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ON CONSTRUCTING BALANCED INCOMPLETE BLOCK DESIGNS
FROM ASSOCIATION MATRICES
WITH SPECIAL REFERENCE TO ASSOCIATION SClIElMES
OF TWO AND THREE CLASSES.·
by
William C. Blackwelder
University of North Carolina
Institute of Statistics Mimeo Series No. 496
November 1966
1
This investigation was supported in part by a Public
Health Service training grant (number GM-38) from the
Institute of General Medical Sciences, Public Health
SerVice, and in part by the Air Force Office of
Scientific Research Contract Number AF-AFOSR-76o-65.
DEPARTMENT OF STATISTICS
UNIVERSITY OF NORTH CAROLINA
Chapel Hill, N. C.
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Introduction
1.
1.1 Balanced Incomplete Block Designs
Balanced incomplete block (BIB) designs were introduced by Bose
[1
J and have since been studied extensively. Connor [6J introduced the
concept of an incidence matrix for a BIB design; in this paper such designs
will be examined via their incidence matrices.
(v, b, r, k,
~),
Suppose we have a BIB
where v is the number of treatments, b is the number of
blocks, r is the number of replications, k is the block Size, and
~
is the
number of blocks in which any pair of treatments occur together.
The
v x b matrix N is the incidence matrix for the BIB design, where
= (n ij ),
(1.1)
N
(1.2)
n ..
=1
=0
otherwise.
~J
and
if the ith treatment occurs in the jth block,
A necessary and sufficient condition for a v x b matrix N of O's and lIs
to be an incidence matrix for a BIB (v, b, r, k, ~) is that the following
relations hold:
o<
N' J
(1.5 )
By J
by I
~
v
< r ;
=K
=r
N Nt
J
b ,v ;
Iv +
~(J
v
- I )
v
[loJ.
we mean the v x v matrix of l's, by J
v
b ,v
the b x v matrix of l's, and
the v x v identity matrix.
v
Now for any m x n matrix A of O's and lIs, let us call the matrix
J
m,n
-
A the complement of A.
BIB (v, b, r, k,
~)
,
Then if N is an incidence matrix for a
and i f k < v -1, the complement of N is an incidence
matrix for a BIB (v, b, r *, k * , I... * ), where r *
~*
=b
- 2r +
~.
1
=b
*
- r, k
=v
- k, and
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1.2 Association Schemes
[2J, [4J
An Association scheme in m associate classes is a set of v elements
(objects, treatments, varieties) which satisfies the following conditions:
(i)
any two treatments are either 1st, 2nd, ••• , or mth associates,
and the relation of association is symmetrical;
(11)
each element has exactly n. ith associates (i
1
where the number n
if ex
(iii)
and
~
i
= 1,
2, ••• , m),
is independent of the element chosen;
are ith associates, then the number of elements
j~
which are
i
-
associates of ex
and kth associates of
~
i
is Pjk '
and Pjk
is independent of the pair of ith associates chosen
(i, j, k
=
1, 2, ••• , m).
The numbers
(1.6)
are called the parameters of the association scheme; all must be positive
integers.
The following relations among the parameters are easily shown:
i
( 1. 7)
p
(L8)
.~
jk
i
kj
= p
ni
= v-l
i=l
(1. 9)
= n. if i
J
= nj
.
(1.10) n. p
1
i
jk
f
- 1 1' f '1
= n
j
j,
= J;.
pj
ik
It is useful to make the convention that each element is the zero-th
associate of itself and of no other elements.
2
Then we must have
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(1.11)
n
(1.12)
P ..
0
0
= 1
,.
0
~J
J~
= n
i
ko
(1.13 )
= P
P
i
j
f
= 0 if i
01:
,
k,
= 1 if i = k.
Then (1. 8) and (1. 9) become
~ n.
(1.14 )
.
~
.'
J.=o
= v,
m
I
(1 15)
i
P'J k . = n.J
k=o
For the case m=2, it is sufficient to specify v, n1 ,
P~l
' and
P~l
and the other parameters are then determined; see, for exaT:1ple, [2]
Given an m-class association scheme, we call the matrices B.
J.
m) the association matrices of the scherae, "There
(i = 0, 1,
••
0
,
,v
a
li
(1.16)
and
1
b .
v~
(1.17)
b
8
ai
= 1 if
a
and
~
are i!£ associates,
= 0 otherwise
Clearly, we have
I
j
if i = j;
I'
I
I
f
= 0 if i
= po.
3
v
b .
1n.
'
I
~18)
B
o
= I
.-I
v
and
B + B + •.• + B
o
1
m
Also, the linear form c
if and only if Co
o
= Jv •
B + C B + ••. + c B is equal to the zero matrix
0
11m m
= c l = •.• = cm = 0;
i.e., the Bi'S are linearly independent.
The association matirces satisfy the relation
(1.20)
BkB j
(j,
= BjBk =
k = 0, 1,
o
1
P'lB
+ Pjk B1 + •.•
J~ 0
" ' , m).
The result (1.20), along with the linear independence of the B. IS, will be
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quite important in the proofs of the theorems of Sections 3 and 4 of this
paper.
Association schemes were introduced by Bose and Shimamoto (5] to aid
in the classification and analysis of partially balanced designs.
In the
present paper association schemes will be used in the construction of BIB
designs by the following tvlO methods, where B. denotes an association
l
Inatrix:
(i)
obtaining a matrix of the form B.
II
+ B. + ••• + B. which
l2
It
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-a
II
II
will be a BIB incidence matrix, and (ii) obtaining a matrix of the
form (B. : B.
ll'
J.
: ••• : B. ] which will be a BIB incidence matrix.
2
•
lS
First, however, let us define a number of known types of association
schemes of two and three classes.
II
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2.
Types of Association Schemes
2.1
Association Schemes of Two Classes
[5J
(a) Group Divisible (GD) Scheme
Suppose, for integers I > 2 and n
of v
= In
with t
elements.
~
2, there is a set
Let the elements be arranged in a rectangular array
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= In
t-n
n2 = n(
1
Pn = n-2
1
P12 =
1
P22 = n( £-1)
(2.1)
°
(b)
2
III = °
2
I1.2 = n-l
2
122
= n( t-2)
Triangular Association Scheme
[5J
Suppose, for some positive integer n, there is a set of
v = (n) = n(n-l) elements. Arrange the v elements in an nxn array as
2
2
follows: leave the leading diagonal positions blank, and fill the n(n-l)
2
positions so as to make the array syrrnnetric 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.
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Such an array is an association scheme, called a triangular association
scheme.
The parameters of the scheme are as follows.
= n(n-l)
v
2
(2.2)
n
l
= 2n
n 2 '"
1
Pu
1
P12
1
P22
I
- 4
(n-2) (n-3)
I
2
2
Pll
2
P12
2
P22
= n-2
- n-3
(n-3) (n-4)
=
2
= 4
= 2n
- 8
= (n-4) (n-5)
2
From the parameter values, we see that n > 4.
(c)
Pseudo-Cyclic Association Scheme
Suppose there is a set of v elements; denote them by the integers
Suppose there is a set of integers (d , d , ••• ,
l
2
satisfying the following conditions:
1, 2, •.. , v.
(i)
(ii)
the d's are distinct, and 0 < d < v (j= 1, 2, ••. , n );
j
l
among the nl{nl-l) differences d i -dj(i~j; i, j=l, 2, ••• , n )
l
reduced (mod v), each of the numbers d , d , ••• ,d
occurs
2
l
nl
a times and each of the numbers e l , e 2 , ••. , e occurs p
n2
times, where d , d , ••• , d , e , e , ••• , e
are all the
2
l
2
l
n2
nl
integers, 1, 2, •.• , v-l. Clearly, n ~n2 p =nl(nil).
l
Given the element k (k
.-I
=
1, 2, ••• , v), define its first associates
as the elements k+d , k+d , ••• , k+d
(mod v); the remaining (v-n -1)
l
2
n
1
l
elements are the second associates of k. Then we have an association scheme,
called a cyclic association scheme, with the following parameters.
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v
n
n
(2.3)
l
2 = v-n1 -1
2
Pll = 13
1
Pll = a
1
P12 =n1 - a-I
2
P12 = n l - 13
2
1
P22 = n2 -n l + a+l
P22 = n2 -n1 + 13-1
We see that, given v, the set of d's completely determines such a
scheme.
A few examples of cyclic association schemes are given below.
Some Cyclic Association Schemes
n
v
n
13
6
6
17
8
29
14
8
14
l
Set of d's
2
2,5,6,7,8, 11
3,5,6,7,10,11, 12, 14
1,4,5,6,7,9, 13, 16, 20,22,23,24,25,28
All the known cyclic association schemes are such that v= 4u+l,
n l = n2 = 2u, and
a = u-l, for some positive integer u.
Then the
association scheme has the following parameters.
(2.4)
v
= 4u
n
=n
l
2
+ 1
= 2u
1
Pll = u-l
2
Pll = u
1
P12 = u
2
P12
=u
2
P22 = u-l
Following the nomenclature in [7J, we will call any association scheme
1
P22 = u
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satisfying the parameters (2.4) pseudo-cyclic, whether or not it is obtainable
by the cyclic method described in [5 J.
(d)
Singly Linked Block
(SLB)
Association Scheme [2J, [5J
Suppose N' is an incidence matrix for a BIB design with b
treatments, v blocks, k replications, block size r, and
pair of treatments occurs together in exactly one block.
b-l = k(r-l); this gives us v = k(rk-k+l)
r
~
= 1; i.e. every
Then bk .. vr and
and b=rk-k+l.
It has been shown that in this case N is an incidence matrix for a
partially balanced incomplete block (PBIB) design with v treatments, b blocks,
r replications, k plots per block,
~l
= 1, and
~2
= O.
Defining first
associates as two treatments which appear together in some block of the
derived PBIB design, we get a two-class association scheme called a singly
linked block (SLB) association scheme, with. the following parameters.
r
l
n2
= r(k-l)
= (k-r)(r-l)(k-l)
r
1
2
Pll = k-2+(r-l)
2
2
Pll = r
2
P
= r(k-r-l)
12
2
2
P22 = (k-r) + 2(r-l)
1
P12 = (r-l) (k-r)
1
(r-l) (k-r) (k-r-l)
P22 =
r
k(k-l)
r
For r=2 the SLB scheme is the same as the triangular scheme with m = k+l.
(e)
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III
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v = k(rk-k+l)
n
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Latin Square (Lg(n) ) and Pseudo-Latin Square Association
Schemes _ 2 J, ~ 5 J, [ 7 J
2
Suppose we have a set of v= n elements, arranged in an nxn array.
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
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be second associates, we can define an L2 (n) association scheme.
For 3 ::: g ::: n+l, if a set of (g-2) mutually orthogonal nxn Latin
squares exists, we can define a Latin square (Lg(n) ) association scheme
from the nxn array of the v elements in the following manner.
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 g=4, n=4, we can take the Latin squares L8 1 and L8 ,
2
where
L8
1
=
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If two
1
2
3
4
2
1
4
3
:>
4
Lj.
J.
C
3
2
1
and
1
2
3
4
3
4
1
2
4
3
2
1
2
1
4
)
I f the 16 elements are arranged in the array
1
2
3
4
5
6
9
10
7 8
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 symbol 3 in L8
symbol 2 in L8 •
2
9
1
and to the
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For 2
~
g
n + 1, the Latin square association scheme has the following
~
parameters.
2
v=n
n
n
(2.6)
= g{n-l)
l
= (n-g+l){n-l)
2
pil = (g-1){g-2) +n-2
2
Pll
= g{g-l)
pi2 = (n-g+l){g-l)
2
P12
= g{n-g)
P~2 = (n-g+l){n-g)
2
2
P22 = (n-g) + g-2
Following the nomenclature in [7
J,
let us call an association scheme
with the parameters (2.6) 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 obtained from any 6 x 6 Latin square;
3
its complement, or the scheme obtained by interchanging first and second
associate classes, has the parameters of an L
4 (6) scheme, but no pair of
mutually orthogonal 6 x 6 Latin squares exists.
(f)
Negative Latin Square (NL {n) ) Association Scheme [7J
8
It has been found that in many cases negative values of g and n
will result in non-negative integers for the L (n) parameters (2.6).
g
The simplest case is for g =-1 and n
= -4;
the following parameters.
= 16
v
n
2
the resulting scheme has
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III
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I
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= 10
1
Pll = 0
1
P12
=4
p
2
12
10
=3
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a-
Substituting -g for g and -n for n in (2.6), we get the following set
of parameters.
v
2
P
11
1
Pll
=0
1
P12
=4
=3
1
P22
2
P12
=6
2
P22
=6
=2
Substituting -g for g and -n for n in (2.6), we get the following set
of parameters.
= n2
v
l
= g(n+l)
2
= (n-g-l)(n+l)
n
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= 16
n
(2.7)
1
Pll
= (g+l) (g+2)-n-2
2
P
ll
= g(g+l)
1
P12
1
P22
= (n-g-l) (g+l)
2
P12
= g(n-g)
= (n-g-l) (n-g)
2
P22
= (n-g) 2 -(g+2)
An association scheme with the parameters (2.7) is called a negative
Latin Square (NLg (n) ) association scheme.
(g)
Pseudo-Geometric Association Scheme
[2J
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;
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each point is incident with r lines;
(ii)
(iii)
each line is incident with k points;
if the point P is not incident with the line l, then there are
(iv)
exactly t lines (t
~
1) which are incident with P and also incident
with some point incident With l .
Clearly, we have
(2.8)
1 ::: t ::: k, 1 ::: t ::: 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 line s b in. a partial
geometry ( r, k, t) satisfy the relations
k[(r-l)(k-l) +tJ
v =
t
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and
(2.10)
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b _ r[(r-l) (k-l)+tJ
t
For convenience, we 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 passes through the point.
P and Q joins P and Q.
that t
A line which contains two points
If a point P lies on two lines t
and m, we say
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
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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
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to a PBIB design with parameters
(2.11)
v, b, r, k, Al = 1, A2 = 0,
where v and b are given by (2.9) and (2.10).
The parameters of the
corresponding association scheme are the following.
=
v
n
1
k[(r-l) (k-l)+tJ
t
= r(k-l)
_ (r-l)(k-l)(k-t)
n2 -
(2.12)
1
Pll
t
2
Pll = rt
= (t-l)(r-l)+k-2
1
P12 = (r-l)(k-t)
2
P12 = r(k-t-l)
1
(r-l) (k-t) (k-t-l)
P22 =
t
2
P22
= (r-l)(k-l)(k-t)
t
-r(k-t-l)-l
We will call any association scheme with the parameters (2.12) and for
which (2.8) holds a pseudo-geometric association scheme, since such a scheme
may exist without being derived from a partial geometry (r, k, t).
However,
if a pseudo-geometric scheme is a scheme derived from a partial geometry, we
will call it a geometric association scheme.
Several of the association schemes mentioned earlier in this chapter are
special cases of pseudo-geometric schemes.
In particUlar, a partial geometry
(r, k, r-l) gives rise to an L (k) scheme.
Thus a pseudo-Latin square scheme
r
is just a special case of a pseudo-geometric scheme.
Also, a partial
geometry (r, k, r) gives us an SLB association scheme; thus we might
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 might
speak of a pseudo-triangular scheme as a special case of a pseudo-geometric
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scheme.
2.2
Association Schemes of Three Classes
(a)
Group Divisible (GD) m-Associate Scheme
.8J
Suppose we have the number of elements
We can denote an element by an ordered m-triple (i , i , ••• , i ), where
l
2
m
i
j
€(o, 1, ••• , Nj-l) for j= 1, 2, ••• , m. Let two elements which have
only the first (m-j) coordinates in common be jth associates (j=l, 2, ••• ,m).
Then we have an association scheme, called a group divisible m-associate
scheme, with the following parameters:
°(i-l)x(i-l)
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for i=l, 2, ••• , m, we have
x i-l
°(i-l)x(m-i)
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-------r------------,
x i-l
I
( j,
k=l, 2, 3),
I
-
1
O(m_1)x(i_l)
:
D(m-i+l)x(m-i+l)
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til,
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I,
I
where 0sxt is the sxt matrix of zeros, x _ is a column vector of order
i l
(i-l) with elements n l , n2 , ••• , n~_l' respectively, and D(
) (
)
...
m-i+l x m-i+l
is a diagonal matrix with diagonal elements
[NmNm- 1'.· Nm-,.,
i~2(Nm- ~. +l-2)J, n.~ +l' n i +2 , .;., nm, respectively.
For a 3-class GD scheme, we have N , N , N
2
l
3
below.
14
~
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II
2, with the parameters given
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nl
= N3
n
= N (N -1)
3 2
2
~
(2.14)
- 1
= N3N2 (Nl -1)
1
=0
3
Pll
=0
2
= N3 -l
p3
=0
3
P13
= N3 -l
Pll
1
P
= 0
12-
P12
1
12
2
P13
=0
P13 = 0
1
P22
= N3 (N2 -1)
1
P23
=0
2
P22
2
P23
P~3 = N3N2 (Nl-l)
(b)
2
Pll = N3 -2
= N3 (N2 -2)
3
P22
=0
=0
3
P23
= N3 (N2 -l)
-1)
3
P33
P~3 = N3N2 (Nl
Tetrahedral Association Scheme
= N3N2
(N
l
-2)
[3J
A three-class association scheme, called a tetrahedral scheme, can
be defined in a manner analogous to the definition of the two-class
triangular scheme.
Suppose there exists a set of v
= (~)
elements, for
some positive integer n; we can denote the v elements by unordered triples
(Xl' X2 ' ~), where Xl ~ x2 ~ ~ and Xl' x2 ' and x range from 0 to n-l.
3
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.
For i=l, 2, 3, call two elements ith associates if
they differ in exactly i coordinates; for example, the elements (1, 2, 3)
I
I
I
and
(1, 4, 2) are first associates. This definition of association gives
us an association scheme in three classes, with the following parameters.
n(n-l)(n-2)
v=
6
15
I
.-I
_ 3(n-3)(n-4)
n2 n
(2.15)
3
2
=
P12
1
P13
1
P22
6
2
1
P
... n-2
11
1
(n-3 )(n-4 )(n-5)
Pll
2
= 2(n-4)
P12
2
=4
3 _
Pll - 0
= 2(n-4)
P12
=0
P13
= n-5
= ( n-4) 2
2
P22
=
1
(n-4) (n-5)
P23 ...
2
p2
23
3
p3
13
3 _ 3(n-6) (n-7)
P23 2
= (n-5) (n-6)
2
P33
6
=
= 3(n-6)
P~2 = 9(n-6)
(n-5)(n+2)
2
= (n-4)(n-5)(n-6)
=9
(n-5)(n-6)(n-7)
6
(n-6)(n-7)(n-8)
6
We see that for such a scheme we must have n > 6.
(c)
Cubic Association Scheme
Suppose we have a set of v
elements, for some integer n ~ 2.
Consider the v elements as ordered triples (Xl' x2 ' ~), where xl' x '
2
and ~ range from 0 to n-l. Call two treatments ith associates if they
have exactly i coordinates different (i
= 1,
2, 3).
Equivalently, we can
consider the elements as points in three-dimensional 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
II
I'
III!
~ 9 ]
= n3
I
I
I
I
ex are the remaining points lying in the three
planes determined by the first associates of ex; other p'Jints are third
associates of ex.
I,
I
I
I
I
I
The resulting scheme is an association scheme in three classes,
called a cubic scheme.
The parameters are the following.
16
I
I
I.
I
I
I
I
I
I
I
lit
I
I
I
I
I
I
1
= 3(n-1)
n2
= 3(n-1) 2
n
• (n_1)3
n
(2.16)
1
11
P
3
2
= n-2
P11
p~ = 2(n-1)
1
P13
2
P12
=0
P~2 = 2(n-1)(n-2)
(d)
1
P23
= (n-1)
2
1
P33
= (n-1) 2 (n-2)
P~3
2
P33
3 _
=2
P11 - 0
= 2(n-2)
p3
12
=3
= n-1
p3
13
= 3(n-2)
= n2 -2n+2
p3
= 6(n-2)
= 2(n-2)(n-1)
p3
23
= 3(n_2)2
= (n-1)(n-2)
2
P~3 = (n_2)3
Rectangular Association Scheme [13]
Suppose we have a set of v =1 n elements for some integers I , n
~
2.
Then we can arrange the v elements in a rectangular array with.t rows and
n co1unms.
If two elements appear in the same row, 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.
n
3
(2.17)
1
P11
1
= (e -1) (n-1)
= n-2
= 0
P12
I
22
17
I
1
P13
(e)
2
=0
Pl~
= n-l
3
P13
= n-2
3
P22
=0
1
P22
=0
2
P22 =.1-2
1
P23
= 1-1
2
P23
=0
3
P23
=
1
P33
=
2
P
33
= (.t-2){n-l)
3
P33
=
(.t-l) (n-2)
.-I
.1-2
(1-2) (n-2)
Three-Class Association Scheme from an Orthogonal Array [12]
An orthogonal array (N, m, s, t) is an mxN rectangular array of N
assembles, with m constraints, in s symbols (e. g., the elements of the array
may be the integers 0, 1, "., s-l), such that in any t-rowed submatrix of
the array each of the st possible column vectors appears exactly ~ times,
As
where
t
= N.
A is called the index of the array.
Suppose we have an orthogonal array (n2'~1 + ~2' n, 2), where
n,
~l'
and
~2
are positive integers such that n
A =1 in this case.
We see that the index
treatments.
a and
Define two treatments
colunms corresponding t:) a
in the first
~l
and
rows; let a and
2: 2 and
~l
+
~2 ~
n.
2
Consider the n assemblies as
~
as first associates if the
a.
are alike in exactly one position
~
be second associates if the columns
corresponding to them coincide in exactly one position in the remaining
~2
rows; otherwise a and
~
will be third associates.
Since the array has
strength 2 and index 1, we see that the definition of association is
unambiguous; for two colunms of the array can be alike in at most one
position.
Then we have a three-class association scheme with the
following parameters.
v
nl
~
= n2
= ~l{n-l)
=
I
I
I
I
I
I
till
I
I
I
I
I
I
~2{n-l)
18
I
I
I.
I
I
I
I
I
I
I
III
I
I
I
I
I
I
I
n = 13 (n-1), where 13 = n+1- (131 + 132 )
3
3
3
(2.18)
1
P11 = n-2+(131- 1 )(131- 2 )
2
P11 = 131 (13 1- 1 )
P12
1
= 132 (131- 1 )
P12
1
= 133 (131- 1)
3
P11 = 131 (131- 1 )
3
= 131132
3
= 131 (133- 1 )
2
= 131 (132- 1 )
P12
P13
2
= 131133
P13
P122 = 132 (132- 1)
2
P22
= n-2+(132- 1 )(132- 2 )
P~3 = 132~3
P~3
=
P1 = 13 ( 13 -1)
33
3 3
P2 = 133 (133- 1 )
33
P
13
19
133 (132-1)
P322 = 132 (132- 1 )
P~3 = 132 (133
P333
-1)
= n-2+(~3-1)(~3-2)
I
3.
Construction of BIB Designs from Linear Combinations of
Association Matrices
In this section we wish to construct BIB designs by obtaining matrices
of the form B. + B.
~l
~2
+ ••• + B which will be BIB incidence matrices,
it
where the B.'s are association matrices.
~
The method is qUite similar
to a method suggested by Shrikhande and Singh [llJ.
We prove the
following theorem.
Theorem 3.1
Suppose we have an m-class association scheme in v elements, with
association matrices Bo = Iv, B , B , ••• , B •
l
2
m
are distinct integers such that i
< m.
+ B +
Then the necessary and sufficient condition for C = B
i
+ B.
~t
l
i
to be an incidence matrix for a BIB
r
= ni
1
and
A
(i , i ,
2
l
(i , i ,
2
is that A l
2
j
Suppose i , i , ••• , it
l
2
€(O, 1,2, ••. , m) for j = 1,2, ••• , t
t
k
L:
P..
~j~ l
j< .t
+ n.
~2
+ ••• + n.
~t
... , it)
... , it)
= r(r-l)
v-l
,
t
be a positive integer and
L:
j=l
k
p. .
~j~j
+
(i , i 2 , ••• , it)
=A l
for k - 1
, 2
, m
,
... .
Proof: (For convenience, we shall denote A
during the proof.)
(i l , i 2 , ••• , it)
by A
Suppose A is a positive integer.
required necessary and sufficient condition is that
(1) C J = r
v
J
Then the
2
.-I
I
I
I
I
I
I
til
I
I
I
I
I
I
v
and
20
I
I
I.
I
I
I
I
I
I
I
= (r-"A)I
v + "A J v = r I v +
"A (Jv - I v ).
NmT
e
J
v
= (B.~1
= n.
=r
+ B.
~2
J
~1
+ ••• + B. ) J
~t
+ n.
+ n. ,J +
~2
v
v
v
~t
J
v
J •
v
Then (1) is always satisfied when the B.1 's
ee'
+ B
= (B.
1
= B.
1
~t
1
+
+ B. )
1
t
Since
~1
P~i = n i
P~j = 0
and
for i
have
t
eel = r
Iv +(
I
1
p ..
~
j=l
.1.
J ;j
t
+
(L
2
p ..
j=l
1.1.
+ 2
J J
t
Lprr: .
1.1.
j=l
.~
p 2. . )
J<...
t
+ ••• + (
j
t
J J
+ 2
I
," '/J,J'-,
1
j
1J,
P~ i
j J,
) Bm.
Now
"A (J v - I v ).
I
1t
+ B. B. + ••• + B. B. + 2 \. B. B.
1
~t 1 t
L" 1,. ~J,
2 12
j< J, J
B.
1
+ ••• + B. ) (B.
i2
1
are association matrices.
=
III
I
I
I
I
I
I
eel
(2)
21
I
Then, since the B. 's are linearly independent, the necessary and
l.
sufficient condition for (2) to hold is that
= f.. , k = 1, 2, .•• , m, and the theorem is
j<.t
proved.
We now apply Theorem 3.1 to the specific schemes discussed in sections
2.1 and 2.2 to determine what designs can be constructed.
Two-Class Schemes
(a)
GD scheme: no BIB designs can be constructed using the method of
Theorem 3.1.
(b)
Triangular scheme:
for n=6, the parameters (2.2) are such that the
matrix E, is an incidence matrix for a BIB (15, 15, 8, 8, 4).
(c)
Pseudo-cyclic scheme: we get no designs.
(d)
SLB scheme:
for k=2r + 1 in (2.5), the matrix B1is an incidence
matrix for a BIB (4r
2
- 1, 4r
2
2
2
2
- 1, 2r , 2r , r ), if the corresponding
SLB scheme exists.
(e)
L
g
(n) scheme:
for the case n = 2g in (2.6), if the scheme eXists,
then B is an incidence matrix for a BIB
l
2
2
(4g , 4g , g(2g-1), g(2g-1), g(g-l) ).
(f)
NL
g
(n) scheme:
if n = 2g in (2.7) and the corresponding NLg (n)
scheme eXists, then B1 is an incidence matrix for a BIB
22
.-I
I
I
I
I
I
I
III
I
I
I
I
I
I
I
I
I
.1
I
I
I
I
I
I
II
I
I
I
I
I
I
I
•I
(g)
Pseudo-geometric scheme:
that t
rk(k-2)
( k- r -1
= k-r-l,
,
then B is an incidence matrix for a BIB
l
rk(k-2)
k-r-l
for the case t
if a scheme with parameters (2.12) is such
, r(k-l), r(k-l), r(k-r-1) ') ;
= k-r+1,
the scheme is such. that B is an incidence
2
matrix for a BIB
k[r (k-2) + 2J
k[r(k-2) +2J
(
k-r+1
'k-r+1
(r_l)2 (k-1)
,
k-r+1
2
(r-1) (k-1)
k-r+l
(r-1/ (r-2) )
k-r+1
Three-Class Schemes
(a)
GD scheme: no designs are obtainable from Theorem 3.1-
(b)
Tetrahedral scheme:
if n
=7
in (2.15), then the scheme is such
that B is an incidence matrix for a BIB (35, 35, 18, 18, 9).
2
(c) Cubic scheme: for the case n = 4 in (2.16), the matrix (B
1
+B )
3
is an incidence matrix for a BIB (64, 64, 36, 36, 20 ).
(d)
Rectangular scheme:
for ~
= n =4
in (2.17), the matrix (B
l
+B )
2
is an incidence matrix for a BIB (16, 16, 6, 6, 2).
(e)
Scheme from an orthogonal array:
in (2.18), if n is even and we
take t3 l = ~ , then if the scheme exists B is an incidence matrix for a
l
2
2
BIB (n , n , ~(n-1), ¥(n-1), ~(n-2) ); if n is even and we have ~l + ~2
=¥
'
n( n-l ) , 2
n( n-l,
) n(n-2)
then (B1 + B2 ) is an incidence matrix for a BIB ( n2 , n2 , 2
4 ).
We may be able to construct a design in the latter case which is not obtainable
in the former case, e.g., for n
= 6•
23
I
4.
.1I
Construction of BIB Designs from Juxtapositions
of Association Matrices
Suppose we wish to find a matrix of the form [B
.
•
i1
B
i2
where the B.'s are association matrices, which will be a BIB incidence
~
matrix; we call such a matrix a juxtaposition of association matrices.
We
can prove the following theorem.
Theorem 4.1
Suppose we have an m-class association scheme in v elements, with
= 1,
are distinct integers such that i j €(l, 2, ••• , m) for j
Then the necessary and sufficient condition for D
to be an incidence matrix for a BIB (v, tv, tk, k,
__
tk....,:('"=-k-_l....)
v-l
~..
i
~l~2··· t
(11)
: B.
1
~2
~..
i)'
~l~2··· t
where
be a positive integer ,
••• =
n.
~t
t 2
\ ' Pi ."1..
J J
(iii)
L
=k
=
,
=
=
j=l
Proof:
--
(For convenience, we shall denote
~
i l i 2 •• .it
by
~.)
Suppose
~
= n i = ... = n. = k. (It is
i1
2
~t
obvious that (i) and (ii) must be satisfied for D to be a BIB incidence
is''a positive integer, and suppose n
matrix.)
Then the required necessary and sufficient condition is that
(1) D'
J
v
= k J
tv,v
and
24
~
m.
... .• Bit ]
, is that
i~1~2··· t
~..
= [Bi
2, .•. , t
I
I
I
I
I
I
.1
I
I
I
I
I
I
I
-.
I
I
I.
I
I
I
I
I
I
I
(2)
= (tk - A) I v
DD'
= tk
+ AJv
I v + A(Jv - I v ).
Now
kJ
·..
D' J v
=
.....
J
...
Then
v
v
·...•
·kJ..
=
=kJtv,v
v
(1) is always satisfied.
DD' = [ B.
1
: B. : . . . : Bi ]
.
t
1 . 12 .
....
II
I
I
I
I
I
I
I
I'
I
= tk
m
+I
c
p. i B
1
1 1 c
c =1
= tk I v + (
+
t
L
j=l
~
c
p. i Bc
l.2 2 .
I
v
+ ..• +
~
c=l
c==l
c
p . . Bc
l.tl.t
t
t
pi.i.) B +
l
J J
(
LP~
j=l
j
i ) B2 + ••• +
j
(L
j=l
P~ .1.
. )Bm•
J J
Now
= A(Jv - I v ).
Then, by the linear independence of' the Bi'S, the necessary and suff'icient
condition f'or (2) to hold is that
25
I
t
t
I
1
p ..
j=l
~'~j
=
J
I P~jij
=
=
••• =
A
j=l
.1I
thus the theorem is proved.
I
I
I
I
I
I
Let us now apply Theorem 4.1 to the schemes of sections 2.1 and 2.2.
Two-class schemes
(a)
GD scheme:
we get no BIB designs.
(b)
Triangular scheme:
(c)
Pseudo-cyclic scheme:
we get no designs.
for any scheme with parameters
: B2 J is an incidence matrix for a
l
BIB (4u + 1, 2(4u+l), 4u, 2u, 2u-l).
(2.4), the matrix [B
(d)
SLB scheme:
we get no designs.
(e)
Lg(n) scheme:
if a scheme with parameters (2.6) is such that
: B J is an incidence matrix for a BIB
2
2
2
2
( (2g-l) , 2(2g-l) , 4g(g-1), 2g(g-1), 2g -2g-l).
n = 2g-l, then [B
(f)
NLg(n) scheme:
II
l
if a scheme with parameters (2.7) has n = 2g+l,
: B J is an incidence matrix for a BIB( (2g+l)2, 2(2g+l)2,
2
l
4g(g+1), 2g(g+1), 2g2 + 2g-l).
then =B
Note that such a design is also obtainable from a scheme with Lg+1(2g+l)
parameters, if such a scheme exists.
(g)
Pseudo-geometric scheme:
if a scheme with. parameters (2.12) is such
: B J is an incidence matrix for
l
2
a BIB ( (2r_l)2, 2(2r_l)2, 4r(r-l), 2r(r-l), 2r2 _2r_l).
that k = 2r-l and t= r-l, then [B
Three-class Schemes
(a)
GD scheme:
(b)
Tetrahedral scheme:
we get no designs.
we get no designs.
26
•
I
I
I
I
I
I
I
I
I
I
I.
I
I
I
I
I
I
I
(c)
Cubic scheme:
we get no designs.
(d)
Rectangular scheme:
(e)
Scheme from an orthogonal array:
we get no designs.
in (2.18), if n > 2 is even and
2
the orthogonal array (n , n, n, 2) eXists, then we take 13
and [B
l
n(~-l) , n(~-2)); if n > 2 and n ~ 2 (mod
the orthogonal array (n2 ,
BIB
n+ 1
d
= 132 = 133 = ~,an
2
2
2
n2 _l
(n , 3n , n -1, -3-'
2(~+1)
132 =
¥'
3) and
, n, 2) eXists, then we take
[B·
1: B2 • B3 J is an
n2 _4)
-3- .
II
I
I
I
I
I
I
I
'I
I
I
=
: B J is an incidence matrix for a BIB
2
(n2 , 2n2 , n(n-l),
131
1
27
in~idence
matrix for a
I
I
5. Acknowledgment
The author expresses his appreciation to Dr. I. M. Chakravarti for
directing this research and going through the manuscript.
I.
I
I
I
I
I
I
II
I
I
I
I,
I
I
I
28
'.I
I
I
.1
I
I
I
I
I
I
II
I
I
I
I
I
I
I
'I
I
I
REFERENCES
[lJ
Bose, R. C. (1939), liOn the construction of balanced incomplete
block designs.1I
[2J
Annals of Eugenics, 9, 353-399.
Bose, R. C. (1963), IICombinatorial properties of partially
balanced designs and association schemes. II
Sankhya,
Series A, 25, 109-136.
C3J
Bose, R. C. and Laskar, Renu (1966), "A characterization of
tetrahedral graphs."
[4J
Unpublished paper.
Bose, R. C. and Mesner, D. M. (1959), "On linear associative
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balanced designs."
Annals of Mathematical Statistics, 30,
21-38.
[5J
Bose, R. C. and Shimamoto, T. (1952), "Classification and analysis
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Journal of the American Statistical
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[6J
Connor,
w.
S. (1952), "On the structure of balanced incomplete
block designs."
Annals of Mathematical Statistics, 23,
57-71.
[7J
Mesner, D. M. (1964), "Negative Latin square designs."
Institute
of Statistics, Mimeo Series No. 410.
[8J
Raghavarao, D. (1960), "A generalization of group divisible
designs. II
Annals of Mathematical Statistics, 31, 756-771.
29
I
I
,-9J
Raghavarao, D. and Chandrasekhararao, K. (1964), "Cubic designs."
Annals of Mathematical Statistics, 35, 389-397.
[lOJ
Ryser, H. J. (1963), Combinatorial Mathematics, Syracuse:
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[llJ
Shrikhande, S. S. and Singh, N. K. (1962), "On a method of
constructing symmetrical balanced incomplete block designs."
Sankhya, Series A, 24, 25-32 •
..12J
Singh, N. K. and Shukla, G. C. (1963), "The non-existence of
some partially balanced incomplete block designs with
three associate classes."
Journal of the Indian Statis-
tical Association, 1, 71-77.
[13J
Vartak, M. N. (1959), "The non-existence of certain partially
balanced incomplete block designs."
Statistics, 30, 1051-1062.
Annals of Mathematical
'.I
I
I
I
I
I
II
I
I
I
I
I
I
I
I.
30
I