Pairwise balanced designs with odd block sizes exceeding five

47

Discrete Mathematics 84 (1990) 47-62 North-Holland

PAIRWISE BALANCED DESIGNS SIZES EXCEEDING FIVE R.C. MULLIN* and D.R. STINSON** *Departmentof Combinatorics and Optimization, N2L 3Gl Canada **Department of Computer Canada

Science,

University

WITH ODD BLOCK

University

of Manitoba,

of Waterloo, Winnipeg,

Waterloo, Manitoba

Ontario

R3T 2N2

Present address of Dr. Stinson: Computer

Science and Engineering,

Universiv

of Nebraska,

Lincoln,

NE 68588-0115,

USA

Received 15 May 1986 Revised 19 October 1988 In this paper we construct pairwise balanced designs (PBDs) having block sizes which are odd prime powers exceeding 5. Such a PBD contains an odd number of points. We show that this condition is sufficient when the number of points is at least 2129, with at most 103 possible exceptions below this value. This is accomplished, in part, by some interesting new recursive constructions for PBDs. Also, we give several applications to the construction of other types of combinatorial designs, such as Room squares, skew Room squares, Room cubes, separable orthogonal arrays, and perpendicular arrays. We prove the new result that there is a perpendicular array PA(n, 7) for all odd n 2 2129.

1. Introduction A pairwise balanced design (or PBD) is a pair (X, Se) such that X is a set of elements called points, and ti is a set of subsets (called blocks) of X, each of cardinality at least two, such that every unordered pair of points is contained in a unique block in &. If v is a positive integer and K is a set of positive integers, each of which is greater than or equal to 2, then we say that (X, &) is a (v, K)-PBD if 1x1 = V, and IAlEKforeveryArzSe. Pairwise balanced designs are of fundamental importance in combinatorial theory, being of interest in their own right, as well as having many applications in the construction of other types of designs. In general, one usually is interested in constructing (v, K)-PBDs for some specified set K. We will denote B(K) = {v: there exists a (v, K)-PBD}. A set K is said to be PBD-closed if B(K) = K. In this paper, we investigate the set B(P,), where P7 is defined to be the set of odd primes powers not less than 7. According to Wilson’s theory of PBD-closed sets ([19], [20], and [23]), there exists a constant N such that, for all u> N, Y E B(P,) if and only if v is odd. Unfortunately, this theory does not yield any reasonable upper bounds on N. However, we are able to give an upper bound on N: N < 2127. Further, there are at most 103 odd integers n > 7 for which an 0012-365X/90/$03.50 @ 1990 - Elsevier Science Publishers B.V. (North-Holland)

R.C. Mullin, D. R. Stinson

48

(n, B(P,))-PBD

d oes not exist. The possible

exceptions

are those IZ in the set

(15, 21, 33, 35, 39, 45, 51, 55, 65, 69, 75, 87, 93, 95, 105, 111, 115, 123, 129, 135, 141,155,159,165, 183,185,195,201,205,213, 215,219,231, 235, 237, 245, 249, 255, 265, 267, 285,291,295, 303, 305, 309, 315, 321, 327, 335, 339, 345, 355, 363, 365, 375, 381, 395,415,445,

447, 453, 455,

465, 471, 483, 485, 501, 507, 519, 525, 543,573,

605, 615, 651,

579,597,

655,699, 717,735, 805, 843, 845, 861,903, 921,933, 945, 951, 957, 1047, 1077, 1119, 1227, 1315, 1383, 1515, 1595, 1623, 1795, 2127). For future reference, we name this set of 103 possible exceptions Q. Note that the elements of Q less than 55 are not in B(P,). Our PBD-result has several implications concerning other types of designs. We mention several corollaries in Section 7.

2. Recursive

constructions

for PBDs

In this section, we describe several recursive constructions for PBDs with odd block sizes. First, we need to define some terminology. A group-divisible design (or GDD) is a triple (X, 3, a), which satisfies the following properties: 1. % is a partition of X into subsets called groups 2. ~4 is a set of subsets of X (called blocks) such that a group and a block contain at most one common point. 3. every pair of points from distinct groups occurs in a unique block. The group-type of a GDD(X, 3, &) is the multiset { ]G( : G E %}. We usually use an “exponential” notation to describe group-types: a group-type 1i2i3k, . . . denotes i occurrences of 1, j occurrences of 2, etc. As with PBDs, we will say that a GDD is a K-GDD if IAl E K for every A E ~4. A transversal design TD(k, n) is a GDD with kn ponts, k groups of size IZ, and n2 blocks of size k. It follows that every group and every block of a transversal design intersect in a point. It is well-known that a TD(k, n) is equivalent to k - 2 mutually orthogonal Latin squares (MOLS) of order n. For a list of lower bounds on the number of MOLS of all orders up to 10000, we refer the reader to Brouwer [3]. A set of blocks (in a PBD or GDD) that partitions the point set is called a parallel class. If a PBD contains a parallel class, then the blocks in the parallel class can be taken as groups in a GDD. If the blocks of a PBD or GDD can be partitioned into parallel classes, then the design is said to be resolvable. We also need to define various types of incomplete designs. First, we discuss incomplete GDDs. Informally, an incomplete GDD, or IGDD, is a GDD from which a sub-GDD is missing (this is the “hole”). We give a formal definition. An

Pairwise balanced designs

49

IGDD is a quadruple (X, Y, 3, a) which satisfies the following properties: 1. X is a set of points, and Y E X, 2. 59is a partition of X into groups, 3. .& is a set of blocks, each of which intersects each group in at most one point, 4. no block contains two members of Y, and 5. every pair of points {x, y} from distinct groups, such that at least one of x, y is in X\Y, occurs in a unique block of d. We say that an IGDD(X, Y, ‘S, a) is a K-IGDD if IAl E K for every block A E d. The type of the IGDD, is defined to be the multiset of ordered pairs {(ICI, IG n YI): G E %}. A s with GDDs, we shall use an exponential notation to describe types. Note that if Y = 0, then the IGDD is a GDD. We define a TD(k, n) - TD(k, m) ( an incomplete transversal design) to be a {k}-IGDD of group-type (n, m)“. (This concept was introduced by J. Horton in

[61.) We also need PBDs containing subdesigns, or flats. Let (X, .&) be a PBD. If a set of points Y c X has the property that, for any A E s4, either IY rl A I s 1 or A G Y, then we say that Y is a&t of the PBD. The order of the flat is IYI. If Y is a flat, then we can delete all blocks A c Y, replacing them by a single block, Y and the result is a PBD. Also, any block or point of a PBD is itself a flat. However, often we do not require that the flat be present, i.e. it can be “missing”. Hence, we define incomplete PBDs, as follows. An incomplete PBD (or IPBD) is a triple (X, Y, &), where X is a set of points, Y G X, and A is a set of blocks which satisfies the following properties: 1. for any A E &, IA fl YI c 1 2. any two points x, z, not both in Y, occur in a unique block. Equivalently, we require that (X, a U {Y}) b e a PBD. We say that (X, Y, a) is a (v, w, K)-IPBD if [XI= V, (YI = w, and IAl E K for every A E &. We obtain PBDs from IGDDs by filling in groups as follows. Lemma 2.1. Let K be a set of positive integers, and let a 2 0. Suppose that the following designs exist: a K-IGDD of type T, where T = {(tI, ul), (t,, u,)}; a (t; +a, ui +a, K)-IPBD, for 1 lion; and a (u +a, K)(t2, uz), . . f > PBD, where u = ClsiG,, ui. Then there exists a (t + a, K)-PBD, containing a flat of order u + a, where t = CIGirn ti. Proof. Let (X, Y, 3, a) 1 =Si s n, let (G; U Q, (Y (Y U 52, 93) be a (u + a, desired (t + a, K)-PBD.

and let Q be a set of cardinality a. For be a (ti + a, ui + a, K)-IPBD. Also, let Then (X U Q, d U (LJlsisn a(i)) U 93) is the

be the IGDD,

fl Gi) U Sz, Se(i))

K)-PBD. 0

Beginning with a GDD, we have Corollary 2.2. Let K be a set of positive integers, and let a 2 0. Suppose that the

50

R.C.

Mullin,

D. R. Stinson

following designs exist: a K-GDD of type T = {tI, t2, . . . , t,} ; a (ti + a, a, K)IPBD, for 1 s i s n; and an (a, K)-PBD. Then there exists a (t + a, K)-PBD, where t = Clsizn tip ~~thning j7& C$ order ti + U, (1 s i c n), and order U. Proof.

In Lemma 2.1, let u1 = u2 = . . . = u, = 0.

The following specialization

0

of Corollary 2.2 is very useful.

Corollary 2.3. Let K be a set of positive integers, and let a = 0 or 1. Suppose that the following designs exist: a K-GDD of type T = {tI, t2, . . . , t,}; and a (ti + a, K)-PBD, for 1 c i
Any PBD trivially contains a flat of order 0 or 1.

Some useful GDDs arise from transversal examples.

0

designs. We will employ the following

Lemma 2.4. Zf there is a TD(k, n), then there is a {k, n}-GDD

having group-type

(k - l)“(n - 1)‘. Proof.

Delete a point x, taking the blocks (and group) through x as groups of a GDD. •i

Lemma 2.5. Zf there is a TD(k, n - l), group-type (k - l)“-‘(n - 1)‘.

then there exists a {k, n}-GDD

of

Proof. First, adjoin a new point COto the groups of the TD, and then delete some other point x, taking as groups the blocks (and group) through x. 0 Lemma 2.6. Zf there is a TD(k, n) which has a parallel class, then there is a {k, n}-GDD of group-type k”. Proof. Take the groups of the TD as blocks, and the parallel class of blocks of the TD as groups of a new GDD. 0 Lemma 2.7. Zf there is a TD(k, n - 1) which has a parallel class, then there is a {k, n}-GDD of group-type k”-lll. Proof. Adjoin a new point m to the groups of the TD and consider these as blocks of a new GDD. The groups of the new GDD consist of the parallel class of the TD and a group (00). 0

These GDDs are useful as ingredients in Wilson’s Fundamental GDDs ([22]). We briefly describe the construction.

Construction

for

Construction 2.8. (Fundamental Construction). Suppose (X, 99, &) is a GDD, and let w : X+ Z+ U (0) be any function (we refer to w as a weighting). For

51

Pairwise balanced designs

every x E X, let S(X) be a set of w(x) “copies” of x. For every A E &, suppose that (LA+), {s(x):x EA}, S(A)) is a GDD. Then (UXsXs(x),

{lJxeG~(x): G E31,UAEdWA)) isa GDD.

We now give some applications of Construction 2.8, using the above designs as ingredients. Using the GDDs in Lemmata 2.4 and 2.5, we obtain Theorem 2.9. Suppose there exists a TD(k, n) a TD(k, n - l), and a TD(n + 1, m), and 0 c t G m. Then there is a {k, n}-GDD having group-type (mk m)“-‘(mn

- m)‘(tk

- t)‘.

Proof. In all groups but two of a TD(n + 1, m), give the points weight k - 1. In the second last group, give the points weight n - 1. In the last group, give t points weight k - 1, and give the remaining points weight 0. We require GDDs of group-types (k - l)“(n - 1)’ and (k - l)“-‘(n - 1)‘. These exist by Lemmata 2.4

and 2.5.

Cl

We now state some useful corollaries of Theorem 2.9. Corollary 2.10

[lo, Theorem 5.21. Suppose Then there is a (7, 9}-GDD with group-type 1 EB{~, 9, 6m + 1, 8m + 1, 6t + l}.

Proof. Take

TD(7,8).

k = 7 and

n = 9,

noting

that

there is a TD(lO, m) and 0
there

exists

a TD(7,9)

and

a

0

Corollary 2.11 [ll, Theorem 2.101. Suppose Then there is a (7, 17}-GDD with group-type 6t + 1 eB(7, 17, 6m + 1, 16m + 1, 6t + l}. Proof. Take k = 7 and n = 17, noting TD(7,16). 0

that

there is a TD(18, m) and 0~ t c m. (6m)‘“(16m)‘(6t)‘. Hence, 112m +

there

exists a TD(7,17)

and a

Theorem 2.12. Suppose there is a TD(n, n), a TD(n, n + l), and a TD(n + 2, m), Then there is an {n, n + 2}-GDD with group-type (mn and OStSm. m)R+‘(tn + t)‘. Proof. In all groups but one of a TD(n + 2, m), give the points the last group, give t points weight n + 1, and give the remaining We require GDDs of group-types (n - l)n+l and (n - l)n+l(n + (respectively) from a TD(n, n) (Lemma 2.4), and from a TD(n,

2.5).

0

weight n - 1. In points weight 0. l)l. These come n + 1) (Lemma

R.C.

52

Mullin,

D. R. Stinson

Corollary 2.13 [ll, p. 4321. Suppose there is a TD(9, m) and 0 < t G m. Then there is a (7, 9}-GDD 1, 8t+

of group-type

(6m)8(8t)‘.

Hence,

48m + 8t + 1 E B(7,

9, 6m +

l}.

Proof. Set n = 7 in Theorem Now we obtain the GDDs

another

2.12. The required

construction,

in Lemmata

TD(7,7)

by applying

and TD(7,8)

the fundamental

exist.

construction

0 to

2.6 and 2.7.

Theorem 2.14. Suppose

there exist TD(k,

n) and TD(k,

n - l),

both

having

a

parallel class. Suppose also that there is a TD(n, m) and 0 < t c m. Then there is a {k, n}-GDD of group-type (mk)“-‘(m + kt - t)‘.

Proof. In all groups group, give t points GDDs of group-types 2.6 and 2.7. 0

but one of a TD(n, m), give points weight k. In the last weight k and the remaining points weight 1. We require k” and k”-‘11, which come (respectively) from Lemmata

Corollary 2.15 [ll, Lemma Then there is a (7, 9}-GDD B(7, 9, 7m, m + 6t).

2.141. Suppose with group-type

there is a TD(9, m) and 0s t G m. (7m)X(m + 6t)‘. Hence, 57m + 6t E

Proof. Take k = 7 and n = 9, noting that there both of which contain parallel classes. 0 We have an interesting

new variation

exists a TD(7,9)

of the above

and a TD(7,8),

constructions.

Theorem 2.16. Suppose

there exist GDDs of group-types ai, a’c’, and amdl, having block-sizes from P. Zf there exists a TD(i + 1, m) and 0 < t G m, then there exists a P-GDD with group-type am’(tc + d)‘.

Proof. Adjoin a new point 00 to the groups of a TD(i + 1, m), and delete m -t old points from one group G. Form a GDD by taking as groups G (with w), and let all other groups be single points. Give 00 weight d, give the points of G weight c, and give all other points weight a. 0 Corollary 2.17. Suppose

there exist a TD(n, n), a TD(n, n + l), and a TD(n + and let 0 < t c m. Then there is an {n, n + 2, m}-GDD with group-type nm+m(nt + t + m l)l. (n - 1)

2, m),

Proof. In Theorem 2.16, let a = n - 1, c = n + 1, d = m - 1, and i = n + 1. We have a GDD of type ui = (n - l)“+’ (Lemma 2.4); one of type (n - l)n+l(n + 1)’ = aicl (Lemma 2.5); and one of type (n - l)“(m - 1)’ = amdl (Lemma 2.4), Cl using a TD(n, m). The result follows.

Pairwbe

balanced designs

53

Corollary 2.18. Suppose there exists a TD(9, m), and 0
2.17, noting that a TD(7,7)

and a TD(7,8)

In a similar vein, we have Theorem 2.19. Suppose there exist GDDs of group-types I’+’ and licl, having block-sizes from P. If there exists a TD(i + 1, m) and 0 < t < m, then there exists a GDD with group-type m’(tc - t + m)‘, having block-sizes from P. Proof. Give all points weight 1, except for t points in the last group, which get weight c. 0 Corollary 2.20 [ll,

Lemma 2.131. Suppose there is a TD(43, m) and 0 c t < m.

Then there exists a {7,43}-GDD B(7, 43, m, m + 6t).

of group-type

m”‘(m + 6t)‘. Hence,

43m + 6t E

We now describe a new construction which is of a more general nature. This construction can be thought of either as a generalization of Wilson’s construction for mutually orthogonal Latin squares ([21]), or as a special case of Stinson’s general construction for GDDs ([17]). Theorem 2.21. Let P be a set of positive integers. Suppose there is a TD(u + 1, m), and let 0 s t < m. If there exist a P-GDD of type sU and a P-IGDD of type (s + 1, l)“(l, l)‘, then there exists a P-IGDD of type (ms + t, t)“(t, t)‘. Proof. Let (X, 3, a) be a TD(u + 1, m), where %= {Gi: 1 s i s u + l}. For x E X\G,+i, let s(x) denote s copies of x. Pick a subset Y E G,+l of cardinality t. For x E Y, let s(x) = {(x, i) : 1 cisu+l}beasetofu+lcopiesofx,andfor x E G,+i\Y, let s(x) = 0. For any subset 2 E X, define s(Z) = lJxEzs(x). We have hypothesized the existence of the following designs. For any block A E _$! such that A n Y = 0, let (s(A), {s(x):x EA}, B(A)) be a P-GDD (of type s”). ForanyblockAE&suchthatAflY={y}, let (s(A), S(Y), {s(x) U {(Y, i)> be a P-IGDD of type (s(X), s(Y), % a,>, of type {i}):l~i~u}U{YX{u+1}}

IX E

Gi, 1
U} U {(y,

u +

l)}}, W(A))

(s + 1, l)“(l, 1)‘. We construct a P-IGDD (ms + t, t)“(t, t)‘, by defining X = {s(Gi) U (Y x and S=IJ,,~%I(A). Cl

R.C. Mullin. D. R. Stinson

54

Corollary 2.22. Suppose there is a TD(n, n), a TD(n, n + 1) with u parallel class, and a TD(n + 2, m), where 0 c t cm. Then there is an {n, n + 2}-IGDD of group-type (nm - m + t, t)““(t, t)‘. Proof. From a TD(n, n), obtain a GDD of group-type (n - l)n+l, having blocks of size IZ, using Lemma 2.4. From a TD(n, n + 1) which has a parallel class, construct an {n, IZ+ 2}-GDD with group-type IZn+lll. Delete a block of size it + 2 to create an IGDD of type (n, l)n+l(l, 1)‘. Now, apply Theorem 2.21, as follows. Let u = n + 1, s = n - 1, and P = {n, n + 2). This yields an {n, n + 2}-IGDD of type (nm - m + t, t)““(t, t)‘, as desired. 0 Corollary 2.23. Suppose there exists a TD(9, m) and 0 G t sm. Zf there is a TD(9, t) or t = 7, then there exists a (7, 9}-GDD of type (6m + t)?‘. Hence, 48m + 9t E B(7, 9, t, 6m + t}. Proof. We apply Corollary 2.22 with n = 7. This produces a {7,9}-IGDD of type (6m + t, t)8(t, t)‘. If there is a TD(9, t), we complete this IGDD by filling in the blocks of this TD. If t = 7, we instead use the blocks of a {7,9}-GDD of type 79,

which exists by Lemma 2.6.

Cl

The remaining constructions are “product” type constructions. We will describe a very general type of product construction which utilizes incomplete transversal designs. The following construction is referred to as the singular indirect product (see [12] and [14]). Theorem 2.24. Suppose K is a set of positive integers and u E K; suppose v, w, and a are integers such that 0 G a s w s v; and suppose the following designs exist:

1. a TD(u, v - a) - TD(u, w - a), 2. u(v, w, K)-IPBD, and 3. a(u(w - a) + a, K)-PBD. Then there is a (u(v - a) + a, K)-PBD u(w - a) + a. Hence,

in particular,

that contains u(v - a) + a E B(K).

Proof. This is an immediate corollary incomplete transversal design. Cl

f&s

of

order

of Lemma 2.1, where the IGDD

If we let w = a in the singular indirect product,

we obtain the singular

u and

is an

direct

product. Theorem 2.25. Suppose K is a set of positive integers and u E K. Suppose v and w are non-negative integers such that w G v; suppose there exists a TD(u, v); suppose there is a (v, w, K)-IPBD; and suppose there is a (w, K)-PBD. Then there is a (u(v - w) + w, K)-PBD that contains f&s of order u, v, and w. Hence, in particular, u(v - w) + w E B(K).

Pairwise balanced designs

If we further specialize this construction

5.5

by letting w = 0, we obtain the direct

product. Theorem 2.26. Suppose K is a set of positive integers and u, v E K. Zf there exists a

TD(u, v), then there is a (uv, K)-PBD that contains

flats

of order u and v. Hence,

in particular, uv E B(K).

In order to apply the singular indirect product, we need incomplete designs. We use constructions given in [4] and [21] to produce these.

transversal

Lemma 2.27. Suppose there exist: a TD(k, m), a TD(k, m + l), a TD(k + 1, t), and 0 < u < t. Then there exists a TD(k, mt + u) - TD(k, u). Lemma 2.28. Suppose there exist: a TD(R, m), a TD(k, m + l), a TD(k, m + 2), a TD(k + 2, t), a TD(k, u), and 0~ v ct. Then there exists a TD(k, mt + u + v) - TD(k, v).

Finally, we also use a well-known result of MacNeish [8]. Lemma

2.29. Suppose that n has prime power factorization n = np?.

Then there

exists a TD(k, n) if k c 1 + min{pT}.

Using MacNeish’s Theorem useful corollary.

and the direct product,

we obtain the following

Lemma 2.30. Zf v is odd and v $ B(P,), then at least one of the following holds: v = 3 or 15 mod 18, or v = 5, 15, 3.5,or 45 mod 50. Proof. Let v have prime power factorization v = UIGiC,pF, where 7
power. If m > 1, then there is a TD(p;‘, vlpq’), by Lemma 2.29. By induction, v/pyl E B(P,), so we also have v E B(P,). 0 Even though 15 $ B(P,),

it is sometimes possible to multiply by 15, as follows.

Lemma 2.31. Suppose 3n E B(P,),

and there exists a TD(7, n). Then 45n E B(P7).

Proof. A {7}-GDD of group-type 315 has been constructed by Baker [l]. Give every point weight n and apply the Fundamental Construction, replacing each block by the blocks of a TD(7, n). Then fill in the groups with (3n, B(P,))PBDs. 0

We use another construction

for PBDs due to Brouwer [3].

56

R.C. Mullin. D. R. Stinson

Theorem

2.32. Suppose

q is a prime power

and 0 < t < q2 - q + 1. Then there is a

(t(q* + q + l), {t, q + t})-PBD. We also make use of the following Theorem 2”-l,

2.33. For

any

construction

positive

integer

of Seiden n,

there

[16].

is a resolvable

(22”-’ -

(2”~‘})-PBD.

Finally,

we note the existence

Theorem

2.34.

of projective

For any positive

planes

of order

a power

of 2.

integer j, there is a (2” + 2j + 1, (2’ + l})-PBD.

3. Some special cases In this section, we construct several “small” PBDs which must be handled as special cases. Many of these are obtained as applications of the singular indirect product construction. Consequently, we need to construct PBDs containing flats. We obtain these in several ways. If ingredients have been obtained by applying one of the product constructions, then they contain flats as indicated in the statements

of these constructions.

We also make

use of a few special

examples.

Lemma 3.1. There exists a (147, (7, ll})-PBD which contains flats of order 7 and 11, and there exists a (189, (9, 13})-PBD which contains flats of order 9 and 13. Proof.

Apply

Theorem

2.32 with q = 4, t = 7, 9.

We have a few applications Lemma

3.2.

(57,

177,

of the singular 357,

399,

561,

El

direct

product.

585,

935,

1205,

1239,

1415,

2013,

2015) G B(P,). Proof. Suitable equations are 57 = 7(9 - 1) + 1, 177 = 11(17 - 1) + 1, 357 = 7(57 - 7) + 7, 399 = 7.57, 561= 7(81- 1) + 1, 585 = 9(73 - 9) + 9, 935 = 7(143 11) + 11 (143 = 11.13) 1205 = 7(173 - 1) + 1, 1239 = 7.177, 1415 = 7(203 - 1) + 1, 2013 = 7(297 - 11) + 11 (297 = 11.27) and 2015 = 7(299 - 13) + 13 (299 = 13.23). The necessary TDs are known to exist (see [2]), so the result follows. (Remark: a 0 (73, {9})-PBD is just a projective plane of order 8 (Theorem 2.34)). Lemma

3.3. {855,1215}

E B(P,).

Proof. Apply Theorem 2.31 with n = 19, 27, observing and TD(7,27), and (57, Sl} zB(P,). 0

that there

exist TD(7,19)

Pairwise balanced designs

57

Lemma 3.4. 1585 E B(P,). Proof. Apply Corollary 2.13 with the equation 1585 = 48.32 + 8 * 6 + 1. The 0 necessary ingredients are TD(9,32) and (49,193) c_B(P,). Lemma 3.5. 685 E B(P,). Proof. Apply Corollary 2.18 with the equation 685 = 49 . 13 + 8 - 6, observing q that there exists a TD(9,13), and {13,61} GB(P,). Lemma 3.6. 745 E B(P,). Proof. We use a special application of the Fundamental Construction for this value. Begin with a resolvable (120, {8})-PBD, which exists by Theorem 2.33. Adjoin three “infinite” points to three parallel classes, thereby constructing an {8,9}-GDD of group-type 311i2’. Observe that each block of size 9 meets G3, the group of size 3, whereas no block of size 8 does. Now, give the points of G3 weight 8, give every other point weight 6, and apply the Fundamental Construction. As ingredients we use a {7}-GDD of type 68, and a {7,9}-GDD of type 6881 (these exist from Lemma 2.4 with k = n = 7, and from Lemma 2.5 with k = 7, IZ= 9). We obtain a {7,9}-GDD of group-type 612’24’, which gives rise to a (745, {7,9,25})-PBD. Cl Lemma 3.7. 273 E B(P,). Proof. Apply Theorem 16). •!

2.34 with i = 4 (i.e. there is a projective

plane of order

Lemma 3.8. {2787,3399} E B(P,). Proof. These PBDs are obtained from Corollary 2.23, using the equations 2787 = 48.56 + 9 ’ 11 and 3399 = 48 * 65 + 9 * 31. The required TD(9, ll), TD(9,31), TD(9,56) and TD(9,65) exist (see [2]); and {11,31,347,421} c B(P,). 0 Since the singular indirect product is a very complicated construction, we apply it only when other constructions fail. It is most convenient to handle these exceptional cases before proceeding further. We present applications of this construction in Table 1, with P = P,.

58

R.C.

Mullin,

D.R.

Stinson

Table 1. Applications of the singular indirect product w

n

Equation

405 411 417 429 435 515 695 755 815 987 1085 1115 1235 1245 1255 1293 1851 1959 2019 2391 3183

7(63 - 6) + 6 7(63 - 5) + 5 7(63 - 4) + 4 7(63 - 2) + 2 7(63 - 1) + 1 7(77 - 4) + 4 7(113 - 16) + 16 7(113 - 6) + 6 7(119-3)+3 7( 147 - 7) + 7 7(161- 7) + 7 7(161- 2) + 2 9(147- 11) + 11 7(189 - 13) + 13 7(187 - 9) + 9 7( 189 - 5) + 5 7(273 - 10) + 10 7(297 - 22) + 22 7(297 - 10) + 10 7(351- 11) + 11 7(459 - 5) + 5

4. Members

17 7 7 7 7 11 13 11 7 17 27 11 13 17

PBD with flat

Incomplete TD

U(W-~)+a

63=7.9 63=7.9 63=7.9 63=7.9 63=7.9 77=7.11 113 = 7(17 - 1) + 1 113=7(17-1)+1 119 = 7. 17 Lemma 3.1 161= 7.23 161= 7 ‘23 Lemma 3.1 Lemma 3.1 187 = 11 17 189 = 7.27 PG(2,16) 297 = 11’ 27 297=11.27 351= 13.27 459=17.27

TD(7,57) 58=7.8+2 59=7.8+3 61=7.8+5 62=7.8+6 73=7.9+7+3 TD(7,97) TD(7,107) 116=7.16+4 TD(7,140) TD(7,154) 159 = 11 .13 + 11+ 5 TD(9,147) TD(7,176) 178 = 7 .25 + 1 + 2 184=7.25+7+2 263 = 8 .32 + 7 275 = 7.37 + 11+ 5 TD(7,287) 340=7.27+9+2 454=7.59+29+12

13 19 25 37 43 25 23 13 31

of B(P,) congruent

to

37 7 13 23 19 59 57 17 25 89

1 module 6

As a starting point, we use the results of [15], in which we investigated the set B(P,,,), where PI,~ is defined to be the set of prime powers congruent to 1 modulo 6. The following result is proved. Theorem 4.1. Zf n = 1 mod 6 is a positive integer, then there is an (n, PI,,)-PBD, except possibly for the 31 values n E (55,

115, 145, 205, 235, 253, 265, 295, 319, 355, 391, 415, 445, 451,

493, 649, 655, 667, 685, 697, 745, 781, 799, 805, 1243, 1255, 1315, 1585, 1795, 1819, 1921). We can eliminate several of these possibilities using block sizes in PT. First, we make a useful observation. Lemma 4.2. (253, 1921) E B(Z’,).

319, 391, 451, 493, 649, 667, 697, 781, 799,

Proof. By Lemma

2.30, if v = 1 mod 6 and n $I?(&),

result follows.

0

1243, 1819,

then v =Omod5.

The

Pairwbe

balanced designs

59

In Section 3, we showed that (145, 685, 745, 1255, 1585) EB(&). have Theorem

Hence,

we

4.3. Zf n = 1 mod 6 is a positive integer, then there is an (n, P,)-PBD,

except possibly for the following 13 values of n: n E (55, 115,205, 235, 265,295,

355,415,445,655,

805, 1315, 1795).

5. PBDs up to order 30001

In this section, we describe the construction of (v, B(P,))-PBDs for odd v s 30001 in preparation for determining the spectrum of these PBDs for v = 3 or 5 modulo 6. This was accomplished with a short computer run, applying several of the constructions given in Section 2. Our computer program had a knowledge of the results of Sections 3 and 4. The only transversal designs employed were those that exist by Lemma 2.29 (MacNeish’s Theorem). Given an integer v = 3 or 5 modulo 6, the program attempted to construct a (v, B(P,))-PBD by applying the following constructions (in the order indicated): Lemma 2.30 (prime power factorization); Corollary 2.13 (the 48m + 8t + 1 construction); Corollary 2.15 (the 57m + 6t construction); Corollary 2.18 (the 49m + 8t construction); Corollary 2.23 (the 48m + 9t construction); Corollary 2.10 (the 56m + 6t + 1 construction); Corollary 2.11 (the 112m + 6t + 1 construction); and Corollary 2.20 (the 43m + 6t construction). The program constructed all remaining PBDs in the stated interval, except for those indicated as exceptions in the Introduction. So, we have Theorem

5.1. Zf 7 c n s 30001, n is odd, and n B Q, then n E B(P,).

For the sake of interest, we mention the frequency of application of these constructions (of course, when we construct a PBD by means of one of these constructions, we do not bother to check if it can also be obtained by means of one of the others). Corollary 2.13 was applied 1062 times; Corollary 2.15 was used 2579 times; Corollary 2.18 was used 547 times; Corollary 2.23 was used 29 times; Corollary 2.10 was used 34 times; Corollary 2.11 was used 6 times; and Corollary 2.20 was required 12 times.

6. The spectrum of B(P,) In this section, we show that all odd values n exceeding 30001 are in B(P,). We accomplish this in two steps. It is simplest to first consider n = 1 modulo 8. Then we handle n = 3 and 5 modulo 6.

R. C. Mullin, D.R. Stirnon

60

Theorem

6.1. For all n = 1 modulo

8, n 2 953, n E B(P,).

Proof. We prove this theorem by induction on n. So, suppose that n E B(P,) if n = 1 modulo 8, 953 G n < N, for some N = 1 modulo 8. In view of Theorem 5.1, we can assume 124, uniquely.

119 < t =G positive

that N 2 30009. We can write N = 48m + 8t + 1, where Now it is easy to see that among any 10 consecutive

integers,

at least one of them is relatively

m’ such

that

m - 9 dm’

G m and

prime

g.c.d.(m,

to 2 * 3 . 5 .7. Hence, 2.3

we can find

* 5 - 7) = 1. Hence,

there

is a

TD(9, m’), by Lemma 2.33. We want to apply Corollary 2.13, with the equation N = 48m’ + 8t’ + 1 (t’= t +6(m -m’)). First, we check that m’> t’. Since N >300093 15785 = and m’s309 -9 = 300. On the other 48 - 309 + 8 .119 + 1, we have m ~309, hand, t s 124, so t’ s 124 + 6.10 = 184. Hence, we find that m’ > t’ 3 119. Then, it follows by induction that 8t + 1 EB(P~). Also, since m’ 2 300, we have 6m’ + 1~ 1801, and hence 6m’ + 1 EB(P~) by Theorem 4.3. Since there is a TD(9, m’), we obtain N EB(P~). By induction, the proof is complete. 0

The other

two cases are handled

in a similar

fashion.

We combine

them

in one

proof.

Theorem 6.2. For all II = 3 modulo n = 5 modulo 6, n 3 1601, n E B(P,).

6,

n 22133,

n E B(P,);

and

for

all

Proof. By Theorem 5.1, we can assume that n 5 30003. We will apply Corollary 2.10. First, we can write n = 56m + 6t + 1, where 300 28625=56~479+6~300+1, we have m 3479, and m’ 2 479 - 12 = 467. On the other hand, t s 355, so t’ G 355 + (28/3). 12 = 467. Hence, we find that m’ 2 t’ 3 300. Since m’ 2 t’ 2 300, we have 6m’ + 13 6t’ + 12 1801, and hence (6m + 1,6t’ + l} E B(P,), by Theorem 4.3. Also, 8m’ + 1 E B(P,) by Theorem 6.1. Hence, we obtain n E B(P,). Theorem

•I

6.3.

Zf n > 5 is odd and n 4 Q, then there is a (n, B(P,))-PBD.

Proof. This is an immediate

consequence

of Theorems

4.3, 5.1, and 6.2.

0

Pairwbe balanced designs

61

7. Applications

In this section we give several applications of the PBD-closure results proved in Section 6. The first problem we consider is that of Room designs. A Room square of side n is an n x n array R of cells, in which each cell either is empty or contains an unordered pair chosen from a set of n + 1 symbols, such that each symbol occurs in exactly one cell of each row and column of R, and each pair of symbols occurs in exactly one cell of R. A Room cube of side n is an n x n x n cube, such that each two-dimensional projection is a Room square of side n. A Room square is skew if one symbol occurs precisely in the cells of the main diagonal, and for every pair of off-diagonal cells symmetrically situated with respect to the main diagonal, precisely one of the two cells is empty. The spectrum for each of these types of Room designs is the set {n : n 2 7 and n is odd}. For Room squares, this was proved in 1975 by Mullin and Wallis [9]; for Room cubes, this was proved in 1981 by Dinitz and Stinson [5], and for skew Room squares, this was proved in 1981 by Stinson [18]. The proofs of these results are quite complicated, requiring various direct and recursive constructions. However, the PBD-closure result we have proved in this paper applies to these problems. This is because direct constructions are known which provide these designs for any side in the set P7, and since the set of sides of any of the three classes of designs is PBD-closed. Hence, we have these designs of any side in B(P,), leaving relatively few special cases to be dealt with. The other problem we consider is that of separable orthogonal arrays. An orthogonal array OA(n, k) is an n2 x k array of n symbols, such that the ordered pairs occurring in any two columns of the array comprise all n2 possibilities. A perpendicular array PA(n, k) is an n(n - 1)/2 x k array of n symbols, such that the unordered pairs occurring in any two columns of the array comprise all n(n - 1)/2 possible unordered pairs of distinct elements. A PA(n, k) can exist for k 3 3 only if n is odd. An OA(n, k) is separable if we can partition the n* rows into three sets of rows A, B, and C, of sizes n, n(n - 1)/2, and n(n - 1)/2 respectively, such that A contains n rows of the form (i, i, . . . , i), and such that B and C are each PA(n, k)‘s. In Lindner, Mullin, and van Rees [7], the following results are proved concerning separable OA(n, k). First, it is shown that there is a separable OA(n, n) for any odd prime power n. Second, for fixed k, the set SOA = {n: there exists a separable OA(n, k)}, is PBD-closed. By using previously known PBD results, the sets SOA were almost completely determined for k = 3, 4, and 5. The results we have proved apply to SOA(6) and SOA(7). We have the following new theorem. Theorem 7.1. Zf n > 5 is odd and n r$Q, then there is a separable OA(n, 7) (and consequently a separable OA(n, 6), a PA(n, 7), and a PA(n, 6)).

62

R.C.

Mullin,

D.R.

Stinson

References [l] R.D. Baker, An elliptic semiplane, J. Combin. Theory Ser. A (1978) 193-195. [2] A.E. Brouwer, The number of mutually orthogonal Latin squares--a table up to order 10000, Research Report ZW 123/79, Math. Centrum, Amsterdam, 1979. [3] A.E. Brouwer, A series of separable designs with application to pairwise orthogonal Latin squares, European. J. Combin. 1 (1980) 39-41. [4] A.E. Brouwer and G.H.J. van Rees, More mutually orthogonal Latin squares, Discrete Math. 39 (1982) 263-281. [S] J.H. Dinitz and D.R. Stinson, The spectrum of Room cubes, European J. Combin. 2 (1981) 221-230. [6] J.D. Horton, Sub-Latin squares and incomplete orthogonal arravs. < J. Combin. Theorv, Ser. A 16 (1974) 23-33. _ [71 C.C. Lindner, R.C. Mullin, and G.H.J. van Rees, Separable orthogonal arrays, Utilitas Math. 31 (1987) 25-32. PI H.F. MacNeish, Euler squares, Ann. Math. 23 (1922) 221-227. I91 R.C. Mullin and W.D. Wallis, The existence of Room squares, Aequationes Math. 13 (1975) l-7. [101 R.C. Mullin, D.R. Stinson and W.D. Wallis, Concerning the spectrum of skew Room squares, Ars Combin. 6 (1978) 277-291. WI R.C. Mullin, D.R. Stinson and W.D. Wallis, Skew squares of low order, Proc. Eighth Manitoba Conf. on Numerical Mathematics and Computing, Winnipeg (1978) 413-434. WI R.C. Mullin, A generalization of the singular direct product with application to skew Room squares, J. Combin. Theory Ser. A 29 (1980) 306-318. 1131R.C. Mullin, P.J. Schellenberg, G.H.J. van Rees and S.A. Vanstone, On the construction of perpendicular arrays, Utilitas Math. 18 (1980) 141-160. 1141R.C. Mullin, P.J. Schellenberg, S.A. Vanstone and W.D. Wallis, On the existence of frames, Discrete Math. 37 (1981) 79-104. WI R.C. Mullin and D.R. Stinson, Pairwise balanced designs with block sixes 6t + 1, Graphs Combin. 3 (1987) 365-377. WI E. Seiden, A method of construction of resolvable BIBD, Sankhya A 25 (1963) 393-394. u71 D.R. Stinson, A general construction for group-divisible designs, Discrete Math. 33 (1981) 89-94. P31 D.R. Stinson, The spectrum of skew Room squares, J. Austral. Math. Sot. Ser. A 31 (1981) 475-480. 1191R.M. Wilson, An existence theory for pairwise balanced designs I, J. Combin. Theory Ser. A 13 (1972) 220-245. PO1 R.M. Wilson, An existence theory for pairwise balanced designs II, J. Combin. Theory Ser. A 13 (1972) 246-273. WI R.M. Wilson, Concerning the number of mutually orthogonal Latin squares, Discrete Math. 9 (1974) 181-198. P21 R.M. Wilson, Constructions and uses of pairwise balanced designs, Math. Centre Tracts 55 (1974) 18-41. [231 R.M. Wilson, An existence theory for pairwise balanced designs III, J. Combin. Theory Ser. A 18 (1975) 71-79.