Kishore Sinha, Dept. Statistics, Birsa Agril. University, Ranchi - 834006
(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.
A. Group divisible designs
Catalogue of new group divisible designs reported
after the publication of Clatworthy (1973) are presented here. This is an update
of Sinha (1991).
Table 1.New Group divisible
designs with r, k £ 10.
No.
v r k b m n
l1
l2
E*
Source**
________________________________________________________________________
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]
________________________________________________________________________
Table 1(cont.…). New Group
divisible designs with r, k £ 10
No.
v r k b m n
l1
l2
E*
Source**
________________________________________________________________________
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]
________________________________________________________________________
*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 Tables
TABLE 2.1
GD 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*
R1 4 4 2 8 2 2 2 1
R2 4 5 2 10 2 2 3 1
R3 4 5 2 10 2 2 1 2
R4 4 6 2 12 2 2 4 1
R5 4 7 2 14 2 2 5 1
R6 4 7 2 14 2 2 3 2
R7 4 7 2 14 2 2 1 3
R8 4 8 2 16 2 2 6 1
R10 4 8 2 16 2 2 2 3
R11 4 9 2 18 2 2 7 1
R12 4 9 2 18 2 2 5 2
R13 4 9 2 18 2 2 1 4
R14 4 10 2 20 2 2 8 1
R16 4 10 2 20 2 2 4 3
R18 6 4 2 12 3 2 0 1
R19 6 6 2 18 3 2 2 1
R20 6 7 2 21 2 3 2 1
R21 6 7 2 21 3 2 3 1
R22 6 8 2 24 3 2 4 1
R24 6 8 2 24 2 3 1 2
R25 6 9 2 27 2 3 3 1
R26 6 9 2 27 3 2 5 1
R27 6 9 2 27 3 2 1 2
R28 6 10 2 30 3 2 6 1
R29 8 6 2 24 4 2 0 1
R30 8 8 2 32 4 2 2 1
R31 8 9 2 36 4 2 3 1
R32 8 10 2 40 2 4 2 1
R33 8 10 2 40 4 2 4 1
R34 9 6 2 27 3 3 0 1
R35 9 10 2 45 3 3 2 1
R36 10 8 2 40 5 2 0 1
R37 10 10 2 50 5 2 2 1
R38 12 8 2 48 3 4 0 1
R39 12 9 2 54 4 3 0 1
R40 12 10 2 60 6 2 0 1
R41 15 10 2 75 3 5 0 1
R42 6 3 3 6 3 2 2 1
R43 6 6 3 12 2 3 3 2
R45 6 7 3 14 2 3 4 2
R46 6 7 3 14 3 2 2 3
R47 6 8 3 16 2 3 5 2
R48 6 8 3 16 3 2 4 3
R49 6 9 3 18 2 3 6 2
R50 6 9 3 18 3 2 6 3
R51 6 9 3 18 3 2 6 3
R52 6 9 3 18 2 3 3 4
R53 6 10 3 20 2 3 7 2
R54 8 3 3 8 4 2 0 1 (124)mod8
R56 8 9 3 24 4 2 6 2
R57 8 9 3 24 4 2 0 3
R58 8 9 3 24 2 4 2 3
R59 9 5 3 15 3 3 2 1
R60 9 6 3 18 3 3 3 1
R61 9 7 3 21 3 3 4 1
R62 9 7 3 21 3 3 1 2
R63 9 8 3 24 3 3 5 1
R64 9 9 3 27 3 3 6 1
R65 9 9 3 27 3 3 3 2
R66 9 10 3 30 3 3 7 1
R68 9 10 3 l 30 3 3 1 3
R69 10 6 3 20 5 2 4 1
R70 12 5 3 20 6 2 0 1
R71 12 6 3 24 6 2 2 1
R72 12 7 3 28 3 4 2 1
R73 12 7 3 28 6 2 4 1
R74 12 8 3 32 2 6 2 1
R75 12 9 3 36 4 3 0 2
R76 12 10 3 40 3 4 4 1
R78 12 10 3 40 4 3 1 2
R79 14 6 3 28 7 2 0 1
R80 14 9 3 42 7 2 6 1
R81 15 6 3 30 5 3 0 1
R82 15 8 3 40 5 3 2 1
R83 15 9 3 45 3 5 2 1
R84 15 9 3 45 5 3 3 1
R85 15 10 3 50 5 3 4 1
R86 16 6 3 32 4 4 0 1
R87 16 9 3 48 4 4 2 1 (1513)(1211)(136) mod16
R88 18 8 3 48 9 2 0 1
R89 18 9 3 54 9 2 2 1
R90 20 9 3 60 10 2 0 1
R91 21 9 3 63 7 3 0 1
R92 24 9 3 72 4 6 0 1
R93 24 10 3 80 6 4 0 1
R94 6 4 4 6 2 3 3 2
R96 6 8 4 12 3 2 4 5
R97 8 5 4 10 4 2 3 2
R98 8 8 4 16 2 4 4 3
R99 8 8 4 16 4 2 6 3
R100 8 9 4 18 2 4 5 3
R101 8 9 4 18 4 2 3 4
R102 8 10 4 20 2 4 6 3
R104 9 4 4 9 3 3 3 1
R106 10 8 4 20 5 2 0 3 (1247)(1258)mod 9
R107 10 10 4 25 2 5 5 2
R108 10 10 4 25 5 2 6 3
R109 12 4 4 12 6 2 2 1
R111 12 10 4 30 3 4 2 3
R112 14 4 4 14 7 2 0 1
R114 15 4 4 15 5 3 0 1 (1 3 4 12) mod 15
R115 15 8 4 30 5 3 6 1
R117 15 8 4 30 3 5 1 2
R118 16 6 4 24 4 4 2 1
R119 16 7 4 28 4 4 3 1
R120 16 8 4 32 4 4 4 1
R121 16 9 4 36 4 4 5 1
R122 16 9 4 36 4 4 1 2
R123 16 10 4 40 4 4 6 1
R124 20 9 4 45 4 5 3 1
R125 24 7 4 42 8 3 0 1
R126 24 9 4 54 8 3 3 1
R127 24 10 4 60 3 8 2 1
R128 26 8 4 52 13 2 0 1
R129 27 8 4 54 9 3 0 1
R130 28 8 4 56 7 4 0 1
R131 28 10 4 70 7 4 2 1
R132 30 10 4 75 15 2 2 1
R133 8 5 5 8 2 4 4 2 ( 1 2 3 5 7 ) mod 8
R134 8 5 5 8 4 2 2 3 ( 1 3 4 5 6 ) mod 8
R135 8 10 5 16 2 4 8 4
R137 9 5 5 9 3 3 4 2 (1 3 4 6 7 ) mod 9
R139 10 5 5 10 5 2 4 2
R140 10 7 5 14 5 2 4 3
R141 10 10 5 20 2 5 5 4
R143 12 5 5 12 3 4 4 1
R144 12 5 5 12 6 2 0 2 ( 1 2 4 9 12) mod 12
R145 12 5 5 12 4 3 1 2 ( 1 2 4 6 7 ) mod12
R149 15 10 5 30 5 3 8 2
R150 15 10 5 30 5 3 2 3
R151 18 10 5 36 9 2 8 2
R152 20 10 5 40 5 4 8 1
R153 24 5 5 24 6 4 0 1
R155 25 7 5 35 5 5 2 1
R156 25 8 5 40 5 5 3 1
R157 25 9 5 45 5 5 4 1
R158 25 10 5 50 5 5 5 1
R159 35 10 5 70 5 7 2 1
R160 39 10 5 78 13 3 2 1
R161 40 9 5 72 10 4 0 1
R162 44 10 5 88 11 4 0 1
R163 45 10 5 90 9 5 0 1
R164 8 9 6 12 2 4 7 6
R165 9 10 6 15 3 3 7 6
R166 10 6 6 10 2 5 5 2
R167 12 9 6 18 3 4 7 3
R168 15 6 6 15 3 5 5 1 ( 1 2 4 7 10 13 ) mod15
R169 18 8 6 24 6 3 5 2
R170 27 6 6 27 9 3 3 1
R171 28 6 6 28 7 4 2 1
R172 9 7 7 9 3 3 6 5 ( 1 2 3 5 6 8 9 ) mod 9
R173 12 7 7 12 2 6 6 2 ( 1 2 3 5 7 9 11) mod 9
R174 12 7 7 12 3 4 6 3 ( 1 2 4 5 7 8 11)mod12
R175 12 7 7 12 6 2 2 4 ( 1 2 3 4 6 7 11 )mod 12
R176 12 7 7 12 4 3 3 4 ( 1 3 6 7 8 9 10 )mod12
R177 14 7 7 14 7 2 6 3 (12348910)mod14
R178 18 7 7 18 3 6 6 1 (1247101316)mod18
R179 20 7 7 20 4 5 3 2 (1346101418)mod20
R180 20 7 7 20 10 2 6 2 (1234111213)mod20
R181 28 10 7 40 4 7 3 2
R182 33 7 7 33 3 11 2 1
R183 48 7 7 48 8 6 0 1 (12511313638)mod48
R184 49 9 7 63 7 7 2 1
R185 49 10 7 70 7 7 3 1
R186 12 8 8 12 6 2 6 5 (14579101112)mod12
R187 14 8 8 14 2 7 7 2 (1235791113)mod14
R188 21 8 8 21 3 7 7 1 (136912151821)mod21
R189 24 8 8 24 4 6 4 2
R190 48 8 8 48 12 4 4 1
R191 63 8 8 63 9 7 0 1 (1681438484952)mod63
R192 64 10 8 80 8 8 2 1
R193 12 9 9 12 3 4 8 6 (12356891112)mod 12
R194 15 9 9 15 3 5 8 4 (124578111314)mod15
R195 16 9 9 16 2 8 8 2 (1246810121416)mod16
R196 18 9 9 18 6 3 6 4 (1247910141516)mod18
R197 18 9 9 18 9 2 8 4 (1236811121517)mod18
R198 24 9 9 24 3 8 8 1 (12471013161922)mod24
R199 26 9 9 26 13 2 0 3
R200 28 9 9 28 4 7 5 2
R201 78 9 9 78 13 6 0 1
R202 80 9 9 80 10 8 0 1 (136102244575875)mod80
R203 12 10 10 12 4 3 9 8 (1234678101112)mod12
R204 14 10 10 14 2 7 8 6 (1234678101214)mod14
R205 14 10 10 14 7 2 6 7 (146791011121314)mod14
R206 18 10 10 18 2 9 9 2 (1246810121416)mod18
R207 27 10 10 27 3 9 9 1
R208 32 10 10 32 4 8 6 2
R209 75 10 10 75 15 5 5 1
* multiples of design are not listed
(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
_______________________________________________________________________
Replications Blocks
________________________________________________________________________
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.
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
.
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)
References
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.
TABLE 2.1
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)
________________________________________________________________________
* Sinha
and Kageyama (1992)
TABLE 2.2
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].
References
Bagchi,
S., Cheng, C.S. (1993) Some optimal designs of block size two.
J.Statist.Plann.Infer. 37,245-253.
Bagchi,
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