Department of Statistics,
Birsa Agricultural University,
E-mail: [email protected]
Home-page: www.geocities.com/kishsinha2000/index.html
(Curriculum-vitae, nearly perfect codes)
For various definitions, see, Clatworthy (1973), Raghavarao (1971),
Dey (1986).
Here, we shall present catalogue of new group divisible designs and new resolvable designs.
No. v r k b m n l1 l2 E* Source**
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1 8 4 4 8 2 4 2 3 - B[2004]
2 12 4 6 8 3 4 2 3 - B[2004]
3 12 6 6 12 3 4 2 3 - B[2004]
4 16 9 3 48 8 2 4 1 0.69 BP[1980]
5 18 10 3 60 9 2 4 1 0.69 F[1976]
6 12 7 4 21 6 2 1 2 0.82 F[1976]
7 12 7 4 21 2 6 3 1 0.79 JT[1977]
8 12 8 4 24 4 3 3 2 0.82 JT[1977]
9 12 9 4 27 2 6 3 2 0.82 JT[1977]
10 12 10 4 30 6 2 0 3 0.81 F[1976]
11 14 10 4 35 7 2 6 2 0.80 F[1976]
12 16 6 4 24 8 2 4 1 0.78 BP[1980]
13 18 10 4 45 6 3 0 2 0.80 F[1976]
14 20 8 4 40 10 2 6 1 0.76 BP[1980]
15 22 8 4 44 11 2 2 4 0.77 F[1976]
16 24 9 4 54 12 2 5 1 0.77 F[1976]
17 26 10 4 65 13 2 6 1 0.76 F[1976]
18 14 10 5 28 7 2 4 3 0.86 JT[1977]
19 15 8 5 24 3 5 3 2 0.86 JT[1977]
20 15 8 5 24 5 3 4 2 0.85 JT[1977]
21 15 10 5 30 3 5 5 2 0.84 S[1989]
22 22 10 5 44 11 2 0 2 0.84 F[1976]
23 12 7 6 14 6 2 5 3 0.91 JT[1977]
24 12 9 6 18 6 2 5 4 0.91 F[1976]
25 12 9 6 18 3 4 7 3 0.89 BP[1982]
26 12 10 6 20 3 4 6 4 0.91 JT[1977]
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No. v r k b m n l1 l2 E* Source**
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27 16 9 6 24 4 4 7 2 0.86 S[1989]
28 12 7 7 12 3 4 6 3 0.92 BP[1982]
30 16 7 7 16 4 4 2 3 0.91 JT[1977]
31 16 7 7 16 8 2 0 3 0.91 D[1977]
32 21 7 7 21 7 3 3 2 0.90 F[1976]
33 24 7 7 24 8 3 0 2 0.89 F[1976]
34 35 7 7 35 7 5 3 1 0.87 F[1976]
35 45 7 7 45 15 3 0 1 0.88 DR[1990]
36 42 8 8 42 7 6 4 1 0.88 F[1976]
37 16 9 9 16 4 4 4 5 0.95 JT[1977]
38 18 10 9 20 3 6 4 5 0.79 JT[1977]
39 20 9 9 20 4 5 3 4 0.94 JT[1977]
40 20 9 9 20 10 2 0 4 0.93 D[1977]
41 24 9 9 24 6 4 4 3 0.93 S[1987]
42 38 9 9 38 19 2 0 2 0.91 DR[1990]
43 40 9 9 40 10 4 0 2 0.91 DN[1985]
44 49 9 9 49 7 7 5 1 0.89 F[1976]
45 21 10 10 21 7 3 9 4 0.94 F[1976]
46 21 10 10 21 3 7 8 3 0.93 BP[1982]
47 24 10 10 24 8 3 3 4 0.94 S[1987]
48 28 10 10 28 7 4 6 3 0.93 F[1976]
49 56 10 10 56 7 8 6 1 0.89 F[1976]
50 35 10 7 50 7 5 0 2 0.88 GD[1995]
51 40 10 8 50 8 5 0 2 0.89 GD [1995]
52 45 10 9 50 9 5 0 2 0.91 GD [1995 ]
53 50 10 10 50 10 5 0 2 0.92 GD [1995]
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*E stands for efficiency factor.
** The abbreviations B,BP, D, DN, DR, F, GD, JT and S stand for Bagchi,Bhagwandas and Parihar, Dey, Dey and Nigam, De and Roy, Freeman, Ghosh and Divecha, John and Turner, and Sinha respectively.
GD designs from Clatworthy's TablesGD designs from Clatworthy's Tables
(i) Semi-regular GD designs
No v r k b m n l1 l2 cyclic solution*
SR1 4 2 2 4 2 2 0 1 (1 2) mod 4
SR6 6 3 2 9 2 3 0 1
SR9 8 4 2 16 2 4 0 1
SR11 10 5 2 25 2 5 0 1
SR13 12 6 2 36 2 6 0 1
SR14 14 7 2 49 2 7 0 1
SR15 16 8 2 64 2 8 0 1
SR16 18 9 2 81 2 9 0 1
SR17 20 10 2 100 2 10 0 1
SR18 6 2 3 4 3 2 0 1
SR23 9 3 3 9 3 3 0 1
SR26 12 4 3 16 3 4 0 1
SR28 15 5 3 25 3 5 0 1
SR30 18 6 3 36 3 6 0 1
SR31 21 7 3 49 3 7 0 1
SR32 24 8 3 64 3 8 0 1
SR33 27 9 3 81 3 9 0 1
SR34 30 10 3 100 3 10 0 1
SR35 6 6 4 9 2 3 3 4
SR36 8 4 4 8 4 2 0 2
SR37 8 4 6 12 4 2 0 3
SR38 8 6 4 12 2 4 2 3
SR40 8 10 4 20 4 2 0 5
SR41 12 3 4 9 4 3 0 1
SR44 16 4 4 16 4 4 0 1
SR46 20 5 4 25 4 5 0 1
SR48 28 7 4 49 4 7 0 1
SR49 32 8 4 64 4 8 0 1
SR50 36 9 4 81 4 9 0 1
SR51 40 10 4 100 4 10 0 1
SR52 10 4 5 8 5 2 0 2
SR53 10 6 5 12 5 2 0 3
SR55 10 10 5 20 5 2 0 5
SR56 15 6 5 18 5 3 0 2
SR57 15 9 5 27 5 3 0 3
SR58 20 4 5 16 5 4 0 1
SR60 25 5 5 25 5 5 0 1
SR62 35 7 5 49 5 7 0 1
SR63 40 8 5 64 5 8 0 1
SR64 45 9 5 81 5 9 0 1
SR65 9 6 6 9 3 3 3 4
SR66 12 4 6 8 6 2 0 2
SR67 12 6 6 12 6 2 0 3
SR68 12 6 6 12 3 4 2 3
SR69 12 8 6 16 6 2 0 4
SR70 12 10 6 20 6 2 0 5
SR71 12 10 6 20 2 6 4 5
SR72 18 6 6 18 6 3 0 2
SR73 18 9 6 27 6 3 0 3
SR74 24 8 6 32 6 4 0 2
SR75 30 5 6 25 6 5 0 1
SR76 30 10 6 50 6 5 0 2
SR77 42 7 6 49 6 7 0 1
SR78 48 8 6 64 6 8 0 1
SR79 54 9 6 81 6 9 0 1
SR80 14 4 7 8 7 2 0 2
SR81 14 6 7 12 7 2 0 3
SR83 14 10 7 20 7 2 0 5
SR84 21 6 7 18 7 3 0 2
SR85 21 9 7 27 7 3 0 3
SR86 28 8 7 32 7 4 0 2
SR87 49 7 7 49 7 7 0 1
SR88 56 8 7 64 7 8 0 1
SR89 63 9 7 81 7 9 0 1
SR90 12 6 8 9 4 3 3 4
SR91 16 6 8 12 8 2 0 3
SR92 16 8 8 16 8 2 0 4
SR93 16 10 8 20 8 2 0 5
SR94 24 9 8 27 8 3 0 3
SR95 32 8 8 32 8 4 0 2
SR96 56 7 8 49 8 7 0 1
SR97 64 8 8 64 8 8 0 1
SR98 72 9 8 81 8 9 0 1
SR99 18 6 9 12 9 2 0 3
SR100 18 8 9 16 9 2 0 4
SR101 18 10 9 20 9 2 0 5
SR102 27 9 9 27 9 3 0 3
SR103 36 8 9 32 9 4 0 2
SR104 72 8 9 64 9 8 0 1
SR105 81 9 9 81 9 9 0 1
SR106 20 6 10 12 10 2 0 3
SR107 20 8 10 16 10 2 0 4
SR108 20 10 10 20 10 2 0 5
SR109 30 9 10 27 10 3 0 3
SR110 90 9 10 81 10 9 0 1
No v r k b m n l1 l2 cyclic solution*
(B) Resolvable designs
The solutions of resolvable designs T3, T17, T21, T46, M8 are found in Sinha (1978),
Sinha and Dey (1982). These designs are duplicate of non-resolvable designs given in
Clatworthy (1973). For the same designs Clatworthy (1973) reported r-resolvable solutions.
Table1.Resolvable design T3: v=10,r=6,k=2,b=30,l1 =0,l2 =2
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Replications Blocks
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I (1,8) (2,7) (3,9) (4,6) (5,10)
II (1,8) (2,10)(3,7) (4,5)(6,9)
III (1,9) (2,6)(3,7)(4,8)(5,10)
IV (1,9)(2,10)(3,5)(4,6)(7,8)
V (1,10) (2,6)(3,9)(4, 5) (7,8)
VI (1,10)(2,7)(3,5)(4,8)(6,9)
Table 2.Resolvable designT17: v=15,r=6,k=3, l1 =0,l2 =2.
Replications Blocks
I (1, 10, 15) (2,8,14) (3, 9, 11) (4, 7, 12) (5, 6, 13)
II (1,10,15) (2, 9, 1 3) (3, 8, 12) (4, 6, 14) (5, 7, 11)
III (1,11,14) (2,7,15) (3,8,12) (4,9,10) (5,6,13)
IV (1,11,14) (2,9,13) (3,6,15) (4,7,12) (5,8,10)
V (1,12,13) (2,8,14) (3,6,15) (4,9,10) (5,7,11)
VI (1,12,13) (2,7,15) (3,9,11) (4,6,14) (5,8,10)
Table 3. Resolvable designT21 : v =21,r=10,k=3,b=70,l1 =0,l2 =2 .
Replications Blocks
I (1, 2, 7)(3,5,17)(4,6,20)(8,9,16)(10, 11, 21)(12, 15, 18)(13, 14, 19)
II (1, 3, 8)(2, 4, 13)(5, 6, 21)(7, 11, 15)(9, 10, 19)(12, 14, 17)(16, 18, 20)
III (1, 2, 7)(3, 5, 17)(4, 6, 20)(8, 11, 18)(9, 10, 19)(12, 13, 16)(14, 15, 21)
IV (1, 3, 8)(2, 4, 13)(5, 6, 21)(7, 10, 14)(9, 11, 20)(12, 15, 18)(16, 17, 19)
V (1, 4, 9)(2, 5, 14)(3, 6, 18)(7, 8, 12)(10, 11, 21)(13, 15, 20)(16, 17, 19)
VI (1, 4, 9)(2, 5, 14)(3, 6, 18)(7, 11, 15)(8, 10, 17)(12, 13, 16)(19, 20, 21)
VII (1, 5, 10)(2, 6, 15)(3, 4, 16)(7, 8, 12)(9, 11, 20)(13, 14, 19)(17, 18 21)
VIII (1, 5, 10)(2, 6, 15)(3, 4, 16)(7, 9, 13)(8, 11, 18)(12, 14, 17)(19, 20,21)
IX (1, 6, 11)(2, 3, 12)(4, 5, 19)(7, 9, 13)(8, 10, 17)(14, 15, 21)(16, 18, 20)
X (1, 6, 11)(2, 3, 12)(4, 5, 19)(7, 10, 14)(8, 9, 16)(13, 15, 20)(17, 18, 21)
Bagchi, S. (2004) Construction of group divisible designs and rectangular designs from
resolvable and almost resolvable balanced incomplete block designs,
J.Statist.Plann.Inference, 119, 401-410.
Bhagwandas and Parihar, J.S., (1980) Some new group divisible designs,
J.Statist.Plan.Inference , 4,321-323.
Bhagwandas and Parihar, J.S. (1982) Some new series of regular group divisible designs
Commun.Statist.Th.Method 11,761-768
Clatworthy, W.H., Cameron, J.M. and Speckman, J.A. (1973) Tables of two-associate –
class partially balanced designs ,Natl.Bur.Stand.(U.S),Appl.Math.Ser.63.
De, A.K. and Roy, B.K., (1990) Computer construction of some group divisible
designs,Sankhya(b)52,82-92
Dey, A. (1977) Construction of regular group divisible designs, Biometrika 64,647-649
Dey, A. and Nigam, A.K., (1985) Construction of group divisible designs,
J.Indian Soc.Agric.Statist.,37,163-166
Dey, A. (1986) Theory of block designs, Wiley Eastern Ltd., New Delhi
Freeman, G.H. (1976) A cyclic method of constructing regular group divisible
Incomplete block designs, Biometrika 63,555-558
Ghosh, D.K. and Divecha, J. (1995) Some new semi-regular GD designs,
Sankhya (B), 57,453-455
John, J.A. and Turner, G. (1977) Some new group divisible designs,
J.Statist.Plann.Inferernce 1,103-107
Raghavarao, D. (1971) Constructions and combinatorial problems in design of
experiments, John Wiley & sons, New York.
Sinha, K. (1978) A resolvable triangular partially balanced incomplete
block designs,Biometerika 65(3)665.
Sinha, K.and Dey, A. (1982) On resolvable PBIB designs, J.Statist.Plann.Inference,
6, (2) 179-182
Sinha, K. (1987) A method of construction of regular group divisible designs,
Biometrika 74,443-4.
Sinha, K. (1989) A method of constructing PBIB designs, J.Indian Soc.Agric.Statist.
41,313-315.
Sinha, K. (1991) A list of new group divisible designs, J.Res.Natl.Inst. Standards
& Technology, USA, 96 (1991) 613 - 615.
(C) E-optimal binary block designs
(i)
E-optimal nested group divisible designs:
A nested GD design with v (=2pt) treatment divided into t sets of 2p treatments
each and b blocks of size k, satisfying the following:
(1) each of t sets consists of 2 groups of size p (³2); and any two treatments
(i)in the same group and same set are called first associates, ( ii) in the different group and same set are called second associates,(iii) otherwise ,called third associates.
(2) Each treatment is repeated r times.
(3) Any two treatments which are i-th associates occur together in li blocks
for i=1, 2, 3, where l1=u-1, l2=u+1,l 3=u for a positive integer u .
Sinha and Shah (1988) established E-optimality of 3-concurrence most balanced
designs. They describe the case with t-groups where the i-th group has 2pi treatments.
Sinha and Kageyama (1992) and we deal with the case where pi = p for all i.
(ii) E-optimal rectangular designs:
Rectangular designs are 3-associate PBIB designs based on a rectangular association scheme of v=mn treatments arranged in a rectangle of m rows and n columns. Cheng and Constantine (1986) and Bagchi and Cheng (1993) proved that a rectangular design with m =2 and l1= l2=l 3 -1 is E-optimal over a class of designs with block size two.
It is clear that the E-optimality property for m=2 and l1= l2 is also preserved for n =2
and l1= l2 , renaming of the first associates and the second associates.
E-optimal block (nested group divisible) designs with r, k £ 10.
No v b r k l1 l2 l3 Source
1 8 32 8 2 0 2 1 no.1,table2.1*
2 18 60 10 3 0 2 1 no.6,table2.1*
3 8 10 5 4 1 3 2 no.1,table 2.2*
4 18 21 7 6 1 3 2 no.3,table 2.2*
5 32 36 9 8 1 3 2 no.4,table 2.2 *
6 12 24 6 3 0 2 1 Sinha(1994)
7 12 10 5 6 1 3 2 Sinha(1999)
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* Sinha and Kageyama (1992)
E-optimal block (rectangular) designs r, k £ 10
________________________________________________________________________
No v b r k l1 l2 l3 Source
1. 6 6 2 2 0 0 1 no.1**
2. 8 12 3 2 0 0 1 no.16 **
3. 10 20 4 2 0 0 1 no.20**
4. 12 30 5 2 0 0 1 no. 25**
5. 14 42 6 2 0 0 1 no.35**
6. 16 56 7 2 0 0 1 no.42**
7. 18 72 8 2 0 0 1 no.48 **
8. 20 90 9 2 0 0 1 no.52**
9. 10 20 8 4 2 4 3 B[2004]
**Sinha, Kageyama, Singh (1993), B [2004] refers to Bagchi [2004].
Bagchi, S., Cheng, C.S. (1993) Some optimal designs of block size two.
J.Statist.Plann.Infer. 37,245-253.
Bagchi, S. (2004) Construction of group divisible designs and rectangular designs from
resolvable and almost resolvable balanced incomplete block designs,
J.Statist.Plann.Infer., 119, 401-410.
Cheng, C.S., Constantine, G.M. (1986) On the efficiency of regular generalized line
graph designs,J.Statist.Plann.Infer.,15,1-10
Sinha, B.K. and Shah, K.R. (1988). Optimality of aspects of 3-concurrence most balanced designs,
J.Statist.Plann.Infer., 20,229-236.
Sinha, K., Kageyama, S., Singh, M.K. (1993) Construction of rectangular designs,
Statistics, 25,63-70.
Sinha, K., Kageyama, S. (1992) Constructions of some E-optimal 3-concurrence most
balanced designs, J.Statist.plann.Infer.,31,127-132.
Sinha, K. (1994) On E-optimal nested group divisible designs,
Sankhya, 56B, 3,447-8.
Sinha, K. (1999) Constructions of nested group divisible designs and rectangular designs,
Statist.Inference. & Design of Experiments, eds., Dixit and Satam,Narosha Publishing
Co., New Delhi, 110-116.