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A METHOD OF CONSTRUCTING PBIB DESIGNS OF T TYPE
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by
S. Ikeda
University of North Carolina
Institute of statistics Mimeo Ser:ies No. 508
January 1967
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This research was supported by the
U.S. Army Research Office-Durham
Grant No. DA-ARO-D-3l-l24-G8l4.
DEPARTMENT OF STATISTICS
University of North Carolina
Chapel Hill, N. C.
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Contents
1.
Introduction
2.
Preliminary results
3.
4.
Construction of PBIB designs of T type
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A class of BIB designs
5.
Construction of triangular PBIB designs
6.
Construction ef PBIB designs of T type
3
References
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1.
Introduction
An association scheme with m associate classes, called T type, has
m
been introduced in [5J as a generalization of the so-called triangular type:
Let nand m be any given positive integers such that 1 < m < n/2, and let
= =
us put v = (n).
m
of all subsets of the set of n integers, (1,2, ••• ,n), which contain c distinct integers; the cardinality of V is clearly (~).
c
Suppose now that we are given a set of v treatments, (¢l' ••• '¢v J• Let
us take and fix a one-to-one correspondence between these v treatments and
the v elements of V in an arbitrary way.
m
(rl, ••• ,r ) and (ri, ••• ,r~), respectively.
m
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Then, ¢ and ¢' are said to
be u-th associates if and only if (rl, ••• ,rm) and (ri, ••• ,r~) have m-u integers in cornmon, u=l, ••• ,m.
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Under this correspondence,
suppose two treatments, ¢ and ¢', correspond to two elements of Vm,
of itself.
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Let, for any given positive integer c, V denote the set
c
In particular, any treatment is O-th associate
This is called the T
association scheme.
--m....;.;....;;..,;....;.,,;;;..;...;;.=~~~~
For later use, it is convenient to say that two elements of V are
c
in s-relation if they have exactly s integers in common.
Then, one can say
that two treatments are u-th associates if the corresponding elements of V
m
are in (m-u)-relation.
Parameters of the T association scheme have been shown [5J to be
m
(1.1)
n
u
= (m ) (nu-m) ,
u
u
= O,l, ••• ,m,
It is evident that these parameters coincide with those of the triangular association scheme when m = 2 •
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An allocation of v treatments with T association scheme to b blocks
m
of size k each in such a way that each treatment occurs in rblocks, each
block contains k different treatments and two treatments which are u-th
associates occur together in
design of
Tm~.
~
u
blocks, u
= l, ••• ,m,
--
In particular, A.o is defined to be r.
vr
= bk
m
and L:
u=o
n
u
~
u
= rk.
In passing, it is noted that non-existence criteria have been given for
regular and synnnetrical PBIB designs of T (triangular) type [3J, of T
2
3
type [4J, and in general of T type [5J.
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In this article, a general method of constructing PBIB designs of T
m
type is devised (sections 2 and 3).
designs when m
=1
(section
4),
This gives a class of trivial BIB
PBIB designs of T type which can be constructed by this method are shown in
3
section 6.
Preliminary results
Let d be any given integer such that 1 < d < n. We say, as before,
= =
that (r , ••• ,r ) in V and (t , ... ,td ) in V are in s-relation to one
m
d
m
l
l
another if and only if they have s integers in common, s
= 0,
1, ••• ,m.
Then the following two results are straightforward.
Lemma 2.1.
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and two classes of triangular PBIB designs
(section 5), most of which have been solved already except for a few designs.
2.
-.
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is called a PBIB
Parameters of PBIB designs of T type satisfy the relations
m
(1.2)
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For any given (rl, ••• ,r ) in V , the number of those
m
m
s-relation to (r1, ••• ,r ) is given by
m
2
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using the convention (x)
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For any given (tl, ••• ,t ) in V , the number of those
d
d
(rl, ••• ,r ) in Vm which are in s-relation to (tl, ••• ,td ) is given by
m
(2.2)
c(s) --
(ds)(mn-_ds)'
= 0 , 1 , ••• ,m ) ,
Note that the above two values, R(s) and C(s), are independent of the
choices of (rl, ••• ,rm) and (tl, ••• ,td ) respectively.
It is also easy to see the following:
Lemma 2.3.
Let (rl, ••• ,rm) and (r l', ••• ,r'm) be any two members of Vm
which are in s-relation to one another.
Then, the number of (tl, ••• ,td)'s
in Vd which are in s-relation to (rl, ••• ,rm) and at the same time in s'relation to (ri, ••• ,r~) is independent of the choices of (rl, ••• ,rm) and
(ri, ••• ,r~) and is given by
I (s Sf)
u '
=
~(m-u)(u)(u)(
p
q q'
n-m-u )
d-p-q-q'
,
(0 <
=
S,
s' <• n),
with the same convention as in Lemma 2.1, where the summation is taken over
all non-negative integers p,q and q'restricted by the conditions p+q = s
and p+q'
= st.
The following two lemmas are. easily proved:
Lemma 2.4.
(2.4)
(~) R(s)
= (~)
C(s) , s
.e
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(s
with the same convention as in the above lemma.
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if x < y or y < O.
Lemma 2.2.
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=0
3
= O,l, ••• ,m.
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Lemma
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2.5.
I u (s,s') = I u (s' ,s) , u,s,s' = O,l, ••• ,m,
and
R(s)
I
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C(s') =R(s') C(s), s,s' =O,l, ••• ,m.
Finally, we have
Lennna
2.6.
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For any given s and s' ,
m
~
(2.7)
n I (s,s') = R(s) C(s').
u=o
Proof.
-
u
U
Form a rectangular array, i.e., a matrix A, with (n) rows and
m
(~) columns, listing the members of V down the row headings and the members
m
of Vd across the column headings. If (rl, ••• ,rm) in Vm is in s-relation to
(tl, ••• ,td ) in Vd , we enter the integer s in the position whose rowcorresponds to (rl, ••• ,rm) and column to (tl, ••• ,td ).
There is no harm in
assuming that the first R(s) positions on the first row of A are occupied
by any given integer s.
Consider first the case s
= s'.
In this case, both sides of (2.7) give
the number of entries "s" contained in the first R(s) columns of A.
°
In this case, I (s,s') = and both
o
sides of (2.7) give the number of entries" s'" contained in the first R(s)
Second, consider the case s ~
Sf.
columns of A.
This proves the lennna.
3.
Construction of PBIB designs of T type.
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Suppose we ~are given b = (~) blocks.
We make a one-to-one correspondence between these b blocks and the
members of· Vd in any way.
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Now, let us divide the set of integers, (0, 1, ••. ,mJ, into two nonempty subsets, J
o
and J , and form a v x b matrix N from the matrix A
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given in the proof of Lemma 2.6 in such a way that the (i,j) position of
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N receives "1" if the (i,j) element of A belongs to J , and "0" otherwise.
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Then, we have the following
Theorem 3.1.
Nis the incidence matrix of a PBIB design of T type
m
with parameters
r = I:
s
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,.
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A = I:
u (s,s')e J
Proof.
R(s),
6 J
x
l
k
=
l
J
I (s,s'), u
u
l
C(s),
I:
S 6
J
l
= l, ••• ,m.
From Lemmas 2.1 and 2.2 it follows that each row and each
column of N contain exactly rand k entries
"1", respectively.
By Lemma 2.3 it is seen that any pair of treatments which are u-th
associates occurs in A blocks.
u
Lemma 2.4 implies that vr
= bk.
Since I (s,s) = R(s), putting A =r,
o
0
we can see from Lemma 2.6 that
m
I:
u=o
n
u
A
U
=
R(s)C(s')
I:
(s,s' )e" J
l
x J
= rk.
l
This proves the theorem.
It is noted that, by Lemma 2.5, AU in (3.1) can be written as
I:
s < s'
s,s'e J l
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I (s,s'),
u
u = 1, ••• ,m.
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4.
A class of BIB designs.
In the case m = 1, (3.1) gives the parameters of BIB designs: Putting
J
o
= to} and J
(4.1)
v
l = tl), we have
= n,
k
= d,
b
n
= (d)'
r
= (n-l)
d-l'
n-2
= (d-2)'
~
(
2 ~ d ~ n-l),
and putting J o = tl} and J l = {oJ,
(4.2)
It is clear that (4.2) is the complementary design of (4.1), and is
obtained by changing d in (4.1) for n-d.
Hence the class of BIB designs
(4.1) is 'self-complementary' in the sense that the complementary design of
any design of this class is also contained in the class.
Designs of this class are quire trivial.
It would be of some interest,
however, to investigate a "splitting propertyH of some of these designs:
For example , if we take n
v
= 6,
k
= 3,
b
BIB design with
=6
and d
=3
= 4,
= 20, r = 10 and ~.
v = 6, k = 3, b = 10,
r
in (4.1), we have a BIB design with
which is a duplicate of the existent
=5
and
~
= 2.
In general, suppose
the last three parameters of (4.1) have a common diviser c(> 1), and put
b = cb', r = crt and
~
= c~'.
Then, what conditions guarantee the existence
of BIB design with parameters v, k, b', r' and
~'?
This is left for further investigations.
5. Construction of triangular PBIB designs
In the case m = 2, PBIB designs with parameters (1.1) and (3.1) are
triangular PBIB designs, for which a general method of construction was
proposed in Section 3.
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All possible choices of J
l
from (0, 1, 2} are
~
among which the last three cases give the complementary designs of the first
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three respectively.
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Hence, in this section, we consider only the first
three cases.
For m = 2, the parameters of the association, (1.1), are
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1
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t
2-t (2-t)( t )( t )( n-2-t )
= E
i
2-s-J.' 2-u-i s+u-2+i ,s,t,u=O,1,2.
psUi=o
5.1.
= (O}
l
In this case, the parameters (3.1) are given by
The case J
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J
J
for 1 < d < n-3.
=
=
= (l}.
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The parameters of the design are given by
5.2. The case J
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for 1 ~ d ~ (n-l)/2; changing d for n-d gives the same design.
1
(
The parameters of the design are given by
J.
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v.(~), k=(~), b=(~), r=(~:~), ~l=(~:;)' ~2=(~:t),
(5.5)
for 3
~
d
~
n-l.
This gives the same class of triangular PBIB designs given
by Masuyama [6J, and it is easily seen that the parameters (5.5) are obtained
by changing d in (5.3) for n-d.
Some of the designs of the first two classes are listed below together
with their complementary designs. Table 5.1 below is the same as that of [6J.
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6.
Construction of PBIB designs of T type.
3
In the case m=3, Section 3 offers a method of construction of T PBIB
3
designs.
In
this case, the parameters of the association are given
by
nl = 3(n-3), ~ = 3(n~3), ~ = (n;3),
(6.1)
t
t
3- (3_t)( t )( t )( n-3-t )
Psu = ~ i 3-s-i 3-u-i s+u-3+i' s,t,u=O,1,2,3,
i=o
with the same convention as in Lemma 2.1.
To give the parameters of the design, we first give the following:
(6.2)
and
Il(0,0)=(n~4), Il(O,l)=(~:i),
I l (0,2)=O, I l (2,2) =
I
l
Il(l,l)
= 2(~:i)+(~:~),
(~:~)+2(~:~), I l (0,3) = 0,
(3,3)=(~:t),
12(0,0)=(n~5), 12(0,1)=2(~:i), 12 (1,1)= (~:i)+4(::~),
(6.3)
12(0,2)=(::~), 12(2,2)=4(&:~)+(~:'), 12 (0,3) = 0,
12 (3,3 )=(~:~),
1 (0,0):z(n;6),
3
13(0,1)=3(~:~), 13(1,1)=9(~:~),
13(0,2)=(::~), 13(2,2)=9(~:~), 13(0,3)=(::~),
1
3
C~,3 )=(~:~).
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. All possible choices of J l from (0,1,2,3) are given by the following
7 pairs, each of which gives a class of T PBIB designs and their comple3
mentary designs:
(6.4)
6.1.
= (0).
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The 'parameters of the design (3.1) are given, by (6.2) and (6.3),
The case J
as follows:
(n-6 )
. (n-4)
(n- 5 )
Al = d ' A2 = d ' A3 = d
'
6.2.
(1 ~ d ~ n-4; n ~
6)
The case J l = (1).
The parameters (3.1) are given by
(6.6)
Al
6.2 1 •
= 2(~:i)+(~:~), A2~(~:i)+4(~:~), A3=9(~~~),
(1
~d~
n-3).
= (2).
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The parameters (3.1) are given by
The case J
(6.6' )
"' _(n-4) (n-4) "' _I,(n- 5 ) (n-5)
(n-6) (
)
I~l - d-2 +2 d-3 ' ~2~ d-3 + d-4 ' A3=9 d-4 ' 2 ~ d ~ n-2 •
12
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It is easily noticed that these parameters are obtained by changing d in
(6.6) for n-3, and hence (6.6') give the same class as (6.6).
6.l~
The case J l =(3).
The parameters (3.1) are given by
(6.5' )
n-4) ~ (n- 5 ) ~ (n-6)
~l= (d-4' 2= d-5' 3 = d-6 '
(4 ~ d ~ n),
which give the same class of T designs as (6.5).
3
6.3.
The case Jl-(O,l).
The parameters (3.1) are given by
3
3
v= (3n) ' k= (n-d)
3 +d (n-d)
2 ,b= (n)
d ,r= (n-d )+3 (nd-l ) '
(6.7)
. (n-5) (n-5) 4(n-5 )
n-4) 4(n-4) (n-4)
~l= ( d + d-l + d-2 '~2= d +5 d-l + d-2'
~3
6.4.
n-2).
k=(n;d)+(n_cl)(~), b.(~),r=(n~3)+3(::~),
~ =(n-4)+(n-4)+2(n-4), ~ =(n-5)+2(n-')+4(n-5) +(n- 5 )
2 d
d-2
d-3
d-4'
1 d
d-2
d-3
~ = (n-6)+6(n-6)+9(n-6),
3
6.5.
(1 ~ d ~
The case J l ={O,2).
The parameters (3.1) are given by
v=(;),
(6.8)
(n-6)
=(n-6)+6(n-6)
d
d-l +9 d-2 '
d
d-2d-4
The case J l = (O,3).
The parameters (3.1) are given by
13
(1 _< d _< n-l).
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~ =(n-4)+(n-4), ~ =(n-5)+(n- 5 ), ~ =(n-6)+ 2(n-6)+(n-6),
1
d
d-4
2
d
d-5
3
d
(1
d-3
d-6
:s d :s n/2),
for which changing d for n-d gives the same design.
Some of the designs of these classes are listed below together with
their complementary designs.
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Table 6.3
T3 designs with parameters (6.7) and their complementary designs.
class is self-complementary. )
n
d
v
k
b
r
6
1
2
3
4
1
2
3
4
5
1
2
3
4
5
6
1
2
3
4
5
6
7
20
20
20
20
35
35
35
35
35
56
56
56
56
56
56
20
16
10
4
35
30
22
13
5
56
50
40
28
16
6
6
15
20
15
7
21
35
35
21
8
28
56
70
56
28
9
36
84
126
126
84
36
6
12
10
3
7
18
22
13
3
8
25
40
35
16
3
9
33
65
75
51
19
3
7
8
9
84 84
84 77
84 65
84 50
84 34
84 19
84 7
A.
A.
1
6
10
6
1
7
16
16
7
1
8
23
32
23
8
1
9
31
55
55
31
9
1
2
6
9
4
0
7
15
13
4
0
8
22
28
17
4
0
9
30
50
45
21
4
0
(This
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A.
3
6
9
0
0
7
15
9
9
0
8
22
24
9
0
0
9
30
46
33
9
0
0
(Rand. Block)
(Rand. Block)
(Rand. BloCk)
(Rand. Block)
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Table 6.4
T~ designs with parameters (6.8) and their complementary designs.
c1 ss is self-complementary) •
n
d
v
k
b
r
6
1
2
3
4
5
1
2
3
4
5
6
1
2
20
20
20
20
20
35
35
35
35
35
35
56
56
10
8
10
12
10
20
15
16
19
20
15
35
26
6
15
20
15
6
7
21
35
35
21
7
8
28
3 ..
6
10
9
3
4
9
16
19
12
3
"5
13
7
8
A.
A.
1 2
2 1
2 2
4 6
5 5
2 1
3 2
4 3
6 8
9 11
7 6
2 1
4 3
7 5
16
A.
3
0
6
0
9
0
1
6
6
9
9
0
2
7
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Fina1~y,
we shall show an actual procedure of constructing a T PBIB
3
design.
Exapp1e 6.1
Construction of T design with parameters
3
v=20, k=8, b=15,
r=6, ,\=2,
\=2,
~=6,
n1=9, n2 =9, n,=1.
(1°)
123
124
125
126
U4
U5
U6
~5
~6
156
234
~5
236
245
~6
256
345
346
356
456
(2°)
Form the matrix A:
12
2
2
2
2
1
1
1
1
1
1
1
1
1
1
1
1
0
0
0
0
13
2
1
1
1
2
2
2
1
1
1
1
1
1
0
0
0
1
1
1
0
14
1
2
1
1
2
1
1
2
2
1
1
0
0
1
1
0
1
1
0
1
15
1
1
2
1
1
2
1
2
1
2
0
1
Q
1
0
1
1
0
1
1
16
1
1
1
2
1
1
2
1
2
2
0
0
1
0
1
1
0
1
1
1
23
2
1
1
1
1
"1
1
0
0
0
2
2
2
1
1
1
1
1
1
0
24
1
2
1
1
1
0
0
1
1
0
2
1
1
2
2
1
1
1
0
1
25
1
1
2
1
0
1
0
1
0
1
1
2
1
2
1
2
1
0
1
1
26
1
1
1
2
0
0
1
0
1
1
1
1
2
1
2
2
0
1
1
1
34
1
1
0
0
2
1
1
1
1
0
2
1
1
1
1
0
2
2
1
1
3 36
1 1
0 0
1 0
0 1
1 1
2 1
1 2
1 0
0 1
1 1
1 1
2 1
1 2
1 0
0 1
1 1
2 1
1 2
2 2
1 1
4
0
1
1
0
1
1
0
2
1
1
1
1
0
2.
1
1
2
1
1
2
46
0
1
0
1
1
0
1
1
2
1
1
0
1
6
0
0
1
1
0
1
1
1
1
2
0
1
1
1', 1
2 1
1 2
1 1
2 1
1 2
2 2
Form the incidence matrix N by changing 0 and 2 for 1 and
1 for 0, and by taking any correspondences (1:1) between treatments and rows, and between blocks and columns of A.
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~
Treatmen
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
(3
0
)
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
2
4
5
6
1 1 0 0 0
1 0 1 0 0
1 0 0 1 0
1 0 0 0 1
0 1 1 0 0
0 1 0 1 0
0 1 0 0 1
0 0 1 1 0
0 0 1 0 1
0 0 0 1 1
0 0 0 1 1
0 0 1 0 1
0 0 1 1 0
0 1 0 0 1
0 1 0 1 0
0 1 1 0 0
1 0 0 0 1
1 0 0 1 0
1 0 1 0 0
1 1 0 0 0
1
0
0
1
3
7
0
1
0
O· 0
0 0
0 1
0 1
1 0
1 0
1 1
1 1
1 0
1 0
0 1
0 1
0 0
0 0
0 0
0 1
1 0
9
10
11
12
13
14
15
0 0
0 0
1 0
0 1
1 1
0 1
1 0
0 1
1 0
0 0
0 0
1 0
0 1
1 0
0 1
1 1
0 1
1 0
0 0
0 0
0
0
1
1
1
0
0
0
0
1
1
0
0
0
0
1
1
1
0
0
0
1
0
1
0
1
0
0
1
0
0
1
0
0
1
0
1
0
1
0
0
1
1
0
0
0
1
1
0
0
0
0
1
1
0
0
0
1
1
0
1
0
0
1
0
0
1
1
0
0
0
0
1
1
0
0
1
0
0
1
1
0
1
0
0
1
0
0
1
0
0
1
0
0
1
0
0
1
0
1
1
1
0
0
1
0
0
0
0
1
1
0
0
0
0
1
0
0
1
1
9
10
11
12
13
14
15
16
17
18
19
20
2 2
1 1
1 2
2 1
1 1
1 2
2 1
0 1
1 0
1 1
2 2
2 3
3 2
1 ,·2
2 1
2 2
1 2
2 1
2 2
1 1
2
2
1
1
2
1
1
1
1
0
3
2
2
2
2
1
2
2
1
1
1
1
2
2
1
2
2
2
2
3
0
1
1
1
1
2
1
1
2
2
1
2
1
2
.2
1
2
2
3
2
1
1
2
2
1
2
2
1
3
2
2
1
1
0
2
1
1
2
1
1
2
2
1
1
2
2
2
3
1
2
2
1
1
2
0
1
1
1
2
2
1
2
1
2
1
2
3
2
2
2
1
1
3
2
2
2
2
1
2
1
1
1
1
0
2
2
1
1
2
2
2
3
1
1
2
1
2
2
1
1
2
1
2
2
0
1
1
1
2
2
3
2
1
2
1
2
1
2
1
2
1
2
1
2
1
0
1
1
2
3
2
2
2
1
1
2
2
1
2
1
1
2
2
1
1
1
0
1
3
2
2
2
2
2
2
1
1
1
2
2
2
1
1
1
1
1
1
0
8
Set up the association scheme:
1
2
3
4
5
6
0
1
1
1
1
1
1
2
2
2
1
1
1
2
2
2
2
2
2
3
1
0
1
1
1
2
2
1
1
2
1
2
2
1
1
2
2
2
3
2
1
1
0
1
2
1
2
1
2
1
2
1
2
1
2
1
2
3
2
2
1
1
1
0
2
2
1
2
1
1
2
2
1
2
1
1
3
2
2
2
1
1
2
2
0
1
1
1
1
2
1
2
2
2
2
3
1
1
2
2
1 1
2 2
1 2
2 1
1 1
0 1
1 0
1 2
2 1
1 1
2 2
1 2
2 1
2 3
3 2
2 2
1 2
2 1
1 1
2 2
7
8
19
0
1
1
2
1
1
2
1
2
2
1
2
1
2
1
1
0
1
2
1
2
1
.'I
From (20 ), we thus have an allocation of 20 treatments with T
3
0
association scheme (3 ) into 15 blocks of size 8 each:
Treatments
Blocks
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
1
1
2
3
4
1
2
3
4
3
2
2
1
1
1
2
5
5
6
7
8
6
5
5
4
4
3
4
3
2
3
6
8
8
9
9
7
7
6
5
6
7
7
6
5
4
7
9
10
10
10
10
9
8
10
9
8
8
9
10
17
14
12
11
11
11
11
12
13
11
12
13
13
12
11
18
15
13
13
12
12
14
14
15
16
15
14
14
15
16
19
16
16
15
14
13
15
16
16
17
17
18
17
18
19
20
20
19
18
17
20
19
18
17
18
19
19
20
20
20
Note that the upper half of the incidence matrix N in (20 ) is the
incidence matrix of a BIB design with parameters
v=10, k=4, b=15 , r=6,
A.=2.
The author is grateful to Mr. T.A. Dowling for his valuable comments
and help given to this work.
20
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,.,
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,
References
L1J
R.C. Bose, W.R. Clatworthyand S.S. Shrikhande (1954), Tables of
Partially Balanced Designs with Two Associate Classes, Inst.
Stat., UNC, Reprint Ser., No. 50.
L2 J W. R. Clatworthy (1956), Contributions on Partially Balanced Incomplete
Block Designs with Two Associate Classes, NBS, App. Math. Ser.,
No. 47.
[3J
J. Ogawa (1959), A necessary condition for eXistence of regular and
symmetrical experimental designs of triangular type, Ann. Math.
Statist., 30, 1063-1071.
[4 J
K. Kusumoto (1965), A necessary condition for eXistence of' regular and
symmetrical PBIB designs of T3 type, Ann. Inst. Stat. Math., Toky?,
17, 149-161.
[5 J M. Ogasawara (1965), A necessary condition for the eXistence of regular
and symmetrical PBIB designs of Tm type, Inst. Stat., UNC, Mimeo
Ser. No. 418.
[6J
M. Masuyama (1965), A class of triangular designs, Rep. Stat. App. Res.,"
JUSE, 12, 32-34.
[7J
M. Masuyama (1965), Cyclic generation of triangular PBIB designs, Rep.
Stat. App. Res., JUSE, 12, 73-81.
[8J
Chang Li-chien, Liu Chang-wen and Liu Wan-ru (1965), Incomplete block
designs with triangular parameters for which k < 10 and r < 10.
Scientia Sin1ca, 14, 329-338.
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21