Biometrika (1990), 77, 1, pp. 193-202

Printed in Great Britain

Some constructions for balanced incomplete block designs with nested rows and columns BY NIZAM UDDIN A N D JOHN P. MORGAN

Department of Mathematics and Statistics, Old Dominion University, Norfolk,

Virginia 23529, U. S.A.

SUMMARY A method of construction of balanced incomplete block designs with nested rows and columns is developed using difference techniques, from which many infinite series of designs are obtained. Some key words: Balanced incomplete block design; Nested row-column design; Supplementary difference set.

Incomplete block designs with nested rows and columns are designs for v treatments in b blocks of size k = pq < v, where each block is composed of p rows and q columns. Such a design is balanced if: (a) a treatment occurs at most once in each block, (b) each treatment occurs in r blocks, and (c) pqrIv - p N I N : -qN2N:+ N N 1 = aI, -AJv. Here I, is the v x v identity matrix, J, is the v x v matrix of ones, N , , N, and N are respectively the treatment-row, treatment-column and treatment-block incidence matrices, and a and A are integers. Nested row-column designs satisfying (a)-(c) are called balanced incomplete block designs with nested rows and columns, and will be denoted by BIBRC (v, b, r, p, q, A), or BIBRC for short. Such designs were introduced by Singh & Dey (1979) for the elimination of heterogeneity in two directions within each block. Constructions have been given by Singh & Dey (1979), Street (1981), Agrawal & Prasad (1982, 1983) and Cheng (1986). A few of these designs show up in Ipinyomi & John's (1985) listing of nested row-column designs. Condition (c) says that the C-matrix for estimation of treatment contrasts has the same form as that of an ordinary balanced incomplete block design, in the absence of nested rows and columns; hence the balance. In this paper we present a technique for construction of BIBRC designs, based on the method of differences, that takes advantage of the fact that if p = q a sufficient condition for (c) to hold is that botk ( N , , N,) and N are incidence matrices for balanced incomplete block designs. For vectors a and b of lengths n, and n, respectively, we shall use B(a, b) to denote a n, x n, array whose (i, j)th element is equal to the sum of the ith element of a and the jth element b. The symbol R(a, b) will be used to denote a vector whose elements are, in some order, those of B(a, b). Also 1 +yA will be used to denote (1 +yz 1 z E A).

The construction method may be summarized as follows. CONSTRUCTION. Let G be an abelian group of order v. Suppose that we canjnd two sets of vectors b,, . . . ,bm and b : , . . . ,bh on G which are m-supplementary difference sets of B I B designs, i.e. ( i ) each bj has p distinct elements of G and each bi has q distinct elements of G, and (ii) each nonzero element of G occurs mp(p - l ) / ( v- 1 ) times among the symmetric diflerences arising from the bj's and mq(q - l ) / ( v- 1 ) times among the symmetric diferences arising from the bj's. Suppose further that (iii) the m vectors R, = R ( d i ,d : ) are together composed of A = mpq(p - l ) ( q- l ) / ( v- 1 ) occurrences of each nonzero element of G, where di and d : are the vectors of symmetric diflerences corresponding to bi and b: respectively. Then there exists a B I B R C (v, b, r, p, q, A ) design with b = mu, r = mpq, p, q and A. ProoJ: Define m initial p x q blocks Bi = B(bi,b:). These m blocks, when developed, give our design. By ( i ) and (ii), N 1 and N, are incidence matrices o f B I B designs. The symmetric differencesarising from Bi are q copies o f di from columns, p copies o f d : from rows, and the elements o f R,, all diagonal differences;so, by (ii)and (iii), N is the incidence matrix o f a B I B design. Example. For v = 19 and G = write b1= (0,2,3, 14),b, = (0,4,6,9),b3= (0, 1,7,1 I ) , b: = (0,6,l o ) , b; = (0, 1 , 12) and bi = (O,2, 5 ) . The initial blocks are

A B I B R C (19, 57, 36, 4, 3, 12) is found by successively adding 0 , . . . , 18 (mod 19) to the initial blocks. I f p = q, it is sufficient for (ii) that the 2mp(p - 1 ) combined symmetric differences from the bi's and b:'s are balanced, a fact which is taken advantage o f in 9 3 below. Applying this technique we obtain several infinite series o f designs as presented in the following theorems and corollaries. In all cases G will be taken as the finite field G F , with primitive element x.

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THEOREM 1. Let v = 2tm 1 be a prime power and write xu(= 1 - xmi. ( a ) I f there exists a positive integer u (ui- u j ) (mod m ) for i,j = 1 , . .. , t, then there exists a B I B R C with b = mu, r = 4mt2,p = q = 2t and A = 2t(2t - I),. ( b ) If there exists a positive integer u $ ui, (ui- uj) (mod m ) for i,j = 1, . .. , t, then there exists a B I B R C with b = mv, r = m(2t I)', p = q = 2t 1 and A = 2t(2t I),. ( c ) If there exists a positive integer u $ u,, (ui- u j ) (mod m ) for i,j = 1, ... , t, then there exists a BIBRC with b = m v , r = 2 m t ( 2 t + l ) , p=2t, q = 2 t + l and A = 2t(2t + 1)(2t- 1 ) .

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Roo$ ( a ) Let bl = ( x Oxm, , .. . , X ( 2 t - l ) m )

bi = xi-lbl ,

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Balanced incomplete block designs with nested rows and columns Sprott (1954) has shown that bl, . .. , b,, and hence b:, difference sets for a BIBD for which

.. . , b;,

195

are m-supplementary

and O means the Kronecker product. Using x'" = -1 we obtain = Xim(l - X(f-i)m)= Xim+u,-t 1+ SO

that dl = blOw where w = (xul,xul, ... , xu", xu!-',xu!). Now Ri = R(di, d:) = xi-'R(d1, d:) = x i - ' ~ ( d , ,x udl),so

Using R(xnlbl,xn2bl)= x n l b l O ( l+xn2-"lb,), which can be verified easily, we obtain R ( d l , xudl)= {wjbl O (1 + w,wjlxubl)I j, 1= 1, . .. , 2 t - I), wj being the jth component of w. Since (xO,x l , . . . ,xm-I)O bl = G - {0), condition (iii) is satisfied and the proof is completed. To prove (b), adjoin 0 to the bi's and b:'s in the proof of (a). To prove (c), adjoin 0 to the bl's in the proof of (a). The conditions on u are needed to give di fl d: = 0, ensuring the binary property, i.e. that a treatment occurs at most once in each block. Sufficient, but often not necessary, conditions for the existence of u, and hence the designs of Theorem 1, are for (a), (b) and (c) respectively. The following two corollaries illustrate the application of this theorem. Note that the sufficient conditions are not necessary in Corollaries l(b), 2(a) and 2(c). COROLLARY 1. Let v = 4m + 1 be a prime power. ( a ) If m 2 4, then there exists a BIBRC with b = v(v - 1)/4, r = 4(v - I), p = q = 4 and A = 36. (b) If m > 7, then there exists a BIBRC with b = V ( V- 1)/4, r = 25(v - 1)/4, p = q = 5 and A = 100. (c) If m 6, then there exists a BIBRC with b = v(v - 1)/4, r = 5(v - I), p = 4, q = 5 and A = 60. Agrawal & Prasad (1982, 1983) have constructed 4 x 4 designs with the same r and prime power v = 16m+ 1, and with r = 8(v - 1) for prime power v = 2m + 1. Similar comparisons for the 4 x 5 and 5 x 5 designs may be made to the results of Agrawal & Prasad (1982, 1983) and Street (1981): in each case our series are for smaller r or a less sparse series of v. COROLLARY 2. Let v = s n be an odd prime power. 2, then there exists a BIBRC with b = s n ( s n- l ) / ( s - I), r = ( s - l ) ( s n- I), ( a ) If n p = q = s - 1 and A = ( ~ - l ) ( s - 2 ) ~ . (b) n 3, then there exists a BIBRC with b = s n ( s n- l ) / ( s - I), r = s2(sn- l ) / ( s - I), p = q = s and A = (s - l)s2. (c) If n 2 2, then there exists a BIBRC with b = s n ( s n- l ) / ( s - I), r = s ( s n- I), p = s - 1, q = s and A = s ( s - l ) ( s -2). Corollary 2 follows upon taking m = (sn - l ) / ( s - I), in which case ui = 0 (mod m) for every i. The condition n 2 3 in (b) is required to give incomplete blocks.

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THEOREM 2. Let v = 2 t m 1 be a prime power where t > 1 is odd, and write xu$= 1 - x2"'. ( a ) If there exists a positive integer u ( u i- u,) (mod m ) for i, j = 1 , .. . ,i ( t - I ) , then there exists a BIBRC with b = mv, r = mt2, p = q = t and A = i t ( t - I ) ~ . ( b ) If there exists a positive integer u 8f u i , ( u i- u j ) (mod m ) for i, j = 1,. .. ,i ( t - I ) , then there exists a BIBRC with b = mv, r = m ( t I ) ~ p, = q = t + 1 and A = i t ( t + 1)'. ( c ) If there exists a positive integer u ui, ( u i - u j ) (mod m ) for i, j = 1, . . . , i ( t - I ) , then there exists a BIBRC with b = mu, r = m t ( t I ) , p = t, q = t 1 and A = i t ( t 2 - 1).

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ProoJ: Here we prove (b) first. Let b , = ( 0 , xO,x2", . . . ,x ( " - ~ ) "= ) ( 0 , bT). The symmetric differences arising from b1 are the nonzero elements of B ( b l ,- b,). Hence

see also Sprott (1954). Using x'"

= - 1,

and for odd i, 1 +xi" = xirn+"(l-i)/2, gives

say. Taking bi = xi-'b1 and bj = xu+'-'b1, for i = 1, .. . , m, it is easily seen that each of rows and columns will be BIBDS. The diagonal differences are Ri = x i - ' ~ ( d , x, u d l ) for i = 1,. .. ,m. Upon noting that ( b f ,xmbT) is the b , of Theorem 1, we obtain

SO

{ R 1 ,. . . , R,) = ( x O x, l , . . . ,x r n p 1 C )3 R ( d l , x u d l ) is a balanced list. To prove (a), delete 0 from the hi's and bj's in the proof of (b). To prove (c), delete 0 from the bi's in the proof of (b). Sufficient conditions for the existence of u in (a), (b) and (c) of Theorem 2 are

respectively. We present three corollaries as applications of Theorem 2. We then show in § 3 how, under certain conditions, the number of blocks in the first two theorems may be reduced.

3. Let v = 6 m + 1 be a prime power. COROLLARY ( a ) If m 2 2, then there exists BIBRC with b = V ( V - 1 ) / 6 , r = 3 ( v - 1 ) / 2 , p = q = 3 and A =6. ( b ) I f m 2 4, then there exists a BIBRC with b = v ( v - 1 ) / 6 , r = 8 ( v - 1 ) / 3 ,p = q = 4 and A = 24. ( c ) If m 2 3, then there exists a BIBRC with b = V(V - 1 ) / 6 , r = 2 ( v - I ) , p = 3, q = 4 and A = 12. The 4 x 4 designs constructed here may be compared to those of Corollary 1 and the previously mentioned designs of Agrawal & Prasad (1982, 1983). Designs with the same parameters as the 3 x 3 and 3 x 4 series are constructed by Street (1981), but for m 2 3 and m 2 4, respectively.

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4. Let v = 10m 1 be a prime power. COROLLARY ( a ) If m 2 4, then there exists a BIBRC with b = V ( V - 1)/10, r = 5 ( v - 1 ) / 2 , p and A = 40. ( b ) If m 2 7, then there exists a BIBRC with b = V ( V - 1 ) / 10, r = 18(v - 1 ) / 5 , p and A = 90.

=q =5 =q

=6

Balanced incomplete block designs with nested rows and columns

197

( c ) I f m > 6, then there exists a B I B R C with b = V ( V - 1 ) / 10, r = 3(v - I ) , p = 5, q = 6 and h = 60. COROLLARY5. Let v = sn ( n 2 2 ) be an odd prime power, and let s = 4 a +3. ( a ) If a 2 1 , then there exists a BIBRC with b = s n ( s n- l ) / ( s- I ) , r = ( s n- l ) ( s- 1)/4, p = q = ( s - 1 ) / 2 and A = ( s-3)'(s - 1 ) / 16. ( b ) If a 2 0 , then there exists a B I B R C with b = s n ( s n- l ) / ( s- I ) , r = ( s I ) ~ ( s-"1 ) / { 4 ( s- I ) ) , p = q = i ( s 1 ) and A = ( s I ) ~ (-s1)/16. ( c ) If a 2 1 , then there exists a B I B R C with b = s n ( s n- 1 ) / ( s- I ) , r = ( S l ) ( s n- 1)/4, p = i ( s - I ) , q = i ( s + 1 ) and A = ( s 2- 1 ) ( s- 3)/16.

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+

+

+

Combining the constructions o f Theorems 1 and 2 gives the following.

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THEOREM 3. Let v = 2tm 1 be a prime power where t > 1 is odd and write xu(= 1 - xmi. ( a ) If there exists a positive integer u $ - u j ) (mod m ) for i = 1 , ... ,i ( t - 1 ) and j = 1 , . .. , t, then there exists a B I B R C with b = mv, r = 2mt2, p = t, q = 2t and A = t ( t - 1)(2t- 1 ) . ( b ) Ifthere exists a positive integer u uZi,( u2i- u j ) (mod m )for i = 1 , . .. ,i ( t - 1 ) and j = 1 , . .. , t, then there exists a B I B R C with b = mv, r = mt(2t 1 ) , p = t, q = 2t 1 and A = t ( t - 1 ) ( 2 t + l ) . ( c ) If there exists a positive integer u -uj, ( u Z i uj) (mod m ) for i = 1 , . . . ,i ( t - 1 ) and j = 1, .. . , t, then there exists a BIBRC with b = mv, r = 2mt(t I ) , p = t 1, q = 2t and A = t ( t l ) ( 2 t- 1 ) . ( d ) Ifthere exists apositive integer u $ u2,,- uj, ( u Z i uj)(mod m )for i = 1 , . . . ,i ( t - 1 ) a n d j = l , . .. , t, t h e n t h e r e e x i s t s a ~ ~ withb=mv, ~~c r = m ( t + 1 ) ( 2 t + l ) , p = t+1, q = 2 t + l and A = t ( t + l ) ( 2 t + l ) .

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+

+

+

+

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Proof: In each case take the bi's from Theorem 2 and the b:'s from Theorem 1. Sufficient conditions for the existence o f u in ( a ) - ( d )o f Theorem 3 are

respectively. All the designs given in this section serve also as nested balanced incomplete block designs (Preece, 1967).

In this section we turn our attention to reducing the number o f blocks for sub-series o f the Theorems 1 and 2 designs.

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THEOREM 4. Let v = 4tm 1 be a prime power and write xul = 1 - x2"'. ( a ) If ui - uj $ m (mod 2 m ) for i,j = 1 , . . . , t, then there exists a B I B R C with b = mv, r = 4mt2,p = q = 2t and A = t(2t - 1)'. ( b ) If u,, ui - uj m (mod 2 m )for i,j = 1, .. . , t, then there exists a B I B R C with b = mv, r=m(2t+l)', p = q = 2 t + l andA=t(2t+l)'.

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Proof: ( a ) In Theorem l ( a ) write v =2tmo+l, where m o = 2 m ; it will be shown that with the conditions o f this theorem and proper choice o f u, it is sufficientto use just the first m initial blocks given there. With b, = ( x O x, 2", .. . , x ' ~ ' - ~)' "take

i.e. take u = m = $moin Theorem l ( a ) ,for i = 1 , .. . , m. Then dl = b,O ( x Y ,x Y,x u ! ) ,where xY = ( x u ' ,. . . , xu[-'),S O the bi's and bj's are together a 2m-supplementary difference set, which by the conditions on the uj's satisfy di fl dj = 0. The diagonal differences are

Now R ( d l ,x m d l ) can be written as four copies of R l l , two copies of each of R,, and R,, , and one copy of R,,, where

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Factor these as follows, using R(xnlbl, xn2bl)= xnlbl0( 1 xn2-nlbl)and xmbl= x-" b, = b: :

Since

(x', X I , . . . , x m - ' ) @ ( b l ,b : ) = G F , - {o), ( x Ox, l , .. . ,x m - ' )O R l 1is a balanced list, as is ( x Ox, l , . .. , x m - ' )O (R,,, R,,). The proof is complete if ( x O x, l , . . . , ~ " - ~ ) O b , C 3+( b1 : ) is a balanced list, which will be the case provided the elements of 1 + b{ = 1 + xmbl can be partitioned into t pairs of the form x91, i = 1 , . . ., t where ki is odd. One such partition is ~

~

g

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~

l

~

To prove (b), adjoin 0 to the b,'s and hi's in (a). As an example of Theorem 4, t = 2 gives 4 x 4 blocks with r = 2(v - I ) , and 5 x 5 blocks with r = 25(v - 1)/8, for v = 8 m 1 a prime power; compare Corollary 1. For v < 500 the conditions fail to hold for v = 81, and for v = 289 in the 5 x 5 case. Setting t = 3, the conditions for v = 12m 1 in 6 x 6 and 7 x 7 blocks also fail for two values of v < 500: v = 37 in 6 x 6 blocks, and v = 169. Corresponding to Corollary 2 we have the following.

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COROLLARY 6 . Let v = sn be an odd prime power where n is even. ( a ) I f n 2 2, then there exists a BIBRC with b = s n ( s n- 1)/{2(s- I ) ) , r = i ( s - l ) ( s n- I ) , p = q = s - 1 and A = ; ( ~ - 2 ) ~ ( s - l ) . ( b ) I f n 2 4 , then there exists a BIBRC with b = s n ( s n- 1 ) / { 2 ( s - I ) ) , r = s2(sn- 1)/{2(s- I ) ) , p = q = s and A = i s 2 ( s- 1 ) .

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THEOREM 5. Let v = 4tm 1 be a prime power, where t > 1 is odd and write xu[= 1 - x4"'. 1 ( a ) If ui- uj 8 m (mod 2 m ) for i, j = 1 , . . . ,,(t - I ) , then there exists a BIBRC with 2 b = m v , r = m t 2 , p = q = t a n d A = t ( t - 1 ) 14.

Balanced incomplete block designs with nested rows and columns

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1 (b) If u,, ui - uj m (mod 2m) for i, j = I, . . . ,z(t - I), then there exists a b=mv, r = m ( t + ~ ) p~=, q = t + l a n d ~ = t ( t + 1 ) ~ / 4 .

199 BIBRC

with

ProoJ: Theorem 5 stands in relation to Theorem 2 as Theorem 4 to Theorem 1, and the proof is similar. The initial blocks are B(bi, b:) where for (a),

and for (b), 0 is adjoined to the bi's and hi's of (a).

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COROLLARY 7. Let v = 12m 1 be a prime power. ( a ) There exists a BIBRC with b = V ( V- 1)/12, r = 3(v - 1)/4, p = q = 3 and A = 3. (b) If m 2 2, then there exists a BIBRC with b = v(v - 1)/12, r = 4(v - 1)/3, p = q = 4 and A = 12. ProoJ: The proof of (a) is immediate from Theorem 5 since there is only a single ui. For (b) the condition is u, m (mod 2m) where xul = 1-x4". If this fails take the bi's as given in the theorem, but bl = xb,. Then dl = (xO,x2", . . . , X ~ ~ " ) O (xXm~) ,showing , that the rows and columns will each be BIBDS. NOWRi = xi-' ~ ( d ]xdl) , simplifies to

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and the diagonal differences are balanced as well. COROLLARY 8. If v = 20m + 1 (m 2 2) is a prime power, then there exists a b = v(v - 1)/20, r = 5(v - 1)/4, p = q = 5 and A = 20.

BIBRC

with

ProoJ: The condition is S = u2- u, 8 m (mod 2m) where xS = 1 + x4". If this fails take the bi's as given in Theorem 5 but bl = xb,. The proof follows easily upon noting that x-Uldl= (XO,x ~x ,~ " ,~ ~bl. ~ ) 0 The conditions for v = 2 0 m + l in 6 x 6 blocks fail twice for v <500 (v =41,61). Corollaries 7 and 8 should be compared to Corollaries 3 and 4, respectively. We also have the following; compare Corollary 5.

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COROLLARY 9. Let v = s nbe an odd prime power where n is even, and s = 4 a 3 ( a 2 1). There exist BIBRCS with: ( a ) b = s n ( s n- 1)/{2(s - I)), r = ( s - 1 ) - 1 8 , p = q = ( s - 1)/2 and A = ( s - 3)2(s- 1)/32; and (b) b = s n ( s n- 1)/{2(s - I)), r = ( s + l)2(sn- 1)/{8(s - I)), p = q = ( s + 1)/2 and A = ( s + I ) ~ (-s 1)/32. Finally, we note that stricter conditions on the ui's can for a given v, p and q, give designs with yet smaller r than obtained in Theorems 1-5, or designs that cannot be constructed by those theorems. Unfortunately such conditions are often not satisfied. One example worthy of note is for v = 169 and p = q = 4, compare Corollary 7, where r = 112 is attained by using the fact that u1= m (mod 2m). The required vectors are bl = (0, xO,x S6,x112),bi = x2jP2bla nd bl = xb,, for i = 1,. . . ,7. For v = 169 in 6 x 6 blocks, where Theorem 4 fails, take bl = (xO,x2', . .. , x 140), bi = x2jP2b and b: = x3bi, for i = 1,. .. , 14; for 7 x 7 blocks adjoin 0 to each of these vectors. For v = 289 in 5 x 5 blocks, where again Theorem 4 fails, take b1= (0, xO,x ~x ~ ~,x216),bi ~ ~ =,x2jP2bland b: = x7bi, for i = 1, ... ,36. We have not been able to get designs for v = 41 or 61 in 6 x 6 blocks by these techniques; see Corollary 8.

Table 1. Constructed r,,, Method Source

45 9 36 32 9 36 99 18 24 32 72 12 84 52 63 112 140 175 45 60 80 90 105 27 36 48 108 180 126 144 225 168 90 80 40 50 63 84 112 126 21 168 196 207 36 48 64 144 240 48 96 150 16

Th. 1 Th. 5 Th.4 Th.4 Th. 2 Th. 2 Th. 1 Th. 5 Th. 2 Th. 5 Th.3 Th. 1 Th. 3 Th. 1 Th. 4 Th.1 Th. 1 Th.1 Th. 2 Th. 2 Th. 2 Th. 3 Th.3 Th. 5 Th. 2 Th. 5 Th. 3 Th.1 Th.3 Th. 3 Th. 1 Th. 3 Th. 4 Th.4 Th. 1 Th. 5 Th. 2 Th. 2 Th. 2 Th.3 Th.3 Th. 3 Th.3 Th. 1 Th. 5 Th. 2 Th. 5 Th. 3 Th. 1 Th.3 Th.3 Th.4 Th.3

AP5 IJ

P

A P ~

A P ~

-

cl S D ~

-

cl(AP1,4~7) -

s6 s6, c l (P, 3 x 5) -

BIBRCS

for v G 101

U

P

9

r

49 49 53 53 53 53 59 61 61 61 61 61 61 61 61 61 61 61 61 61 67 67 67 67

6 6 3 4 4 5 3 3 3 4 3 4 3 4 5 4 5 6 6 7 3 3 4 3

6 7 3 4 5 5 3 3 4 4 6 5 7 6 5 7 6 6 7 7 3 4 4 6

144 336 117 208 260 325 261 45 120 80 180 300 210 240 75 280 180 180 420 245 99 132 176 198

67 67 67 67 67 67 71 71 71 71 73 73 73 73 73 73 73 73 73 73 73 73 79 79

3 4 4 6 6 7 3 5 5 6 3 3 4 3 4 3 4 5 4 6 6 7 3 3

7 6 7 6 7 7 3 5 6 6 3 4 4 6 5 7 6 5 7 6 7 7 3 4

231 264 308 396 462 539 315 175 210 252 54 144 96 216 360 252 288 225 336 216 504 294 117 156

79 79 79

4 3 3

4 6 7

208 234 273

A P ~

-

A P ~~6 , -

S D ~ A P ~

-

~ 6c ,~ ( A P 3~X7) , s6, c 1 ( ~ ~ 2x,73) -

s6,c1(3x7) A P ~

c ~ ( A P ~ 7) ,~x A P ~

A P ~

-

s6 -

cl(4x7) cl(4x7) -

cl

vIethod Source Th.4 Th.1 Th.4 Th.l Th. 1 Th. 1 Th. 1 Th. 5 Th. 2 Th.5 Th. 3 Th. 1 Th. 3 Th.3 Th. 5 Th. 3 Th. 2 Th.4 Th. 1 Th. 4 Th. 2 Th. 2 Th. 2 Th. 3

cl(6x7)

cl -

cl(~~1,4x13) A P ~

A P ~

S D ~

s6 A P ~

s6 cl(5x6) A P ~

-

s6 s6 -

s6, C l ( A P ~3, x 11) Th. 3 s 6 Th. 3 Th. 3 Th.l ~ l ( A P 1 , 6 ~ 1 1 ) Th. 1 Th. 1 Th. 1 AP5, c l ( s 6 , 3 x 7) Th. 2 s6 Th. 2 ~ 6c,1( A P ~5, X 7) Th.2 Th. 5 S D ~ Th. 2 A P ~ Th. 5 Th. 3 AP4, S6 Th. 1 Th. 3 s6 Th. 3 A P ~ Th.4 Th.3 Th. 4 Th. 1 Th. 4 Th. 2 s 6 Th. 2 s6, c ~ ( A P 3~ , x 13) Th.2 Th. 3 s 6 Th. 3 s 6

Balanced incomplete block designs with nested rows and columns

20 1

Table 1 (cont.) r,,, Method Source

312 364 468 546 637 40 80 80 100 240 144 400 440 480 320 528 40 369 198 176 440 275 72

Th.3 Th. 3 Th. 1 Th. 1 Th. 1 Th.4 Th. 1 Th. 1 Th. 5 Th. 2 Th. 5 Th. 3 Th. 3 Th. 3 Th.4 Th. 3 Th. 1 Th. 1 Th. 4 Th. 4 Th. 1 Th. 4 Th. 5

-

SD2 S D ~ A P ~

s6

-

-

cl(8x9) -

cl A P ~

c ~ ( A P ~11) ,~x

-

U

97 97 97 97 97 97 97 97 97 97 97 97 97 97 101 101 101 101 101 101 101 101

P

9

3 4 4 4 3 6 4 5 3 7 4 6 5 5 4 7 6 6 6 7 7 7 8 8 8 9 9 9 3 3 4 4 4 5 5 5 5 6 6 6 5 10 5 11

r,,, Method Source

96 128 288 480 336 384 300 448 288 672 392 384 864 486 225 400 100 125 300 180 500 550

Th. 2 Th. 5 Th. 3 Th. 1 Th. 3 Th. 3 Th.4 Th. 3 Th. 4 Th. 1 Th.4 Th. 4 Th. 1 Th. 4 Th. 4 Th.1 Th. 1 Th. 5 Th. 2 Th.5 Th. 3 Th. 3

AP1 -

s6

s6 AP4 -

-

-

cl(4x5) A P ~

-

s6 -

AP4, S6 -

P, Preece (1967); S D ~Singh , & Dey (1979, Th. 2); s6, Street (1981, Th. 6); AP1, A P and ~ AP4, Theorems 1, 2 and 4 of Agrawal & Prasad (1982); AP5, Agrawal & Prasad (1983, Th. 5); IJ, Ipinyomi & John (1985); Cl, Cheng (1986, I%. 2.1). Cheng's result combines a BIBD with a BIBRC to give a new BIBRC, both row-column designs having the same v; when the referenced design does not explicitly appear in his paper, the dimensions p x q of the required initial B I B R C are in parentheses, as well as a source if that design does not appear in this table.

Table 1 lists the designs constructed in this paper for v G 101 and 3 c p G q. The series of 2 x 2 designs of Theorems 1 and 4 have the same r as those constructed by Agrawal & Prasad (1982, 1983), and Theorem 1 gives 2 x 3 designs with the same r as Agrawal & Prasad (1983, Th. 5), so those designs are not included. Only the construction with the smallest r is given, and r, is the smallest replication for given v, p and q we have found in the literature, including this paper. Sources are listed for designs with replications less than or equal to that constructed here; of the 149 designs in this range, 80 appear to be new. An extensive list of BIBDS is given by Mathon & Rosa (1985). ACKNOWLEDGEMENT The authors thank the referee for comments leading to improvement of the presentation.

AGRAWAL,H. L. & PRASAD,J. (1982). Some methods of construction of balanced incomplete block designs with nested rows and columns. Biometrika 69, 481-3. AGRAWAL,H. L. & PRASAD,J. (1983). On construction of balanced incomplete block designs with nested rows and columns. Sankhyd B 45, 345-50.

CHENG, C.-S. (1986). A method for constructing balanced incomplete-block designs with nested rows and columns. Biometrika 73, 695-700. IPINYOMI,R. A. & JOHN, J. A. (1985). Nested generalized cyclic row-column designs. Biometrika 72,403-9. MATHON,R. & ROSA, A. (1985).Tables of parameters of BIBDS with r 41 including existence, enumeration, and resolvability results. In Algorithms in Combinatorial Design Theory, Ed. C. J . Colbourn and M. J. Colbourn, Annals of Discrete Mathematics 26, 275-308. Amsterdam: North-Holland. PREECE,D. A. (1967). Nested balanced incomplete block designs. Biometrika 54, 479-86. SINGH, M. & DEY, A. (1979). Block designs with nested rows and columns. Biometrika 66, 321-6. SPROTT, D. A. (1954). A note on balanced incomplete block designs. Can. J. Math. 6, 341-6. STREET, D. J. (1981). Graeco-Latin and nested row and column designs. In Combinatorial Mathematics, 8, Lecture Notes in Mathematics, 884, Ed. K. L. McAvaney, pp. 304-13. Berlin: Springer-Verlag.

[Received August 1988. Revised March 19891