Some constructions of two-associate class PBIB designs

Ann. Inst. Statist. Math. 39 (1987), Part A, 671-679

SOME CONSTRUCTIONS OF TWO-ASSOCIATE CLASS PBIB DESIGNS

SNIGDHA BANERJEE, SANPEI KAGEYAMA AND BHAGWANDAS

(Received Jan. 27, 1986; revised June 11, 1986)

Summary

Two-associate class PBIB designs, having association schemes of GD or L2 types, are constructed by using pat terned matrices and by methods of taking unions of sets of blocks.

1. Introduction

A large number of methods of construction of two-associate partially balanced incomplete block (PBIB) designs are available in liter- a ture (cf. Clatworthy [1] and Raghavarao [3]). We here present fu r ther methods of construction of semi-regular group divisible (GD) designs from self-complementary balanced incomplete block (BIB) designs. The method of block extensions applied to incidence matrices of certain BIB designs gives SRGD designs and L2 designs. Series of SRGD and L2 designs are constructed by a method of taking unions of blocks in affine resolvable block designs. Some methods are explained in Section 2, while our constructions are presented in Sections 3 and 4.

2. Definitions and methods

For most of definitions of incidence s t ructures discussed here, refer to Raghavarao [3]. We describe only some definitions to avoid confu- sion of available notations here.

An incomplete block design with parameters v (number of t reatments), b (number of blocks) and r (replication number of each treat- ment) is said to be a-resolvable, if the blocks can be parti t ioned into t resolution classes, $1, S~, . . . , S~, each of p blocks (b=~t) such tha t every t r ea tmen t is repeated a t imes in each class S~ (r=at). An a- resolvable block design is said to be affine a-resolvable, if any pair of blocks in the same class S, intersect in x t rea tments (say) and any pair

Key words and phrases: BIB design, group divisible (GD), semi-regular (SR), self-complementary, affine u-resolvable, Lz.

671

672 SNIGDHA BANERJEE, SANPEI KAGEYAMA AND BHAGWANDAS

of blocks from different classes S~ and Sj ( i4: j) intersect in y treatments (say). I t is well-known tha t if a=l (and then t--r), the design is affine resolvable in the usual sense with x = 0 .

If N is the incidence matr ix of a binary block design D, then 2V

=J,,b--N is the incidence matr ix of a design /) called the complement of D where J,,b is a v• matr ix with unit elements everywhere. A

design D is said to be self-complementary, if its complement D has precisely the same parameters as D. If D is self-complementary and

if D and /) are also isomorphic, then D is said to be truly self-complementary.

A BIB design is an incomplete block design with parameters v, b, r in which every block is of size k and every pair of distinct t reat- mea ts occurs exactly in ~ blocks. We shall denote such a design as a BIBD (v, b, r, k, ~). If v=b, such a BIB design is said to be symmetric, and in tha t case r=k. Therefore, a symmetr ic BIB design can be wr i t t en as an SBIBD (v, k, ~). I t is well-known tha t any two distinct blocks of an SBIBD (v, k, ~) have ~ t rea tments in common. The complement of a BIBD (v, b, r, k, ~) is a BIBD (v, b, b--r, v--k, b--2r+~) . In order tha t a BIB design is self-complementary, it is necessary and suf- ficient tha t v=2k. It is known (cf. Shrikhande and Raghavarao [4]) tha t for an affine a-resolvable BIBD (v, b, r, k, ~), x=k--r+~l and y=k2/v.

Let N=[N, N2,..., Nb] be the v • incidence matr ix of an incomplete block design D, where N~, i - -1, 2 , . . . , b, are the i-th column of N. A method of block extensions used in this paper for some s con- sists of choosing s new blocks each of two blocks out of the original b blocks of an incomplete block design with equal block size k and keeping one block over the other, thus making s new blocks of size 2k, i.e., the incidence matr ix N* of the new design is of the form N*

where N ~ * = | ~ / ] for i = 1 , 2 , . . . , s; l<l , l '~_b. ~ [ Y l ~ , N.* , . . . , N,*], L~.Je

A block design derivable from D by a method of block unions is a design in which any block is of the form B~UBj for some i e j ; B~, Bj being blocks in D.

3. Construction based on patterned matrices

Let O~,s be an i x j matr ix whose all elements are zero.

THEOREM 3.1. A truly self-complementary SRGD design D* with incidence matrix N* and parameters v*=4k=b*, r*=2k=k*; ~*=0, 2*=k; m*=2k, n * = 2 exists i f and only i f

N . = [ N J2~,1 02,.1~ 0,.,~,1 &,,,,J

SOME CONSTRUCTIONS OF TWO-ASSOCIATE CLASS PBII3 DESIGNS 673

where N is the incidence matr ix o f a self-complementary BIBD (2k, 4/r

--2, 2k--1, k, k - - l ) and N, its complement.

PROOF. Suppose that a self-complementary BIBD (2k, 4 k - 2 , 2 k - l , k, k - l ) exists. Let t rea tments corresponding to the i-th row and (2k + i ) - th row in N* constitute the i- th group of a GD association scheme for l<_i~2k. Then it follows tha t N* is the incidence matr ix of a t ruly self-complementary SRGD design with the required parameters. Conversely, s tar t ing from a t ruly self-complementary symmetric SRGD design with parameters v*=4k, k*---2k, ~*=0, ~*=k, m*--2k, n*--2, it is easy to see that we can ge t a pair of disjoint blocks, to form at least one resolution class each of size 2k such tha t one t r ea tmen t of one block paired with the t r ea tmen t from the other block forms a group. Then the incidence matr ix of the SRGD design admits the following natural matrix decomposition on wri t ing t rea tments of the i- th group as the i-th and (2k+i)- th t r ea tmen t s :

J2~,1

Now, it is obvious to show the values of parameters as v '=2k, b'---4k --2, r ' = 2 k - - 1 and ~'--k--1 in N. Let b, be the number of uni ty in the i- th block of N. Then it follows tha t

4k--2 4 k ~ 2

b~=2k(k--1) , 5~, b , ( b ~ - l ) = 2 k ( 2 k - - 1 ) ( k - 1 ) . ~:=1 i = 1

4k--2 4k--2

In this case, let t ing b={I/(4k--2)} ~ b~:/~, we can show Z (b~--b)~=O Z = I Z = l

af ter some calculation. Then b~=b=k for all i. Hence, N is the incidence matr ix of a self-complementary BIBD (2k, 4 k - 2 , 2k-- l , k, k-- l ) , and M is the incidence matr ix of its complement, because of ~* =0 and the grouping of t reatments . Thus, the proof is completed.

Theorem 3.1 can be generalized by replacing J~.~ and 02~.1 by J2~,~ and 02~,~, respectively, for some positive in teger I. This process yields some repeated blocks.

A GD design with parameters v--ran, b, r, k, ~, ~2, m, n is de- noted by a GD design (v, b, r, k; ~,, ~2; m, n).

THEOREM 3.2. I f N is the incidence ma t r i x o f a self-complementary BIBD (2k, l (4k -2 ) , / (2k - - l ) , k , / (k - - l ) ) f o r a positive integer l, then


I N _AT J2,,u 02,.u ] _3, 0 , . . , . _ 2 , [],,,:0,,~] [ 4 , , : 0 , , ]

N * = Jl , l ; N O~,u J~,,~,'

LO,,,.~-2, J1,~,,~-2, [01,,: J1,~1 [01,~: d,,,]J

is the incidence matrix of a truly self-complementary SRGD design (4k+2, 8kl, 4kl, 2k+1; 0, 2kl; 2k+1, 2).

PROOF. Let the treatments corresponding to the i-th and (2k+1 + i ) - th rows define new groups for l ~ i K 2 k + l . Clearly ~*=0 and in N* any inner product of any two rows corresponding to different groups has value 2kl. Therefore, 2"=2kl. The other relations for v*, b*, r*, k* and the semi-regularity condition of r*k*=v*2~* can easily be veri- fied, along with the truly self-complement.

We now mention some combinatorial meaning of Theorems 3.1 and 3.2. Preece [2] gave various non-isomorphic solutions of self-complementary BIB designs with v=2k, and hence each of such solutions would give a solution to the symmetric SRGD design, and therefore our theorems are powerful enough to produce as many non-isomorphic solutions to the SRGD design as there are non-isomorphic solutions to the BIB design. For example, Preece [2] presented 4 non-isomorphic solutions of a BIBD (8, 14, 7, 4, 3) which, from Theorem 3.1, yield 4 non- isomorphic solutions for an SRGD design with parameters v=b=16, r= k----8, 2~=0, ~2-=4, m=8, n=2, though Clatworthy [1] listed only one solution as SR 92. Similarly, 3 non-isomorphic solutions of a BIBD (10, 18, 9, 5, 4) yield 3 non-isomorphic solutions for an SRGD design with parameters v=b=20, r=k=lO, 21=0, 2~-=5, m=lO, n=2, one of them is known as SR 108.

An old method of juxtaposing incidence matrices of some available designs will yield other designs. This is fundamental and sometimes useful. For example, if N is the incidence matrix of a BIBD (Ik, b, r, k, 2) for a positive integer l, then N*=[N: I~QJ~.~] is the incidence matr ix of an RGD design (lk, b+lp, r+p, k; ~+p, 2 ; l, k) for any positive integer p. As an individual example, an existing BIBD (27, 39, 13, 9, 4) with l=3 and p--2 yields the RGD design (27, 45, 15, 9; 6, 4; 3, 9), which may be new.

4. Method of block extension THEOREM 4.1. The existence of a BIBD (v, b, r, k, 2) implies the ex-

istece of an SRGD design (Iv, b ~, rb ~-~, lk; 2b ~-', ~ab~-2; l, v) for any positive integer l>= 2.

SOME CONSTRUCTIONS OF TWO-ASSOCIATE CLASS PBIB DESIGNS 675

PROOF. Le t N be the v x b incidence mat r ix of a BIBD (v, b, r , k, 2) and let N, be the i- th column of N, as N = [N, N2 , . . . , Nb]. Call

I-N,,-1

l

where ( i , i2 , . . . , i~) is a permutat ion on blocks of order l. We consider all possible b ~ vectors of the type N * . �9 and let N * - r N . * . .

Zl,Z2, . . . ,Z~. - - L ~l , Z 2 , " ' , ~ t ,

N~.~,. . . , j t , . . . , N,*8~....,,~] of size l v x b k follows :

If we wr i t e the lv t r e a tmen t s as

1 v + l �9 �9 �9 ( l - - 1 ) v + l

2 v + 2 �9 �9 �9 ( l - -1 )v+2

v 2v �9 �9 �9 lv

then the i - th column forms the i- th group, i=1 , 2 , . . . , l , i.e., m * = l and n*=v . Now, consider a pair of first associates, which belongs to the same group. It was occurring originally in 2 blocks, each of which is now replicated b ~-1 times. Hence 2"=2b H . A pair of second associates {0, r belonging to different groups, occurs in the resul t ing design as many t imes as the pair of columns N, , N~, say, containing t~ and r respectively, occurs toge ther as columns of N*, i.e., 2*=r2b ~-2. Other constants are easy to check, besides verification of the relation r*k*--v*2*=O for the semi-regulari ty of the GD design, since r * - 2 * =bZ-l( r - -2)>O.

THEOREM 4.2. The existence o f an SBIBD (v, k, 2) implies the existence o f an L2(v) design with parameters v * = v 2, b*=2v, r*---2k, k*= vk, 2 * = k + L 2*=22.

PROOF. In the incidence matr ix N * of Theorem 4.1 wi th l=2 , when v=b, N * and N*, for j~=j' will have k + 2 t r ea tmen t s in common. The same is t rue for N * and N~*j for i r whereas N * and NZ, for iq=i', j C j ' have 22 t r ea tmen t s in common. If B*j is a block corresponding to N ~ and if we arrange these blocks as follows:

B * = I B*~ B * B * . . . B ' f ]

. . .

�9 . . B ~ J

Then i t follows tha t the dual of the SRGD design N * constructed in


Theorem 4.1 with l=2 gives an L~(v) design with the required parameters, where the usual association scheme of the L2(v) design is the dual to the block arrangement B* given above. Thus, the proof is completed.

Remark . A trivial SBIBD (v, v - - l , v--2) th rough Theorem 4.2 yields the complement of a well-known simple lattice with v * : v ~, b*=2v, r* =2, k*=v , ~*=1, 2*=0.

THEOREM 4.3. The existence o f an a ~ n e a-resolvable BIBD (v, b=flt, r = a t , k, ~) implies the existence o f an ap-resolvable SRGD design (2v, t/~ 2, ta/~, 2k; p~, a2t; 2, v).

PROOF. Let the incidence matr ix N of the affine a-resolvable BIBD (v, b=~t, r=a t , k, ~) be wr i t ten as :

S, S~ S~

N = [ N ) , N L . . . , N ] : N ? , N ~ , . . . , N ] : . . . : N : , N . ; , , . . . , N ] ]

where S,, i = 1 , 2 , . . . , t, is the i- th resolution class. Now consider

s~* s* S*

N * = [NJ, N? ~, " . , N ~ : N,~, N L " " , N]~ : "'" : N~i, N,i, " " ]

,_[N:] where N ~ 3 - 1 N } | , i, / : 1 , 2 , . . . , /~; l : l , 2 , . . . , t . Then N* is the 2v•

p2t incidence matr ix of a design with v*=2v t rea tments , t sets of /~2 blocks each of size 2k. Every t r ea tmen t occurs in each set S* a~ t imes and hence apt t imes in N*. Any pair of t rea tments , u, s, occurs toge ther in p~ blocks if l < u , s < v or v + l ~ _ u , s~2v . If l ~ u ~ v and v + l ~ s < 2 v , then the pair u, s occur together in a2t blocks. Considering the following ar rangement of t rea tments

Group Trea tments

1 2 - . . v

v + l v + 2 . - . 2v

we obtain an SRGD design, since r * - - 2 * = a p t - - ] p = p ( r - 2 ) > O and r ' k * - v * ~ * = 2 a p t k - 2 v a 2 t = 2 v a ( r - a t ) = O . Thus, the proof is completed.

THEOREM 4.4. The existence o f an a ~ n e a-resolvable BIBD (v, b : p t , r : a t , k, ~) implies the existence o f t L~(p) designs wi th parameters v*= ~2, b*=2v, r*=2k , k*=ap, ] * = 2 k + ~ - r , ] * = 2 ( k + ] - r ) .

PROOF. With the same notations as in Theorem 4.3 and the pro-


cedure used in Theorem 4.2, t ake any set S*, i=1, 2 , . . . , t, of N*, say S*. Then N ~ and N~, for j C j ' have k + x t r ea tmen t s in common. Note tha t x = k + 2 - - r is a block intersection number in the affine a- resolvabili ty. Similarly, N ~ and N~,~ for ig=i' have k + x t r ea tmen t s in common, while N ~ and N~,~s, for i r j C j ' in tersect in 2x t rea tments . Le t us ar range these blocks as follows:

B~I B ~ . . . B ~ B*= 22 .2~ .

Then the dual of B*, for each of m = l , 2 , . . . , t , gives the required association scheme, i.e. B ~. and B ~ , , ~,~, are first associates iff t hey lie ei ther in the same row or the same column of the above a r rangement , o t h e r w i s e , t hey are second associates. Thus, the proof is completed.

Remark. The L2 design with the same paramete r s as in Theorem 4.4 can be given by considering new blocks of the following type also:

kY)'J

lg=l' ; l, I '=l , 2,. �9 t ;

l, l' fixed for each des ign.

5. Method of block unions

THEOREM 5.1. The existence of an a~ne resolvable block design (v, b=flr, r, k) implies the existence of an SRGD design (b/1, v, Ik, r; O, 12y; r, ~/l) with y= k2/v i f fl/l is a positive integer (=p, say) for any positive integer l satisfying lk<v.

PROOF. Let the blocks of the block design part i t ioned into r resolution classes be wr i t t en as follows:

Here,

S~ S~ S~

[BmBl~ , ' " , f f~ : 2 ~ , B n , " ' , B v : "'" : B~ ,B~2, . . . ,B) , ] .

lB . f-I B.,[= O, j ~ j ' ;

]B~ NB,,j,l=y=k2/v, i~=i'; i, i ' = 1 , 2 , . - . , r ,

j , j ' = l , 2 , . . . , ~ .

Since p=lp, every set St can be divided into p distinct subsets of l blocks each. Form the unions of the I blocks in each of the p subsets for every set S~. Then in the result ing design, the re are v t r ea tmen t s


and pr=bfl blocks. Letting B*j be the j - th block in the i-th class in B* the resulting design, ]B*~NB*j,I=O for j c j ' and I ~JNB*rI=I2Y, i r

since any two blocks from the same class S~ are disjoint, while any two blocks from different resolution classes intersect in y treatments. Every t rea tment is repeated once in every class of both the original design and the resulting design, and then the resulting block size is Ik. In this case, the dual of the above design can be shown to be the required GD design with blocks in the i-th resolution class corresponding to the i-th group. Thus, m=r and n=~/1. The proof is completed.

THEOREM 5.2. The existence of an a~ne resolvable block design (v, b=~r, r, k) for fl>=2, implies the existence of an L~(~) design with parameters v*=/~ 2, b*=v, r*=2k-k']v, k*=2p--1, ~*=k, ~*=2k2/v.

PROOF. Let S~ and $2 denote any two resolution classes of the affine resolvable block design, and let BI~, i=1, 2 , . . . , ~, and B2j, 3"= 1, 2 , . - - , p, be the ~ blocks in Si and $2, respectively. Consider BI~U B2j, i, 3"=1, 2 , . - - , p, as new blocks. Then in the resulting /~ blocks, each of the v treatments occurs 2 p - 1 times. Since [Bl~NB2j[=k2/v, IB~UB~jl=2k--k~/v for every i and j . If we write B~UB2j as i j and arrange the resulting blocks in the following p • p a r ray :

I i 12 �9 �9 �9 I#

21 22 - �9 �9 2#

and call any two blocks first associates if they lie either in the same row or in the same column of the array, otherwise call them second associates, then it is easy to see that any two blocks which are first associates intersect in k t reatments and any two blocks which are second associates intersect in 2k~/v t reatments. Now, dualizing the above arrangement, we get the required result.

Acknowledgement

The first version of this work was wri t ten while the second author was visiting Department of Statistics, University of Indore. Research of the second author was supported in part by the University Grants Commission, India, and the Grant 59540043 (C), Japan. The authors thank the referee for his useful comments.

UNIVERSITY OF INDORE, INDIA HIROSHIMA UNIVERSITY UNIVERSITY OF INDORE, INDIA


~EFERENCES

[ 1 ] Clatworthy, W. H. (1973). Tables of Two.Associate-Class Partially Balanced Designs, NBS Applied Mathematics Series, No. 63.

[2] Preece, D. A. (1967). Incomplete block designs with v=2k, Sankhya, A29, 305-316. [ 3 ] Raghavarao, D. (1971). Constructions and Combinatorial Problems in Design of Experi-

ments, Wiley, New York. [4] Shrikhande, S. S. and Raghavarao, D. (1963). Affine a-resolvable incomplete block

designs, Contributions to Statistics, Pergamon Press, Calcutta, 471-480.

Documents

Some constructions of two-associate class PBIB designs