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INSTITUTE OF STA11S1lCI
80X 54S7
STATE COl,.:LEGE STATION
RALEIGH, NORTH CAROUIa
CONSTRUCTION OF BALANCED INCOMPLETE
BLOCK DESIGNS FROM ASSOCIATION MATRICES
by
William C. Blackwelder
University of North Carolina
Institute of Statistics Mimeo Series No. 481
June 1966
In Chapter I, balanced incomplete block designs and m-class
association schemes are defined. Necessary and sufficient
conditions are found in Chapter II for a BIB incidence matrix
to be (i) a linear combination of association matrices or
(ii) a matrix formed by placing association matrices side by
side (this is referred to as a juxtaposition of association
matrices). In Chapters III and IV, various methods of constructing association schemes with two and three classes
are discussed; using the results from Chapter II, the schemes
are then examined to determine whether BIB designs can be
constructed from them.
This research was supported by National Institutes of Health
Institute of General Medical Sciences Grant No. 5-Tl-GM-38-l3.
DEPARTMENT OF S'fATISTICS
University of North Carolina
Chapel Hill, N. C.
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TABLE OF CONTENTS
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Chapter
...·. • • ·.
LIST OF 51MBOLS. • • • • • . . . . • • • • · . . . . . . .
INTRODUCTION • • • • • . . . . . . . . . . . · . .
1.1. Balanced IncoIll:Plete Blocks (Bm) Designs • • • · . .
Association Schemes. • •.• . . . . . . . . . . . . .
ACKNOWLEDGMENTS. • • • • • • • •
I.
]
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1.3. Partially Balanced Incomplete Block (PBm) Designs.
II.
THE PROBLEM AND FUNDA.MENTAL RESULTS ••
·..•
·...
• •
III. TWO-CLASS ASSOCIATION SCHEMES. • • • • • • • • • • • •
Je
1
1
1
Introduction. • • • • • • • • • • • • • • • • • • •
3.2. Group Divisible (GD) Scheme• • •
]
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-1
1
• • • •
.• ···.• ····
Pseudo-Cyclic Association Scheme •
·•
·• ·
v
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6
9
10
19
19
20
3.3. Triangular Association Scheme.
21
3.4.
24
3.5. Singly
Liru~ed
Block (SLB) Association Scheme •
27
3.6. Latin Square (Lg(n) ) and Pseudo-Latin Square
Association Schemes. • •
]
1
·.·.
iv
.•
....• . • ·•
• •
30
3.7. Negative Latin Square (Ntg (n)) Association Scheme.. 35
3.8. Pseudo-Geometric Association Scheme. • • • • • • •• 37
IV.
THREE-CLASS ASSOCIATION SCHEMES. • • • •
·. • • • · ..
43
4.1.
• •
43
Introduction. •
. .. . ....
.
•
· . . . . ••
4.2. Group Divisible (GD) m-Associate Scheme. •
4.3.
Tetrahedral Association Scheme •
4.4. Cubic Association Scheme. • • •
•
·.
· . . · . . • • ••
·..·... .
44
49
52
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Chapter
. . . . . . . . . ..
4.6. Three-Class Association Scheme from an
Orthogonal Array. • • • • •• •••• . . . . . • •
BIBLIOORAPHY. • • • • • . . . . . . . . . . . . . • • ••
4.5. Rectangular Association Scheme.
Page
56
58
611.
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ACKNOWLEDGMENTS
I would like to express my gratitude to Dr. I. M. Chakravarti,
Who proposed the problem discussed in this thesis and guided my work
throughout.
For their willingness to participate on my master's commi.ttee,
my thanks go to Dr. Dana. Quade and Dr. Elizabeth Coulter.
I wish to thank Mrs. Judy Zeuner for her conscientious and
patient work in typing this thesis.
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LIST OF SYMBOIB
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Meaning
vxv matrix of l's.
vxb matrix of l's.
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vxv i.dentity matrix.
x
transpose of the vector x.
N
transpose of the matrix N.
deter:md.na.nt of the matrix N.
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CHAPTER
I
INTRODUCTION
1.1.
Balanced Incomplete Block (BIB) Designs
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A balanced incomplete block (BIB) design is an arrangement of
elements (objects, varieties, treatments) in b
v
sUbsets, called blocks,
such that
(i) each block contains exactly k elements, all distinct,
(ii)
(iii)
each element appears in exactly r
blocks,
each pair of distinct elements occur together in exactly A
blocks.
(1.1.1)
The numbers
v, b, r, k, A
are called the parameters of the design; all mDSt be positive integers.
The parameters k and r
are referred
number of replications, respectively.
vdth the parameters (1.1.1) as a BIB
t~
as the block size and the
We shall refer to a BIB design
(v, b, r, k,A).
It is easy to see that the parameters (1.1.1) must satisfy the
relations
bk
(1.1.2)
11
= vr
Numbers in square brackets refer to bibliography entries.
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-,
and
(1.1.3)
=
A(v-l)
Suppose ex and f3
r(k-l)
are two distinct treatments of a BIB design.
Then ex and f3 oc'cur together in A blocks, and ex
occurs in a total of
In case A = r, we see from (1.1.3) that v
r blocks; thus A~ r.
= k,
and the design is a randomized blocks design; then for a "proper" BIB
design, Le., one that is not a randomized blocks design, we have
(1.1.4)
O<A<r
Now consider the vxb matrix N = ( (n .. ) ) (i = 1,2, •.• , v;
~J
j =
1,2, •.• , b), Where
(1.1.5)
=
nij
1
if the
i~
treatment occurs in the
-'I
j~
block,
=
0
otherwise.
We shall call N t11e incidence matrix for the BIB (v, b, r, k, A).
is clear from (1.1.5) that
(1.1.6)
t
J
N
=
v
kJ
b ,v
and
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(1.1. 7)
NU
=
r
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I v + A(Jv - I v )
It
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Conversely, if (1.1.4) holds and if N is a vxb matrix of O's and lIs
satisfying (1.1.6) and (1.1.7), then N is an incidence matrix for a
BIB (v, b, r, k, A)
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[20] •
Thus we have the fo 1lowing lemma..
Lemma 1.1.1
Suppose
N is a vxb matrix of O's and lIs, and suppose A and r
are integers such that 0 < A < r.
Then the necessary and sufficient
condition "for N to be an incidence matrix for a BIB
is that
(i)
,
N
J
,
(ii)
=
v
NN
=
r
b,v
I v + 'A{Jv - I").
V
,
implies that I NN
b, r, k, A)
and
kJ
Now from (1.1.7)
,
(v,
'VTe
I>0
see that INN
I=
rk ( r-A )
for a proper BIB design.
v.
a vxv matrix, has rank
V-l
•
N.
Then b
,
Hence NN , which is
We know that the rank of
ceed b, the number of columns of
Then (1.1.4)
N can not ex-
> rank N
,
=rank NN =v;
thus for a proper BIB design, we have the ineq'lll.lity
(1.1. 8)
b
> v
,
which was first proved by R. A. Fisher [11].
For any mxn matrix A of O's and lIs, the matrix J
be called the complement of A.
m,n
- A shall
We now prove the following lennna.
Lemma 101.2
'SupposeN
is an incidence matrix for a BIB (v, b, r, k, A),
and suppose k < v-1.
matrix for a BIB
(v,
*
Then the complement of
* * * *
b , r , k , 'A),
Where"
N is an incidence
4
v*
= v, b.* = b, r * = b-r, k* = v-k,
""A* = b-2r + ""A.
Denote the complement of N by N*; i.e.,
Proof:
* =J
N
v,b
-N.
From (1.1.5), N I J v = k J,b,v ;
then
*
(N)
I
J v = (Jv, b-N) J v = J,b ,v J v
-kJ,b
, v =( v-Ie) J,b ,v = k*J,b ,v •
I
From (1.1. 7),
NN
=
r I v + ""A(Jv - I v );
then
=
(Jv, b - N) (Jv, b - N)
=
J
I
J.
v,b b,v
= (b - 2r
= (b-r)
= r*
I
v
I
-NJ.
b,v
+ ""A) J
v
+
+
v
+
-J
v,b
N
I
+NN
(r-""A) I v
(b - 2r + ""A) (Jv - I v )
""A*(Jv - I v ).
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From (1.1.4) we see that ""A < r; adding (b-2r) to both sides of
* .*
*( * \
the inequality, we have ""A < r. Now ""1\*= r ~ -1.( = (b-rHv-k-1 l
v -1
v-1
Then if k < v-1, ""A* > 0, and by Lemma 1.1.1 N* is an incidence matrix
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for a BIB(v*, b*, r *, k*, A* ).
(Note:
The only case not allowed in Lemma 1.1.2, '\-Then k
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gives
a trivial example of a BIB design; some references, for example [20],
omit the case k
= v-l altogether in considering BIB designs.)
A BIB design ivith v
design.
= band
r
=k
is known as a symmetric BIB
In this paper we shall be concerned largely with such de-
signs.
Now the relations (1.1.2) and (1.1.3) are necessary, but not in
general sUfficient, for the existence of a BIB(v, b, r, k, A).
ever, Hanani has shown [14] that (i) for k
(ii) for k
=5
How-
= 3,4, and any A or
and A = 1,4,20, the relations (1.1.2) and (1.1.3) are
both necessary and sufficient for the existence of a BIB (v, b, r, k,A).
Fisher and Yates [J2] have tabled BIB designs with r
II
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= v-l,
~
15.
Rao
[19] has studied BIB designs with r = 11 to 15. Sprott [25] lists
designs with r
= 16 to 20. The literature contains a large number
of articles showing the impossibility of certain BIB designs:
for
example, [21] and [22].
Numerous methods may be employed in the construction of BIB
designs.
A few examples are the rethod of differences [1], finite
geometries [1], construction from known BIB designs [3], orthogonal
arrays [9], Hadamard matrices [23], and association schemes [23].
The last method has been used to construct some symmetric BIB designs.
This work will be concerned with an analysis of methods of constructing BIB designs from association matrices.
employed.
Two methods will be
The first, linear combinations of association matrices, is
<luite similar to a method suggested by Shrikhande and Singh in [23];
6
however, the result in the present work was arrived at independently.
The second method, jU):taposition of association raatrices, is believed
to be new.
1.2. Association Schemes [2]
An association scheme in
m associate classes is a set of
v
elements (objects, treatments, varieties) Which satisfies the following conditions:
(i)
any two treatlrents are either
l~ 2~ ••• ,
or m.:!?h associates,
and the relation of association is symmetrical;
(ii)
-
each element has exactly n. ith associates (i
~
= 1,2, •.• ,
m),
Where the number n. is independent of the element chosen;
~
(iii)
if a and
~
are
i~
associates, then the number of elements
j~ associates of a and k~ associates of ~ is P~k '
which are
and P~k is independent of the pair of i.:!?h associates chosen
(i, j, k
= 1,2,
... , m).
The numbers
(1.2.1)
are called the parameters of the association scheme; all must be
positive integers.
The following relations among the parameters (1.2.1) are easily
shown:
(1.2.2)
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m
(1.2.3 )
I
= v-l
ni
i=l
m
(1.2.4)
I
Pjk
= nj
if i
rj
,
k=l
= nj-l if i = j
i
n.~ P'1
J):
It is useful to :make the convention
that each element is the
zero-th associate of itself and of no other elements.
have
(1.2.6)
n0
(1.2.7)
Pij
0
=1
0
= Pji = 0 if i
rj ,
= n j if i = j
(1.2.8)
i
Pko
i
= Pok
= 0 if i r k ,
= 1 if i =k
Then (1.2.3) and (1.2.4) become
m
~
i=o
(1.2.10)
ni
=
v,
.
Then we must
8
For the caaem
1
= 2, it is sufficient to specify v, nl' Pll' and
2
P11, and the other parameters are then determined; see, for example,
[2] •
Given an m-class association scheme, we call the matrices B
i
(i = 0, 1, .•• , m) the association matrices of the scheme, where
b1
li
(1.2.11)
~
) )
B =( ( bai
i
b~
·..
v
b1i
·..
=
1
bvi
(1.2.12)
2
bU
= 1 if a and
b
•
• ••
2
• ••
vi
and
v
b
vi
~ are i~ associates,
= 0 otherwise.
(1.2.13 )
and
(1.2.14)
AlsO, the linear form Co B + c B + ••• + c m Bm is equal to the
o
1 1
zero matrix if and only if Co = c1 = ••• = cm = 0; i.e., the Bi'S are
lit
Bj = Bj
(j, k
= 0,
The association matrices satisfy the relation
J1t
=
o
1
Pjk Bo + Pjk B1 +
1, ••• , m).
-.
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Clearly, we have
linearly independent.
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i
~.
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The result (1.2.15), along with the linear independence of the B. 's ,
l.
will be quite important in the proofs of the theorems in Chapter II •
1.3.
Partially Balanced Incomplete Block (PBIB) Designs [2J
Given an m-class association scheme of
treatments, a partially
v
balanced incomplete block (PBIB) design is an arrangement of the
treatments in b blocks such that
(i)
each block contains exactly k
treatments, all distinct;
(ii)
each treatment occurs in exactly r
iii)
if two treatments a and .13 are ith associates, they occur
together in exactly
~
blocks, where
the choice of the particular pair of
(i
= 1,
Clearly, i f \
blocks;
~
is independent of
i~
associates a and
j
j
.~
~l
1
13
2, ••• , m).
= '2 = ••• = \n, the design is a BIB design.
PBIB designs were introduced by Bose and Nair [6J; a slightly
generalized definition
.~s
given by Nair and Rao [16J.
in the preceding paragraph is that of Nair and Rao.
The definition
Bose and Shima-
moto [8J introduced the concept of association schemes and based the
definition of PBIB designs on these Schemes.
The present work will
not be concerned "With PBIB designs, but they are rrentioned because
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CHAPTER
II
THE PROBLEM AND FUNDAMENTAL RESULTS
The problem considered here is that of construction of BIB designs
from the association matrices B.~ by the folloidng two methods:
(i)
to obtain a matrix of the form B. + B. +
~l
~2
...
+
B.
~t
'Which
will be a BIB incidence matrix, and
(ii)
to obtain a matrix of the form [ B. : B. :
~l·
Which will be a BIB
~2·
...
incidence matrix.
A matrix of type (i) will be referred to as a linear combination of
association matrices, and a matrix of type (ii) vdll be called a
juxtaposition of association matrices.
The two theorems Which follow
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give the necessary and sufficient conditions for (i) and (ii).
Theorem 2.1
(Linear Combinations)
Suppose we have an m-class association scheme in v
with association .matrices Bo
elements,
= Iv,
B , B , ••• , B.
Suppose
m
l
2
i l , i , ••• , it are distinct integers such that i j €(O,1,2, ••• ,m) for
2
j = l,2, ••• , t < m. Then the necessary and sufficient condition for
C = B. + B. + ••• + B.
~l
~2
~t
to be an incidence matrix for a BIB
(i , i 2 , • '" it)
), where
{v, v, r, r, A l
r = n.
~l
+ n.
~2
+ ••• + n.
~t
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I,
I
),
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and
is that
2
+
t
E
j<.e
~:
(iI' i 2,···, it)
= A
= 1,2,
for k
""
m.
(i ,i , ••• ,i )
t by
(For convenience, we shall denote A . l 2
during the proof.)
Suppose" is a positive integer.
A
Then the
required necessary and sufficient condition is that
(1)
C Jv = r J
(2)
CC
and
v
,
(r-A) I
=
v
+ AJ
=
v
r I v + A(Jv - I v ).
Now
C Jv =
+ B.
(B.
~l
n.
=
~1
=r
+ ••• + B. ) J
~2
J
v
~t
+ n.
~2
J
v
+
...
v
+ n.
J
~t
v
Jv
Then (1) is always satisfied when the B.~ 's are association matrices.
,
CC
=
,
(B.
~l
+ B.
~2
-1- •••
+ B. ) (B.
~t
...
~l
+
+ 2
+ B. )
~t
12
k
p.
•••
4
~t""t
-.
B
k
k=o
Since P~i = ni and P~j = 0 for i
,
CC
f
j, we
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have
t
=r
I
v +
(I.1=1 p~.i.
J J
t
+
..
L P~~.~.
( \
j=l
2 \
L
j<t
J J
t
+ ••• + (
t
+
I
j=l
P~.i.
t
+
2
J J
I
j<J
P~.ilt
J
_.
)
XI
Now
Then, since the B.~ 's are linearly independent, the necessary and
sufficient condition for (2) to hold is that
t
I
P~.i.
j=l J J
Theorem 2.2
t
p~.
= A,
L
~'~J
j<J
J
+ 2 \
k
= 1,
2, "', m •
(Juxtaposition)
Suppose we have an m-class association scheme in v
with association matrices Bo = Iv,
~, ~,
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I
••• , Bm•
elements,
Suppose
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3
]
i , i 2 , ••• , it are distinct integers such that i j £(1,2, ••• , m)
1
for j = 1, 2, '.', t
~
m.
Then the necessary and sufficient con-
dition for D = [B.1 ,: B.:
: B.1 ] to be an incidence matrix for
1 .
t
1
2
a BIB (v, tv, tk, 1\:, A..
'1. 1 ••. 1. ) , where
t
1 2
= tk(k-l)
v-l
(i)
A..
. be a positive integer ,
1112 ···1t
n.1 = n.1 =
2
1
t
t
.
(iii) I p.
l
1.1.. =
j=l J J
(ii)
=n
it
=k
j~pih =
Proof:
J
-,
=
Ira
p. ,
1.1.
j=l J ,1
= A..
1 1 , •. 1.
1 2
t
= n. = ••• = n. = k.
1
1
t
1
2
(It is obvious that (i) and (ii) must be satisfied for D to be a
1
BIB incidence matrix.)
Then the required necessary and sufficient
condition is that
(1)
D J
(2)
DD
=
v
k
J
tv, v
and
le
t
A is a positive integer, and suppose n.
]
-,
,
(For convenience, we shall denote A..
. by A.) Suppose
.
'1.112···1t
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]
is that
= (tk =
A) I v + A J v
tk I v + A( J v - I v ).
14
B.
...
• ••
B.
I
DJ
kJ
~2
=
v
-,
kJ
~l
J
• ••
.•
v
v
=
=
v
·..B.
kJ
~t
'.
Jt
h
I
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V,V
v
Then (1) is always satisfied.
B.
• •••• B. ]
.
~2
~t
...
B.
~2
....
-'I
B.
~t
=B. B.
~l ~l
m
-I-
\
L
+ B. B.
-I-
c
p ..
B
B. B.
~t ~t
= tk
Iv
m
p~
. B
~l~l c
c.l
t
= tk
-1- •••
~2 ~2
I
v
-I- ( \
I
+
_._.
~
2 ~2
c
c=l
p~
t
: ) B
~ ~.~.
j.l
J J
l
I
I
p~~.~.
. )
L
j=l J J
-I- ( \
t
B +••• +( \ p~ . ) Bm •
2
~'~j
j=l J
L
Now
= 'A(Jv
- I v) •
Then, by the linear independence of the B.~ IS, the necessary and
sufficient condition for (2) to hold is that
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15
t
P~
\
/
t
1
j=l
\
.
=L
~j~J'
m
Pi,i.
j=l
J J
=
If we want an n-class association scheme in v
1\ •
elements such
that [ B. : Bi : ••• : B. ] and [B j + B. + ••• + Bj ] are
~l'
2'
. ~t
1
J2
s
both BIB incidence tntrices, where s
.5 t and the Bj ' s form a sub-
set of the B. 's, then the necessary and sufficient condition is
~
found by combining the conditions from Theorems 2.1 and 2.2.
If we want the scheme such that [ B.: B. ] and [B. + Bj ] are
~.
J
~
both BIB incidence matrices, we have
=
2k(1\:-1)
and
v-I
.
Note that 1\(ij) - 2 A.
.~
J
~(ij)
1\
_ 2k(2k-l)
h
v-l
,were ni
= n j = k.
2k ; but 2k < v-I if the scheme has
= V:l
more than two associate classes, and in any event 2k is not greater
than v-I.
Then
(1\(ij) - 2 \j) is an integer only if 2k=n
l
+~=v-I.
In this case there are only two associate classes and Bl + B is
2
trivially an incidence matrix for a BIB, since B]+B
. 2 = J v - I.
v
In Theorem 2.1, let t = 2 and m = .3. Then 1're see that the
necessary and sufficient condition for Bi + Bj to be a BIB incidence
matrix is that 1\(ij) be an integer and
16
t = 1 in Theorem 2. 1, we get the condition for Bi to
be a BIB incidence matrix; viz., that
Lettin~
12m
p ~~.
.. = P~~
••
= ••• = p ..
~~
where f.. (i) is a positive integer.
The following corollary shows
that in this case it is enough to show that P~.
~~
the connnon value :must then be A(i) •
= p2i . =••. = prr:.;
~
~~
Shrikhande and Singh [23]
proved that the condition in Theorem 2.1 allows an m-class scheme
to be collapsed into a two-class scheme such that pil = pil or
1
2
P22 = P22· However, Theorem 2.1 and the folloi'ltng corollary
.i"ere proved independently.
Corollary 2.3 (to Theorem 2.1)
For an m-class association scheme in v elements with association matrices Bo = Iv' Bl , B , •• ·, Bm, the necessary and sufficient
2
i) ),
condition for B.~ to be an incidence mtrix for a BIBv,v,n.,n.,A
«
~
~
(i)
n i (ni -1)
, is that
where A
=
1
v-
...
Proof:
m
= Pii = x,
for some positive integer x.
1
2m
(
Suppose Pii
= ~ii = ••• = Pii far some i €.1,2, ••• ,m);
let x be the connnon value.
Now
m
I
i
.. -- n i -1'
_u_. P ~J
since Piio -- 1:,
j=l
also,
i
j
n.~ Pij = n. Pii'
J
i
or Pij
=~
n
i
j
Pii
I
I
-.
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_.
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17
Then
lil
=
I
~j
1' ••
n
j=l
.
~
~~
,
1
m
and we have. n p~~
+ n2
•• + •••
+ n I'
2 1'~~
m ii
l ••
x(n + n + ••• + nm) = n.(n.
- 1). Since
~
~
l
2
x
=
)
= n.~ ( n.~ - 1,
or
n.~ (n.~ -1)
=
v-I
and the condition in Theorem 1 is satisfied if x is a positive
integer.
In Theorem 2.2, t&cing t = 2 and m = 3, we see that the condition
for [ Bi : Bj ] to b a BIB incidence matrix is that n i
Aij
= 2~~~-1)
1'3.. + p 3 .
jJ
~~
be a positive integer, .and pii +
"]23
=
p~i
+
(saYh
P~j =
= A.~J,.
For the case t = 3 and m = 3, we must have
~_
P~j
= nj = k
~
= n
2
=
~
= k,
= 3k(k-l) a positive integer, and
v-I
In this case 3k
If t
= v-I
and \23
= k-l.
=2 and m = 2,
" dence
~nc~
· ·~s
rnatr~x
the condition for [ B1 : B2 ] to be a BIB
tha t n = ~ an d I'll
1 + 1'22
1 = I'll
2 + 1'22
2 = A
"J.2.
l
For this case,
~
'1.2
(nl +n2 ) (nl-l)
n +n
1 2
=
= n1 -1
(equivalently, A
"I2 = n2 -1).
18
In Chapte.rs III and IV various methods of constructing association schemes of two and three associate classes are discussed.
These
association schemes are examined for their possible use in constructing BIB designs in the special cases considered in Theorems 2.1
and 2.2.
Whenever the parameter A(i) or
A(ij) appears it will be
understood to be defined as in Theorem 2.1; a parameter of the form
~j
or
~jk
is defined as in Theorem 2.2.
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CHAPTER
III
TWO-CLASS ASSOCIATION SCHEMES
3.1.
Introduction
Several methods of constructing two-class association schemes
will be discussed.
The parameters resulting from each method will be
examined in order to determine whether
(a)
Bl'
(b)
B2 "
(c) [Bl : B2 ]
can be BIB incidence matrices.
[2], [8], and [15].
The methods discussed can be found in
The necessary and sufficient conditions in each
case are as follows, from Theorems 2.1 and 2.2:
(g)
(a)
2
1
Pn = Pll
(b)
1
P22
(c)
n
l
=
=
n
2
2
P22
1
2
2
and P1 + P = P + P22
= n l - l.
ll
n
22
We note that if N b is a BIB incidence matrix, then the pairs
v,
must occur an equal nwnber of times for any two rows, as well as
the pairs (~).
Then the complement of N, or Jv,b- N, is also a BIB
incidence matrix.
Thus it is unnecessary to consider the combinations
20
3.2.
Group Divisible (GD) Scheme
3.2.1.
The scheme
Suppose, for integers
elenents.
J,
[8]
J,
~
2 and n
~
2, there is a set of v
= .In
Let the elements be arranged in a rectangular array with
rows and n columns.
Call any two elements 'Which appear together
in a row first associates; if two elements are in different rows
they are second associates.
Then the rectangular array gives us a
two-class association scheme with the following parameters.
v
1
= J,n
I,
I
n2 = n(J,-l)
(3.2.1)pl
11
pl
12
1
P22
I
I
eI
I
I
1
= n-2
Pll
=0
=0
p2
= n-l
= n(J,-l)
P22
2
12
2
I:
eJ
I·
= n(J,-2)
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3.2.2. DesignS from B _
122
We need Pll = Pll; but Pll
= O. Then Bl can not be a BIB in-
cidence matrix.
3.2.3. Designs from Be_
I'
I
I'1
Ii
1
1
i
'1
2
The condition is that P = P22 ; then n(J,-l)
22
Thus B can not be a BIB incidence matrix.
2
= n(J,-2),
and n=O.
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3.2.4. Designs from
~l:~-l
122
= Pu + P22 = nl -1. But
2n-l
nl = n2 for n -1 = n(.t-l), or .t =----.
There are no integers
n
We must have n
l
~,
,n
= n2 and
1
Pn + P22
~
2 which satisfy the above; thus [ B : B2 ] can not be a BIB
l
incidence matrix.
Then GD schemes can not be used to construct any BIB design such
that Bl , B2 , or [ B : B ] is the incidence matrix.
l
2
3.3.
Tria.ngularAssociation Scheme [8]
3.3.1. Definition of the scheme
Suppose, for some positive integer n, there is a set of
v
= (2n) _- n(n-l)
2
as follows:
DJ~-l)
elements.
Arrange the
v
elements in an nxn array
leave the leading diagonal positions blank, and fill the
pcsitions above the leading diagonal· with the v elements; fill
the remaining
n~-l)
positions so as to make the array symmetric with
respect to the diagonal.
Define first associates as two elements
which appear in the same row (equivalently, the same column) of. the
resulting array; if two treatments do not appear in the same row,
they are second associates.
The v
elements might also be considered as unordered pairs
(i,j), where i
f
j and i,j
= 0,1,
••• , n-l.
Then two elements are
first associates if they differ in exactly one coordinate; otherwise
they are second associates.
Such an array is an association scheme, called a triangular
association scheme.
The parameters of the scheme are as follows.
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22
_ n(n-l)
v 2
n
.
1
= 2n
-
4
_ (n-2)(n-d)
n2 2·
Pll
1
= n-2
2
Pll
pl
= n-3
P
1
P22
= (n-3)(n-4)
12
=4
2
=2n-8
12
2
P22
2
From the parameter values , we see that n
(n-4)(n-5)
2
=
~
4.
An example of a tri-
angular scheme 1 with n=4, is the following.
x123
1 x 4 5
2 4 x 6
.-I
3 56 x
Element
1
2
3
4
5
6
First associates
Second associates
2 3
1 3
4 5
6
4
6
1 2
5
1 2 5
1 3 4
2 3 4
6
5
4
6
6
3
2
5
1
An examination of the parameters (;.;.1) shows that in one ease 1
will be a BIB incidence matrix; B and [B
2
1
::52] can never be BIB in-
cidence.matrices.
3.3.2. Designs from B _
1 l
2
12
We must have P = P1l ; then n=6 and P1l = Pll = 4• So for n=6,
11
B is a BIB incidence matrix•. The parameters and. scheme for this
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23
case are the following.
= 15
n =8
1
v
= 0,-
n
2
pI
11
=h
2
pU
=4
pI
=3
p
2
=4
=3
P22
12
1
P22
12
2
=1
2 3 4 5
6 7· 8 9
x 10 11 12
3 7 10 x 13 14
4 8 11 13
x 15
5 9 12 14 15 x
x 1
1 x
2 6
Element
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
First associates
2
1
1
1
1
1
1
1
1
2
2
2
3
3
4
3
3
2
2
2
2
3
4
5
3
4
5
4
5
5
lj.
5
4 5
5
3 5
3 4
7 8
6 8
6 7
6 7
6,- 7
(J
8
6 9
'l 8
7 9
8 9
lj.
6
6
7
8
9
9
9
9
8
11
10
10
10
10
11
7
10
10
11
12
10
10
11
12
12
12
11
11
12
12
B is an incidence .:matrix for a
1
8
11
13
13
14
11
13
13
14
13
13
14
14
13
13
Second associates
9
12
14
15
15
12
14
15
15
14
15
15
15
15
14
10
7
6
6
6
3
2
2
2
1
1
1
1
1
1
11
8
8
7
7
4
lj.
3
3
4
3
3
2
2
2
12
13
13
11
10
10
13
9
9
9
8
5
5 11
5 10
4 10
5 8
5 7
4 7
5 6
4 6
3 6
BIB(15, 15, 8, 8, 4).
14 15
14 15
12 15
12 14
11 13
14 15
12 15
12 14
11 13
9 15
9 14
8 13
9 12
8 11
7 10
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3.3.3. Designs from Be_
The condition is
(n-3)(n-4) _
2
that
(n-4)(n-5)
-
-.
P~ = P~2 ; then
2
'
or
n=4.
But in this case the common valu.e of P~2 and P~2 is o.
Then B2 can
be a BIB incidence matrix.
3.3.4. Designs from (,]1:4.1
We must have n l
n = n for 2(n-2) =
2
l
v=2l;
\2 = 2(~g)(9)
12
2
= n2 and P1
ll + P22 = Pll + P22 = \2.
(n-2~(n-3) , or n=7. Then n l = n2 = 10
= 9·
For n=7, ph +
P~2
=5 +
4~3)
Now
and
= 11
F '\2'
Then [ Bl : B2 ] can not be a BIB,incidence matrix.
e
3.4. Pseudo-Cyclic Association Scheme [8], [15]
3.4.1. Definition
Suppose there is a set of v
integers 1, 2, "', v.
(C\,
~,
(i)
... , d
n
1
elements; denote them by the
Suppose there is a set of integers
) satisfying the following conditions:
the d's are distinct, and 0 < d < v (j
j
reduced (mod v), each of the numbers dl ,
= 1,2, ••• ,
~,
••• , d occurs
n1
o times and each of the numbers e l , e2 , "" e~ occurs ~
times, where d , d , ••• , d , e , e , ••• , e
are all the
1 2
2
nl l
n2
integers 1, 2, "', v-l. Clearly, nl o+n2 ~ = nl(nl-l).
Given the element k (k = 1; 2, "', v), define its first
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I'
I'
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e
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25
,.
associates as the elements k+d l , k+~, ••• , k+d
(mod v); the renl
maining (v~nl-l) elements are the second associates of k. Then we
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I
the following parameters.
I
I
n2 = v-nl-l
I
I
have an association shceme, called a cyclic association scheme, vnth
v
nl
1
I'll
1
1'12
1
1'22
2
I'll
= n -0:-1
1'12
2
l
n2-nl -I(X+1
=
= f3
= n l -f3
2
1'22 = n2 -nl +f3 -l
We see that, given v, the set of d's completely determines such
a scheme.
I.
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I
I
I
I
I
Ie
I
I
= 0:
A few' examples of cyclic association schemes are given
below.
Some Cyclic Association Schemes
v
~
~
13
17
29
6
8
14
6
All the
8
14
kn010ID
Set of d's
5, 6, 7, 8, 11
3, 5, 6, 7,10, ll, 12, 14
2,
1, 4, 5, 6, 7,
9, 13, 16,20,22,23,24,25,28
cyclic association schemes are such that
v = 4u+l, n
= n = 2u, and 0: = u-l, for some positive integer u.
l
2
Then the association scheme has the following parameters.
n
v
= 4u
l
= n2 = 2u
+ 1
1'1= u-l
11
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26
1
P12
1
P22
=u
=u
=u
= u-l
-,
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Following the nomenclature in [15], we will call a;ny association
scheme satisfying the parameters (3.4.2)
~seudo-cyclic,
whether or
not it is obtainable by the cyclic method described in [8].
I
Let us examine the parameters (3.4.2) to determine whether B ,
l
B2 , or [ Bl : B ] can be a BIB incidence matrix.
2
I
3.4.2. Designs from B _
1
1
2
1
We must have Pll = Pll; but Pll
not be a BIB incidence matrix.
= u-l
2
and Pll
= u. Then Bl can
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3.4.3. Designs from B2
2
2
2
We need Pll = P22 but P1l
be a BIB incidence matrix.
=u
I
2
and P22
= u-l. Then B2 can not
I
The condition is that
nl-l.
Now
~
= n2 = 2u
~
= n2 and P1ll
1
+ P22
= P2ll
2
+ P22
=
~
=
for any pseudo-cyclic scheme, and
1
1
2"2
Pll + P = 2u-l = Pll + P22 • Thus for any pseudo-cyclic association
22
scheme, with parameters (3.4.2), [B l : B ] is an incidence matrix
2
for a BIB (4u +1, 2(4u + 1), 4u, 2u, 2u-l). .An example is the
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following:
u
= 2,
v =
9, n l
= ~ =
4.
(Note that this scheme is not obtainable by the cyclic :tOOthod.)
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27
Element
First associates
2
1
1
1
1
2
2
:5
1
2
3
4
5
6
7
8
9
:5
3
3
2
5
4
4
5
4
5
4 5
6
8
6
7
7
6
6
7
6 7 8
4 5 8
4 5 6
2 " 7
2 3'" 6
1 3 5
1 5 h
1 2 5
1 2 4
7
9
8
9
8
9
9
8
1
P11
=1
2
P11
=2
1
P12
=2
2
P12
=2
1
P22
=2
2
P22= 1
-
011110000
1 0 100 110 0
110 0 000 1 1
100011010
1 0 0 1 0 0 1 01
o 1 0 1 0 01 1 0
o 1 0 0 1 1 0 01
o 0 1 1 0 1 0 01
o0 1 0 1 0 1 1 0
Second associates
o0 0 0 0 1 1 1 1
000 110 0 1 1
000 111 100
o 110 0 010 1
o 110 0 1 0 1 0
101010001
1 0 1 100 0 1 0
110010100
110101000
[B1 : B J is a incidence matrix for a BIB(9, 18,
2
3.5. Singly Linked Block (SLB)
9
9
7
9
8
9
8
7
6
8, 4, 3).
Association Scheme [2], [8]
3,.5.1. The scheme
,
Suppose N is an incidence matrix for a BIB design with b
treatments,
v blocks,
k
replications, block size r, and
A = l;i.e.
every pair of treatments occurs together in exactly one block.
bk = vr and b-1 = 1~(r-1); this gives us v = },,(rk-k+1)
r
It has been shown that in this case
Then
and b=rk-k+l.
N 'is an incidence matrix
28
for a PBIB design 'With v treatments, b blocks, r replications, k
plots per block,
"J..
= 1, and '\ =
o.
Defining first associates as
two treatments which appear together in some block of'tihe derived
PBIB design, we get a two-class association scheme called a
si~
linked block .(SLB) association scheme, with the following parameters.
v = k(rl;:-k+J.)
r
nl = r(k-l)
~ = (k-r) (~-l) (k-l)
1
11 = k-2+(r-l)
P
1
2
2
P12
= (r-l) (k-r)
1
P22
=
P
11
=r2
P~ = r(k-r-l)
(r-l) (k-r) (k-r-l)
r
P~ = (k_r)2+ 2(r-l) k~-l)
For r=2 the SLB scheme is the same as the triangular scheme
with m = k+l.
An
SLB
examination of the parameters (3.5.1) shows that one series of
schemes
is such that one of the association matrices is a
BIB incidence matrix.
3.5.2. DeSignS from Bl _
The condition is that P~ = pil; then (k-2) + (r_l)2 = r 2, or
k = 2r +1.
in
In this case the
terms of r.
SLB
scheme has the following parameters,
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29
v
= (21'-1)(21'+1)
n1
= 21'2
n
2
= 2(r-1)(r+1)
= l'
2
P11
P12
1
= (1'-1) (r+1)
P12-r
1
= (r-1)(r+1)
P
22
1
P11
P22
2
2
2
2
2
= r 2 -3
2
Then B is an incidence matrix of a BIB (41'2_1, 4r _1, 2r2 ,
1
2r2 , r 2 ). In this case the original BIB of interest, from Which
2
the SLB scheme is obtainab1~ has r(2r-1) treatn~nts, (4r _1) blocks,
(2r+1) replications,
l'
plots per block, and A = 1.
Designs from Bel
2
We must have P22 = P22 ; hence
3.5.3.
= (k_r)2 + 2(r-1) - k(k-1)
r
,or k = 2r-1-
(r-1)(k-r)(k-r-1)
r
In this case the SLB scheme has the following parameters.
v
= (2r-1) (2r2 -3r+2)
r
n1
= 2r(r-1)
n = 2(1'-1)3
2
l'
1
2
P
11 = r -2
2
1
I.
I
I
=r
P12 = (1'-1)
2
1
(r-1) (r-2)
P22
=
l'
2
P11
2
P12
2
P22
= l'2
= 1'(1'-2)
= (r-1)2(r_2)
r
30
1
2
But note that in (:3.5.,), P22 and P22 can never be integers fa
r > 2, and for r = 2 the common value is o. Then B2 can never be
a BIB incidence matrix.
I
I
eI
I
I
I·
for r(k-l)
(k-l)
or k = r(2r-l)
= (k-r) (r-l)
,
r-l
r
and
'1.2 =
In this case
2r3 -2r2+1
(r-l)
k-2 + (r_l)2 + (r-l)(k-r)(k-r-l)
= 2r3 -2r2+r+l for k =r(2r-l) •
r-l
r-l'
r
then pil + P~2 = ~ When 2~-2r2+1 = 2r3-2r2 + r+l, i.e., for
r
= o. But there can be no such association scheme, and so [Bl :B2 ]
can not be a BIB incidence matrix.
Latin Square (Lg en) ) and Pseudo-Latin Square Association
Schemes [2], [8], [15]
.3.6.1. Definition of the scheme
array.
= n2
elements, arranged in an nxn
Letting two elements which appear in the same row or the
same column be first associates and two elements which do not
appear together in a row or column be second associates, we can
define an
~(n)
association scheme.
.-I
I
.3.6.
Suppose we have a set of v
I
I
I
I
I
I
I
.-I'
I
I
I
,
I
I
I
I
I
I
I
f
I
I
I
I
I
I
.I
31
For 3
.s g oS n+1,
if a set of (g-2) mutually orthogonal nxn Latin
squares exists, we can define a Latin square (Lg(n) ) association
scheme from the
array of the
m:l1
v elements in the following manner.
If two elements appear in the same row or column of the array, or if
they correspond to the same symbol in one of the (g-2) Latin squares,
they are first associates; otherwise the two elements are second
associates.
For the case gdt, n=4, we can take the Latin squares LS, and
LS , Where
2
1
2
3
4
2
1
4
3
3
4
1
2
!~
3
2
1
and
1
2
3
4
3
4
1
2
1.~
3
2
1
2
1
4
3
=
If the 16 elements are arranged in the array
1
2
3
4
5
6
7
8
9
10
11 12
13
14
15
,
16
then the first associates of the element 8 are 5,6,7,4,12,16,3,9,14,
2,11,and 13; for 8 corresponds to the symbo13 in LS 1 and to the
32
to the symbol.2 in L8 •
2
For 2 S g S n + 1, the Latin square association scheme has the
following parameters.
v=n
nl
2
= g(n-l)
p11
2
= g(g-l)
= (n-g+l)(g-l)
p12
2
= g(n-g)
= (n-g+l)(n-g)
2
P22
= (g-l) (g-2)+.n-2
Pi2
1
P22
= (n-g) 2
I
I
+ g-2
Following the nomenclature in [15], let us call an association
scheme with the parameters (3.6.1) a pseudo-Latin square association
scheme, whether or not it is obtainable from a set of (g-2) mutually
orthogonal Latin squares.
For example, an L (6) scheme can be ob3
tained from any 6 x 6 Latin square; its complement, or the scheme
obtained by interchanging first and second associate classes, has the
parameters of an L4 (6) scheme, but no pair of mutually orthogonal
6 x 6 Latin squares exists.
An examination of the parameters (3.6.1) shows that two distinct
series of pseudo-Latin square association schemes are such that BIB
designs can be constructed from the association matrices.
3.6.2. Designs from B _
l
The condition is that pil
or n
=2g.
as follows.
= P~
•I
I
I
~ = (n-g+l)(n-l)
pil
I
I
; then (g-1)(g-2) + n-2
= g(g-l),
In this case the parameters of the association scheme are
I
I
..
I
I
I
I
I
I
-.
I,
I
I
33
,
I
I
I
I
I
I
I
f
I
-•
•
= 4g2
v
(3.6.2)
nl
= c(2g-1)
~
= (g+1)(2g-1)
1
Pll
Pi2 : (g-l)(g+l)
1
P22
I
= g(g-l)
2
P
= g2
12
2
P22
= (g+2) (g-l)
hi,
g
= 2,3, and 4, respectively) exist, and they give the following
BIB designs:
(16; 16, 6, 6, 2), (36, 36, 15, 15, 6) and
(64, 64, 28, 28, 12).
3.6.3. Designs
fl~m
Be
The condition is that P~2
g-2, or n
= P~; then (n-g+l)(n-g) = (n_g)2
+
= 2(g-1). In this case the parameters are the following,
v
= 4(g_1)2
nl
= g(2g-3)
~ = (g-1)(2g-3)
1
Pll
= (g-2) (g+l)
p 112- (g_1)2
1
.e
= g(g+l)
2
Pll
2
Then Bl is an incidence matrix for a BIB(4g ,
g(2g-1), g(2g-1),
g(g-l». The first three members of the series n = 2g (letting
P22
I
= g(g-l)
= (g-l) (g-2)
2
p.
= g(g-l)
2
= g(g-2)
11
P12
2
P22 = (g-1)(g-2)
It should be noted that the scheme with the parameters (3.6.3) is the
complement of the scheme with parameters (3.6.2), Which was shown
to be such
Bl is a BIB incidence. matrix.
designs by considering B
2
for g(n-l)
and \2
t~t
= (n-g+l)
(n-l), or n
= 2g
Then we get no new
-1; then nl
•I
= n2 = 2g(g-l),
= 2g2 -2g-1. For
n
= 2g-1
f
121
Pll = g -g-l and P22 = g(g-l);
then
1
1
P ll + P22 =
2
Also, Pll
~
•
= g(g-l) and P222 = (n-g)(n-g-l) + n-2
= g2 -g-l
2
2
= 2g2 -2g-1 = \2 .
Then for any Lg(n) association scheme with n=2g-1, [B l :B2 ] is
a BIB incidence matrix. In this case the parameters of the scheme
are as follows.
= (2g-1) 2
~ = ~ = 2g(g-1)
v
pil
= g2_g _1
2
p 11
1
= g(g-l)
P12 = g(g-l)
P12
1
P22
= g(g-l)
= g(g-l)
2
2
I
I
I
I
I
I
..
then
Pll + P22
.1
I
2
P22 = g -g-l
For such a scheme, [B :B ] is an incidence matrix for a BIB( (2g_1)2,
l 2
I
I
I
I
I;
Ii
-.!
,
1
1
"
I
I
,
I
I
I
I
I
I
I
f
I
I
I
•
35
2
2
2(2g-1) , 4g(g-1), 2g(g-1), 2g -g-l).
Letting g
= 2,
3, and 4, re-
spectively, we see that the corresponding schemes exist, and we obtain the following BIB designs:
(9, 18,
8, 4, 3), (25, 50, 24, 12, 11),
and (49, 98, 48, 24, 23).
3.7.
Negative Latin Square (NLg(n) ) Association Scheme [15]
3.7.1. Definition
It has been found that in many cases negative values of g and n
will result in non-negative integers for the Lg (n) parameters (3.6.1).
The simplest case is for g
= -1
and n
= -4;
the resulting scheme
has the following parameters.
v = 16
5
11
=
11
= 10
1
2
2
11
=2
=4
2
P12
=3
=6
2
P22
=6
1
Pll
=
°
p
Substituting -g for g and -n for n in (3.6.1), we get the
following set of parameters.
= n2
v
nl
= g(n+l)
n
= (n-g-l)(n+l)
2
1
Pll
=
(g+1)(g+2)-n-2
P1l
2
= g(g+l)
2
= 'g(n-g)
P12
1
= (n-g-l)(g+l)
P12
P~2
= (n-g-l)(n-g)
P~2 = (n_g)2_(g+2)
36
An
,
association scheme vTi th the parameters (3.7.1) 1'Till be called a
negative Latin square (NLg(n» association. scheme.
Let us examine the parameters (3.7.1) to determine whether B , B ,
l
2
and [B l :B ] can be BIB incidence matrices.
2
3.7.2.
Designs from B _
l
We must have pil = pil; then (g+1)(g+2)-n-2= g(g+l), or n=2g.
this case the
NL
g
In
(n) parameters are the following, in terms of g.
2
v = 4g
n = g(2g+1)
l
~
1
1
p12 = (g-l)(g+l)
pl
22
= g(g-l)
2
I'll = g(g+l)
2
p12 =
2
g
2
p22 = (g-2) (g+l)
Then B is an incidence matrix for a BIB (4g2 , 4g2 , g(2g+l), g(2g+1),
l
g(g+l». For g=2, the Ntg (2g) scheme exists [15], and we get the
BIB (16, 16, 10, 10,
known [15].
BIB
6). For n=3, the existence of the scheme is un-
Letting n=4, the scheme exists [15], and we get the
(64, 64, 36, 36, 20).
3.7.3. Designs from BeThe condition is that p~2 = p~2; then (n-g-l)(n-g) = (n_g)2 - (g+2),
or n=2(g+1).
It is easily verified that the resulting scheme is just
the complement of the NLg(n) scheme with ,n=2g.
any further BIB designs by considering B2 •
I
I
I
I
I
= (g-1)(2g+l)
I'll = g(g+l)
I
I
Then we can not obtain
I
I
..
I
I
I
I
I
I
-.
I
I
I
,
I
I
I
I
I
I
I
f
I
-•
37
3.7.4. Designs from f!l:~L
1
1
2
2
We need nl = n2 and Pll + P22 = PH + P22 = \2· Now nl = n2 for
g(n+1) = (n-g-l) (n+l), or n = 2g+1; then n = n2 = 2g(g+1), and
1
2
A12 = 2g +2g-1. For n = 2g+1, the parameters are as follows.
= (2g+1)2
v
= n2 = 2g(g+1)
n
P
l
1
2
n = g +g-l
1
P
12
=
g(g+l)
1
1
for n
2
= g(g+l)
2
= g(g+l)
P
12
2
P22
P~2 = g(g+l)
Then P1l + P22
Pll
= g
222
2
= 2g +2g-1 = \2 and Pll + P22 = 2g
2
+g-l
+ 2g-1
= \2. Then
= 2g+1, a negative Latin square association scheme is such that
(B :B ] is an incidence matrix for a BIB«2g+1)2, 2(2g+1)2, 4g(g+1),
l 2
2
2g(g+1), 2g +2g-1).
Note that a member of this series of BIB designs
is also obtainable from a scheme with Lg + (2g+1) parameters, if such a
1
scheme exists.
3.8. Pseudo-Geometric Association Scheme (2]
3.8.1 Partial Geometry
A partial geometry (r,k,t) is a system of points and lines, and a
relation of incidence which satisfies the following axioms:
(i)
any two distinct points are incident with not more than one
line;
(ii)
each point
~s
incident with r
lines;
(iii) each line is incident with k points;
(i v)
if the point P is not incident with the line
.e,
then there are
·1
38
exactly t lines (t > 1) which are incident wi. th P and also incident with
.
-
some point incident with J.
Clearly, we have
( 3 . 8 . 1 ) 1 S t S k, 1 S t s r, where
r
and k are > 2.
It is easily seen from an examination of the four axioms above that
given a partial geometry (r,k,t), we can obtain a dual partial geometry
(k,r,t) by changing points to lines and lines to points.
The number of points v and the number of lines
b
in a partial
geometry (r,k,t) satisfy the relations
v
_ !L(r-l)(k-l)+tl
t
b
_ !l(r-l)(k-l)+t]
t
and
For convenience, ,ve may use the ordinary geometric language when
referring to partial geometries.
Thus if a point and line are incident,
we say that the point lies on the line (is contained in the line) and
that the line passe.s through the point.
points P and Q.
joins P and Q..
J, and m, we say that
A line Which contains two
If a point P lies on two lines
J, and m intersect at P.
Let us call the points of a partial geometry treatments and call
the lines blocks.
The relation of incidence will then be that of a
treatment's being contained in a block.
Call two treatments first
associates if they occur together in a block; otherwise they are second
associates.
Thus we see that a partial geometry (r,k,t) is equivalent
I
I
,
I
I
I
I
I
I
I
..
I
I
I
I
·1
I
-.
I:
I
I
,
I
I
I
I
I
I
I
f
I
I
I
-
39
to a PBIB design with parameters
(3.8.4)
vlhere
v, b, r, k,
v
~
= 1,
and b are given by (3.8.2) and (3.8.3).
The parameters of
the corresponding association scheme are the follovling.
= l~[(r-l)(k-l)+t]
v
t
nl
= r(k-l)
~
= ir-l)(~-l)(k-t)
1
= rt
Pl1 = (t-l) (r-l)+k-2
1
P12
= r(lc-t-l)
= (r-l) (k-t)
(r-l)(k-t)(k-t-l)
t
2 _ (r-l)(k-l)(k-t)
P22 t
- r(k-t-l)-l
We will call any association scheme with the parameters (3.8.5) and
for which (3.8.1) holds a pseudo-geometric association scheme, since
such a scheme may
(r,k,t).
e~dst
without bring derived from a partial geometry
However, if a pseudo-geometric scheme is a scheme derived from
a partial geometry, we will call it a geometric association scheme.
In this
secti~n
\,c consider no methods of constructing partial
geometries and the corresponding schemes.
However, several of the
association schemes ruentioned earlier in this chapter are special cases
of pseudo-geometric schemes.
gives rise to an Lr (1:) .scheme.
In particular, a partial geometry (r,k,r-l)
Thus a pseudo-Latin sCluare scheme is
just a special case of a pseudo-geometric scheme.
AlSO, a partial
geometry (r,k,r) gives us an SLB association sclremej thus we might
40
introduce the term pseudo-SLB scheme, corresponding to a pseudo- geometric
scheme with the appropriate parameters.
It has been noted previously
that a triangular scheme is a special case of an SLB scheme; hence we
r.1ight speak of a pseudo-triangular scheme as a special case of' a pseudo-
I
I
•I
geometric sche.me.
Let us now
ex~oune
the parameters (3.8.5) for possible BIB designs.
From the preceding paragraph, we knmv that we i,rill :)btain some; however,
there may be BIB designs obtainable from pseudo-geometric schemes Which
are not pseudo-Latin square or pseudo-SLB shcemes.
3.8.2. Designs from B _
l
··
.
1=2
For Bl t 0 be a BIB ~nc~dence
rnat r~x,
we nee d I'll
I'll; then
(t-l)(r-l)+k-2 = rt, or t = l{-r-l.
In this case tIle scheme has the
following parameters.
v
nl
..
!1~(l:-2)
k-r-l
=
= r(k-l)
_ i!:l)(r+l)(k-l)
n2 k-r-l
(3.8.6)
pil
=:
r(l;:-r-l)
2
I'll
pi2
=
(r-l)(r+l)
P12
1
_ r(r-l)(r+l)
p22 -
l;:-r-l
2
=
r(l\:-r-l)
=:
r
2
P22 -
2
;~
r-'+r - 2k+2
l;:-r-l
Thus a pseudo-geometric association scheme with the parameters
(3.8.6) is such that BJ. is a BIB incidence matrix.
that in this case
- -
r+2 < k < 2r+l
I
I
I
I
I
I
From (3).8.1), we see
I
I
I
I
I
I
-.
I
I
I
41
Then B1 is an incJ.dence matrix for a BIB
"I
r{1~-l), r(k-r-1»).
I
I
I
I
I
I
and the BIB design is given in section 3.5.2.
.e
I
I
I
I
I
I
.e
I
NO";'T
far the case k
= 21',
q~:~~~2)
~~!~i2) ,r(k-l),
,
the SCheljle (:3.8.6) is of
pseudo-Latin square type} and the BIB design abtaJ.ned from B is given
1
If J: = 21' + 1, we get a sche~,nc lr.tth SLB parameters,
in section 3.6.2.
~ ~ ~
:J.O.).
D'
eSJ.gns f ron B2-
1
We nee d P22
-1, or t
2
= P22
;
= k-r+l.
.1.1
lJ1en
(r-1)(k-t)(k-t- 11 = (r-1)(k-1)(k-t~ -r(k-t-l)
t
t
In this case the pseudo-geometric scheme has the
following parameters.
v = ~[r(k-2)+2]
1:-1'+1
n
1
(3.8.8)
= :c (l:-1)
_ (r-1)2(k-1)
k-r+1
~ -
1
2
P11 == dJ~.+1)-r -2
2
P 11 == .• (l~_r+])
1
2
P12 ::: (:'-1)
P2 ::: :c ( 1'-2)
J2
2
2 := i."-::-1) . (1'-2)
P22
::-1'+1
1 _ Ji:~JJ2(r-2)
P22 ::-1'+1
.L
.,.
•
The inequalities (3.8.1) impose the further restriction
l'
< 1: < 21'-1 •
Then a pseudo-geometric scheme vdth parameters (3.3.8) is such that
the association .matrix B is an incidence matrix for a BIB
2
(
1;:[1'(11:-2)+21 ~r(l:-2)+2J
(r_1)2(k_1) (r-1)2(l:-1~
(r-1)2(r-2))
k-r+1
}
1:-1'+1
}
k-r+1
} k-r+1
} 1:-1'+1
•
Note that for k
scheme; for k
= 2(r-l)}
= 2r-l}
I
I
..
the scheme (3.8.8) is a psuedo-Latin square
it has SLB parameters.
For [B :B ] to be a BIB incidence matrix} the condition is that
l 2
_
1
1
_
2
2
_.?
(
)
_(r-1){l~-1)(h:::.U
n l - n2 and Pll + P22 - Pll + P22 • Now nl - n2 Ior r k-l or t _ k(r-l)
- 2r-l
if t
k(r-l)
In this case ~
= n2 = r(k-l)} and
~2
t
= rk-r-l.
)
Now
pl + 1)1
= 2r-l } 1122
association scheme is as follows.
v
= (2r_l)2
n1
= 2r(r-l)
n2
= 2r(r-l)
~I
Pi1
= r 2 _r_l
2
Pu
= r(r-l)
1
P12
= r(r-l)
P12
2
= r(r-l)
P~2 = r(r-l)
P22
2
= r 2 -r-l
For a scheme with parameters {3.8.lOL [B l : B2 ] is an incidence matrix
for a BIB«2r-l)2} 2(2r_l)2} 4r(r-l)} 2r(r-l)} 2r2 -r-l). Note that
this series of designs is included in the series obtained from pseudocyclic schemes such that [B l :B ] is an BIB incidence matrix. The
2
members of the present series are the members of the series obtained
2
from pseudo-cyclic schemes for which u = r _r for some r > 2 (see
section 3.4).
I
I
I
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I
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I
I
I
I
I
I
-.
I
I
I
('
I
I
I
I
I
I
I
.-
CHAPrER
IV
THREE-CLASS ASSOCIATION SCHEMES
It.l.
Introduction
In this chapter vffi shall discuss some methJQs
t;ll~ee-class
J~
constructing
associati:m schemes and determine whether linear combinations
or juxtapositions of the association matrices Jbtained can be incidence
x~~Grices
for BIB designs.
The necessary and sufficient conditions
for the cases consiQered are the following:
for
for
I
for
[B.J..:B.J,
J
and
for
and
I
.1
44
It should be noted that it is unnecessar,rto consider the combinations
B
o
+ B ; for B + B lr.i.ll be a BIB incidence matrix if and only if
k
k
o
B. + B. is, since B + B} = J-(B. + B ) (i
J.
J
..
0
~
J.
j
4.2.
f
j
f
k; i,j,k = 1,2,3).
Group Divisible (GD) m-Associate Scheme [17]
4.2.1.
Definition of the scheme
Suppose we have the number of elements
v = Nl N2 •.• Nm, where all the N.J. 's are > 2.
He can denote an element by an ordered m-triple (iI' i , ••• , i m),
2
where i j €(O,l, ••. ,Nj-l) for j • 1,2, ••• , m.
Let two elements which
have only the first (X;l-j) coordinates in common be j:!?h,associates
Then vie have an association scheme, called a crollp
(j=1,2, ••. ,m).
divisible m-associate scheme, with the following parameters: for i=1,2,
••• , m,
1'Te
have
(4.2.1)
I
I
I
x J... 1
o(m-i)x(i-l)
-II
I
I
I
I
I
x.J.- 1
O(i-l)x(m-i)
I
I.
_ _ _ _ _ _ _
(j,k=1,2,3) ,
D(m-i+l)x(m-i+l)
I
I
..
I
I
I
I
I
I
I
-.
I
I
I
I
I
I'
-.
I
I
I
,
I
I
I
I
I
I
I
45
'I"here 0 xt is the sxt Y:Jatrix of zeros, x. 1 is a colur:m vector of order
s
~-
(i-I) with elements n l , n2 , •.. , n i _l , respectively, and D(m-i+l)x(m-i+l)
is a diagonal matri::
"'iTit11
diagonal elements
[NmNm- 1 ••• lit.c1-~"+2(Nm-~'+1-2 )], n.~+ l' n.+
, ••• , n m, respectively.
~ 2
2,
F·:Jr a 3-class GD schene, 'I'le ha.ve N , N
l
given below.
n
n
3
= N (N
-1)
3 2
2
= N3N2
(N
1
- 1)
2
P
.e
I
I
I
I
I
I
.e
I
N
3
n =0
Pi2
2
P 13
P~2 = N3 (N2-l)
1
=0
P23
P~3 = N3N2 (N1 -l)
2
P22
2
·P23
2
P33
An example is the following.
v
= 12
n 1 -- 1
~
=2
~
=8
= N3 -1·
=0
= N3 (N2 -2)
=0
.
= N3N2 (N l -l)
~
2, lr.i.. tIl the parameters
I
I
46
1
P11
1
p12
1
P13
=0
=0
=0
2
Pn
=0
p2
12
2
P13
=1
3
P12
=0
=0
3
P13
=1
=0
3
p22
=0
=0
3
P23 = 2
=8
P333
First
associates
Second
associates
1
P22 = 2
1
P23
1
P33
=0
=8
3-coordinate
representation
Element
1
2
3
2
1
4
3
6
5
8
7
10
9
12
11
(0,0,0)
(0,0,1)
(O,l,Oj
!~
(0,1,1
(1,0,0
5
6
7
(1,0,1~
8
a./
10
11
12
2
P22
2
P23
2
P33
I
(1,1,0
(1,1,1)
(2,0,0)
(2,0,1)
(2,1,0~
(2,1,1
p3
11
3,4
3,4
1,2
1,2
7,8
7,8
5,6
5,6
11,12
11,12
9,10
9,10
=0
..
=4
Third
associates
5, 6, 7, 8, 9, 10, 11, J2
5,6,7,8,9,10,11, 12
5,6,7,8,9,10,11, 12
5,6,7,8,9,10,11, J2
1,2,3,!~,9,10,1l,12
1, 2, 3, 4, 9, 10, 11, J2
1,2,3,4,9,10,11, 12
1,2,3,4,9,10,11, 12
1,2,3,4,5, 6, 7, 8
1,2,3,4,5, 6, 7, 8
1,2,3,4,5, 6, 7, 8
1,2,3,4,5, 6, 7, 8
Let us investigate the possibility of constructing BIB design from 3-class
GD scl1.emes.
h.2.2. Linear combinations
If B + B is
1
2
2
2
2 _ 3
Pl1 + P22 + 2P 12 - P1l +
1 + P1 + 2P1 =
a BIB incidence matrix, then Pl1
22
12
3
3 _ (12)
333
P22 + 2P 12 - A
• But Pl1 +P 22 + 2p12 = 0;
I
I
I
I
I
I
I
-.
I
I
I
I
I
I
-.
I
---------~
I
I
,
I
I
I
I
I
I
I
.-
therefore, B1 + B2 can not be a BIB incidence matrix.
(ii)
•
1
+ B_
)
for
N;<: - 2 + N_ N (n -1) = N N (N -1),
)
2 1
./
3 2 1
./
3
3
Pu + P3~) + 21...J r )
P
1
1
1
+ P
+ 2D J 11
33
~ .-5
= N3
= P211
or It.
:;
= 2;
N (N -2) + N -1
2 I
3
2
2
+ P
+ 2p
33
13
= 2N1
:N - 2N •
2
2
Then
Then B + B can not be a BIB incidence matrix.
1
3
(iii)
I
I
B
B + B
2
3
1
1
Nmv P
+P
22
33
for
Then B + B can not be a BIB incidence matrix.
2
3
3
Since P2n = 0 = PU'
B1 can not be a BIB incidence matrix.
3
B2 can not be a BIB incidence matrix, for P22
= o.
48
(vi)
~
..
matrix.
4.2.3.
Juxtapositions
We have nl
= n2
for N -1
3
= N3 (N2 -1),
or N
2
=
2 N -1
i .;
3
2 which satisfy the above relation.
there are no integers N ,N ~
2 3
[B l :B ] can not be a BIB incidence matrix.
2
N3
= 1
Then
1
+ (N -N NJ! •
2 2
We see that N - N2Nl = N (1-N2 )
-2; for, Nl , N2 ?: 2 and l-Nl :s -12
2
Then the above expression for N is negative; therefore [B :B ] can not
l 3
3
be a BIB incidence raatrix.
:s
Now n2
I
I
= n3
for N (N2-1)
3
= N3N2 (N l -l),
or Nl
=
+- ;
2N -1
thus
2
we have the same situation as for [B :B2 J, and [B2 :B ] can not be a
l
3
BIB incidence matrix.
Since no two of the n 's can be the same, certainly all three
i
I
I
I
I
I
I
I
-.
I
I
I
I
I
_.
I
I
I
I
"I
I
I
I
I
I
I
,I
I
I
I
I
I
,I
4.3. Tetrahedral Association Scheme [4]
4.3 .1.
Defin!tion of scheme
A three-class association scheme, called a tetrahedral scheme,
can be defined in a manner analogous to the definition of the twoclass triangular scheme.
Suppose there exists a set of v
= (;)
elements, for some positive integer n; we can denote the v elements
r r
by unordered triples (xl' x2' "3), where Xl x2 "3 and xl ,x2' and
"3 range from 0 to n-l. The elements can then be considered as points
in three-dimensional Euclidean space; two elements with the same
coordinates, in any order, will be considered the same.
call two elements
i~
For i=1,2,3,
associates if they differ in exactly i coordi-
nates; for example, the elements (1,2,3) and (1,4,2) are first associates.
This definition of association gives us an association scheme
in three classes, with the following parameters.
v
= n{n-lg{n-2)
nl = 3{n-3)
_ 3{n-' )(n-4)
n2 -
2
(4.3.1)
1
.
Pll : n-2
p
1
: 2(n-4)
12
1
P
~
: o·
ll
p3
12
: 9
Pl~
,.. 0
pi3 : 3(n-6)
1
P22
= (n-4) 2
P~ = 9(n-6)
I
I
50
P~3 = (n~4£(n-5)
P~3 = 3(n-6£(n- Zl
P~3 • (n-5)(n-6)
1 = (n-4)(n-5)(n-6) 2 = (n-5)(n-6)(n-7)
P33.
6
P33
6
p3 _ (n-6)(n-Z)(n-8)
336
-
We see that tor such a scheme to exist we must have n > 6.
Let us examine the parameters ot the tetrahedral scheme and attempt
to construct BIB designs.
4.3.2.
Linear combinations
(i)
B + B
l
2
Now pil +
P~2
p2 + p2 + 2p2
11
22
12
+
=4
2p~ = (n-2)
+ (n-5)(n+2) + 4(n-4)
2
2
2
have n -3n-2
2
+ (n_4)2 + 4(n-4) ... n -3n-2, and
= n +~n-34
, or n2 -lln+30
= 0;
2
=n
+5n-34 Then we must
2··
in order tor tetrahedral
scheme to exist, we must take n = 6. But pil + P~ + 2p~ = 9(n-6)+18
111
= 18 tor n = 6, whereas Pll
+P22 + 2p12 = 16; then B +B can not be
l 2
a BIB incidence mtrix.
Bl + ~
1
1
1
(n-4){n-5) (n-6)
We have Pll + P33 + 2p13 = n-2 +
6
(ii)
2
3
n -15n + Bon-1,32
6
3
2
... n - lBn
'
=
and p2 + p2 + 2p2
4 + (n-5)(n-6)(n-7) + 2(n 5)
11
33
1,3 ...
6
-
6119n - 246.
The condition then requires
2
2
n3 _ l5n2 + Bon - 132 = n3 - lBn + ll9n - 246, or n - 13n + 38 ...
o.
..
I
I
I
I
I·
I
I
-.
I
I
I
I
I
I
-.
I
I
I
51
(iii)
('
I
I
I
I
I
I
I
•e
I
I
I
I
I
I
.e
I
B +
2
P~2 + P~3
Now
= n3
:a,
2
- 3n - 28n + 96
6
)(6-5 ){n-6) + (n-4 )(n-5)
+
2p~3 = (n_4)2 + (n-4
'
and p2 + 2 + 2 2 _ (n-5)(n+2)
22 P33
P23 2
2
- 3n
(n- 5 )(n 6)(n-7) + 2(n-5)(n-6)
= n3
2
have n3 - 3n - 28n + 96
2
- 3n - 34n + 120, or n
6
Since P11
(v)
For n
Then we must
= 4.
Since a
Bl can not be a BIB incidence matrix.
B
2
-
Now P~
Then n
= 0,
= n3
634n + 120.
+
=6
= 7,
or n
1
P22
= P~
for (n_4)2
= 7.
For n
2
= P22
= 9,
= (n-5 )!n+2)
, or n2 -13n+42
= 6, P~ = P~2 = 4;
3
and P22
= 9.
but
= O.
P~2 = 9(n-6) = o.
In this case B2 is an incidence
matrix for a BIB(35, 35, 18, 18, 9).
(vi) ~
1
2
Now P33 = P33
f
or
Ln-4)tn - 5)(n-6) _ (n-5)(n-6)(n-7) • t~-6
' -.....
n must be 6 for a tetrahedral scheme to exist.
2
p3 - 0
P33 = 33 - • Then
4.3.3.
~
Juxtapositions
But for n = 6, P~3
can not be a BIB incidence matrix.
=
52
and v = 20.
integer.
~t
for n = 6,
A.t2 • 2(i~(8)
=
~~,
and
\2
is not an
Then [B :B ] can not be a BIB incidence matrix.
l 2
~ii) [Bl:~]
Now
~ = n;
for 3(n-3) = (n-3)(6-4 )(n-5) , or
~
- 911+2 = O.
Since the above has no integral solution, [Bl:~] can not be a BIB
incidence matrix.
(iii)
[B :B ]
2 3
Now" = n., for 3(n-3)(n-4)
.~:J
then
~
3~90
, and
=
~
... !n-3)(n-4)(n-5)
6'
2
... 165 and v
= 364.
But for n ... 14,
'23 is not an integer.
Then
[B2:~]
or n
= 14,·
\.3'" 33~t;29)
=
can not be a BIB
incidence matrix.
[Bl :B2 :B ]
3
From <til above we see that n
(iv)
~
for all integral n.
3
[B :B :B ] can not be a BIB incidence matrix.
1 2 3
4.4.
l
n
Then
Cubic Association Scheme [18]
4.4.1.
The scheme
Suppose we have a set of v =n' elements, for some integer n
~ 2.
Consider the v elements as ordered triples (Xl' x2 ' x,), where xl'~'
and X, range from 0 to n-l. Call two treatments i~ associates if
they have exactly i
coordinates different (i
= 1,2,
,).
EqUivalently,
we can consider the elements as points in tbree-d..imensional Euclidean
space.
Then the first associates of a point ex are those points lying
on the three lines through ex which are perpendicular to the coordinate planes; the second associates of ex are the remaining points
I
I
..
I
.1
I
I
I
I
I
-.
I
I
I
I
I
I
-.
I
I
:1
'·e
I
I
I
I
I
I
I
I
53
lying in the t~eeplanes determined by the first associatesaf ex;
other points are third a.ssociates of ex.
The resulting scheme is an association scheme in three classes,
called a cubic scheme.
= n3
v
nl
= 3(n-l)
n
= 3(n-1)
2
n,
(4.4.1)
2
= (n_l)3
1
= n-2
2
P11 = 2
1
= 2(n-l)
P12
Pl1
P12
1
P13
=0
...; 0
P3
li3
12
2
= 2(n-2)
p
2
= n-1
P~
2
P13
=3
= 3(n-2)
P22 = 2(n-1)(n-2)
P22
= n2 -2n+2
P~2 = 6(n-2)
1
2
P23 = (n-1)
P~3
= 2(n-2)(n-l)
P23 = 3(n-2)
1
k33
2
133
= (n-1) (n-2)
1
.I
I
I
I
I
I
e
I
I
The parameters are the following.
= (n-l) 2 (n-2)
3
3
P33
2
2
= (n-2) 3
An examination of the above 'parameters shows that we can obtain
one BIB design from the a.ssociation matrices of a cubic scheme.-
4.4.2.
Linear combinations
(i) B1 + B
2
Now P~ + P~2 + 2Pi2 = (n-2) + 2(n-1)(n-2) + 4(n-1)
and
P~ + P~2 + 2p~
2n2 -n-2
= n2+2n_4, ~r
2
= 2n2 -n-2,
2
= 2 + n -2n+2+4n-8 = n +2n-4; then we must have
n2 -3n+2=Q. The only n 2: 2 which satisfies this
relation is n = 2; then
But in this case ph +
P~
+
P~2
+
2p~ = pil
P~ + 2Pi, = 6n - 12 + 6
+
P~
+
2p~
= 4.
= 6; therefore B1+B
2
I
I
-.
can not be a BIB incidence matrix.
(ii)
11. + :a,
1 2
Now P11 + P + 2P = (n-2) + (n-1) (n-2)
33
13
1
1
= n3 -4n2~n-4, and
2
2
2
2
3 2
P11 + P + 2p13 = 2 + (n-1)(n-2) + 2n-2 = n -5n + 100-4; then we
33
need n3 -4n2~n-4= n3 -5n2+10n-4, or n2 -4n-o.In this case n = 4 and
1
1
122
3
P11 + P33 + 2p13 = Pu + P + 2P13
33
(n_2)3 + 6(n-2) = 20 for n
A(13) = 366~5= 20.
III
4; v
= 20.
= 64, ~
:3
3
3
Now P11 + P33 + 2P13
= 9, . ~= 27,
n,
=
= 27, and
Sp B1 + B is an incidence matrix for a BIB
3
I-
(64, 64, 36, 36, 20).
B + B,
(iii)
2
1
1
1
2
We have P22 + P + 2p23 = 2(n-1)(n-2) + (n-1) (n-2) + 2(n-1)
33
2
3
2
2
2
2
2
= n -5n+4, and P22 + P" + 2p23 = n -2n+2+(n-1)(n-2) + 4(n-2)(n-1)
= n3 -6n:+6;
our condition requires that n' -5n+4 = n' -6n~, or n = 2.
1
1
1
2
2
2
In this case P22 + P3, + 2p23 = P22 + ~, + 2P23
= 2;
2P~3
Then B2 +
= 6(n-2)
+ (n_2)3 +6(n_2)2
=0
for n
= 2.
'....3
but P22 + .!:33 +
~.
can not
be a BIB incidence matrix.
Now Pt. = 0; then B1 can not be a BIB incidence natrix.
(v) ~
1
2
2
2
We have P22 = P22 for 2(n-1)(n-2) = n -2n+2, or n -4n+2 = 0;
this relation has no integral solution, and B can not be a BIB in2
cidence matrix.
I
I
I
I
I
I
-.
I
I
I
I
I
I
-.
I
I
I
.I
I
I
I
I
I
I
.-
I
I
I
I
I
55
(vi) ~
1
2
2
2
Now P33 = P33 for (n-l) (n-2) = (n-l)(n-2) , or n = 2.
in this case
P~3
=
P~3
=
P~3
= o.
Then
B:3
But
can not be a BIB incidence
matrix.
4.4.3.
Juxtapositions
(i)
[B :B2 ]
l
We have nl
v
= 8.
= n2
for 3(n-l)
But in this case
= 3(n-l) 2 ,
~ = 2(3~(2) =
or n = 2; then ~=n2=3 and
¥-, and \2 is not an integer.
Thus [B :B ] can not be a BIB incidence matrix.
l 2
(ii)
[B :B ]
l 3
Now ~ = ~ for 3(n-l) = (n_l)3, or n -2n-2 =
2
o.
Since this
relation has no integral solution, [Bl :B:3] can not be a BIB incidence
matrix.
(iii)
[B2 :B ]
3
Now ~
~ =~
= n3
= 27 and v
= (n_l)3, or
for n = 4, \3
for 3(n_l)2
n = 4; in this case
= 64.
=
But
2(2t~(26) = 1~6
, and
\3 is not an integer. Thus [~:~] can not be a BIB incidence natrix.
The condition requires nl = n2
that n 1 and
= ~;
but we saw in (ii) above
11:; can not be equal. Then [B1 :B2 :B3 ] can not be a BIB
incidence matrix.
56
Rectangul~ Association
4.5.
4.5.1.
Scheme [26]
The scheme
Suppose we have a set of v = .en elements for some integers .e, ~ 2.
Then we can arrange the v elements in a rectangular array with .e rows
and n columns.
If two elements appear in the same
:roY,
call them
first associates; if they appear in the same column, call them second
associates; otherwise call them third associates.
Then we have a three-
class association scheme, called a rectangular scheme, with the following parameters.
v
~
1
Pll
p
l
12
= n-2
= .en
= .e-l
n =. (.e-l) (n-l)
3
2
Pll = 0
= 0
2
PJ2 = 0
2
;11 = 0
;12 :: 1
pl = 0
13
P13
= n-l
;13 = n-2
P~
2
P22
=.e-2
;22 = 0
= 0
P23 = .e-l
2
P23 = 0
~3
= .e-2
~3
2
P
~3
-= (.t-2)(n-2)
1
= (.e-l) (n-2)
33
= (2-2)(...1)
Let us examine the pa;rameters of the rectangular scheme to determine Whether linear combinations or partitions of the association
matrices can give us BIB designs.
4.5.2.
Linear combinations
(i) B1 + B2
I
I
-.
I
I
I
I
I
I
I
-.
I
I
I
I
I
I
-.
I
I
I
.e
I
I
I
I
I
I
I
.e
57
11122
2
Now Pll + P22 + 2P12 = n-2, and Pll + P22 + 2p12
we must have .t=n.
Now pil +
common value of.t
and
I
+
2P~
n must be 4.
In this case v = 16, n =
- 2.
-
(ii)
B +B
l
3
P~3
Now pil +
pil +
P~3
+ 2Pi,
t, n.
Then B1 +
(iii)
B +
2
3 = n-2 + (.t-l)(n-2) = £(n-2), and
= (t-2)(n-l)
= t(n-l),
that t(n-2)
+ 2pi
B.3
+ 2(n-l)
= .t(n-1);
the condition requires
which is an impossible relation for positive
cannot be a B IB incidence matrix.
:a,
1
We have P22
+1
P" +1
2P2, = (.t-l)(n-2 ) + 2 ( t-l) = n(t-l), and
P~2
+ P;,
+2p~, = t-2
+ (.t-2)(n-l)
= n(.t-2).
Then we
II1\B t
have
= n(.t-2), which can not be satisfied for positive .t, n. So
B2 + B, can not be a BIB incidence matrix.
2
,
= Pll = 0,
Bl can not be a BIB incidence matrix.
P~2 = P~2 = 0,
B2 can not be a BIB incidence matrix.
Since Pll
(v) ~
Since
~
12,1,
()
We need P" = P" = P,,; but P" = P" for £-1 (n-2)
.e
I
3,
incidence matrix for a BIB(16, 16, 6, 6, 2).
(vi)
I
then
= 2, which tells us that the
- 9, and "n1 -- 3, n2 -- 3 , n3 "12 -- (6){5)
15
n(.t-1)
I
I
I
P~
= .t-2;
(£-2)(n-2), or n
= 2.
a BIB incidence matrix.
In this case P;,=
P~,
= 0, so
~
=
can not be
58
4.5.3.
JUxtapositions
(i)
[B :B ]
1 2
We need
o. Then
P~
+
P~
=
P~l
+
P~
a
~l
+
~
[~:~] can not be a BIB incidence matrix.
or
.e=2.
112
2
3
3
P11 + P33 = Pll + P33 ... Pn + P33; but for
.e=2,
Now n1 = n,for n-l
= o.
;
= (.e-l)(n-l),
The oondition requires
223
3
Pll + P33 = P11 + P
33
So [B1:B;] can not be a BIB incidence natrix.
We have ~
= n,
for .e-l
= (.e-l)(n-l),
or n
= 2. 'The
condition
1
1
2
23
3
1
1
requires P22 + P33 ... P22 + P33 = P22 + P33; but for n = 2, P22 + P
33
=~+Pg3=O' Therei'ore
(iv)
Now
[B2 :B;] can not bea BIB incidence matrix.
[B1 :B2 :B;]
~ = ~ = n;
for .e=n=2; but then
2
2
2
3
3
3
P11 + P22 + 133 = PI1 + P22 + P33 =
BIB incidence matrix.
ph + P~2 + P~3 ...
o. So [B1 :B2 :B;] can not be a
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59
the array.
4.6.2.
Definition of the scheme
Suppose we have an orthogonal array (n2 , 131 + 132 , n, 2), where
~,
+ 13 oS n.
2
2
Consider the n assemblies
131 , and 13 are positive integers such that n
2
=1
We see that the index"
as treatments.
in this case.
~
2 and 13
1
Define two treatments a and 13 as first associates if
the columns corresponding to a
and 13 are alike in exactl:y one position.
in the first 131 rows; let ex and 13 be second associates if the columns
corresponding to them coincide in exactl:y one" position in the remaining
132 rows; otherwise a and 13 will be third associates. Since the array has
strength 2 and index 1, we see that the definition of association is
unambiguous; for two columns of the array can be alike in at most one
position.
Then we have a three-class association scheme with the
following parameters.
2
v=n
n
1 = 131 (n-1)
~ = 132 (n-1)
D:? = 133 (n-1), where 133 = n+1 - (131 + 132 )
(4.6.1)
1 2 "
3
P11 = n-2+(t31- 1 )(l3l - 2 )
P11 = 131 (131 - 1 )
Pll = 131 (131 - 1 )
P~ = 132 (131 - 1 )
P~ = 131 (132 - 1 )
P~
1 2 3
P13 = 133 (13 1 - 1 )
P13 = 131 133
P13
P22
1
= 132 (132 - 1 )
P22 = n-2+(132 - 1 ) (132 - 2 )
2
Pk3
= 133 (133 - 1 )
P~3 = 133 (133 - 1 )
= 131 132
= 131 (133 - 1 )
3
P22 = 132 (132 - 1 )
1 2 3
P23 = 132 (13 -1)
P23 = 132 13
P23 = 13 (132 - 1 )
3
3
3
P~3
= n-2+(133 -1)(13 -2)
3
60
Now let us attempt to construct BIB designs from linear oombinations
and juxtapositions of the association matrices B.•
J.
4.6.'.
Linear combinationj
(i)
B
1
.
1
2'
The requirement is that P11 = Pll = Pll •
2'
Now P11 = Pll =
131 (13 ..1);
1
pi! = n-2 + (131 - 1)(131-2) = 131 (131 -1)
for 131
= n/2
•
Then n must be even and we must take 13 = n/2 ; in this case the
1
association scheme with parameters (4.6.1) is such that B:t is an
2 2
incidence matrix for a BIB(n , n , [<n-1), ~n..1), ~n-2».
(ii)
B
2
-
We need
P~2
n-2+(13 -1)(13 -2).
2
2
=
P~ = J22
• Now
P~
=
P~ = 132 (132 - 1),
and
P~
=
Then our condition is satisfied for 13 = n/2, in
2
2
2 n
which case ~ will be an incidence matrix for a BIB(n ,n ,~n..1),
thus we get no new designs from considering B2 •
[<n-1), t(n-2»;
(iii)
~
We see that pk, =
~,
=
13,(~-1), and~,
= n-2 + (13,-1)(13,-2);
thus if 13, = n/2, we get the same series of BIB designs as obtained
from
~
and B2 •
(iv) B1 + B
2
The condition in this case is that
P~
+
2p~
=
ph + P~2 + 2p~ = P~ +
ph + P~2 + 2p~ = 7\(12), Where 7\(J2) =
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61
-
~
~
(nl+n2)(nl+~-1)
is a positive integer.
v-I
n-2+(~1-1)(~1-2)
+
~2(~2-l)
P~
+
P~
the value of
~l(~l-l)
+
+2
2P~
~2(~1-1) =
is the same.
~2(~2-l) + 2~1~2 = (~l +~2)2
+
III
Now Pll + P22 + 2p12 =
-
n +
(~1+~2)2_3(~1~2);
We have
(~1+~2)'
ph + P~ + 2P~
Then we must have
n
or ~1+~2 = 2'
iii
n(N-2)
In this case the comroon value of Pll + P22 + 2p12 is
n + (~1~2)
2
- 3(~1~2)
)
= (~1~2)2
- (~1~2 '
is also the value of A(12).
2 2'
n n, 2 ) exists, the derived association
the orthogonal array ( n,
scheme with parameters given by (4.6.1), where 131 and ~2are any positive
integers such that ~1+~2 =~, is such that ~-+~ is an incidence matrix
for a BIB(n2 , n2 ,
~n-l), ~n-l)"
n(4-2l ).
Again, we have the same
series of designs as we got from considering Bi alone.
1-
worthwhile to consider B +B also, since it may allow us to
l 2
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~
, which
Thus if n is an even positive integer and
~
I
=
designs not obtainable from the B alone.
i
However, it is
const~ct
For example, let n = 6.
Then to construct the BIB(.36, 36, 15, 15, 6) from B , B , or B.3 w~d
l
2
require the existence of an orthogonal array (.36, 4,6,2); we see this
by letting ~1'~2' or 13.3 be~, or 3.
But the existence of such an array
is equivaient to the existence of a pair of orthogonal Latin squares of
side 6, which is impossible; then the orthogonal array (.36,4,6,2) does
J
not exist, and B (i=1,2,.3) can not be an incidence matrix for the
i
BIB(36,36,15,15,6) • However, the orthogonal array (.36,.3,6,2), which
J
J
~
I
is equivalent to one Latin square of side 6, does exist.
~l
= 1,
~2
Letting
= 2, we see that the association scheme with parameters
(4.6.1) is then such that B +B is an incidence matrix for a BIB(.36,36,
l 2
15,15,6) •
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The requirement for
~+~
to be a BIB incidence matrix is that
A 11ttle manipulation like that done when considering B +B will show
l 2
that the requirement is satisfied when 13 1' ~
=~
for same even positive
integer n; then we get the same BIB deligns by considering B
l
got from Bl +B •
2
(vi)
+:a,
as we
B +:B,
2
As in parts (iv) and (v), B +B, will be an incidence matrix for
2
2
2
n(·
)
n(
)
n(4-2»
.
n and the
the BIB ( n , n , 2' n-l , '2 n-l ,
when ~2"~3
= 2'
2
orthogonal array (n ,
132"~' n, 2) exists.
The requirement for [B :B ] to be a BIB incidence matrix is that
l 2
1122,
3
nl = ~ and Pll + P22 = Pll + P22 = Pll + P22 = \2' where \2 ==
2n (n -l)
l
v-l
and pil +
P~2
2
n + 2x -
4x
~
2
and
tied.
= ~ for 131 =13 ; let x be the
2
l
11222
1'hen P1l + P22 == Pu + P22 == n+2x -4x,
is a positive integer.
conmon value of 131 and 132 •
= n
=
Now n
= 2-?- - 2x; we see that for equality we must have
==
2
2x - 2x, or x
n(~-l); then
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4.6.4. Juxtapositions
1
-.
\2
==
n
2·
i
i
n(n-2)
In this case P1l + P22 == .. 2
'
==n(~-2)._ and our requireJlSnt is satis-
Then if n is a.n even' integer > 2 and the orthogonal array
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2
(n , n, n, 2) exists, the scheme with parameters (4.6.1) is such that
[B ·B ] is an incidence matrix for a BIB(n2 2n2 n(n-l) n(n-l) n(n-2»
1: 2
"
'2' 2
•
The simplest example is for n :::: 4; then we get the BIB(16, ,2, 12,6, 4).
(ii)
[Bl:~]
,
[B2 :B,]
For [Bl:~]' the requirement is that nl :::: ~ and pil + P~,
2
2
. Pll + P"
,3
:::: Pll + P22::::
::::
.
tu·
By the same reasoning as for [B :B ],
l
2
we see that if 131 :::: 13 :::: ~ and 132 :::: 1 and if the orthogonal array
3
(n2 , n;2, n, 2) exists, then the scheme 'With parameters (4.6.1) is such
that [B :B ] is an incidence matrix fo~ a BIB (n2 , 2n2, n(n_l),n(~-l) ,
l 3
n(n-2)
) • Therefore we gain nothing extra by considering [Bl:B,]'
2
Similarly, we get the same series of designs by considering [B2:B,]'
We need nl :::: n2 ::::
~;
J~
: : "'3.
then 131 :::: 132 :::: 13 ::::
3
n+l
T
'
and n1 :::: n2 ::::
n,
. also that P 1 + P1 + P 1 :::: P2 + P2 + P2 ::::
We reqUJ.re
11
11
22
22
33
33
2
3
3,
n -4
Pl1 + P22 + P" :::: n1-1 :::: ""3. We see from (4.6.1) that for
131 :::: 132 :::: 13, :::: n;1 the requirement is satisfied. Then if n is greater
than 2 and n == 2 (mod 3) and if the orthogonal array (n2,2(~+1),n,2)
exists, then the association scheme with parameters (4.6.1) is such
2 2 2 n2 _1
that [B :B2 :B ] is an incidence matrix far a. BIB(n ,3n ,n -1,~,
l
3
2 .
n -4).
3
The simplest case, n
=
5, gives us the BIB(25,75,24,8,7).
-I-
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-.
BI:8LIOORAPHY
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e.
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[6]
Bose, R. C. and Nair, K. R.
(19'9),
complete block designs."
[7]
Bose, R. C. and Nair, K. R.
''Partial1:y balanced in-
sankb;yi', 4, 337-'72.
(1962),
'~esolvable
balanced in-
complete block designs with two replications."
Series A,
[8]
Sankb;yi',
24, 9-24.
Bose, R. C. and Shimamoto, T. (1952), "Classification and ana.J.ysis
of partially balanced incomplete block designs with two
associate classes."
Journal of the .American Statistical
Association, 47, 151-184.
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65
[9]
Bose, R. C. and Shrikhande, S. S. (1960),
"On the composition
of balanced incomplete block designs."
Canadian Journal
of Mathematics, 12, 177-188.
[10]
Connor, W. S. (1952),
block designs."
"On the structure of balanced incomplete
Annals of Mathematical Statistics, 23,
57-71.
[11]
Fisher, R. A. (1940),
"An examination of the different possible
solutions of a problem in incomplete blocks." Annals of
Eugenics, 10, 52-75.
[12]
Fisher, R. A. and Yates, F.
edition, Edinburgh:
[13]
Guerin, R. (1963),
(1963),
~tistica1
Tables, Sixth
Oliver and Boyd.
"Survey of balanced and partially balanced
incomplete block designs."
International Statistical
Institute, 34th Session.
[14]
Hanani, Haim
(1961),
"The existence and construction of
balanced incomplete block designs." Annals of Mathematical
Statistics, 32, 361-386.
[15]
Mesner, D. M. (1964),
"Negative Latin square designs."
Institute
of Statistics, Mimeo Series No. 410.
[16]
Nair, K. R. and Rae, C. R. (1942),
incomplete block designs."
[17]
Raghavarao, D. (1960),
"A note on partially balanced
Science and Culture, 7, 568-569.
"A generalization of group divisible
designs." Annals of Mathematical Statistics, 31, 756-771[18]
Raghavarao, D. and Chandrasekhararao, K. (1954), "Cubic designs."
Annals of Mathelnatica.1 Statistics, 35, 389-397.
66
[19] Rao, C. R. (1961),
with
"A study of balanced incomplete block designs
replications 11 to 15."
S~
Series A, 23, 117-
127.
[20] Ryser, H. J. (1963),
Combinatorial Mathematics, Syracuse:
Mathematical Association of America.
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