Two new D-optimal designs (83,56,12), (83,55,12)
Descrição do Produto
Journal of Statistical Planning and Inference 15 (1987) 247-257 North-Holland
TWO NEW D-OPTIMAL
247
D E S I G N S (83,56,12), (83, 55,12)
Nikos FARMAKIS and Stratis KOUNIAS Department of Mathematics, University of Thessaloniki, Greece Received 18 December 1985 Recommended by J. Seberry
Abstract: Two new D-optimal first order designs (83,56,12), (83,55,12) are constructed. The construction method is the same as in Kounias and Farmakis (1984). A list of all the known results for n---3 mod4, n < 100 is presented. AMS Subject Classification: Primary 62K05, 62K10; Secondary 05B20. Keywords: Matrix; Construction; Blocks; Hadamard.
1. Introduction
The construction of D-optimal weighing designs for n-- 3 mod 4 has been originated by Enlich (1964). Specific construction methods have been given by Galil and Kiefer (1982), Kounias and Hadjipantelis (1983) and Kounias and Farmakis (1984). In this note we follow the method of Kounias and Farmakis (1984), using circulant matrices of block size 7. These give the new D-optimal weighing designs (83, 56, 12) and (83, 55, 12). Here (n, k , s ) denotes the D-optimal design with n observations, k factors and s blocks, i.e., o = k - s [ k / s ] blocks are of size r = [k/s] + 1 and u = s - o blocks are of size r. We have also tried circulant matrices of block size 8,9,10,11 but without new results. In the appendix a list is given with all the known results for n --- 3 mod 4, n < 100. This updates a similar list by Galil and Kiefer (1982).
2. The construction
The symmetric circulant matrices of size 7 are the following: AI=(+++++++) A5=(+-++++-
A2=(-++++++ ) A6=(--++++-
) A3=(+++--++
) A4=(++-++-+
) AT=(-+-++-+
) As=(-++--++)
0378-3758/87/$3.50 © 1987, Elsevier Science Publishers B.V. (North-Holland)
)
248
N. Farmakis, S. Kounias / Two new D-optimal designs
where + stands for +1 and - stands for - 1 . We use G' for the transpose of G and R for the matrix with back-diagonal 1 and other elements 0. Consider the matrix of the Goethals-Seidel type:
I D1
D2R D3R D4R1
~_D4R D~R -D~R -D l If D~, D 2, D 3, D 4 are block circulant matrices with m blocks, each of size 7, then for m = 1,2,3 we expect to construct corresponding designs with sample size n = 27, 55, 83. The values m = 1, 2 give no new D-optimal designs. For m = 3 we take:
Dt=(A3,A4,A5), D3 = (Z3, -A4, As),
D2=(-A3,A4,As),
D 4 = (A3, A4, - A s ) .
Now:
(i) Multiply by - 1 the 3 block rows which have - A j, j = 1, 2, 3, in the first block column of G'. (ii) Delete the 24 rows with -1 in the first column of G'. (iii) Delete the first column of G'. The resulting 60x 83 array G~ has the property that is a 60× 60 block matrix with 12 blocks of size 5. Each diagonal element equals 3, the off-diagonal elements in every block are equal to 3 and every other element equals - 1 . To construct the D-optimal design (83, 55, 12) which has 5 blocks of size 4 and 7 blocks of size 5, it is enough to delete from G; 5 rows, one from each of 5 blocks of size 5. To construct the D-optimal design (83, 56, 12) which has 4 blocks of size 4 and 8 blocks of size 5, it is enough to delete from G; 4 rows, one from each of 4 blocks of size 5.
G~G1
References Ehlich, H. (1964). Determinantenabsch~itzungen fiir binare Matrizen mit n m 3 m o d 4 . Math. Z. 84, 438--447. Galil, Z. and J. Kiefer (1980). D-optimum weighing designs. Ann. Statist. 8(6), 1293-1306. Galil, Z. and J. Kiefer (1982). Construction methods for D-optimum weighing designs when n = 3 (mod4). Ann. Statist. 10, 502-510. Kounias, S. and T. Chadjipantelis (1983). Some D-optimal weighing designs for n m 3 (mod 4). J. Statist. Plann. Inference 8, 117-127. Kounias, S. and N. Farmakis (1984). A construction for D-optimal weighing designs when n - 3 mod 4. J. Statist. Piann. Inference 10, 177-187.
N. Farmakis, S. Kounias / Two new D-optimal designs
249
Mitchell, T. (1974). C o m p u t e r construction o f ' D - o p t i m a l ' first order designs. Technometrics 16(2), 211-220. Williamson, J. (1946). Determinants whose elements are 0 and 1. Amer. Math. Monthly 53, 427-434.
Appendix In weighing designs, when the number of observations is n -= 3 mod 4, it has been proved by Ehlich (1964) that, for k factors, the information matrix with the maximum determinant is a k x k block matrix with s blocks. If we know the optimal value of s, then o = k - s [ k / s ] blocks are of size [k/s] + 1 and u = s - v blocks are of size r = [k/s] ([k/s] being the integral part of k / s ) . In Table 1 are listed the optimal value o f s for every n - 3 mod 4, n < 100 and every ½(n + 5)_< k _ n and also the values of r = [k/s], u, o and d = 2 k - n. It has been proved by Galil and Kiefer (1980), that if k < ½ ( n + 5 ) , then $opt=k and one construction of the optimal design is achieved by taking any k x ( n - 1) submatrix o f a H a d a m a r d matrix o f order n. Also in Table 1 are indicated the D-optimal designs for which the construction is known and also the method of their construction.
Table 1 Values o f the parameters o f the D-optimal weighing designs for n - 3 m o d 4, n < 100. n : n u m b e r o f observations. k : n u m b e r o f factors. s : o p t i m a l value o f the number o f blocks. r : block size. u : n u m b e r o f blocks o f size r. 0 : n u m b e r o f blocks o f size r + I.
d:2k-n. e : c a n n o t be constructed but o p t i m u m design is known. x : c a n n o t be constructed. GK :constructed by Galil and Kiefer. K H :constructed by Kounias and Hadjipantelis. K F : c o n s t r u c t e d by Kounias and Farmakis. FK :constructed by Farmakis and Kounias. M : constructed by Mitchell. W : constructed by Williamson. W h e n there is no indication the construction is unknown. n
k
s
r
u
o
d
7 7 7
6 6 7
6 5 5
1 1 1
6 4 3
0 1 2
5 5 7
GK GK e, W
II 11
8 8
8 7
1 1
8 6
0 1
5 5
GK GK
n
k
s
r
u
o
d
11 11 11 11 11 11
9 9 10 10 11 11
7 6 6 5 6 5
1 1 1 2 1 2
5 3 2 5 I 4
2 3 4 0 5 1
7 7 9 9 11 11
x GK GK M e, GK e, GK
N. Farmakis, S. Kounias / Two new D-optimal designs
250
Table 1 (continued) n
k
s
r
u
o
d
15 15 15 15 15 15 15 15 15
10 10 11 12 12 13 13 14 15
10 9 8 7 6 7 6 6 6
1 1 1 1 2 1 2 2 2
10 8 5 2 6 1 5 4 3
0 1 3 5 0 6 1 2 3
5 5 7 9 9 11 11 13 15
GK GK GK, KH GK, KH GK, KH x x x x
19 19 19 19 19 19 19 19 19 19 19 19
12 12 13 13 14 14 15 15 16 17 18 19
12 11 10 9 8 7 8 7 7 6 6 6
l l 1 1 I 2 1 2 2 2 3 3
12 10 7 5 2 7 l 6 5 1 6 5
0 l 3 4 6 0 7 1 2 5 0 1
5 5 7 7 9 9 11 11 13 15 17 19
GK GK KH
23 23 23 23 23 23 23 23 23 23 23 23 23
14 14 15 16 16 17 17 18 19 20 21 22 23
14 13 11 9 8 9 8 8 7 7 7 6 6
1 l 1 l 2 1 2 2 2 2 3 3 3
14 12 .7 2 8 l 7 6 2 1 7 2 1
0 1 4 7 0 8 1 2 5 6 0 4 5
5 5 7 9 9 11 11 13 15 17 19 21 23
27 27 27 27 27 27 27 27 27 27
16 16 17 17 18 18 19 19 20 21
16 15 13 12 l0 9 l0 9 9' 8
1 1 I 1 1 2 1 2 2 2
16 14 9 7 2 9 1 8 7 3
0 1 4 5 8 0 9 1 2 5
5 5 7 7 9 9 11 11 13 15
GK, KH GK, KH GK, KH KH KH KH x GK GK KF KF KH KH KH
X
GK GK GK GK GK GK
"
n
k
s
r
u
o
d
27 27 27 27 27 27 27
22 22 23 24 25 26 27
8 7 7 7 7 6 6
2 3 3 3 3 4 4
2 6 5 4 3 4 3
6 1 2 3 4 2 3
17 17 19 21 23 25 27
31 31 31 31 31 31 31 31 31 31 31 31 31 31 31 31 31
18 18 19 20 20 21 21 22 23 24 25 26 27 28 29 30 31
18 17 14 11 l0 11 10 10 9 8 8 7 7 7 7 6 6
1 1 1 1 2 l 2 2 2 3 3 3 3 4 4 5 5
18 16 9 2 l0 1 9 8 4 8 ~7 2 1 7 6 6 5
0 l~ 5 9 0 10 1 2 5 0 1 5 6 0 1 0 l
5 5 7 9 9 II 11 13 15 17 19 21 23 25 27 29 31
35 35 35 35 35 35 35 35 35 35 35 35 35 35 35 35 35 35 35 35
20 20 21 21 22 22 23 23 24 25 26 27 28 29 30 31 32 33 34 35
20 19 16 15 12 11 12 11 11 9 9 9 8 8 7 7 7 7 6 6
1 1 1 1 1 2 1 2 2 2 2 3 3 3 4 4 4 4 5 5
20 18 11 9 2 11 1 10 9 2 1 9 4 3 5 4 3 2 2 1
0 1 5 6 10 0 11 1 2 7 8 0 4 5 2 3 4 5 4 5
5 5 7 7 9 9 11 11 13 15 17 19 21 23 25 27 29 31 33 35
X
GK GK KH KH KH
KH KH
GK GK GK GK GK GK GK KF KF
GK GK
N. Farmakis, S. Kounias / Two new D-optimal designs
251
Table 1 (continued) n
k
s
r
u
o
d
39 39 39 39 39 39 39 39 39 39 39 39 39 39 39 39 39 39 39 39 39 39
22 22 23 24 24 25 25 26 27 28 28 29 30 31 32 33 34 35 36 37 38 39
22 21 17 13 12 13 12 11 10 10 9 9 8 8 8 8 7 7 7 7 6 6
1 1 l l 2 1 2 2 2 2 3 3 3 3 4 4 4 5 5 5 6 6
22 20 11 2 12 1 11 7 3 2 8 7 2 1 8 7 1 7 6 5 4 3
0 1 6 11 0 12 I 4 7 8 l 2 6 7 0 l 6 0 1 2 2 3
5 5 7 9 9 II 11 13 15 17 17 19 21 23 25 27 29 31 33 35 37 39
43 43 43 43 43 43 43 43 43 43 43 43 43 43 43 43 43 43 43 43 43 43 43 43
24 24 25 25 26 26 27 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43
24 23 19 18 14 13 14 13 12 11 10 10 9 9 8 8 8 7 7 7 7 7 6 6
1 1 l 1 1 2 1 2 2 2 3 3 3 3 4 4 4 5 5 5 5 5 7 7
24 22 13 11 2 13 1 12 8 4 10 9 4 3 6 5 4 5 4 3 2 1 6 5
0 1 6 7 12 0 13 1 4 7 0 1 5 6 2 3 4 2 3 4 5 6 0 1
.5 5 7 7 9 9 11 II 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43
GK GK KF KF KF
KH
x GK GK GK GK GK GK GK
x
n
k
s
r
u
o
d
47 47 47 47 47 47 47 47 47 47 47 47 47 47 47 47 47 47 47 47 47 47 47 47 47
26 26 27 28 28 29 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47
26 25 20 15 14 15 14 13 11 11 10 I0 9 9 8 8 8 8 7 7 7 7 7 7 6
1 1 1 1 2 1 2 2 2 2 3 3 3 4 4 4 4 5 5 6 6 6 6 6 7
26 24 13 2 14 1 13 9 2 1 7 6 1 9 3 2 1 8 1 7 6 5 4 3 l
0 1 7 13 0 14 l 4 9 i0 3 4 8 0 5 6 7 0 6 0 I 2 3 4 5
5 5 7 9 9 ll II 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47
GK GK KH KH KH KF KF KF KH KH
51 51 51 51 51 51 51 51 51 51 51 51 51 51 51 51 51 51 51 51 51
28 28 29 29 30 30 31 31 32 33 34 34 35 36 37 38 39 40 41 42 43
28 27 22 21 16 15 16 15 14 12 12 II 11 10 10 9 9 8 8 8 8
1 l 1 1 1 2 1 2 2 2 2 3 3 3 3 4 4 5 5 5 5
28 26 15 13 2 15 1 14 10 3 2 10 9 4 3 7 6 8 7 6 5
0 1 7 8 14 0 15 1 4 9 10 1 2 6 7 2 3 0 1 2 3
5 5 7 7 9 9 ll 11 13 15 17 17 19 21 23 25 27 29 3I 33 35
GK GK GK GK GK GK GK
x
GK GK GK GK GK GK
N. Farmakis, S. Kounias / Two new D-optimal designs
252
Table 1 (continued) n
k
s
r
u
o
d
n
k
s
r
u
o
d
51 51 51 51 51 51 51 51
44 45 46 47 48 49 50 51
7 7 7 7 7 7 7 6
6 6 6 6 6 7 7 8
5 4 3 2 1 7 6 3
2 3 4 5 6 0 1 3
37 39 41 43 45 47 49 51
55 55 55 55 55 55 55 55 55 55 55 55 55 55 55 55 55 55 55 55 55 55 55 55 55 55 55 55 55
30 30 31 32 32 33 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55
30 29 23 17 16 17 16 15 13 12 12 II 10 10 9 9 9 8 8 8 8 7 7 7 7 7 7 7 6
1 1 l 1 2 1 2 2 2 3 3 3 3 4 4 4 4 5 5 5 5 6 7 7 7 7 7 7 9
30 28 15 2 16 1 15 11 4 12 11 6 1 10 4 3 2 4 3 2 1 1 7 6 5 4 3 2 5
0 1 8 15 0 16 1 4 9 0 1 5 9 0 5 6 7 4 5 6 7 6 0 1 2 3 4 5 1
5 5 7 9 9 11 II 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55
GK GK KF KF KF
59 59 59 59 59 59 59 59 59 59 59 59 59 59 59 59 59 59 59 59 59 59 59 59
36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59
16 14 13 12 11 11 10 10 9 9 9 8 8 8 8 8 7 7 7 7 7 7 7 6
2 2 2 3 3 3 4 4 4 5 5 5 6 6 6 6 7 7 7 7 8 8 8 9
12 5 1 9 4 3 8 7 1 9 8 1 8 7 6 5 4 3 2 1 7 6 5 1
4 9 12 3 7 8 2 3 8 0 1 7 0 1 2 3 3 4 5 6 0 1 2 5
13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59
59 59 59 59 59 59 59 59
32 32 33 33 34 34 35 35
32 31 25 ' 24 18 17 18 17
1 1 1 1 1 2 1 2
32 30 17 15 2 17 1 16
0 1 8 9 16 0 17 1
5 5 7 7 9 9 11 11
GK GK GK GK GK GK GK KF
63 63 63 63 63 63 63 63 63 63 63 63 63 63 63 63 63 63 63 63 63 63
34 34 35 36 36 37 37 38 39 40 40 41 42 43 44 45 46 47 4849 50 51
34 33 26 19 18 19 18 17 14 14 13 13 12 11 11 l0 lO 9 9 9 8 8
1 1 1 1 2 1 2 2 2 2 3 3 3 3 4 4 4 5 5 5 6 6
34 32 17 2 18 1 17 13 3 2 12 11 6 1 11 5 4 7 6 5 6 5
0 1 9 17 0 18 1 4 11 12 I 2 6 10 0 5 6 2 3 4 2 3
5 5 7 9 9 ll 11 13 15 17 17 19 21 23 25 27 29 31 33 35 37 39
X
X
KF
KF KF KF
X
GK GK KH KH KH
KH KH KH
N. Farmakis, S. Kounias / Two new D-optimal designs
253
Table 1 (continued) n
k
s
r
u
o
d
n
k
s
r
u
o
d
63 63 63 63 63 63 63 63 63 63 63 63
52 53 54 55 56 57 58 59 60 61 62 63
8 8 8 7 7 7 7 7 7 7 7 7
6 6 6 7 8 8 8 8 8 8 8 9
4 3 2 1 7 6 5 4 3 2 1 7
4 5 6 6 0 1 2 3 4 5 6 0
41 43 45 47 49 51 53 55 57 59 61 63
67 67
66 67
7 7
9 9
4 3
3 4
65 67
36 36 37 37 38 38 39 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 52 63 64 65
36 35 28 27 20 19 20 19 18 15 14 14 12 12 11 11 10 10 10 9 9 9 8 8 8 8 8 7 7 7 7 7 7 7
1 1 1 l 1 2 1 2 2 2 3 3 3 3 4 4 4 4 5 5 5 5 6 6 7 7 7 8 8 8 8 9 9 9
36 34 19 17 2 19 1 18 14 4 14 13 4 3 9 8 2 1 10 3 2 1 2 1 8 7 6 4 3 2 1 7 6 5
0 1 9 l0 18 0 19 1 4 11 0 l 8 9 2 3 8 9 0 6 7 8 6 7 0 1 2 3 4 5 6 0 1 2
5 5 7 7 9 9 11 ll 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63
38 38 39 40 40 41 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71
38 37 29 21 20 21 20 19 16 15 14 13 12 12 11 10 l0 l0 9 9 9 9 8 8 8 8 8 8 7 7 7 7 7 7 7 7 7
1 1 1 1 2 1 2 2 2 2 3 3 3 4 4 5 5 5 5 6 6 6 7 7 7 7 7 7 9 9 9 9 9 9 9 10 10
38 36 19 2 20 1 19 15 5 1 11 6 1 12 6 10 9 8 1 9 8 7 7 6 5 4 3 2 7 6 5 4 3 2 1 7 6
0 1 10 19 0 20 1 4 I1 14 3 7 11 0 5 0 1 2 8 0 1 2 1 2 3 4 5 6 0 1 2 3 4 5 6 0 1
5 5 7 9 9 11 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 67 69 71
GK GK KF KF KF KF KF KF
67 67 67 67 67 67 67 67 67 67 67 67 67 67 67 67 67 67 67 67 67 67 67 67 67 67 67 67 67 67 67 67 67 67
71 71 71 71 71 71 71 71 71 71 71 71 71 71 71 71 71 71 71 71 71 71 71 71 71 71 71 71 71 71 71 71 71 71 71 71 71 75 75 75 75 75 75
40 40 41 41 42 42
40 39 31 30 22 21
1 1 1 1 1 2
40 38 21 19 2 21
0 1 10 11 20 0
5 5 7 7 9 9
GK GK GK GK GK GK
GK GK GK GK GK GK GK
GK GK GK GK GK
GK GK GK GK GK
x
N. Farmakis, S. Kounias / Two new D-optimal designs
254
Table 1 (continued) n
k
s
r
u
o
d
75 75 75 75 75 75 75 75 75 75 75 75 75 75 75 75 75 75 75 75 75 75 75 75 75 75 75 75 75 75 75
43 43 44 45 46 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71
22 21 19 16 16 15 15 13 13 12 12 11 11 10 10 9 9 9 9 8 8 8 8 8 8 7 7 7 7 7 7
1 2 2 2 2 3 3 3 3 4 4 4 4 5 5 6 6 6 6 7 7 7 7 8' 8 9 9 9 9 10 10
1 20 13 3 2 14 13 4 3 10 9 3 2 6 5 7 6 5 4 4 3 2 1 8 7 4 3 2 1 7 6
21 1 6 13 14 1 2 9 10 2 3 8 9 4 5 2 3 4 5 4 5 6 7 0 1 3 4 5 6 0 1
11 11 13 15 17 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 67
75 75 75 75
72 73 74 75
7 7 7 7
10 I0 10 10
5 4 3 2
2 3 4 5
69 71 73 75
79 79 79 79 79 79 79 79 79 79 79
42 42 43 44 44 45 45 46 47 48 49
42 41 32 23 22 23 22 20 17 16 16
l 1 1 1 2 1 2 2 2 3 3
42 40 21 2 22 1 21 14 4 16 15
0 1 11 21 0 22 1 6 13 0 1
5 5 7 9 9 11 11 13 15 17 19
GK
X GK GK KH KH KH
KH KH KH KH
n
k
s
r
u
o
d
79 79 79 79 79 79 79 79 79 79 79 79 79 79 79 79 79 79 79 79 79 79 79 79 79 79 79 79 79 79
50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79
14 13 13 12 11 11 11 10 10 10 9 9 9 9 8 8 8 8 8 8 7 7 7 7 7 7 7 7 7 7
3 3 4 4 4 5 5 5 5 5 6 6 6 7 8 8 8 8 8 8 10 10 10 10 10 10 10 11 11 11
6 1 13 7 1 11 10 3 2 1 3 2 1 9 8 7 6 5 4 3 7 6 5 4 3 2 1 7 6 5
8 12 0 5 10 0 1 7 8 9 6 7 8 0 0 1 2 3 4 5 0 l 2 3 4 5 6 0 1 2
21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 67 69 71 73 75 77 79
83 83 83 83
44 44 45 45
44 43 34 33
1 1 1 1
44 42 23 21
0 1 11 12
5 5 7 7
GK
83 83 83 83 83 83 83 83 83 83 83 83
46 46 47 47 48 49 50 51 52 53 54 55
24 23 24 23 21 18 17 16 15 14 13 12
1
2
22
9
2 1 2 2 2 2 3 3 3 4 4
23 1 22 15 5 1 13 8 3 11 5
0 23 1 6 13 16 3 7 11 2 7
9 11 ll 13 15 17 19 21 23 25 27
GK GK
GK GK GK
GK
KF KF GK GK GK GK GK FK
N. Farmakis, S. Kounias / Two new D-optimal designs
255
Table 1 (continued) n
k
s
r
u
o
d
83 83 83 83 83 83 83 83 83 83 83 83 83 83 83 83 83 83 83 83 83 83 83 83 83 83 83 83
56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83
12 11 11 10 10 10 9 9 9 9 9 8 8 8 8 8 8 8 7 7 7 7 7 7 7 7 7 7
4 5 5 5 6 6 6 7 7 7 7 8 8 8 8 8 9 9 10 10 10 11 11 11 11 11 11 11
4 9 8 1 10 9 1 9 8 7 6 5 4 3 2 1 8 7 3 2 1 7 6 5 4 3 2 1
8 2 3 9 0 1 8 0 1 2 3 3 4 5 6 7 0 1 4 5 6 0 1 2 3 4 5 6
29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 67 69 71 73 75 77 79 81 83
FK
87 87 87 87 87 87 87 87 87 87 87 87 87 87 87 87 87 87
46 46 47 48 48 49 49 50 51 52 52 53 54 55 56 57 58 59
46 45 35 25 24 25 24 22 19 18 17 17 15 14 14 13 12 12
I 1 1 1 2 1 2 2 2 2 3 3 3 3 4 4 4 4
46 44 23 2 24 1 23 16 6 2 16 15 6 1 14 8 2 1
0 1 i2 23 0 24 1 6 13 I6 1 2 9 13 0 5 I0 11
5 5 7 9 9 11 11 13 15 17 17 19 21 23 25 27 29 31
GK GK KF KF KF
x
n
k
s
r
u
o
d
87 87 87 87 87 87 87 87 87 87 87 87 87 87 87 87 87 87 87 87 87 87 87 87 87 87 87 87
60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87
11 11 10 10 10 10 9 9 9 9 8 8 8 8 8 8 8 7 7 7 7 7 7 7 7 7 7 7
5 5 6 6 6 6 7 7 7 7 8 8 9 9 9 9 9 11 11 11 11 11 11 11 12 12 12 12
6 5 8 7 6 5 6 5 4 3 2 1 8 7 6 5 4 7 6 5 4 3 2 1 7 6 5 4
5 6 2 3 4 5 3 4 5 6 6 7 0 1 2 3 4 0 1 2 3 4 5 6 0 1 2 3
33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 67 69 71 73 75 77 79 81 83 85 87
91 91 91 91 91 91 91 91 91 91 91 91 91 91 91 91 91 91
48 48 49 49 50 50 51 51 52 53 54 55 56 57 58 59 60 61
48 47 37 36 26 25 26 25 23 19 18 18 16 15 14 13 12 12
1 1 1 1 1 2 1 2 2 2 3 3 3 3 4 4 5 5
48 46 25 23 2 25 1 24 17 4 18 17 8 3 12 6 12 11
0 1 12 13 24 0 25 1 6 15 0 1 8 12 2 7 0 1
5 5 7 7 9 9 11 11 13 15 17 19 21 23 25 27 29 31
x GK GK GK GK GK GK GK
N. Farmakis, S. Kounias / Two new D-optimal designs
256
Table l (continued) n
k
s
91 91 91 91 91 91 91 91 91 91 91 91 91 91 91 91 91 91 91 9! 91 91 91 91 91 91 91 91 91 91
62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91
12 11 11 ll l0 10 10 9 9 9 9 9 8 8 8 8 8 8 8 7 7 7 7 7 7 7 7 7 7 7
95 95 95 95 95 95 95 95 95 95 95 95 95 95 95 95
50 50 51 52 52 53 53 54 55 56 57 58 59 60 61 62
50 49 38' 27 26 27 26 24 20 19 18 16 15 15 14 13
r
u
5 10 5 " 3 5 2 5 l 6 4 6 3 6 2 7 3 7 2 7 1 8 9 8 8 9 6 9 5 9 4 9 3 9 2 9 1 10 8 11 3 11 2 11 1 12 7 12 6 12 5 12 4 12 3 12 2 12 1 13 7 1 1 1 1 2 1 2 2 2 2 3 3 3 4 4 4
50 48 25 2 26 1 25 18 5 1 I5 6 1 15 9 3
o
d
2" 8 9 l0 6 7 8 6 7 8 0 1 2 3 4 5 6 7 0 4 5 6 0 1 2 3 4 5 6 0
33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 67 69 71 73 75 77 79 81 83 85 87 89 91
0 1 13 25 0 26 1 6 15 18 3 10 I4 0 5 10
5 5 7 9 9 11 11 13 15 17 19 21 23 25 27 29
n
GK GK KH KH KH KF KF KH KH KH
k
s
r
u
o
d
95 63 95 64 95 65 95 66 95 67 95 68 95 69 95 -70 95 71 95 72 95 73 95 74 95 75 95 76 95 77 95 78 95 79 95 80 95 81 95 82 95 83 95 84 95 85 95 86 95 87 95 88 95 89 95 90 95 91 95 92 95 93 95 94 95 95
13 12 11 II 11 10 10 10 l0 9 9 9 9 9 8 8 8 8 8 8 8 7 7 7 7 7 7 7 7 7 7 7 7
4 5 5 6 6 6 6 7 7 8 8 8 8 8 9 9 9 10 l0 10 10 12 12 12 12 12 12 12 13 13 13 13 13
2 8 1 11 10 2 1 10 9 9 8 7 6 5 3 2 1 8 7 6 5 7 6 5 4 3 2 1 7 6 5 4 3
11 4 10 0 l 8 9 0 1 0 1 2 3 4 5 6 7 0 1 2 3 0 1 2 3 4 5 6 0 1 2 3 4
31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 67 69 71 73 75 77 79 81 83 85 87 89 91 93 95
99 99 99 99 99 99 99 99 99 99 99 99 99
52 51 40 39 28 27 28 27 25 21 20 19 19
1 1 1 1 1 2 1 2 2 2 2 3 3
52 50 27 25 2 27 1 26 19 6 2 18 17
0 1 13 14 26 0 27 1 6 15 18 1 2
5 5 7 7 9 9 ll ll 13 15 17 17 19
52 52 53 53 54 54 55 55 56 57 58 58 59
GK GK GK GK GK, KH GK, KH GK
GK GK GK (3K
N. Farmakis, S. Kounias / Two new D-optimal designs
257
Table 1 (continued) n
k
s
r
u
o
d
99 99 99 99 99 99 99 99 99 99 99 99 99 99 99 99 99 99 99 99
60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79
17 16 15 14 13 13 13 12 11 I1 11 10 10 10 10 9 9 9 9 9
3 3 4 4 4 5 5 5 6 6 6 7 7 7 7 8 8 8 8 8
8 3 13 7 1 13 12 5 9 8 7 9 8 7 6 6 5 4 3 2
9 13 2 7 12 0 1 7 2 3 4 1 2 3 4 3 4 5 6 7
21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59
GK GK
GK GK GK GK GK GK GK GK
n
k
s
r
u
o
d
99 99 99 99 99 99 99 99 99 99 99 99 99 99 99 99 99 99 99 99
80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99
8 8 8 8 8 8 8 8 7 7 7 7 7 7 7 7 7 7 7 7
10 10 10 10 10 10 10 10 12 12 12 13 13 13 13 13 13 13 14 14
8 7 6 5 4 3 2 1 3 2 1 7 6 5 4 3 2 1 7 6
0 1 2 3 4 5 6 7 4 5 6 0 1 2 3 4 5 6 0 1
61 63 65 67 69 71 73 75 77 79 81 83 85 87 89 91 93 95 97 99
GK GK GK GK GK GK GK GK
x
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