric orthogonal mate (SOLSSOM), in: C.J. Colbourn, J.H. Dinitz (Eds.), The CRC Handbook of Combinatorial Designs, CRC Press, Boca Raton, FL, 1996, pp. 447–452. [12] R. Fuji-Hara, S. Kuriki, Mutually balanced nested designs, Discrete Math. 97 (1991) 167–176. [13] R. Fuji-Hara, S. Kuriki, Y. Miao, S. Shinohara, Balanced nested designs and balanced n-ary designs, J. Statist. Plann. Inference, to appear. [14] S. Furino, S. Kageyama, A.C.H. Ling, Y. Miao, J. Yin, Frames with block size four and index three, J. Statist. Plann. Inference, to appear. [15] S. Furino, Y. Miao, J. Yin, Frames and Resolvable Designs: Uses, Constructions, and Existence, CRC Press, Boca Raton, FL, 1996. [16] H. Hanani, Balanced incomplete block designs and related designs, Discrete Math. 11 (1975) 255–369.

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