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More balanced ternary designs with block size four
Journal of Statistical Planning and Inference 17 (1987) 109-133 North-Holland
109
M O R E B A L A N C E D T E R N A R Y DESIGNS WITH BLOCK SIZE FOUR Diane DONOVAN Department of Mathematics, University of Queensland, St. Lucia, 4067, Australia Received 2 July 1986 Recommended by Jennifer Seberry
Abstract: This paper gives necessary and sufficient conditions for the existence of balanced ternary designs with block size four and A = 2 in the cases 02 = 3, 4, 5 and 6; the cases 02 = 1 and 2 have appeared earlier.
AMS Subject Classification: Primary 05[305; Secondary 62K10. Key words and phrases: Block design; Balanced ternary design; Group divisible design; frame.
1. Introduction Balanced ternary designs (BTDs) were first introduced by Tocher (1952). He defined them to be a collection of B multisets of size K, called blocks, chosen from a set of V elements where any element may occur 0, 1 or 2 times in any one block and furthermore each of the (v) pairs of distinct elements occurs A times throughout the blocks of the design. We shall further stipulate that each element must occur a constant n u m b e r of times, say R, throughout the design. It follows that VR = BK.
(1)
If we let 0/, l = 1 or 2, denote the number of blocks in which an element occurs l times then R = 01 + 202
(2)
and A(V-
1) = R ( K - 1 ) - 202.
(3)
We shall write the parameters o f a balanced ternary design as (V, B; Q1, 02, R; K, A). In this paper we consider the existence of balanced ternary designs with K = 4, A = 2 and small 02, that is Q2 = 3, 4, 5 and 6. For 02 = 1 and 2 see Donovan (1986b) and Assaf, H a r t m a n and Mendelsohn (1985). We shall give necessary and sufficient conditions for the existence of such balanced ternary designs. 0378-3758/87/$3.50 © 1987, Elsevier Science Publishers B.V. (North-Holland)
110
D. Donovan / M o r e balanced ternary designs
Since we are considering balanced ternary designs with K = 4 and A = 2 identities (1), (2) and (3) allow us to write the parameters as (V, V ( V - 1 + P2). 2 ( V - 1 - 2Q2) 2 ( V - 1 + ~o2) "4, 2) 6 ' 3 , Q2, 3 ' "
(4)
In addition since A = 2 the blocks of such balanced ternary designs must be of the form wxyz or xxyz and not xxyy; hence B>_o2V. The techniques used in this paper are based on those introduced by Hanani (1975), (see also Hanani (1961)), to give the existence of certain balanced incomplete block designs. We also employ a variation of this method introudced by Assaf, Hartman and Mendelsohn (1985). These techniques rely on the existence of certain group divisible designs and frames. A group divisible design is a collection of subsets of size k, called blocks, chosen from a set of o =mn elements, where this set can be partitioned into n groups of size m, and where the blocks are selected in such a way that each block contains at most one element from each group and any two elements from distinct groups occur together in 2 blocks. The parameters of a group divisible design are written GD(k, 2, m; o). For the purpose of this paper we shall define a frame to be a collection of multisets of size K, called blocks, chosen from a set of V elements such that the following conditions are satisfied: (i) {ooi, i= 1, . . . , f } is a subset of the set of elements, called a hole; (ii) any element not belonging to the hole occurs at most twice in any one block and precisely twice in #2 blocks; (iii) each 0% i = 1, ... , f , occurs at most once in any one block; (iv) the pair xy, where x and y are distinct elements not both belonging to the hole, occurs A times throughout the blocks of the frame. We shall write the parameter as (V[ f ] , Q2,K,A). Note that a balanced ternary design is a frame with f = 0. This paper is divided into six sections. They are 1. Introduction; 2. The case Q2=4; 3. The case Q2=3; 4. The case ~2=5; 5. The case ~2=6; 6. Summary remarks; and an Appendix. The techniques used are summarized in the following l e m m a and theorem, which are an analogue of Hanani's L e m m a 2.25 (1975). We refer the reader to Donovan (1986b) for the proofs. Lemma 1. I f there exist group divisible designs G D ( k ; 2 ' , t : o),
OD(k,2';m; k'm)
and
GD(k, 2", m; ( k ' - 1)m)
then there exists a group divisible design GD(k, 2, {mr, mh* }; (o - t + h)m)
f o r O
D. Donovan / More balanced ternary designs
111
Theorem 2. I f there exists a group divisible design
GD(K,A, { V - f , ( V ' - f ) * } ;
(n- 1)(V-f) + V'-f),
and if there exists a (V[f],#z,K,A) frame and also a BTD with parameters (V;B; Ql, o2,R; K,A) then there exists a B T D with parameters
( ( n - 1 ) ( V - f ) + V',B'; p~,Qz, R'; K,A). We shall require the following results and so we state them here and refer the reader to the relevant papers for their proofs. Theorem 3 (Brouwer, Schrijver and Hanani (1977), Theorem 6.3). For all positive
integers m, A and o the group divisible design GD(4, A, m; o) exists if and only if the design is not GD(4, 1, 2; 8) and not GD(4, 1, 6; 24) and provided that o = 0 (mod m), ;t(o-m)=-O (mod 3), 2o(o-m)=--O (mod 12) and o>_4m or o=m. Theorem 4 (Brouwer (1979), Theorem 4). A group divisible design
GD(4, 1, {2,5*}; 6 q + 5) exists for all positive integers q :/: 1, 2.
The following results refer to transversal designs. A transversal design, T(s, A, t), is a group divisible design, GD(s, A, t; st), in which each block contains precisely one element from each group. Lemma 5 (Hanani (1975), Lemma 3.5). For every prime power q there exists a trans-
versal design T(q + 1, 1, q). Lemma 6 (Hanani (1975), Lemma 3.1). I f a transversal design T(s,A,t) exists then
a transversal design T(s, 2, t) exists f o r all s'<_s.
The following lemma follows from the above result and Hanani (1975), Theorem 3.11. Lemma 7. A transversal design T(5,2,n) exists for all positive integers n. Lemma 8. (Hanarti (1975), Lemma 3.21). A transversal design T(6, 1, 12) exists.
2. The case Q2 = 4
The constructions given here for balanced ternary designs with •2 = 4 are similar to Hanani's original methods.
D. Donovan / More balanced ternary designs
112
If we substitute Q2= 4 into identity (4), the parameters of the balanced ternary designs can be written V ( V + 3 ) . 2 ( V - 9 ) , 4 , 2(V+3) "4,2). V, 6 ' 3 3 ' It follows that necessary conditions for the existence of such balanced ternary designs are that V - 0 (mod3) and V>_21 (since B>_Q2V). We give methods for constructing balanced ternary designs satisfying these parameters and so show that the conditions are in fact sufficient. Balanced ternary designs with the above parameters and V=21,24, ..., 138 are given in the Appendix. For the remaining balanced ternary designs we give the following recursive construction. BTDs with V-O (mod3) and V_>141. By Lemma 7, T(5,2,n), n_>10, exists and GD(4, 1, 3; 12) and GD(4, 1, 3; 15) exist by Theorem 3. Hence by Lemma 1, GD(4, 2, {3n,3h*}; 12n+3h) exists for n___10 and h = 7 , 8 , 9 and 10. So by applying Theorem 2 to these group divisible designs and balanced ternary designs with V=21,24,27, 30 and 3n for n>_ 10 we obtain balanced ternary designs with V= 12n + 21, 12n + 24, 12n + 27 and 12n + 30 for n_> 10. Having given methods for constructing all possible balanced ternary designs with K = 4, A - - 2 and Q2= 4 we state the following theorem. Theorem
9. Necessary and sufficient conditions for the existence of balanced ter-
nary designs with parameters V, V(V+ 3 ) . 2 ( V - 9 ) 4, 2(V+ 3) 6
'
3
'
3
) ;4,2
are that VmO (mod 3) and V_>21. In the remaining three cases to be considered, namely Q2 3, 5 and 6, all constructions rely on Lemma 1 and Theorem 2 and these are applied in the same way as above. Hence for brevity the above construction is the only one we shall list in detail. In all other cases we shall either list the blocks of the balanced ternary design or simply give the relevant designs which can be used in conjunction with Lemma 1 and Theorem 2. =
3 . T h e c a s e ~o2 = 3
Substituting ~2 3 into identity (4) the parameters of the balanced ternary design can be written as =
V(V+2).2(V_7),3,2(V+2) V,
6
'
3
3
) ;4,2 .
D. Donovan / More balanced ternary designs
113
It follows that necessary conditions for the existence of such balanced ternary designs are that V - 4 (rood 6) and V_> 16 (since B_> V2). Balanced ternary designs with the above parameters and V= 16, 22, ..., 58, 70, 82, 118 and 142 are listed in the Appendix. For the remaining designs we consider Vmodulo 72 and divide the designs into the following cases: (i) V--4 (mod 18), V>76; (ii) V - 4 (mod24), V_> 100; (iii) V--- 10 (mod 24), V_> 106; (iv) V-- 16 (mod 24), V_>64; (v) V-- 10 (mod 36), V_> 154; (vi) Vm 70 (mod 72), V_> 286. The above six cases cover all remaining values for V and in each case the following recursive construction establishes the existence of these balanced ternary designs. (i) V=18n+4, n>4. GD(4,2,18; 18n), n > 4 , (22,88; 10,3,16;4,2) BTD and (2214], 3, 4, 2) frame. (ii) V= 24n + 4, n > 4. GD(4, 2, 24; 24n), n _>4, (28, 140; 14, 3, 20; 4, 2) BTD and (28 [4], 3, 4, 2) frame. (iii) V = 2 4 n + 10, n_>4. GD(4,2,24; 24n), n_>4, (34,204; 18,3,24; 4,2) BTD and (34110], 3, 4, 2) frame. (iv) V = 2 4 n + 16, n_>2. GD(4,2,m; 4m), m_> 16, and (m,-~m(m+2); 2 ( m - 7 ) , 3 , 2(m + 2); 4, 2) BTD m _> 16. (v) V= 36n + 10, n _ 4. GD(4, 2, 36; 36n), n _>4, (46, 368; 26, 3, 32; 4, 2) BTD and (46110], 3, 4, 2) frame. (vi) V=72n + 70, n_>3. GD(4, 2, {24,60*}; 72n + 60), n _ 3 , (70,840; 42,3,48; 4,2) BTD and (34110], 3, 4, 2) frame. Hence we have shown that the above conditions are in fact sufficient and so we state the following theorem.
Theorem 10. Necessary and sufficient conditions for the existence o f balanced ternary designs with parameters
V(V+2).2(V-7) 3 , 2 ( V + 2 ) . 4 , 2 ) V,
6
'
3
'
3
'
are that V - 4 (mod 6) and V>_ 16.
4. The case ~o2 --'--5 Given ~o2 5 and identity (4) we can write the parameters of the balanced ternary designs as =
V,
V(V+4). 2 ( V - 11) 2(V+4) ) 6 ' 3 ,5, 3 ;4,2 .
D. Donovan / More balanced ternary designs
114
It follows that V = 2 (mod 6) and since B > V2, V> 26. Balanced ternary designs withthe above parameters and V= 26, 32,..., 74, 86, 92, 140 and 212 are given in the Appendix. Once again if we consider V modulo 72 we can divide the remaining balanced ternary design into six cases. These cases are: (i) V---8 (mod 18), V> 80; (ii) V =- 2 (mod 24), V__>98; (iii) V = 8 (mod 24), V_> 104; (iv) V - 14 (mod 24), V> 110; (v) V - 2 0 (rood 36), V> 164; (vi) V-- 68 (mod 72), V> 284. The following recursive constructions based on the above six cases establishes the existence of these balanced ternary designs. (i) V = 1 8 n + 8 , n > 4 . GD(4,2,18; 18n), n > 4 , (26, 130; 10,5,20;4,2) BTD and (2618], 5, 4, 2) frame. (ii) V = 2 4 n + 2 , n > 4 . GD(4,2,24; 24n), n > 4 , (26, 130; 10,5,20; 4,2) BTD and (2612], 5, 4, 2) frame. (iii) V = 2 4 n + 8 , n>4. GD(4,2,m; 4m), m_>26, and (m,-~m(m+4); 2 ( m - 11),5, 2(m + 4); 4, 2) BTD m >__26. (iv) V = 2 4 n + 14, n>4. GD(4,2,24; 24n), n > 4 , (38,266; 18,5,28; 4,2) BTD and (38[14], 5, 4, 2) frame. (v) V= 36n + 20, n > 4. GD(4, 2, 36; 36n), n > 4, (56, 560; 30, 5, 40; 4, 2) BTD and (56120], 5, 4, 2) frame. (vi) V=72n + 68, n>3. GD(4,2, {24,60*}; 72n + 60), n > 3 , (68,816; 38,5,48; 4,2) BTD and (3218], 5, 4, 2) frame. Thus we have constructed all possible balanced ternary designs with the above parameters and therefore established that the given conditions are sufficient. We therefore state the following theorem.
Necessary and sufficient conditions f o r the existence o f balanced ternary designs with parameters
T h e o r e m 11.
V,
V(V+4). 2 ( V - 11) 2(V+4) ) 6 ' 3 ,5, 3 ;4,2
are that V =- 2 (mod 6) and V> 26.
5. T h e case ~o2 - ~ 6
Given Lo2- 6, identity (4) allows us to write the parameters of the balanced ternary designs as
v ( v + 5) 2 ( v - 13),6, V,
6
'
3
• 4,2 3
'
"
D. Donovan / More balanced ternary designs
115
Necessary conditions for the existence of such balanced ternary designs are V - 1 (rood 3) and V>_ 31 (since B _ V#2). Balanced ternary designs with V= 31, 34, ..., 100, 112, 115, 118, 121, 139, 142, 166, 169, 211,214, 238 and 241 are given in the Appendix. We divide the remaining balanced ternary designs into 13 cases, which cover all possible values of Vmodulo 72. (i) V - 4 (mod 12) and V_> 124; (ii) V - 7 (mod 24) and V_ 103; (iii) V - 10 (mod 24) and V_> 106; (iv) V =- 13 (mod 24) and V>_ 109; (v) V - 1 (mod 36) and V_ 145; (vi) V=- 7 (rood 36) and V_ 151; (vii) V - 10 (rood 36) and V_> 154; (viii) V - 13 (mod 36) and V_> 157; (ix) V - 19 (mod 36) and V_> 163; (x) V - 2 2 (mod 72) and V>_310; (xi) I/-=25 (mod 72) and V_>313; (xii) V - 67 (mod 72) and V>_ 283; (xiii) V - 70 (mod 72) and V>_286. We now consider each of the above classes of balanced ternary designs separately. (i) V= 12n+4, n_>5. GD(4,2,m; 4m), m_31 and (m,~m(m+ 5); 2 ( m - 13);6; 2(m + 5); 4,2) BTD. (ii) V = 2 4 n + 7 , n>_4. GD(4,2,24; 24n), n_>4, (31, 186, 12,6,24; 4,2) BTD and (31 [7], 6, 4, 2) frame. (iii) V=24n + 10, n_>4. GD(4,2,24; 24n), n_>4, (34,221; 14,6,26; 4,2) BTD and (34[10], 6, 4, 2) frame. (iv) V = 2 4 n + 13, n_>4. GD(4,2,24; 24n), n_>4, (37,259; 16,6,28; 4,2) BTD and (37113], 6, 4, 2) frame. (v) V = 3 6 n + 1, n _ 4 . GD(4,2,36; 36n), n>_4, (37,259; 16,6,28;4,2)BTD and (37[1], 6, 4, 2) frame. (vi) V= 36n + 7, n _>4. GD(4, 2, 36; 36n), n >__4, (43, 344; 20, 6, 32; 4, 2) BTD and (43 [7], 6, 4, 2) frame. (vii) V=36n + 10, n_>4. GD(4,2, 36; 36n), n _ 4 , (46,391; 22,6,34; 4,2) BTD and (46110], 6, 4, 2) frame. (viii) V=36n + 13, n_>4. GD(4, 2, 36; 36n), n>_4, (49,441; 24,6,36; 4,2) BTD and (49113],6, 4,2) frame. (ix) V=36n + 19, n_>4. GD(4,2, 36; 36n), n_>4, (55,550; 28,6,40; 4,2) BTD and (55119], 6, 4, 2) frame. (x) V=72n+22, n_>4. GD(4,2,72; 72n), n_>4, (94, 1551; 54,6,66; 4,2) BTD and (94 [22], 6, 4, 2) frame. (xi) V=72n+25, n_>4. GD(4,2,72; 72n), n___4, (97, 1649; 56,6,68; 4,2) BTD and (97 [25], 6, 4, 2) frame. (xii) V=72n+67, n_>3. GD(4,2,{24,60*};72n+60), n_>3, (67,804;36,6,48; 4, 2) BTD and (31 [7], 6, 4, 2) frame.
D. Donovan / More balanced ternary designs
116
(xiii) V = 7 2 n + 7 0 , n > 3 . GD(4,2,{24,60*};72n+60), n > 3 , (70,875;38,6,50; 4, 2) BTD and (34[10], 6, 4, 2) frame. We have therefore shown the sufficiency of the above conditions and so we state the following theorem.
Theorem 12. Necessary and sufficient conditions f o r the existence o f balanced ternary designs with parameters
V(V+5) 2(V-13) V,
6
;
3
2(V+5) .4,2~ ,6,
3
'
are that V=-1 (mod 3 ) a n d V_>31.
6. Summary remarks If we consider balanced ternary designs with K = 4, A = 2 and general Q2 then necessary conditions for the existence of such designs are V _ 5Q2 + 1, since B_> VO2, and V must be congruent modulo 6 to the value given in Table 1. Donovan (1986b) and Assaf, Hartman and Mendelsohn (1985) prove the sufficiency of these conditions for balanced ternary designs with K = 4, A = 2 and P2 = 1, 2. This paper proves the sufficiency of these conditions for balanced ternary designs with K = 4, A = 2 and Q2 = 3, 4, 5, 6. We know that V_> 5~) 2 + 1 is a definite lower bound as Donovan (1986a) has constructed the family of balanced ternary designs with K = 4, A = 2, V= 5~2 + 1 for all positive integers Q2- Further, since the transversal design T(4,2,t) exists for all positive integers t (see Hanani (1975), Lemma 3.11 and Lemma 3.1) we can use Theorem 2 to construct balanced ternary designs with K = 4 , A = 2 and V=4n(5Q2+l) for any positive integer n and Q2. Likewise the existence of certain other group divisible designs can be used in conjunction with these designs to construct various balanced ternary design, but in general the sufficiency of the values given in the above table is still to be investigated. Table 1 BTDs with K = 4, A = 2 ~02 taken modulo 6
0
1
2
3
4
5
V taken modulo 6
1,4
0
2,5
4
0,3
2
Appendix Whilst the general cases have been dealt with in the main body of the text there are a few balanced ternary designs which need to be listed individually. Where it has been necessary to construct the blocks of a specific balanced ternary design we have
D. Donovan / More balanced ternary designs
117
done so by the use of a computer program supplied by Robinson (private communication). In these cases we list the initial block which can be cycled modulo V to achieve all blocks of the design. Our constructions rely heavily on the existence of certain frames and the initial blocks of these have been listed and can be cycled modulo the given permutation to achieve all blocks of the frame. In the cases 62=3 and V=82; 6 2 = 4 and V=66,69,72,81; 6 2 = 5 and V=74,86,82; 6 2 = 6 and V=91,94,97, 100, 121, we list the blocks of the frame ( V[f], 62, 4, 2) and since we have already listed balanced ternary designs with parameters (f, B; 61, 62, R; 4, 2) we simply adjoin the blocks of these designs to the blocks of the frames to obtain balanced ternary designs on the appropriate V. For the remaining values of V we have used Lemma 1, Theorem 2 and the given designs.
Balanced ternary design with K=4, A = 2, 6 2 = 4 and V as follows BTD, V=21: 00112
0027
0038
00415.
These are cycled under the permutation (0 1 --- 20). Frame, V[f] =2113]: 0017
0058
AI002
A l l 15
with short block 0 3 9 12 taken once. These are cycled under the permutation (0 1 -.- 17)(AIA2A3). BTD, V= 24: 0018
00211
00310
00419
with short block 0 6 12 18 taken once. These are cycled under the permutation (0 1 -.. 23). Frame, VLf] = 24[3]: 00114
00210
03612
Al005
A l l 15.
These are cycled under the permutation (0 1 .-. 20)(AIA2A3). Frame, V[f] = 24[6]: 00613
A1123
A1 0 0 3
A 117177
A 116162
with short block 0 2 9 11 taken once. These are cycled under the permutation (0 1 -.- 17)(AIA2AaA4AsA6). BTD, V=27: 00115
00220
00311
00421
051422.
These are cycled under the permutation (0 1 --- 26).
118
D. Donovan / More balanced ternary designs
Frame, V[f] = 27[3]: 0037
00211
051014
A1008
A1778
with short block 0 6 12 18 taken once. These are cycled under the permutation (0 1 --. 23)(A1A2A3). Frame, V[f] = 27 [6]: 01410
01410
All13
A1229
B1005
B14417.
These are cycled under the permutation (0 1 --- 20)(A1AzA3)(BIBzB3). BTD, V= 30: 0016
00212
00311
00716
04817
with short block 0 5 15 20 taken once. These are cycled under the permutation (0 1 -.. 29). Frame, V[f] = 30[3]: 0017
00810
021316
031218
A1005
A l l 15.
These are cycled under the permutation O 1 --- 26)(AlA2A3). Frame, V[f] = 30161: 0035
00711
A1221
Al17177
AI0915
A101622
with short block 0 4 12 16 taken once. These are cycled under the permutation O 1 -.- 23)(AIA2A3A4AsA6). BTD, V= 33: 00118
00225
00322
00424
052128
062027.
These are cycled under the permutation (0 1 -.. 32). BTD, V= 36 0017
00214
00316
00819
04913
051526
with short block 0 6 18 24 cycled once. These are cycled under the permutation (0 1 --- 35). BTD, V= 39: 00121
00230
00326
00431
051934
062233
072232.
041029
052025
These are cycled under the pemutation (0 1 --- 38). BTD, V = 4 2 : 0018
00216
00312
001124
041019
D. Donovan / More balanced ternary designs
119
with short block 0 7 21 28 cycled once. These are cycled under the permutation (0 1 --- 41). BTD, V = 45: 00124
00235
00330
00436
051219
051632
061725
0 6 20 28.
These are cycled under the permutation (0 1 -.. 44). BTD, V = 4 8 : 0019
00218
00322
00411
051528
051536
061229
with short block 0 8 24 32 cycled once. These are cycled under the permutation (0 1 -.- 47). BTD, V = 5 1 : 0 0 127 051019
00 240 061231
00 334 071543
0 0 433 071637
0102238.
These are cycled under the permutation (0 1 ... 50). BTD, V = 54. 0 0 110 051124
00 220 052137
0 0 325 061437
00 412 071433
0112439
with short block 0 9 27 36 cycled once. These are cycled under the permutation (0 1 --. 53). BTD, V= 57: 0 0 130 061320
00 245 082341
0 0 338 082333
00 436 092240
051016 092637.
These are cycled under the permutation (0 1 --. 56). BTD, V = 60: 0 0 111 051231
00 222 061429
0 0 328 062339
0 0 413 082443
051226 092442
with short block 0 10 30 40 cycled once. These are cycled under the permutation (0 1 .-- 59). BTD, V = 63: 0 0 111 061533
00 222 062440
0 0 328 082342
0 0 413 082539
0 51226 0102743.
These are cycled under the permutation (0 1 --. 62). Frame, V [ f ] = 66121]:
051231
0 7 20 34
D. Donovan / More balanced ternary designs
120 0 91536 CI 0 2 3 E l 1 35 40
Al0 019 C l 1 13 29 F 1 0 18 32
A12323 1 Bl0 0 4 D 1 0 2 3 D 1 1 13 29 F l 1 14 25 G l 0 7 14
B12 210 E 1 0 15 20 G I 0 10 20.
These are cycled under the permutation (0 1 -" 44)((A1A2Aa)(BIBEB3)"'"
(G1GEG3).
Frame, V[f] = 69121]: 0 9 15 36 C l 0 4 11 E 1 0 31 46
A I0020 C 1 0411 F 1 0 3 16
A I 1 126 D10822 F 1 1 2 32
B1 00 5 D10822 G l 0 3 16
B 1 1 1 20 E 1 0238 G 1 1 2 11
with short block 0 6 24 30 cycled once. These are cycled under the permutation (0 1 --- 47)(AlA2Aa)(B1BEB3)-.. (G1G2G3). Frame, V[f] = 72121]: 0 91527 B 1 13 13 2 E 1 0 7 21
0 91527 C 1 0 222 E 1 31 47 50
A10 0 1 A15 5 1 C 1 0 2 2 2 D 10 7 2 1 F 1 0 10 23 F ! 0 10 23
Bl 0 0 5 D 1314750 G 1 0 8 25
G 1 0 8 25.
These are cycled under the permutation (0 1 ..- 50)(AIAEA3)(B1BEB3).-- (GIG2G3). BTD, V= 75, 93, 111 and 129: GD(4,2, 18; 18n)
for n=4,5,6 and 7,
BTD(21,84; 8,4, 16; 4,2)
and
Frame(21[3],4,4,2).
BTD, V= 78 and 114: GD(4,2, 18; 18n)
for n =4 and 6,
BTD(24, 108; 10, 4, 18; 4, 2)
and
Frame(24[6], 4, 4, 2).
Frame, V[f] = 81124]: 0 91527 CI0 222 F 1 0 1024
0 91527 C l 0 222 F 1 32 46 56
A10 0 1 Al 5 5 1 D 1 0 11 19 D 1 0 11 19 GI0 526 G103152
B 1 0 0 13 E l 0 7 23 H10328
B 1 2 2 19 E l 0 7 23 H1285356.
These are cycled under the permutation (0 1 --- 56)(AIA2A3)(B1B2B3)-.. (HIH2H3), BTD, V= 84, 96, 108, 120 and 132: GD(4, 2, n; 4n)
for n = 21, 24, 27, 30, 33,
BTD(V,B; ~l,4,R; 4,2) with V--21,24,27,30,33. BTD, V=87: GD(4,2,21; 84), BTD(24, 108; 10,4, 18; 4,2) and Frame(24[3],4,4, 2).
D. Donovan / More balanced ternary designs
121
BTD, V= 90: GD(4, 2, 21; 84), BTD(27, 135; 12, 4, 20; 4, 2) and Frame(27[6], 4, 4, 2). BTD, V= 99 and 123: GD(4, 2, 24; 24n)
for n = 4 and 5,
BTD(27,135; 12,4,20; 4,2)
and
Frame(27[3],4,4,2).
BTD, V= 102 and 126: GD(4, 2, 24; 24n)
for n = 4 and 5,
BTD(30, 165; 14,4,22; 4,2)
and
Frame(30[6],4,4,2).
BTD, V= 105: GD(4,2,21; 105) and
BTD(21,84; 8,4, 16;4,2).
BTD, V= 117: GD(4,2,{24,21"}; 117)
and
BTD(21,84; 8,4, 16; 4,2)
and
BTD(24, 108; 10, 4, 18; 4, 2). BTD, V= 135: GD(4,2,27; 135) and
BTD(27, 135; 12,4,20;4,2).
BTD, V= 138: GD(4, 2, 27; 135), BTD(30, 165; 14, 4, 22; 4, 2) and Frame(30[3], 4, 4, 2). Balanced ternary designs with K = 4 , A = 2, Q2 = 3 and V as f o l l o w s
BTD, V= 16: 0017
00212
00311.
These are cycled under the permutation (0 1 --- 15). BTD, V= 22: 0013
00414
0079
051016.
These are cycled under the permutation (0 1 -.- 21). Frame, V[f] - 22[4]: 00311
04914
A1002
A l l 12
with short block X 0 6 12 cycled twice. These are cycled under the permutation (0 1 -.. 17)(A1A2A3), the element X is fixed.
D. Donovan / More balanced ternary designs
122
BTD, V= 28: 0013
00411
00515
02918
061220.
These are cycled under the permutation (0 1 .-- 27). Frame, V[f] = 28 [4]: 00113
02520
02520
A1007
Al14144
with short block X 0 8 16 cycled twice. These are cycled under the permutation (0 1 --- 23)(A1A2A3), the dement X is fixed. BTD, V= 34: 0013
00421
00511
021220
061524
071422.
These are cycled under the permutation (0 1 --- 33). Frame, V[f] = 34[10]: 0 0 9 21
A I 1 1 12
A 14 4 3
A l 2 8 12 A 15 11 15
B 12 4 21
B I 2 4 21
with short block X 0 8 16 cycled twice. These are cycled under the permutation (0 1 ... 23)(A1AaA3A4AsA6)(BIB2B3), the element X is fixed. BTD, V= 40: 0013
00415
00512
021323
061624
061926
081726.
These are cycled under the permutation (0 1 --- 39). BTD, V= 46: 0013
00417
00514
02820
061930
071928
072230
0102031.
These are cycled under the permutation (0 1 -.. 45). Frame, V[f] = 46110]: 0 1 2 17 A 1 11 11 15
0 2 1127 A 130 30 7
0 3 1029 A 1 0 8 14
A 1101015 A ! 0 8 14
A l 4 7 22
with short block X 0 12 24 cycled twice. These are cycled under the permutation (0 1 -.- 35)(A1A2--- Ag), the element X is fixed. BTD, V= 52: 00 1 3 061338
00 419 072331
00 517 091830
0 2 822 0102334
0102536.
These are cycled under the permutation (0 1 -.- 51).
D. Donovan / More balanced ternary designs
123
BTD, V=58: O0 1 3 072635
O0 421 082034
O0 518 092838
0 2 824 0112536
0 61323 0122743.
These are cycled under the permutation (0 1 -.- 57). BTD, V= 70: oo 1 3 oo 425 0 0 523 0 2 828 0 61320 0 83242 091949 092238 0113344 0122739 0152946 0163451. These are cycled under the permutation (0 1 -.- 69). Frame, V[f] = 82[28]: 0 0 125 B 1 0 411 D 1013 22 Fl 0 326
0 Bl Dl F1
152739 0 411 0 3241 12952
A10010 C 10 219 E 1 0 6 14 G10521
A I 2 222 C l 0 219 E 1 1 41 49 G 1 1 34 50
with short block X 0 18 36 cycled twice. These are cycled under the permutation (GIG2G3), the element X is fixed.
(0 1 .-. 53)(A1A2A3)(BIB2B3).-'
BTD, V= 118: GD(4, 2, {24, 18" }; 114}, BTD(22, 88; 10, 3, 16; 4, 2) and Frame(28[4], 3, 4, 2). BTD, V= 142: GD(4, 2, {24, 18 * }; 138), BTD(22, 88; 10, 3, 16; 4, 2) and Frame(28[4], 3, 4, 2). Balanced ternary designs with K = 4, A = 2 a n d Q2 = 5 and V as f o l l o w s
BTD, V= 26: 00112
00218
00319
00417
00520.
These are cycled under the permutation (0 1 --- 25). Frame, V[f] = 2612]: 00210
00315
00518
00417
X001
with short block X 1 9 17 cycled once. These are cycled under the permutation (0 1 -.- 23)(X Y). Frame, V[f] = 26[8]: 0025
Alll
113
Al13132
Al16162
A1069
X001
with short block X 1 7 13 cycled once. These are cycled under the permutation (0 1 --. 17)(AIA2A3A4AsA6)(X Y).
D. Donovan / More balanced ternary designs
124
BTD, V= 32: 0013
0046
00520
001019
001118
071523.
These are cycled under the permutation (0 1 --. 31). Frame, V[f] = 32[8]: 00318
0101419
A113132
A 110103
A1553
A10812
X001
with short block X 1 9 17 cycled once. These are cycled under the permutation (0 1 .-. 23)(A1AzA3A4AsA6)(X Y). BTD, V = 38: 0013
0046
00517
00824
001323
071626
071827.
These are cycled under the permutation (0 1 ... 37). Frame, V[f] = 38114]: A 199 2 A11710
A 1111116 A 1 1 710
A133 5 A16617 A10820 A10414
X001
with short block X 1 9 17 cycled once. These are cycled under the permutation (0 1 ... 23)(AIA 2 ..-AI2)(X Y). BTD, V= 44: 00518
00720
001032
001115
001619
093036
093036
These are cycled under the permutation (0 1 ... 43). BTD, V= 50: 0 0 145 001630
0 0 238 081831
00 3 9 081831
0 0 429 0112633
0112633.
These are cycled under the permutation (0 1 --. 49). BTD, V= 56: 0 0 3 5 0 0 933 061927 061927
001036 071822
001240 071822
001439 01 226.
These are cycled under the permutation (0 1 --. 55). Frame, V[f] = 56120]: A 19 9 5 A 1292921 A 13131 6 A 12525 5 A 12 16 33 A 1 2 16 33 A 1 4 14 17 A 1 4 14 17 A 1 1 8 10 A l 1 8 10 A 1 0 12 30 A 1 0 621 X 0 0 1
with short block X 1 13 25 cycled once. These are cycled under the permutation (0 1 --- 35)(A1A 2 .--AIs)(X Y).
0124
D. Donovan / More balanced ternary designs
125
BTD, V= 62: 0 0 132 061927
00 320 061927
0 0 539 0102426
0 0 937 0102426
001229 071822
0 7 1822.
These are cycled under the permutation (0 1 .-- 61). BTD, V= 68: 0 0 130 061927
0 0 351 0102426
0 01237 0102426
002332 071822
002833 071822
0 6 1927 0 5 25 34.
These are cycled under the permutation (0 1 -.- 67). Frame, V[f] = 74[26]: A10011 D10720 G13630
A1 1 1 5 O10 720 G141322
B1 0 0 2 E 1 01022 G15 826
B1 1 1 2 E 1 02638 Gll 917
C 1 0 5 19
F 1 0623 X 0015.
C 1 0 5 19 F 1 1 26 43
with short block X 1 17 33 cycled once. These are cycled under the permutation (0 1 --- 47)(A1A2A3)--" (F1F2F3)(G1G2 --" G6)(X Y). Frame, VLf] = 86[26]: 0 5 B1 1 E1 0 G 11
1324 1 30 7 28 11 21
0
5 C10 El 0 G1 1
1324 2 14 32 53 4 10
A1 C1 Fl Gl
1 1 17 0 46 58 0 1 26 11 14 20
A10017 B1 2323 0 D 1 0 4 22 D 1 0 38 56 F 1 0 1 26 G I 0 15 30 X 0 0 27.
with short block X 1 21 41 cycled once. These are cycled under the permutation (0 1 -.- 59)(AIA2A3)---(F1F2Fa)(GIG2G3G4GsG6)(X Y). Frame, V[f] = 92126]: 0 9 BI0 Dl 0 G10
12 30 0 9 1 2 3 0 0 1 B12 216 5 28 E l 0 10 29 631 G 1 2 3 7 6 2
0 133353 C 1 01127 E l 0 10 29 H 1 0 224
A10 0 4 Cl 1 4 0 5 6 F 1 0 15 32 H12 428
A 12 D10 F1 1 X 0
210 528 35 52 0 7.
with short block X 1 23 44 cycled once. These are cycled under the permutation (0 1 .-. 65)(A1AEA3)(BIB2B3)-.-(H1HEH3)(X Y). Whilst cOnstructing the following balanced ternary design we use a slight variation o f Lemma 1 and Theorem 2. Hanani in his original paper used group divisible designs with a number of different group sizes. Since there is only one situation where we require a group divisible design with several groups of different sizes we
126
D. Donovan / More balanced ternary designs
simply give the group divisible design and refer the reader to Hanani (1975), Lemma 2.25. BTD, V= 140: GD(4,2,S; 132)
where S denotes the multiset {24,24,24,24, 18, 18},
BTD(26, 130; 10, 5, 20; 4, 2), Frame(26[8], 5, 4, 2) and Frame(32[8], 5, 4, 2). BTD, V= 212: GD(4, 2, {36, 12"}; 192), BTD(32, 192; 14, 5, 24; 4, 2) and Frame(56[20], 5, 4, 2). Balanced ternary designs with K = 4 , A = 2, and ~2=6 and V as follows BTD, V= 31: 00117
00220
00321
00423
00524
00622.
These are cycled under the permutation (0 1 -.. 30). Frame, V[f] = 31 [4]: 00115
00222
00310
00821
Ai004
A12213.
with short block X 0 9 18 cycled twice. These are cycled under the permutation (0 1 .-. 26)(AIA2A3), the dement X is fixed. Frame, V[f] = 3117]: 0016
0029
A1114
A1884
A10011
A115155
with short blocks 0 5 12 17 cycled once and X 0 8 16 cycled twice. These are cycled under the permutation (0 1 ---23)(AIA2A3A4AsA6), the dement X is fixed. Frame, V[f] = 31 [10]: 06912
Al001
A1227
A1335
A14417
BI0010
B1551
with short block X 0 7 14 cycled twice. These are cycled under the permutation (0 1 ---20)(A1AEA3A4A5A~)(B1B2B3), the dement X is fixed. BTD, V= 34: 0013
0057
00614
00922
001018
001115
with short block 0 4 17 21 cycled once. These are cycled under the permutation (0 1 --- 33). Frame, V[f] = 34110]:
127
D. Donovan / More balanced ternary designs
00115
A1002
A1115
B19920
B1 1 1 1 1 6
B1 4 4 1
B1 0 7 1 4
with short blocks 0 6 12 18 cycled once and X 0 8 16 cycled twice. These are cycled under permutation (0 1--. 23)(A1A2A3)(B1B2B3B4BsB6), element X is fixed.
the
BTD, V= 37: 0013
0057
00614
001115
001221
001317
081827.
These are cycled under the permutation (0 1 .-- 36). Frame, VLF] = 3711]: 00117
00210
0037
00513
00622
001115
with short blocks 0 9 18 27 cycled once and X 0 12 24 cycled twice. These are cycled under the permutation (0 1 --- 35), the element X is fixed. Frame, VLF] = 37110]: 00310
00512
A1001
A 1551
B1 0 0 1 1
B1 1 1 1 4
C1028
C101925
with short block X 0 9 18 cycled twice. These are cycled under the permutation (0 1 -.- 26)(A1A2A3)(BIB2B3)(C1C2C3), the element X is fixed. Frame, VLf] = 37113]: A1224
A 112121
B1 0 0 5
B1 1 1 5
C1009
C1223
C11411
C11411
with short blocks 0 6 12 18 cycled once and X 0 8 16 cycled twice. These are cycled under the permutation (0 1
--- 23)(AIA2A3)(B1B2B3)(C1C2C3C4CsC6)
,
the element X is fixed. BTD, V= 40: 0013
0057
00614
00925
001119
001222
041727
with short block 0 4 20 24 cycled once. These are cycled under the permutation (0 1 --- 39). BTD, V= 43: 0013
0057
00414
001119
001323
001725
061527
061527.
These are cycled under the permutation (0 1 -.- 42).
Frame, V[f] = 43 [7]: 0023
00728
011122
061622
Al005
A l l 15
B l l 114
B10017.
D. Donovan / More balanced ternary designs
128
with short blocks 0 9 18 27 cycled once and X 0 12 24 cycled twice. These are cycled under the permutation (0 1 -.- 35)(AIA2A3)(B1B2B3), the element X is fixed. BTD, V= 46: 0023
00417
00619
00828
001034
001131
0 7 1621
0 7 1621
with short block 0 1 23 24 cycled once. These are cycled under the permutation (0 1 --- 45). Frame, V[f] = 46110]: 0 01 7 A 11 1 17
0 21015 B 1 20 20 9
0 21015 Al0014 B 1 34 34 17 B l 1 1 5
B 1 03 6
with short blocks 0 9 18 27 cycled once and X 0 12 24 cycled twice. These are cycled under the permutation (0 1 --- 35)(AIA2A3)(BIB2B3B4BsB6), the element X is fixed. BTD, V= 49: 00 1 4 001729
00
6
8
001830
001036 071621
001134 071621
0222446.
These are cycled under the permutation (0 1 ... 48). Frame, V[f] =49[131: 0 61419 B I 1 111
0 61419 C 14 4 5
A 10016 C12217
A1224 C1037
B1 0 0 1 1 C103 7
with short blocks 0 9 18 27 cycled once and X 0 12 24 cycled twice. These are cycled under the permutation (0 1 .-- 35)(AIA2A3)(B1BEB3)(CICEC3C4CsC6) , the element X is fixed. BTD, V= 52: 0 0 210 001237
0 0 323 001718
0 0 639 071621
001130 071621
0242832
with short block 0 1 26 27 cycled once. These are cycled under the permutation (0 1 -.. 51). BTD, V= 55: 00 1 4 001922
0 0 635 071621
0 0 837 071621
0 01223 0102527
0 01324 0102527.
These are cycled under the permutation (0 1 -.- 54).
129
D. Donovan / More balanced ternary designs
Frame, V[f] = 55119]: A 1002 D1881
A 1 1010 2 D 1 5 528
B 1 0 1 11 D17711
B 1 0 1 11 D12222 2
C 101419 D101530
C 1 01419 DI0 3 6
with short blocks 0 9 18 27 cycled once and X 0 12 24 cycled twice. These are cycled under the permutation (0 1 ..- 35)(AIA2Aa)(BIB2B3)(CIC2C3)(D1D2DaD4DsD6D7DsD9), the element X is fixed. BTD, V= 58: 00 1 4 002230
00 640 071621
001135 071621
0 01220 0102527
0 01339 0102527
with short block 0 3 29 32 cycled once. These are cycled under the permutation (0 1 --- 57). BTD, V=61: 0 0 1 13 001941
00 335 071621
00 643 071621
0 0 830 0102527
0 01123 0102527
042832.
These are cycled under the permutation (0 1 --- 60). BTD, V= 64: 00 129 002234
00 323 071621
00 440 071621
0 01119 0102527
0 01331 0102527
082026
with short block 0 6 32 38 cycled once. These are cycled under the permutation (0 1 --- 63). BTD, V= 67: 00 1 4 071621
001824 071621
0 02235 0102527
0 02329 0102527
0 02639 0111931
0 03033 0111931.
These are cycled under the permutation (0 1 .-. 66). BTD, V= 70: 00 123 071621
00 332 071621
0 0 426 0102527
0 01337 0102527
0 01842 0111931
0 03036 0111931
with short block 0 6 35 41 cycled once. These are cycled under the permutation (0 1 -.- 69). F r a m e , V[f] = 70122]: 0 18 21 27 Cl 1 8 9 D 1 2 7 17
A l 0 0 20 C l 18 9 E l 0 0 23
A 1 1 1 20 D 14415 E1 1 3 5
B 1 0 0 14 D l0 6 9 E 1 4 22 26
B l 19 19 2 D 1 2 7 17 E l 1 14 27
D. Donovan / More balanced ternary designs
130
with short block 0 12 24 36 cycled once and X 0 16 32 cycled twice. These are cycled under the permutation (0 1 -.. 47)(A1AEA3)(B IBEB3)(C 1C2C3 )(D 1DED3D4DsD6)(21E2E3E4E5E6 ), the element X is fixed. BTD, V= 73: 0 0 124 071621
0 0 441 071621
0 0 645 0102527
0 01844 0102527
0 02235 0111931
0 03033 0111931
0133639.
0 02352 0111931
0262932
These are cycled under the permutation (0 1 --- 72). BTD, V= 76: 0 0 127 071621
00 236 071621
0 0 446 0102861
0 01335 0102861
0 01739 0111931
with short block 0 6 38 44 cycled once. These are cycled under the permutation (0 1 .-- 75). BTD, V= 79: 0 0 1 30 0 03541 011 1931
0 0 437 0 7 16 21 011 1931
0 01847 0 7 16 21 0133639
0 02255 0102527 0133639.
0 02834 0102527
These are cycled under the permutation (0 1 --. 78). BTD, V= 82: 0 0 138 0 03440 0111931
0 0 449 0 71621 0111931
0 01853 0 71621 0133639
0 02250 0102527 0133639
with short block 0 6 41 47 cycled once. These are cycled under the permutation (0 1 --.
0 02452 0102527
81).
BTD, V= 85: 0 0 1 30 0 03247 011 1931
0 0 445 0 71621 011 1931
0 0 650 0 71621 0133639
0 0 17 42 0102861 0133639
These are cycled under the permutation (0 1 -.-
0 0 2 2 37 0102861 0 22729.
84).
BTD, V= 88: 0 0 148 0 03045 011 1931
0 0 653 0 7 16 21 011 1931
0 01732 0 7 16 21 0133639
with short block 0 2 44 46 cycled once.
0 02224 0102861 0133639
0 0 2 5 59 0 102861 0 43842
D. Donovan / More balanced ternary designs
131
These are cycled under the permutation (0 1 ... 87). Frame, V[f] = 91 [31 ]: Al
0 C 1 14 Fl 0 H1 0
017 14 1 826 723
A 11 1 2 6 D 1 0 2 29 F103452 Il 3 615
B 1 0 014 D 1 0 31 58 G 1 0 122 I l 4 7 16
B l 1 1 20 E 1 0 4 28 G103859 I 1 0 5 11
C l 0 010 E l 0 32 56 HI0 723 11 1 5 0 5 6
with short blocks 0 15 30 45 cycled once and X 0 20 40 cycled twice. These are cycled under the permutation (0 1 ... 59)(A1A2A3)(B1B2B3)--. (H1H2H3)(IlI21314IsI6), the element X is fixed. BTD, V= 94: GD(4, 2, 21; 84), BTD(31, 186; 12, 6, 24; 4, 2) and Frame(31 [ 10], 6, 4, 2). Frame, V[f] = 94122]: 0 1 7 8 A l0 023 C 101431 E 1 1 127
0 3 Al 1 D 14 El 2
16 43 120 415 4 6
0 82150 B1 0 044 D l0 6 9 E 1 53539
0 B1 DI E1
12 33 45 37 37 2 2 7 17 02040
0 16 25 50 C l 01431 D 1 2 7 17
with short blocks 0 18 36 54 cycled once and X 0 24 48 cycled twice. These are cycled under the permutation (0 1 --- 71)(A1A2A3)(BIB2B3)(CIC2C3)(D1DED3D4DsD6)(EIE2E3E4EsE6), the element X is fixed. Frame, VLf] = 97125]: 0
1 7 8
0
3 1643
A l 37 37 :E B 1 0 14 31 C 1 01233 C l 13446 D 1 2 7 17 E 1 1 1 27
0
82150
B 1 0 14 31 D 14 415 El 2 4 6
0
16 25 50
C 1 5 5 49 D1 0 6 9 E l 5 35 39
A 10
023
C 1 2 2 21 D 12 7 17 E 1 02040
with short blocks 0 18 36 54 cycled once and X 0 24 28 cycled twice. These are cycled under the permutation (0 1 ... 71)(A1A2A3)(B1B2B3)(C1C2C3C4CsC6)(D1D2D3D4DsD6)(E1E2E3E4E5E6), the element X is fixed. Frame, V[f] = 97[34] A 10 020 D 101123 G 10 428 11 0 5 22
A 1 1 120 D 104052 G 103559 JI 1 7 10
Bl 0 014 E l 01025 H l0 229 Jl 3 9 12
B l 1 1 17 E 1 03853 H 103461 Jl 4 5 35
C l0 Fl 0 11 0 Jl 0
07 826 522 32 62
C 1 14 14 1 F l 03755
D. Donovan / More balanced ternary designs
132
with short block X 0 21 42 cycled twice. These are cycled under the permutation (0 1 --- 62)(A1A2A3)(BIB2B3)'--(IlI213)(J1J2J3J4JsJ6), the element X is fixed. Frame, VLf] = 100131]: 0
61533
C l 0 0 19 F 1 0 14 26 11 0 2 3 2
0 6 15 33 C12323 1 F l 04355 I l 0 37 67
A~ 0 Dl0 G l0 Jl 0
0 1 420 728 31 34
A 1 11 11 1 D 1 0 420 G 1 04162 J1 0 35 38
with short block X 0 23 46 cycled twice. These are cycled under the permutation (0 1 the element X is fixed.
B 1 0 0 11 E1 0 8 2 5 H 10529
B 1 1 1 14 El 0 8 2 5 H 104064
... 68)(AIA2A3)(B1B2B3)-.-(J1J2J3),
BTD, V= 112: GD(4, 2, 27; 108), BTD(31, 186; 12, 6, 24; 4, 2) and Frame(31 [4], 6, 4, 2). BTD, V= 115: GD(4, 2, 21; 105), BTD(31, 186; 12, 6, 24; 4, 2) and Frame(31 [101, 6, 4, 2). BTD, V= 1 18: GD(4, 2, 27; 108), BTD(37, 259; 16, 6, 28; 4, 2) and Frame(37[10], 6, 4, 2). Frame, V[f] = 121134]: 0 12 A 1 19 D1 0 F1 0 Il 0
15 36 19 2 7 32 1434 8 26
0 Bl D1 Gl I1
12 0 0 0 0
15 36 028 7 32 l 43 61 79
0 2 639 B1 2 240 E I 0 16 35 G 104486 J1 0 11 41
0 2 639 C10 0 5 E l 0 16 35 H 102231 Jl 0 46 76
A 10 023 C 1 1 1 11 F l 0 14 34 H 105565 K 1 0 13 40
K 1 0 47 74
with short block X 0 29 58 cycled twice. These are cycled under the permutation (0 1 -.- 86)(AIA2A3)(B1B2B3)-.-(KIK2K3) , the element X is fixed. BTD, V= 139: GD(4, 2, 27; 135), BTD(31, 186; 12, 6, 24; 4, 2) and Frame(31 [4], 6, 4, 2). BTD, V= 142: GD(4, 2, {27, 24" }; 132), BTD(34, 221; 14, 6, 26; 4, 2) and Frame(37 [ 10], 6, 4, 2). BTD, V= 166: GD(4, 2, 27; 162), BTD(31; 186; 12, 6, 24; 4, 2) and Frame(31 [4], 6, 4, 2).
D. Donovan / M o r e balanced ternary designs
133
BTD, V= 169: GD(4, 2, { 27, 24" }; 159), BTD(34, 221; 14, 6, 26; 4, 2) and Frame(37110], 6, 4, 2). BTD, V=221: GD(4, 2, { 36, 30"}; 210), BTD(31, 186; 12, 6, 24; 4, 2) and Frame(37[1 ], 6, 4, 2). BTD, V= 214: GD(4, 2, 48; 192), BTD(70, 875; 38, 6, 50; 4, 2) and Frame(70[22], 6, 4, 2). BTD, V= 238: GD(4, 2, {48, 24" }; 216), BTD(46, 391; 22, 6, 34; 4, 2) and Frame(70[22], 6, 4, 2). BTD, V= 241: GD(4, 2, {48, 27" }; 219), BTD(49, 441; 24, 6, 36; 4, 2) and Frame(70[22], 6, 4, 2).
Acknowledgement I wish to thank my supervisor Dr. Elizabeth J. BiUington for suggesting the problem and for her help in the preparation of the manuscript.
References Assaf, A., A. Hartman and E. Mendelsohn (1985). Multiset designs - designs having blocks with repeated elements. Canad. Numer. 48, 7-24. Brouwer, A.E. (1979). Optimal packings of K4's into a K n. J. Combin. Theory Ser. A 26, 278-297. Brouwer, A.E., A. Schrijver and H. Hanani (1977). Group divisible designs with block size four. Discrete Math. 20, 1-10. Donovan, D. (1986a). A family of balanced ternary designs with block size four. Bull. Austral. Math. Soc. 33, 321-327. Donovan, D. (1986b). Balanced ternary designs with block size four. Ars Combin. 21A, 81-88. Hanani, H. (1961). The existence and construction of balanced incomplete block designs. Ann. Math. Statist. 32, 361-386. Hanani, H. (1975). Balanced incomplete block designs and related designs. Discrete Math. 11,255-369. Robinson, P.J. (1984). Computer program for the construction of cyclic balanced ternary designs. Private communication. Tocher, K.D. (1952). The design and analysis of block experiments. J. Roy. Statist. Soc. Ser. B 14, 45-100.
109
M O R E B A L A N C E D T E R N A R Y DESIGNS WITH BLOCK SIZE FOUR Diane DONOVAN Department of Mathematics, University of Queensland, St. Lucia, 4067, Australia Received 2 July 1986 Recommended by Jennifer Seberry
Abstract: This paper gives necessary and sufficient conditions for the existence of balanced ternary designs with block size four and A = 2 in the cases 02 = 3, 4, 5 and 6; the cases 02 = 1 and 2 have appeared earlier.
AMS Subject Classification: Primary 05[305; Secondary 62K10. Key words and phrases: Block design; Balanced ternary design; Group divisible design; frame.
1. Introduction Balanced ternary designs (BTDs) were first introduced by Tocher (1952). He defined them to be a collection of B multisets of size K, called blocks, chosen from a set of V elements where any element may occur 0, 1 or 2 times in any one block and furthermore each of the (v) pairs of distinct elements occurs A times throughout the blocks of the design. We shall further stipulate that each element must occur a constant n u m b e r of times, say R, throughout the design. It follows that VR = BK.
(1)
If we let 0/, l = 1 or 2, denote the number of blocks in which an element occurs l times then R = 01 + 202
(2)
and A(V-
1) = R ( K - 1 ) - 202.
(3)
We shall write the parameters o f a balanced ternary design as (V, B; Q1, 02, R; K, A). In this paper we consider the existence of balanced ternary designs with K = 4, A = 2 and small 02, that is Q2 = 3, 4, 5 and 6. For 02 = 1 and 2 see Donovan (1986b) and Assaf, H a r t m a n and Mendelsohn (1985). We shall give necessary and sufficient conditions for the existence of such balanced ternary designs. 0378-3758/87/$3.50 © 1987, Elsevier Science Publishers B.V. (North-Holland)
110
D. Donovan / M o r e balanced ternary designs
Since we are considering balanced ternary designs with K = 4 and A = 2 identities (1), (2) and (3) allow us to write the parameters as (V, V ( V - 1 + P2). 2 ( V - 1 - 2Q2) 2 ( V - 1 + ~o2) "4, 2) 6 ' 3 , Q2, 3 ' "
(4)
In addition since A = 2 the blocks of such balanced ternary designs must be of the form wxyz or xxyz and not xxyy; hence B>_o2V. The techniques used in this paper are based on those introduced by Hanani (1975), (see also Hanani (1961)), to give the existence of certain balanced incomplete block designs. We also employ a variation of this method introudced by Assaf, Hartman and Mendelsohn (1985). These techniques rely on the existence of certain group divisible designs and frames. A group divisible design is a collection of subsets of size k, called blocks, chosen from a set of o =mn elements, where this set can be partitioned into n groups of size m, and where the blocks are selected in such a way that each block contains at most one element from each group and any two elements from distinct groups occur together in 2 blocks. The parameters of a group divisible design are written GD(k, 2, m; o). For the purpose of this paper we shall define a frame to be a collection of multisets of size K, called blocks, chosen from a set of V elements such that the following conditions are satisfied: (i) {ooi, i= 1, . . . , f } is a subset of the set of elements, called a hole; (ii) any element not belonging to the hole occurs at most twice in any one block and precisely twice in #2 blocks; (iii) each 0% i = 1, ... , f , occurs at most once in any one block; (iv) the pair xy, where x and y are distinct elements not both belonging to the hole, occurs A times throughout the blocks of the frame. We shall write the parameter as (V[ f ] , Q2,K,A). Note that a balanced ternary design is a frame with f = 0. This paper is divided into six sections. They are 1. Introduction; 2. The case Q2=4; 3. The case Q2=3; 4. The case ~2=5; 5. The case ~2=6; 6. Summary remarks; and an Appendix. The techniques used are summarized in the following l e m m a and theorem, which are an analogue of Hanani's L e m m a 2.25 (1975). We refer the reader to Donovan (1986b) for the proofs. Lemma 1. I f there exist group divisible designs G D ( k ; 2 ' , t : o),
OD(k,2';m; k'm)
and
GD(k, 2", m; ( k ' - 1)m)
then there exists a group divisible design GD(k, 2, {mr, mh* }; (o - t + h)m)
f o r O
D. Donovan / More balanced ternary designs
111
Theorem 2. I f there exists a group divisible design
GD(K,A, { V - f , ( V ' - f ) * } ;
(n- 1)(V-f) + V'-f),
and if there exists a (V[f],#z,K,A) frame and also a BTD with parameters (V;B; Ql, o2,R; K,A) then there exists a B T D with parameters
( ( n - 1 ) ( V - f ) + V',B'; p~,Qz, R'; K,A). We shall require the following results and so we state them here and refer the reader to the relevant papers for their proofs. Theorem 3 (Brouwer, Schrijver and Hanani (1977), Theorem 6.3). For all positive
integers m, A and o the group divisible design GD(4, A, m; o) exists if and only if the design is not GD(4, 1, 2; 8) and not GD(4, 1, 6; 24) and provided that o = 0 (mod m), ;t(o-m)=-O (mod 3), 2o(o-m)=--O (mod 12) and o>_4m or o=m. Theorem 4 (Brouwer (1979), Theorem 4). A group divisible design
GD(4, 1, {2,5*}; 6 q + 5) exists for all positive integers q :/: 1, 2.
The following results refer to transversal designs. A transversal design, T(s, A, t), is a group divisible design, GD(s, A, t; st), in which each block contains precisely one element from each group. Lemma 5 (Hanani (1975), Lemma 3.5). For every prime power q there exists a trans-
versal design T(q + 1, 1, q). Lemma 6 (Hanani (1975), Lemma 3.1). I f a transversal design T(s,A,t) exists then
a transversal design T(s, 2, t) exists f o r all s'<_s.
The following lemma follows from the above result and Hanani (1975), Theorem 3.11. Lemma 7. A transversal design T(5,2,n) exists for all positive integers n. Lemma 8. (Hanarti (1975), Lemma 3.21). A transversal design T(6, 1, 12) exists.
2. The case Q2 = 4
The constructions given here for balanced ternary designs with •2 = 4 are similar to Hanani's original methods.
D. Donovan / More balanced ternary designs
112
If we substitute Q2= 4 into identity (4), the parameters of the balanced ternary designs can be written V ( V + 3 ) . 2 ( V - 9 ) , 4 , 2(V+3) "4,2). V, 6 ' 3 3 ' It follows that necessary conditions for the existence of such balanced ternary designs are that V - 0 (mod3) and V>_21 (since B>_Q2V). We give methods for constructing balanced ternary designs satisfying these parameters and so show that the conditions are in fact sufficient. Balanced ternary designs with the above parameters and V=21,24, ..., 138 are given in the Appendix. For the remaining balanced ternary designs we give the following recursive construction. BTDs with V-O (mod3) and V_>141. By Lemma 7, T(5,2,n), n_>10, exists and GD(4, 1, 3; 12) and GD(4, 1, 3; 15) exist by Theorem 3. Hence by Lemma 1, GD(4, 2, {3n,3h*}; 12n+3h) exists for n___10 and h = 7 , 8 , 9 and 10. So by applying Theorem 2 to these group divisible designs and balanced ternary designs with V=21,24,27, 30 and 3n for n>_ 10 we obtain balanced ternary designs with V= 12n + 21, 12n + 24, 12n + 27 and 12n + 30 for n_> 10. Having given methods for constructing all possible balanced ternary designs with K = 4, A - - 2 and Q2= 4 we state the following theorem. Theorem
9. Necessary and sufficient conditions for the existence of balanced ter-
nary designs with parameters V, V(V+ 3 ) . 2 ( V - 9 ) 4, 2(V+ 3) 6
'
3
'
3
) ;4,2
are that VmO (mod 3) and V_>21. In the remaining three cases to be considered, namely Q2 3, 5 and 6, all constructions rely on Lemma 1 and Theorem 2 and these are applied in the same way as above. Hence for brevity the above construction is the only one we shall list in detail. In all other cases we shall either list the blocks of the balanced ternary design or simply give the relevant designs which can be used in conjunction with Lemma 1 and Theorem 2. =
3 . T h e c a s e ~o2 = 3
Substituting ~2 3 into identity (4) the parameters of the balanced ternary design can be written as =
V(V+2).2(V_7),3,2(V+2) V,
6
'
3
3
) ;4,2 .
D. Donovan / More balanced ternary designs
113
It follows that necessary conditions for the existence of such balanced ternary designs are that V - 4 (rood 6) and V_> 16 (since B_> V2). Balanced ternary designs with the above parameters and V= 16, 22, ..., 58, 70, 82, 118 and 142 are listed in the Appendix. For the remaining designs we consider Vmodulo 72 and divide the designs into the following cases: (i) V--4 (mod 18), V>76; (ii) V - 4 (mod24), V_> 100; (iii) V--- 10 (mod 24), V_> 106; (iv) V-- 16 (mod 24), V_>64; (v) V-- 10 (mod 36), V_> 154; (vi) Vm 70 (mod 72), V_> 286. The above six cases cover all remaining values for V and in each case the following recursive construction establishes the existence of these balanced ternary designs. (i) V=18n+4, n>4. GD(4,2,18; 18n), n > 4 , (22,88; 10,3,16;4,2) BTD and (2214], 3, 4, 2) frame. (ii) V= 24n + 4, n > 4. GD(4, 2, 24; 24n), n _>4, (28, 140; 14, 3, 20; 4, 2) BTD and (28 [4], 3, 4, 2) frame. (iii) V = 2 4 n + 10, n_>4. GD(4,2,24; 24n), n_>4, (34,204; 18,3,24; 4,2) BTD and (34110], 3, 4, 2) frame. (iv) V = 2 4 n + 16, n_>2. GD(4,2,m; 4m), m_> 16, and (m,-~m(m+2); 2 ( m - 7 ) , 3 , 2(m + 2); 4, 2) BTD m _> 16. (v) V= 36n + 10, n _ 4. GD(4, 2, 36; 36n), n _>4, (46, 368; 26, 3, 32; 4, 2) BTD and (46110], 3, 4, 2) frame. (vi) V=72n + 70, n_>3. GD(4, 2, {24,60*}; 72n + 60), n _ 3 , (70,840; 42,3,48; 4,2) BTD and (34110], 3, 4, 2) frame. Hence we have shown that the above conditions are in fact sufficient and so we state the following theorem.
Theorem 10. Necessary and sufficient conditions for the existence o f balanced ternary designs with parameters
V(V+2).2(V-7) 3 , 2 ( V + 2 ) . 4 , 2 ) V,
6
'
3
'
3
'
are that V - 4 (mod 6) and V>_ 16.
4. The case ~o2 --'--5 Given ~o2 5 and identity (4) we can write the parameters of the balanced ternary designs as =
V,
V(V+4). 2 ( V - 11) 2(V+4) ) 6 ' 3 ,5, 3 ;4,2 .
D. Donovan / More balanced ternary designs
114
It follows that V = 2 (mod 6) and since B > V2, V> 26. Balanced ternary designs withthe above parameters and V= 26, 32,..., 74, 86, 92, 140 and 212 are given in the Appendix. Once again if we consider V modulo 72 we can divide the remaining balanced ternary design into six cases. These cases are: (i) V---8 (mod 18), V> 80; (ii) V =- 2 (mod 24), V__>98; (iii) V = 8 (mod 24), V_> 104; (iv) V - 14 (mod 24), V> 110; (v) V - 2 0 (rood 36), V> 164; (vi) V-- 68 (mod 72), V> 284. The following recursive constructions based on the above six cases establishes the existence of these balanced ternary designs. (i) V = 1 8 n + 8 , n > 4 . GD(4,2,18; 18n), n > 4 , (26, 130; 10,5,20;4,2) BTD and (2618], 5, 4, 2) frame. (ii) V = 2 4 n + 2 , n > 4 . GD(4,2,24; 24n), n > 4 , (26, 130; 10,5,20; 4,2) BTD and (2612], 5, 4, 2) frame. (iii) V = 2 4 n + 8 , n>4. GD(4,2,m; 4m), m_>26, and (m,-~m(m+4); 2 ( m - 11),5, 2(m + 4); 4, 2) BTD m >__26. (iv) V = 2 4 n + 14, n>4. GD(4,2,24; 24n), n > 4 , (38,266; 18,5,28; 4,2) BTD and (38[14], 5, 4, 2) frame. (v) V= 36n + 20, n > 4. GD(4, 2, 36; 36n), n > 4, (56, 560; 30, 5, 40; 4, 2) BTD and (56120], 5, 4, 2) frame. (vi) V=72n + 68, n>3. GD(4,2, {24,60*}; 72n + 60), n > 3 , (68,816; 38,5,48; 4,2) BTD and (3218], 5, 4, 2) frame. Thus we have constructed all possible balanced ternary designs with the above parameters and therefore established that the given conditions are sufficient. We therefore state the following theorem.
Necessary and sufficient conditions f o r the existence o f balanced ternary designs with parameters
T h e o r e m 11.
V,
V(V+4). 2 ( V - 11) 2(V+4) ) 6 ' 3 ,5, 3 ;4,2
are that V =- 2 (mod 6) and V> 26.
5. T h e case ~o2 - ~ 6
Given Lo2- 6, identity (4) allows us to write the parameters of the balanced ternary designs as
v ( v + 5) 2 ( v - 13),6, V,
6
'
3
• 4,2 3
'
"
D. Donovan / More balanced ternary designs
115
Necessary conditions for the existence of such balanced ternary designs are V - 1 (rood 3) and V>_ 31 (since B _ V#2). Balanced ternary designs with V= 31, 34, ..., 100, 112, 115, 118, 121, 139, 142, 166, 169, 211,214, 238 and 241 are given in the Appendix. We divide the remaining balanced ternary designs into 13 cases, which cover all possible values of Vmodulo 72. (i) V - 4 (mod 12) and V_> 124; (ii) V - 7 (mod 24) and V_ 103; (iii) V - 10 (mod 24) and V_> 106; (iv) V =- 13 (mod 24) and V>_ 109; (v) V - 1 (mod 36) and V_ 145; (vi) V=- 7 (rood 36) and V_ 151; (vii) V - 10 (rood 36) and V_> 154; (viii) V - 13 (mod 36) and V_> 157; (ix) V - 19 (mod 36) and V_> 163; (x) V - 2 2 (mod 72) and V>_310; (xi) I/-=25 (mod 72) and V_>313; (xii) V - 67 (mod 72) and V>_ 283; (xiii) V - 70 (mod 72) and V>_286. We now consider each of the above classes of balanced ternary designs separately. (i) V= 12n+4, n_>5. GD(4,2,m; 4m), m_31 and (m,~m(m+ 5); 2 ( m - 13);6; 2(m + 5); 4,2) BTD. (ii) V = 2 4 n + 7 , n>_4. GD(4,2,24; 24n), n_>4, (31, 186, 12,6,24; 4,2) BTD and (31 [7], 6, 4, 2) frame. (iii) V=24n + 10, n_>4. GD(4,2,24; 24n), n_>4, (34,221; 14,6,26; 4,2) BTD and (34[10], 6, 4, 2) frame. (iv) V = 2 4 n + 13, n_>4. GD(4,2,24; 24n), n_>4, (37,259; 16,6,28; 4,2) BTD and (37113], 6, 4, 2) frame. (v) V = 3 6 n + 1, n _ 4 . GD(4,2,36; 36n), n>_4, (37,259; 16,6,28;4,2)BTD and (37[1], 6, 4, 2) frame. (vi) V= 36n + 7, n _>4. GD(4, 2, 36; 36n), n >__4, (43, 344; 20, 6, 32; 4, 2) BTD and (43 [7], 6, 4, 2) frame. (vii) V=36n + 10, n_>4. GD(4,2, 36; 36n), n _ 4 , (46,391; 22,6,34; 4,2) BTD and (46110], 6, 4, 2) frame. (viii) V=36n + 13, n_>4. GD(4, 2, 36; 36n), n>_4, (49,441; 24,6,36; 4,2) BTD and (49113],6, 4,2) frame. (ix) V=36n + 19, n_>4. GD(4,2, 36; 36n), n_>4, (55,550; 28,6,40; 4,2) BTD and (55119], 6, 4, 2) frame. (x) V=72n+22, n_>4. GD(4,2,72; 72n), n_>4, (94, 1551; 54,6,66; 4,2) BTD and (94 [22], 6, 4, 2) frame. (xi) V=72n+25, n_>4. GD(4,2,72; 72n), n___4, (97, 1649; 56,6,68; 4,2) BTD and (97 [25], 6, 4, 2) frame. (xii) V=72n+67, n_>3. GD(4,2,{24,60*};72n+60), n_>3, (67,804;36,6,48; 4, 2) BTD and (31 [7], 6, 4, 2) frame.
D. Donovan / More balanced ternary designs
116
(xiii) V = 7 2 n + 7 0 , n > 3 . GD(4,2,{24,60*};72n+60), n > 3 , (70,875;38,6,50; 4, 2) BTD and (34[10], 6, 4, 2) frame. We have therefore shown the sufficiency of the above conditions and so we state the following theorem.
Theorem 12. Necessary and sufficient conditions f o r the existence o f balanced ternary designs with parameters
V(V+5) 2(V-13) V,
6
;
3
2(V+5) .4,2~ ,6,
3
'
are that V=-1 (mod 3 ) a n d V_>31.
6. Summary remarks If we consider balanced ternary designs with K = 4, A = 2 and general Q2 then necessary conditions for the existence of such designs are V _ 5Q2 + 1, since B_> VO2, and V must be congruent modulo 6 to the value given in Table 1. Donovan (1986b) and Assaf, Hartman and Mendelsohn (1985) prove the sufficiency of these conditions for balanced ternary designs with K = 4, A = 2 and P2 = 1, 2. This paper proves the sufficiency of these conditions for balanced ternary designs with K = 4, A = 2 and Q2 = 3, 4, 5, 6. We know that V_> 5~) 2 + 1 is a definite lower bound as Donovan (1986a) has constructed the family of balanced ternary designs with K = 4, A = 2, V= 5~2 + 1 for all positive integers Q2- Further, since the transversal design T(4,2,t) exists for all positive integers t (see Hanani (1975), Lemma 3.11 and Lemma 3.1) we can use Theorem 2 to construct balanced ternary designs with K = 4 , A = 2 and V=4n(5Q2+l) for any positive integer n and Q2. Likewise the existence of certain other group divisible designs can be used in conjunction with these designs to construct various balanced ternary design, but in general the sufficiency of the values given in the above table is still to be investigated. Table 1 BTDs with K = 4, A = 2 ~02 taken modulo 6
0
1
2
3
4
5
V taken modulo 6
1,4
0
2,5
4
0,3
2
Appendix Whilst the general cases have been dealt with in the main body of the text there are a few balanced ternary designs which need to be listed individually. Where it has been necessary to construct the blocks of a specific balanced ternary design we have
D. Donovan / More balanced ternary designs
117
done so by the use of a computer program supplied by Robinson (private communication). In these cases we list the initial block which can be cycled modulo V to achieve all blocks of the design. Our constructions rely heavily on the existence of certain frames and the initial blocks of these have been listed and can be cycled modulo the given permutation to achieve all blocks of the frame. In the cases 62=3 and V=82; 6 2 = 4 and V=66,69,72,81; 6 2 = 5 and V=74,86,82; 6 2 = 6 and V=91,94,97, 100, 121, we list the blocks of the frame ( V[f], 62, 4, 2) and since we have already listed balanced ternary designs with parameters (f, B; 61, 62, R; 4, 2) we simply adjoin the blocks of these designs to the blocks of the frames to obtain balanced ternary designs on the appropriate V. For the remaining values of V we have used Lemma 1, Theorem 2 and the given designs.
Balanced ternary design with K=4, A = 2, 6 2 = 4 and V as follows BTD, V=21: 00112
0027
0038
00415.
These are cycled under the permutation (0 1 --- 20). Frame, V[f] =2113]: 0017
0058
AI002
A l l 15
with short block 0 3 9 12 taken once. These are cycled under the permutation (0 1 -.- 17)(AIA2A3). BTD, V= 24: 0018
00211
00310
00419
with short block 0 6 12 18 taken once. These are cycled under the permutation (0 1 -.. 23). Frame, VLf] = 24[3]: 00114
00210
03612
Al005
A l l 15.
These are cycled under the permutation (0 1 .-. 20)(AIA2A3). Frame, V[f] = 24[6]: 00613
A1123
A1 0 0 3
A 117177
A 116162
with short block 0 2 9 11 taken once. These are cycled under the permutation (0 1 -.- 17)(AIA2AaA4AsA6). BTD, V=27: 00115
00220
00311
00421
051422.
These are cycled under the permutation (0 1 --- 26).
118
D. Donovan / More balanced ternary designs
Frame, V[f] = 27[3]: 0037
00211
051014
A1008
A1778
with short block 0 6 12 18 taken once. These are cycled under the permutation (0 1 --. 23)(A1A2A3). Frame, V[f] = 27 [6]: 01410
01410
All13
A1229
B1005
B14417.
These are cycled under the permutation (0 1 --- 20)(A1AzA3)(BIBzB3). BTD, V= 30: 0016
00212
00311
00716
04817
with short block 0 5 15 20 taken once. These are cycled under the permutation (0 1 -.. 29). Frame, V[f] = 30[3]: 0017
00810
021316
031218
A1005
A l l 15.
These are cycled under the permutation O 1 --- 26)(AlA2A3). Frame, V[f] = 30161: 0035
00711
A1221
Al17177
AI0915
A101622
with short block 0 4 12 16 taken once. These are cycled under the permutation O 1 -.- 23)(AIA2A3A4AsA6). BTD, V= 33: 00118
00225
00322
00424
052128
062027.
These are cycled under the permutation (0 1 -.. 32). BTD, V= 36 0017
00214
00316
00819
04913
051526
with short block 0 6 18 24 cycled once. These are cycled under the permutation (0 1 --- 35). BTD, V= 39: 00121
00230
00326
00431
051934
062233
072232.
041029
052025
These are cycled under the pemutation (0 1 --- 38). BTD, V = 4 2 : 0018
00216
00312
001124
041019
D. Donovan / More balanced ternary designs
119
with short block 0 7 21 28 cycled once. These are cycled under the permutation (0 1 --- 41). BTD, V = 45: 00124
00235
00330
00436
051219
051632
061725
0 6 20 28.
These are cycled under the permutation (0 1 -.. 44). BTD, V = 4 8 : 0019
00218
00322
00411
051528
051536
061229
with short block 0 8 24 32 cycled once. These are cycled under the permutation (0 1 -.- 47). BTD, V = 5 1 : 0 0 127 051019
00 240 061231
00 334 071543
0 0 433 071637
0102238.
These are cycled under the permutation (0 1 ... 50). BTD, V = 54. 0 0 110 051124
00 220 052137
0 0 325 061437
00 412 071433
0112439
with short block 0 9 27 36 cycled once. These are cycled under the permutation (0 1 --. 53). BTD, V= 57: 0 0 130 061320
00 245 082341
0 0 338 082333
00 436 092240
051016 092637.
These are cycled under the permutation (0 1 --. 56). BTD, V = 60: 0 0 111 051231
00 222 061429
0 0 328 062339
0 0 413 082443
051226 092442
with short block 0 10 30 40 cycled once. These are cycled under the permutation (0 1 .-- 59). BTD, V = 63: 0 0 111 061533
00 222 062440
0 0 328 082342
0 0 413 082539
0 51226 0102743.
These are cycled under the permutation (0 1 --. 62). Frame, V [ f ] = 66121]:
051231
0 7 20 34
D. Donovan / More balanced ternary designs
120 0 91536 CI 0 2 3 E l 1 35 40
Al0 019 C l 1 13 29 F 1 0 18 32
A12323 1 Bl0 0 4 D 1 0 2 3 D 1 1 13 29 F l 1 14 25 G l 0 7 14
B12 210 E 1 0 15 20 G I 0 10 20.
These are cycled under the permutation (0 1 -" 44)((A1A2Aa)(BIBEB3)"'"
(G1GEG3).
Frame, V[f] = 69121]: 0 9 15 36 C l 0 4 11 E 1 0 31 46
A I0020 C 1 0411 F 1 0 3 16
A I 1 126 D10822 F 1 1 2 32
B1 00 5 D10822 G l 0 3 16
B 1 1 1 20 E 1 0238 G 1 1 2 11
with short block 0 6 24 30 cycled once. These are cycled under the permutation (0 1 --- 47)(AlA2Aa)(B1BEB3)-.. (G1G2G3). Frame, V[f] = 72121]: 0 91527 B 1 13 13 2 E 1 0 7 21
0 91527 C 1 0 222 E 1 31 47 50
A10 0 1 A15 5 1 C 1 0 2 2 2 D 10 7 2 1 F 1 0 10 23 F ! 0 10 23
Bl 0 0 5 D 1314750 G 1 0 8 25
G 1 0 8 25.
These are cycled under the permutation (0 1 ..- 50)(AIAEA3)(B1BEB3).-- (GIG2G3). BTD, V= 75, 93, 111 and 129: GD(4,2, 18; 18n)
for n=4,5,6 and 7,
BTD(21,84; 8,4, 16; 4,2)
and
Frame(21[3],4,4,2).
BTD, V= 78 and 114: GD(4,2, 18; 18n)
for n =4 and 6,
BTD(24, 108; 10, 4, 18; 4, 2)
and
Frame(24[6], 4, 4, 2).
Frame, V[f] = 81124]: 0 91527 CI0 222 F 1 0 1024
0 91527 C l 0 222 F 1 32 46 56
A10 0 1 Al 5 5 1 D 1 0 11 19 D 1 0 11 19 GI0 526 G103152
B 1 0 0 13 E l 0 7 23 H10328
B 1 2 2 19 E l 0 7 23 H1285356.
These are cycled under the permutation (0 1 --- 56)(AIA2A3)(B1B2B3)-.. (HIH2H3), BTD, V= 84, 96, 108, 120 and 132: GD(4, 2, n; 4n)
for n = 21, 24, 27, 30, 33,
BTD(V,B; ~l,4,R; 4,2) with V--21,24,27,30,33. BTD, V=87: GD(4,2,21; 84), BTD(24, 108; 10,4, 18; 4,2) and Frame(24[3],4,4, 2).
D. Donovan / More balanced ternary designs
121
BTD, V= 90: GD(4, 2, 21; 84), BTD(27, 135; 12, 4, 20; 4, 2) and Frame(27[6], 4, 4, 2). BTD, V= 99 and 123: GD(4, 2, 24; 24n)
for n = 4 and 5,
BTD(27,135; 12,4,20; 4,2)
and
Frame(27[3],4,4,2).
BTD, V= 102 and 126: GD(4, 2, 24; 24n)
for n = 4 and 5,
BTD(30, 165; 14,4,22; 4,2)
and
Frame(30[6],4,4,2).
BTD, V= 105: GD(4,2,21; 105) and
BTD(21,84; 8,4, 16;4,2).
BTD, V= 117: GD(4,2,{24,21"}; 117)
and
BTD(21,84; 8,4, 16; 4,2)
and
BTD(24, 108; 10, 4, 18; 4, 2). BTD, V= 135: GD(4,2,27; 135) and
BTD(27, 135; 12,4,20;4,2).
BTD, V= 138: GD(4, 2, 27; 135), BTD(30, 165; 14, 4, 22; 4, 2) and Frame(30[3], 4, 4, 2). Balanced ternary designs with K = 4 , A = 2, Q2 = 3 and V as f o l l o w s
BTD, V= 16: 0017
00212
00311.
These are cycled under the permutation (0 1 --- 15). BTD, V= 22: 0013
00414
0079
051016.
These are cycled under the permutation (0 1 -.- 21). Frame, V[f] - 22[4]: 00311
04914
A1002
A l l 12
with short block X 0 6 12 cycled twice. These are cycled under the permutation (0 1 -.. 17)(A1A2A3), the element X is fixed.
D. Donovan / More balanced ternary designs
122
BTD, V= 28: 0013
00411
00515
02918
061220.
These are cycled under the permutation (0 1 .-- 27). Frame, V[f] = 28 [4]: 00113
02520
02520
A1007
Al14144
with short block X 0 8 16 cycled twice. These are cycled under the permutation (0 1 --- 23)(A1A2A3), the dement X is fixed. BTD, V= 34: 0013
00421
00511
021220
061524
071422.
These are cycled under the permutation (0 1 --- 33). Frame, V[f] = 34[10]: 0 0 9 21
A I 1 1 12
A 14 4 3
A l 2 8 12 A 15 11 15
B 12 4 21
B I 2 4 21
with short block X 0 8 16 cycled twice. These are cycled under the permutation (0 1 ... 23)(A1AaA3A4AsA6)(BIB2B3), the element X is fixed. BTD, V= 40: 0013
00415
00512
021323
061624
061926
081726.
These are cycled under the permutation (0 1 --- 39). BTD, V= 46: 0013
00417
00514
02820
061930
071928
072230
0102031.
These are cycled under the permutation (0 1 -.. 45). Frame, V[f] = 46110]: 0 1 2 17 A 1 11 11 15
0 2 1127 A 130 30 7
0 3 1029 A 1 0 8 14
A 1101015 A ! 0 8 14
A l 4 7 22
with short block X 0 12 24 cycled twice. These are cycled under the permutation (0 1 -.- 35)(A1A2--- Ag), the element X is fixed. BTD, V= 52: 00 1 3 061338
00 419 072331
00 517 091830
0 2 822 0102334
0102536.
These are cycled under the permutation (0 1 -.- 51).
D. Donovan / More balanced ternary designs
123
BTD, V=58: O0 1 3 072635
O0 421 082034
O0 518 092838
0 2 824 0112536
0 61323 0122743.
These are cycled under the permutation (0 1 -.- 57). BTD, V= 70: oo 1 3 oo 425 0 0 523 0 2 828 0 61320 0 83242 091949 092238 0113344 0122739 0152946 0163451. These are cycled under the permutation (0 1 -.- 69). Frame, V[f] = 82[28]: 0 0 125 B 1 0 411 D 1013 22 Fl 0 326
0 Bl Dl F1
152739 0 411 0 3241 12952
A10010 C 10 219 E 1 0 6 14 G10521
A I 2 222 C l 0 219 E 1 1 41 49 G 1 1 34 50
with short block X 0 18 36 cycled twice. These are cycled under the permutation (GIG2G3), the element X is fixed.
(0 1 .-. 53)(A1A2A3)(BIB2B3).-'
BTD, V= 118: GD(4, 2, {24, 18" }; 114}, BTD(22, 88; 10, 3, 16; 4, 2) and Frame(28[4], 3, 4, 2). BTD, V= 142: GD(4, 2, {24, 18 * }; 138), BTD(22, 88; 10, 3, 16; 4, 2) and Frame(28[4], 3, 4, 2). Balanced ternary designs with K = 4, A = 2 a n d Q2 = 5 and V as f o l l o w s
BTD, V= 26: 00112
00218
00319
00417
00520.
These are cycled under the permutation (0 1 --- 25). Frame, V[f] = 2612]: 00210
00315
00518
00417
X001
with short block X 1 9 17 cycled once. These are cycled under the permutation (0 1 -.- 23)(X Y). Frame, V[f] = 26[8]: 0025
Alll
113
Al13132
Al16162
A1069
X001
with short block X 1 7 13 cycled once. These are cycled under the permutation (0 1 --. 17)(AIA2A3A4AsA6)(X Y).
D. Donovan / More balanced ternary designs
124
BTD, V= 32: 0013
0046
00520
001019
001118
071523.
These are cycled under the permutation (0 1 --. 31). Frame, V[f] = 32[8]: 00318
0101419
A113132
A 110103
A1553
A10812
X001
with short block X 1 9 17 cycled once. These are cycled under the permutation (0 1 .-. 23)(A1AzA3A4AsA6)(X Y). BTD, V = 38: 0013
0046
00517
00824
001323
071626
071827.
These are cycled under the permutation (0 1 ... 37). Frame, V[f] = 38114]: A 199 2 A11710
A 1111116 A 1 1 710
A133 5 A16617 A10820 A10414
X001
with short block X 1 9 17 cycled once. These are cycled under the permutation (0 1 ... 23)(AIA 2 ..-AI2)(X Y). BTD, V= 44: 00518
00720
001032
001115
001619
093036
093036
These are cycled under the permutation (0 1 ... 43). BTD, V= 50: 0 0 145 001630
0 0 238 081831
00 3 9 081831
0 0 429 0112633
0112633.
These are cycled under the permutation (0 1 --. 49). BTD, V= 56: 0 0 3 5 0 0 933 061927 061927
001036 071822
001240 071822
001439 01 226.
These are cycled under the permutation (0 1 --. 55). Frame, V[f] = 56120]: A 19 9 5 A 1292921 A 13131 6 A 12525 5 A 12 16 33 A 1 2 16 33 A 1 4 14 17 A 1 4 14 17 A 1 1 8 10 A l 1 8 10 A 1 0 12 30 A 1 0 621 X 0 0 1
with short block X 1 13 25 cycled once. These are cycled under the permutation (0 1 --- 35)(A1A 2 .--AIs)(X Y).
0124
D. Donovan / More balanced ternary designs
125
BTD, V= 62: 0 0 132 061927
00 320 061927
0 0 539 0102426
0 0 937 0102426
001229 071822
0 7 1822.
These are cycled under the permutation (0 1 .-- 61). BTD, V= 68: 0 0 130 061927
0 0 351 0102426
0 01237 0102426
002332 071822
002833 071822
0 6 1927 0 5 25 34.
These are cycled under the permutation (0 1 -.- 67). Frame, V[f] = 74[26]: A10011 D10720 G13630
A1 1 1 5 O10 720 G141322
B1 0 0 2 E 1 01022 G15 826
B1 1 1 2 E 1 02638 Gll 917
C 1 0 5 19
F 1 0623 X 0015.
C 1 0 5 19 F 1 1 26 43
with short block X 1 17 33 cycled once. These are cycled under the permutation (0 1 --- 47)(A1A2A3)--" (F1F2F3)(G1G2 --" G6)(X Y). Frame, VLf] = 86[26]: 0 5 B1 1 E1 0 G 11
1324 1 30 7 28 11 21
0
5 C10 El 0 G1 1
1324 2 14 32 53 4 10
A1 C1 Fl Gl
1 1 17 0 46 58 0 1 26 11 14 20
A10017 B1 2323 0 D 1 0 4 22 D 1 0 38 56 F 1 0 1 26 G I 0 15 30 X 0 0 27.
with short block X 1 21 41 cycled once. These are cycled under the permutation (0 1 -.- 59)(AIA2A3)---(F1F2Fa)(GIG2G3G4GsG6)(X Y). Frame, V[f] = 92126]: 0 9 BI0 Dl 0 G10
12 30 0 9 1 2 3 0 0 1 B12 216 5 28 E l 0 10 29 631 G 1 2 3 7 6 2
0 133353 C 1 01127 E l 0 10 29 H 1 0 224
A10 0 4 Cl 1 4 0 5 6 F 1 0 15 32 H12 428
A 12 D10 F1 1 X 0
210 528 35 52 0 7.
with short block X 1 23 44 cycled once. These are cycled under the permutation (0 1 .-. 65)(A1AEA3)(BIB2B3)-.-(H1HEH3)(X Y). Whilst cOnstructing the following balanced ternary design we use a slight variation o f Lemma 1 and Theorem 2. Hanani in his original paper used group divisible designs with a number of different group sizes. Since there is only one situation where we require a group divisible design with several groups of different sizes we
126
D. Donovan / More balanced ternary designs
simply give the group divisible design and refer the reader to Hanani (1975), Lemma 2.25. BTD, V= 140: GD(4,2,S; 132)
where S denotes the multiset {24,24,24,24, 18, 18},
BTD(26, 130; 10, 5, 20; 4, 2), Frame(26[8], 5, 4, 2) and Frame(32[8], 5, 4, 2). BTD, V= 212: GD(4, 2, {36, 12"}; 192), BTD(32, 192; 14, 5, 24; 4, 2) and Frame(56[20], 5, 4, 2). Balanced ternary designs with K = 4 , A = 2, and ~2=6 and V as follows BTD, V= 31: 00117
00220
00321
00423
00524
00622.
These are cycled under the permutation (0 1 -.. 30). Frame, V[f] = 31 [4]: 00115
00222
00310
00821
Ai004
A12213.
with short block X 0 9 18 cycled twice. These are cycled under the permutation (0 1 .-. 26)(AIA2A3), the dement X is fixed. Frame, V[f] = 3117]: 0016
0029
A1114
A1884
A10011
A115155
with short blocks 0 5 12 17 cycled once and X 0 8 16 cycled twice. These are cycled under the permutation (0 1 ---23)(AIA2A3A4AsA6), the dement X is fixed. Frame, V[f] = 31 [10]: 06912
Al001
A1227
A1335
A14417
BI0010
B1551
with short block X 0 7 14 cycled twice. These are cycled under the permutation (0 1 ---20)(A1AEA3A4A5A~)(B1B2B3), the dement X is fixed. BTD, V= 34: 0013
0057
00614
00922
001018
001115
with short block 0 4 17 21 cycled once. These are cycled under the permutation (0 1 --- 33). Frame, V[f] = 34110]:
127
D. Donovan / More balanced ternary designs
00115
A1002
A1115
B19920
B1 1 1 1 1 6
B1 4 4 1
B1 0 7 1 4
with short blocks 0 6 12 18 cycled once and X 0 8 16 cycled twice. These are cycled under permutation (0 1--. 23)(A1A2A3)(B1B2B3B4BsB6), element X is fixed.
the
BTD, V= 37: 0013
0057
00614
001115
001221
001317
081827.
These are cycled under the permutation (0 1 .-- 36). Frame, VLF] = 3711]: 00117
00210
0037
00513
00622
001115
with short blocks 0 9 18 27 cycled once and X 0 12 24 cycled twice. These are cycled under the permutation (0 1 --- 35), the element X is fixed. Frame, VLF] = 37110]: 00310
00512
A1001
A 1551
B1 0 0 1 1
B1 1 1 1 4
C1028
C101925
with short block X 0 9 18 cycled twice. These are cycled under the permutation (0 1 -.- 26)(A1A2A3)(BIB2B3)(C1C2C3), the element X is fixed. Frame, VLf] = 37113]: A1224
A 112121
B1 0 0 5
B1 1 1 5
C1009
C1223
C11411
C11411
with short blocks 0 6 12 18 cycled once and X 0 8 16 cycled twice. These are cycled under the permutation (0 1
--- 23)(AIA2A3)(B1B2B3)(C1C2C3C4CsC6)
,
the element X is fixed. BTD, V= 40: 0013
0057
00614
00925
001119
001222
041727
with short block 0 4 20 24 cycled once. These are cycled under the permutation (0 1 --- 39). BTD, V= 43: 0013
0057
00414
001119
001323
001725
061527
061527.
These are cycled under the permutation (0 1 -.- 42).
Frame, V[f] = 43 [7]: 0023
00728
011122
061622
Al005
A l l 15
B l l 114
B10017.
D. Donovan / More balanced ternary designs
128
with short blocks 0 9 18 27 cycled once and X 0 12 24 cycled twice. These are cycled under the permutation (0 1 -.- 35)(AIA2A3)(B1B2B3), the element X is fixed. BTD, V= 46: 0023
00417
00619
00828
001034
001131
0 7 1621
0 7 1621
with short block 0 1 23 24 cycled once. These are cycled under the permutation (0 1 --- 45). Frame, V[f] = 46110]: 0 01 7 A 11 1 17
0 21015 B 1 20 20 9
0 21015 Al0014 B 1 34 34 17 B l 1 1 5
B 1 03 6
with short blocks 0 9 18 27 cycled once and X 0 12 24 cycled twice. These are cycled under the permutation (0 1 --- 35)(AIA2A3)(BIB2B3B4BsB6), the element X is fixed. BTD, V= 49: 00 1 4 001729
00
6
8
001830
001036 071621
001134 071621
0222446.
These are cycled under the permutation (0 1 ... 48). Frame, V[f] =49[131: 0 61419 B I 1 111
0 61419 C 14 4 5
A 10016 C12217
A1224 C1037
B1 0 0 1 1 C103 7
with short blocks 0 9 18 27 cycled once and X 0 12 24 cycled twice. These are cycled under the permutation (0 1 .-- 35)(AIA2A3)(B1BEB3)(CICEC3C4CsC6) , the element X is fixed. BTD, V= 52: 0 0 210 001237
0 0 323 001718
0 0 639 071621
001130 071621
0242832
with short block 0 1 26 27 cycled once. These are cycled under the permutation (0 1 -.. 51). BTD, V= 55: 00 1 4 001922
0 0 635 071621
0 0 837 071621
0 01223 0102527
0 01324 0102527.
These are cycled under the permutation (0 1 -.- 54).
129
D. Donovan / More balanced ternary designs
Frame, V[f] = 55119]: A 1002 D1881
A 1 1010 2 D 1 5 528
B 1 0 1 11 D17711
B 1 0 1 11 D12222 2
C 101419 D101530
C 1 01419 DI0 3 6
with short blocks 0 9 18 27 cycled once and X 0 12 24 cycled twice. These are cycled under the permutation (0 1 ..- 35)(AIA2Aa)(BIB2B3)(CIC2C3)(D1D2DaD4DsD6D7DsD9), the element X is fixed. BTD, V= 58: 00 1 4 002230
00 640 071621
001135 071621
0 01220 0102527
0 01339 0102527
with short block 0 3 29 32 cycled once. These are cycled under the permutation (0 1 --- 57). BTD, V=61: 0 0 1 13 001941
00 335 071621
00 643 071621
0 0 830 0102527
0 01123 0102527
042832.
These are cycled under the permutation (0 1 --- 60). BTD, V= 64: 00 129 002234
00 323 071621
00 440 071621
0 01119 0102527
0 01331 0102527
082026
with short block 0 6 32 38 cycled once. These are cycled under the permutation (0 1 --- 63). BTD, V= 67: 00 1 4 071621
001824 071621
0 02235 0102527
0 02329 0102527
0 02639 0111931
0 03033 0111931.
These are cycled under the permutation (0 1 .-. 66). BTD, V= 70: 00 123 071621
00 332 071621
0 0 426 0102527
0 01337 0102527
0 01842 0111931
0 03036 0111931
with short block 0 6 35 41 cycled once. These are cycled under the permutation (0 1 -.- 69). F r a m e , V[f] = 70122]: 0 18 21 27 Cl 1 8 9 D 1 2 7 17
A l 0 0 20 C l 18 9 E l 0 0 23
A 1 1 1 20 D 14415 E1 1 3 5
B 1 0 0 14 D l0 6 9 E 1 4 22 26
B l 19 19 2 D 1 2 7 17 E l 1 14 27
D. Donovan / More balanced ternary designs
130
with short block 0 12 24 36 cycled once and X 0 16 32 cycled twice. These are cycled under the permutation (0 1 -.. 47)(A1AEA3)(B IBEB3)(C 1C2C3 )(D 1DED3D4DsD6)(21E2E3E4E5E6 ), the element X is fixed. BTD, V= 73: 0 0 124 071621
0 0 441 071621
0 0 645 0102527
0 01844 0102527
0 02235 0111931
0 03033 0111931
0133639.
0 02352 0111931
0262932
These are cycled under the permutation (0 1 --- 72). BTD, V= 76: 0 0 127 071621
00 236 071621
0 0 446 0102861
0 01335 0102861
0 01739 0111931
with short block 0 6 38 44 cycled once. These are cycled under the permutation (0 1 .-- 75). BTD, V= 79: 0 0 1 30 0 03541 011 1931
0 0 437 0 7 16 21 011 1931
0 01847 0 7 16 21 0133639
0 02255 0102527 0133639.
0 02834 0102527
These are cycled under the permutation (0 1 --. 78). BTD, V= 82: 0 0 138 0 03440 0111931
0 0 449 0 71621 0111931
0 01853 0 71621 0133639
0 02250 0102527 0133639
with short block 0 6 41 47 cycled once. These are cycled under the permutation (0 1 --.
0 02452 0102527
81).
BTD, V= 85: 0 0 1 30 0 03247 011 1931
0 0 445 0 71621 011 1931
0 0 650 0 71621 0133639
0 0 17 42 0102861 0133639
These are cycled under the permutation (0 1 -.-
0 0 2 2 37 0102861 0 22729.
84).
BTD, V= 88: 0 0 148 0 03045 011 1931
0 0 653 0 7 16 21 011 1931
0 01732 0 7 16 21 0133639
with short block 0 2 44 46 cycled once.
0 02224 0102861 0133639
0 0 2 5 59 0 102861 0 43842
D. Donovan / More balanced ternary designs
131
These are cycled under the permutation (0 1 ... 87). Frame, V[f] = 91 [31 ]: Al
0 C 1 14 Fl 0 H1 0
017 14 1 826 723
A 11 1 2 6 D 1 0 2 29 F103452 Il 3 615
B 1 0 014 D 1 0 31 58 G 1 0 122 I l 4 7 16
B l 1 1 20 E 1 0 4 28 G103859 I 1 0 5 11
C l 0 010 E l 0 32 56 HI0 723 11 1 5 0 5 6
with short blocks 0 15 30 45 cycled once and X 0 20 40 cycled twice. These are cycled under the permutation (0 1 ... 59)(A1A2A3)(B1B2B3)--. (H1H2H3)(IlI21314IsI6), the element X is fixed. BTD, V= 94: GD(4, 2, 21; 84), BTD(31, 186; 12, 6, 24; 4, 2) and Frame(31 [ 10], 6, 4, 2). Frame, V[f] = 94122]: 0 1 7 8 A l0 023 C 101431 E 1 1 127
0 3 Al 1 D 14 El 2
16 43 120 415 4 6
0 82150 B1 0 044 D l0 6 9 E 1 53539
0 B1 DI E1
12 33 45 37 37 2 2 7 17 02040
0 16 25 50 C l 01431 D 1 2 7 17
with short blocks 0 18 36 54 cycled once and X 0 24 48 cycled twice. These are cycled under the permutation (0 1 --- 71)(A1A2A3)(BIB2B3)(CIC2C3)(D1DED3D4DsD6)(EIE2E3E4EsE6), the element X is fixed. Frame, VLf] = 97125]: 0
1 7 8
0
3 1643
A l 37 37 :E B 1 0 14 31 C 1 01233 C l 13446 D 1 2 7 17 E 1 1 1 27
0
82150
B 1 0 14 31 D 14 415 El 2 4 6
0
16 25 50
C 1 5 5 49 D1 0 6 9 E l 5 35 39
A 10
023
C 1 2 2 21 D 12 7 17 E 1 02040
with short blocks 0 18 36 54 cycled once and X 0 24 28 cycled twice. These are cycled under the permutation (0 1 ... 71)(A1A2A3)(B1B2B3)(C1C2C3C4CsC6)(D1D2D3D4DsD6)(E1E2E3E4E5E6), the element X is fixed. Frame, V[f] = 97[34] A 10 020 D 101123 G 10 428 11 0 5 22
A 1 1 120 D 104052 G 103559 JI 1 7 10
Bl 0 014 E l 01025 H l0 229 Jl 3 9 12
B l 1 1 17 E 1 03853 H 103461 Jl 4 5 35
C l0 Fl 0 11 0 Jl 0
07 826 522 32 62
C 1 14 14 1 F l 03755
D. Donovan / More balanced ternary designs
132
with short block X 0 21 42 cycled twice. These are cycled under the permutation (0 1 --- 62)(A1A2A3)(BIB2B3)'--(IlI213)(J1J2J3J4JsJ6), the element X is fixed. Frame, VLf] = 100131]: 0
61533
C l 0 0 19 F 1 0 14 26 11 0 2 3 2
0 6 15 33 C12323 1 F l 04355 I l 0 37 67
A~ 0 Dl0 G l0 Jl 0
0 1 420 728 31 34
A 1 11 11 1 D 1 0 420 G 1 04162 J1 0 35 38
with short block X 0 23 46 cycled twice. These are cycled under the permutation (0 1 the element X is fixed.
B 1 0 0 11 E1 0 8 2 5 H 10529
B 1 1 1 14 El 0 8 2 5 H 104064
... 68)(AIA2A3)(B1B2B3)-.-(J1J2J3),
BTD, V= 112: GD(4, 2, 27; 108), BTD(31, 186; 12, 6, 24; 4, 2) and Frame(31 [4], 6, 4, 2). BTD, V= 115: GD(4, 2, 21; 105), BTD(31, 186; 12, 6, 24; 4, 2) and Frame(31 [101, 6, 4, 2). BTD, V= 1 18: GD(4, 2, 27; 108), BTD(37, 259; 16, 6, 28; 4, 2) and Frame(37[10], 6, 4, 2). Frame, V[f] = 121134]: 0 12 A 1 19 D1 0 F1 0 Il 0
15 36 19 2 7 32 1434 8 26
0 Bl D1 Gl I1
12 0 0 0 0
15 36 028 7 32 l 43 61 79
0 2 639 B1 2 240 E I 0 16 35 G 104486 J1 0 11 41
0 2 639 C10 0 5 E l 0 16 35 H 102231 Jl 0 46 76
A 10 023 C 1 1 1 11 F l 0 14 34 H 105565 K 1 0 13 40
K 1 0 47 74
with short block X 0 29 58 cycled twice. These are cycled under the permutation (0 1 -.- 86)(AIA2A3)(B1B2B3)-.-(KIK2K3) , the element X is fixed. BTD, V= 139: GD(4, 2, 27; 135), BTD(31, 186; 12, 6, 24; 4, 2) and Frame(31 [4], 6, 4, 2). BTD, V= 142: GD(4, 2, {27, 24" }; 132), BTD(34, 221; 14, 6, 26; 4, 2) and Frame(37 [ 10], 6, 4, 2). BTD, V= 166: GD(4, 2, 27; 162), BTD(31; 186; 12, 6, 24; 4, 2) and Frame(31 [4], 6, 4, 2).
D. Donovan / M o r e balanced ternary designs
133
BTD, V= 169: GD(4, 2, { 27, 24" }; 159), BTD(34, 221; 14, 6, 26; 4, 2) and Frame(37110], 6, 4, 2). BTD, V=221: GD(4, 2, { 36, 30"}; 210), BTD(31, 186; 12, 6, 24; 4, 2) and Frame(37[1 ], 6, 4, 2). BTD, V= 214: GD(4, 2, 48; 192), BTD(70, 875; 38, 6, 50; 4, 2) and Frame(70[22], 6, 4, 2). BTD, V= 238: GD(4, 2, {48, 24" }; 216), BTD(46, 391; 22, 6, 34; 4, 2) and Frame(70[22], 6, 4, 2). BTD, V= 241: GD(4, 2, {48, 27" }; 219), BTD(49, 441; 24, 6, 36; 4, 2) and Frame(70[22], 6, 4, 2).
Acknowledgement I wish to thank my supervisor Dr. Elizabeth J. BiUington for suggesting the problem and for her help in the preparation of the manuscript.
References Assaf, A., A. Hartman and E. Mendelsohn (1985). Multiset designs - designs having blocks with repeated elements. Canad. Numer. 48, 7-24. Brouwer, A.E. (1979). Optimal packings of K4's into a K n. J. Combin. Theory Ser. A 26, 278-297. Brouwer, A.E., A. Schrijver and H. Hanani (1977). Group divisible designs with block size four. Discrete Math. 20, 1-10. Donovan, D. (1986a). A family of balanced ternary designs with block size four. Bull. Austral. Math. Soc. 33, 321-327. Donovan, D. (1986b). Balanced ternary designs with block size four. Ars Combin. 21A, 81-88. Hanani, H. (1961). The existence and construction of balanced incomplete block designs. Ann. Math. Statist. 32, 361-386. Hanani, H. (1975). Balanced incomplete block designs and related designs. Discrete Math. 11,255-369. Robinson, P.J. (1984). Computer program for the construction of cyclic balanced ternary designs. Private communication. Tocher, K.D. (1952). The design and analysis of block experiments. J. Roy. Statist. Soc. Ser. B 14, 45-100.