CONTRIBUTIONS TO PARTIALLY BALANCED WEIGHING DESIGNS
by
K. V. SURYANARAYANA
Department of Statisti~s
University of North Carolina at Chapel Hill
Institute of Statistics Mimeo Series No. 621
It
MAY 1969
This research was supported by the Army Research office~ Durham~
Grant No. DA-ARD-D-31-124-G910~ and Air Force Grant No. AFOSR 68-1415.
ABSTRACT
Much work has been done on the "weighing problem" since it was originally
considered by Yates [48] and Rotelling [28].
A class of weighing designs
called "calibration designs" are of special importance in some practical
type of studies where one wishes to compare the value of the t1 unknown lt
objects in terms of accepted standards.
The solution to this problem
which is the same as the arrangements for s:chedlllles for a tournament problem,
has beenLimlestigatedby R.C.Bose and J.M.Cameron [9, 10].
Some results are obtained on the use of association matrices and
orthogonal latin squares to construct such designs.
In an attempt to
reduce the number of weighings required, association schemes are introduced to a weighing situation.
Some methods of constructing these new
typesot designs-- It partially Balanced
been developed.
Weighin;~
Designs (PBWD) t1--have
Results have been obtained, in the study of the structure
and properties of perfect and partial difference sets in relation to
weighing designs.
Analysis has also been developed for these new designs.
Lastly the tables of parameter combinations o.f'the.,'PBWD.':s':are given.
ii
ACKNOWLEDGMENTS
I gratefully acknowledge the guidance and encouragement given to me
by my adviser, Professor I. M. Chakravarti, during my research.
I wish to thank the other members of my doctoral committee, Professor
R. C. Bose, Professor N. L.Johnson, Professor Gertrude M. Cox and
Professor R. L. Davis for their helpful suggestions and comments.
I wish
also to thank the other members of the Department of Statistics who have
contributed to my graduate training.
I sincerely thank the Institute of International Education for their
financial support of my travel from India.
For financial support during
my graduate training, I sincerely thank the Department of Statistics,
the National Science Foundation and the U.S. Army Research office,
Durham.
To these institutions I gratefully express my appreciation.
I wish to express my deep indebtedness to Professor R. C. Bose
who is responsible for my deciding to come over to Chapel Hill.
It is
a priviledge to have attended his stimulating lectures and to study under
him in the area of combinatorial mathematics.
For her careful typing of the final manuscript, I extend my thanks
to Miss Sandy Peckham.
.,
.. -..
ERRATA. ...".
:.
Paae
Line
iv
3
1
9
_ _~---!~~;,;;;d
_
106
1f5
applicable
applicable
(Example 3:)
Starting with the signs of Cab) ,
(a), (1), (b),
A, B and Ai of
a 22-fa~torial [Table 5.1 p [22]] and
in
'+' and ,_V with
identifying
and
y
all
g
14
51'
14
4
all
16
11
As in
17
.3
h
h
(bi"b i )
a e
a f
(b
17
6t
h
b
(b i ' bf3i )
(b
..
..
18
g.
t~e
a 1
previous theorem, let
t
(1, 1) ...
and the common value is
Let
h
h
ail
h
13 i-f
, b:
,., i 2
(1, 1).
i
t
4
b
ai'
e
hypothesis (b)
19
9
application
19
9+
{bbi
34
8
<X,
0)
(X,
35
4
e
B
e
35
11
if
if
35
12
n 1 • p-1 "
n
35
13
A ... 0
12
>'21 ... 0 •
35
4t
that
that
a e
~
ajf
AU ... 0 "
00
n q
1
h
)
Pi i
19
• 1, b
)
g
ct""l
hypothesis (a)
• 1,
1
+>
B •
belongs to
A
i2
III
x
we have
p
~
(p-l)
0
and
and
A:21 "" 0 "
).21 ..
o•
AI"
,
.
36
7
).
36
5+
3.6 and Corollary 3.3
3.7 and Corollary 3.1.
37
4
r • All
r • All and
38
5+
(1) and (3)
(1) and (2)
39
12
lead
leads
44
9+
8 •
4t + 1
44
8+
8 •
4t - 1
45
6+
(4u + 1» • 17)
47
11
S3
11
case ~
case of A
57
2
valud
valid
58
2
{x2 • x6 , x4u-2}
{x2 , x 6 , ••• , x 4 u-2}
68
9
.).
12
)"
2
then
t
A
12· 22
P • 4t + 1
8
P • (4t - 1) ,
n
t
•
(4u
+ 1)(
• 17)
belongs to Ai
1
7
7
68
9
7
7
6
71
6
81
9+
•
86
.8.9+
blocks these
90·.
4
91
s
mxm
8+
e
X
<rie)l - J
[
91
and
k'
then
n
8 •
(y - x)
6
A12 • 0,
and
blocks.
~~
k
o
-n-1
-1
E
~J
)
•
Theee
t
,
91
3+
9S
2
:B
B
96
St
when
where
96
4t
96
4t
96 last
line
97
nn.ij
l1I1.i1
T:]·
[
-8
k2
101
4'
(2nd mw of matrix)
t01
'4
(last row of matrix)
-2
3,4
_.T
J~
~2(ah
By Lemma 6.2.8,
2
2,+"
l'QJ:
'
[1l
or
8j
Vi
J.
-.8
u
..
1.
-B .~!i -8
)
~ ···k:~1
kv-1
)
21:22 - bBZ21
~&
...
-!
vB
ve
0'0
..!
0.·0
-! ..!
vB
2
•••
1
vlJ'
·•• .. ·••
1 -! ...! .... ..!
.'VB va vB
v8
0
0
0 ... 0
0
0
0 ... 0
vB vB vB
Q
• vB1
T
v
Tj -~
0
0
0
0
0
0
·,';1:1
....
, .T
T'2
·v
'. tD
1
A
.v 2
8j • ;ji'(~2) [(T··....
j Tv> (>:.k1 )
.
1 :1
1;
v
+ (kj-kv)(vsm-tkiTi)]
.
) ..
'
{j-l,2, ••• , (v-l»
A
8v - 0
. ,.
'
103
5
103
Proof: Just q in Theorem 6.2.1,
the estimate ! of ! is obtained
by solving
(Now the balance of the page 103 and
104 can be disregarded)
Proof: Let \IS take without
loss of generality that
kv ~ 0, then the proof follows
by interchanging the last two
equations and the variables
ev and • of the normal
equations and applying the
lemma (6.2.4) with
B •
(r+s)I-SJ
(!1 )
'1-1
(kl k 2 • .kv - I ) 0
AnD. Math. Stat., 17
126
14
134
(Page 134 must be disregarded).
Ann. Math. Stat., 25
iii
TABLE OF CONTENTS
CHAPTER
PAGE
ACKNOWLEDGMENTS
SUMMARY
I.
II.
III.
.
IV.
ii
v
INTRODUCTION
1.1
Weighing Designs
1
1.2
Calibration Designs
2
1.3
Application of the Weighing Type of Designs
2
1.4
Construction of Weighing Designs
3
1.5
Partially Balanced Weighing Designs
3
BALANCED WEIGHING DESIGNS (BWD) AND THEIR CONSTRUCTIO~ FROM
THE KNOWN COMBINATORIAL CONFIGURATIONS
2.1
Introduction
5
2.2
Use of Orthogonal Arrays and Partially Balanced
Arrays
6
2.3
Use of Association Matrices
14
2.4
Some Miscellaneous Methods of Construction of
BWD's
24
THE COMBINATORIAL PROPERTIES OF PARTIALLY BALANCED
WEIGHING DESIGNS
3.1
Introduction
27
3.2
Some Preliminary Results on Partially Balanced
Weighing Designs
28
PARTIAL DIFFERENCE SETS AND PARTIALLY BALANCED WEIGHING
DESIGNS (PBWD)
4.1
Introduction
39
iv
PAGE
CHAPTER
V.
VI.
4.2
Perfect Difference Sets
41
4.3
Partial Difference Sets and their Construction
43
4.4
The Construction of PBWD's from Difference Sets
54
CONSTRUCTION OF' PBWD WITH· TWO ASSOCIATE CLASSES.
5.1
Introduction
70
5.2
Some General Theorems
72
5.3
PBWD with Triangular Association Scheme
75
5.4
PBWD with Latin Square Association Scheme
80
5.5
PBWD with Group Divisible Association Scheme
85
5.6
PBWD with some Miscellaneous Association Schemes
87
ANALYSIS OF BALANCED AND PARTIALLY BALANCED WEIGHING DESIGNS
6.1
Introduction
88
6.2
Analysis of Balanced Weighing Designs with
One Restraint
89
6.3
Analysis of Partially Balanced Weighing Designs
Under a Linear Restraint
106
BIBLIOGRAPHY
122
APPENDIX
127
v
SUMMARY
The weighing problem originally considered by Yates [48] and Rote11ing
[28] is concerned with finding the weights of
v
objects in
N -weighings.
Several authors have considered various aspects of this problem, both
for spring balances and for chemical balances.
As a consequence of these
investigations, i t is known that the weights may be determined with much
greater precision by weighing the objects in combination rather than individually.
The calibration designs investigated by R.C. Bose and
J.M. Cameron [9, 10] are analogous to the designs
cor~esponding
classical tournament problem, which calls for arranging
.
into teams of
p
v
to the
individuals
players so that a player is teamed the same number
of times with each of the other players and also that each player is
pitted equally often against each of the other players.
They provide
balanced designs for scheduling the measurements.
This dissertation is mainly concerned with the extension of the
tournament designs or balanced weighing designs, by introducing association
schemes to the weighing situation.
are:
The main objectives of this dissertation
(i) to develop some new methods of constructing balanced weighing
designs (BWD);
(ii) to extend the analysis of BWD's to the case of
one or more general restraints;
(iii) to construct a new class of designs
called "partially balanced weighing designs", so as to get considerable
reduction in the number of weighings required to estimate the same number
of objects as in the case of
BWD~s;(iv)
to develop the analysis and
a measure of efficiency of these designs as against BWD's.
vi
Chapter [ serves as
an
introduction of the weighing problem, calibration
designs and a brief review of the methods of construction of the latter.
It
also describes the practical applications of weighing designs in general
and in particular, those of the calibration designs.
Chapterl~
deals with
some new methods of constructing the BWD's.
Chapter III deals with the definition and preliminary results on
"Partially Ba1ancl!d Weighing Designs (PBWD)".
Chapter IV deals with the
use and properties of perfect and partial difference sets, particularly
with reference to finding cyclic association schemes and to the cyclic
generation of PBWD' s with this type df'aS'soCiation scheme.
Chapter V is
concerned with some methods of construction of the PBWD's with the association
schemes of the types: (i) triangular;
(ii) Latin-Square; and
(iii)
group divisible.
The methods of construction used in Chapter V can be classified as:
(i) general methods;
(ii) methods depending directly on the structure of
the association schemes.
Recurrent method (constructing PBWD's of smaller
block sizes from those of larger block sizes), method of composition
(for example from resolvable PBIB designs), and miscellaneous methods
(like obtaining
GD -type PBWD from BWD) belong to the first category.
Finally, the last chapter deals with some generalizations of the
analysis of BWD's and with the development of a separate analysis for the
PBWD's.
It is also concerned with the investigations on "a measure of
efficiency of PBWD's as compared to BWD's".
The appendix gives the summary tables of the parameter combinations
as well as the plans of the various types of PBWD's constructed in this
dissertation.
CHAPTER I
INTRODUCTION.
1.1
Weighing
d(~signs.
The weighing problem originally was considered by Yates [48] and
Ho,t:e~lI.ing [28].
In the latter developments, attention has been in the
direction of obtaining "optimum" weighing designs.
been· determined by means of "efficiency".
The pptimality has
Essentially there are three
types of definitions of "efficiency of weighing designs".
definition is due to Mood [34].
The first
The second one is due to Ehrenfeld
[23] and the last one is due to Kishen [29].
If
x
=
«x
ij
)
is the
design matrix (1) below, the solution of the normal equations to
determine the unknown weights depends on singularity or non-singularity
of
(xx') .
The corresponding weighing design is called accordingly
a singular or non-singular weighing design.
1
x ..
~J
=
-1
0
i f the i-th object is placed in the left pan
in the j -th weighing.
if the i""th object is placed in the right
pan in the j-th weighing.
i f the i-th object is not weighed in the
j -th weighing.
(1)
The best weighing designs (best in the sense of Mood [34]) are shown
to be obtainable from the two types of matrices (each of order
N by N)
which are the
P
and
N
(2) and (3) respectively [39]:
SN -matrices defined by the conditions
2
e
P P
N N
= (N-1)IN + I N
(2)
SN
= (N-1) IN
(3)
SN
Although there have been many investigations of this type, the weighing
designs thus constructed arose from some
t~pe
of efficiency considerations.
The concern of this dissertation work is the construction of calibration
designs, which are discussed in Section 1.2.
1.2.
Calibration designs.
The study of the type of weighing designs considered above is not
restricted to weighing nominally equal objects nor to the case where
there are equal number of objects on each pan.
:a.e -,
Some assumptions
that of equal variance in all weighings are deemed to be more valid
when the objects are nominally of equal weight and when the number of
objects is the same on each pan.
Such tt¥pesof designs arise, for
example, in high precision calibration where only the differences
between nominally equal objects can be measured, and the process of calibration consists of assigning the value for the "unknown" objects in
terms of "known" or accepted standards.
This situation
is
the classical tournament problem, which calls for arranging
into teams of
p
similar to
v
individuals
players so that a player is teamed the same number of
times with each of the other players and also that each player is
pitted equally often against each of the players.
1.3.
Applications of the weighing type of designs.
The weighing designs, either the balanced ones or the others are
3
Oipp1.i.covle.. to a great variety of problems of measurement, not only of
weights, but of lengths, voltages and resistances, concentration of
chemicals in solutions, or in general to any situation with additive
effects.
Some special instances of balanced weighing designs, like
in calabration, are already mentioned.
1.4.
Construction of weighing designs.
The construction methods used by Bose and Cameron [9, 10] are
mainly:
(i) the method of symmetrically repeated differences
(ii) the method of composition and
Hadamard matrices.
(iii) the method involving
Chapter IIof this dissertation is mainly concerned
with some new methods, of constructing balanced weighing designs other than
those considered by Bose and Cameron.
Part of Chapter VI is concerned with
the extension of analysis given by Bose and Cameron.
1.5.
Partially Balanced Weighing Designs.
The major part of this dissertation work is concerend with the
definition, properties, construction methods and analysis of "Partially
Balanced Weighing Designs (PBWD)".
Since the adopting of association
schemes is what makes these designs different from BWD, Chapter tIl
deals with the properties and parametric conditions involving the
parameters of both the design as well as the association schemes.
part of Chapter
scheme.
V)1
A
deals with the analysis of PBWD' s with any association
As the PBWD's are of several types depending on the type of
association-scheme, the methods of construction also vary from one type
4
of association to the other.
While the methods of constructing cyclic
PBWD's considered in Chapter LV involve the structure and properties of
...
perfect and partial difference sets, both general and particular methods
are used in Chapter V to construct other types of PBWD' s.
CHAPTER II
BALANCED WEIGHING DESIGNS (BWD) AND THEIR CONSTRUCTION FROM
THE KNOWN COMBINATORIAL CONFIGURATIONS
2.1.
Introduction
Balanced weighing designs have already been defined [9, 10]
Some methods of construction have also been developed.
and
This chapter mainly
deals with the construction of Balanced Weighing Designs from orthogonal
arrays and association matrices.
Some miscellaneous methods are alsn
discussed.
Orthogonal arrays were defined by C.R. Rao [41, 42] .
designs given by Plackett and Burnam [38] are
of strength two.
orthogonal arrays
The problems of construction have been considered in
several papers [8, 17, etc.] .
in [17, 18].
essen~ially
The multifactorial
Partially balanced arrays are considered
Bose, Shrikhande and Bhattacharya [11] and Shrikhande
(Can. J. Math. 16, 736-40) have used and studied the interrelations
between affine resolvable designs, semi-regular group divisible designs,
orthogonal arrays and Hadamard matrices.
In this chapter, the attempt
is to use them for the construction of balanced weighing designs.
These
constructions are discussed in Section 2.2.
The association matrices, which arise from the association schemes
cf partially balanced incomplete block designs, were used by Blackwelder,
W.C. [5] and Chakravarti, I.M. and Blackwelder [Symposium on Combinatorial
Mathematics held at Chapel Hill' in constructing balanced incomplete block
designs.
Few results are developed in Section 2.3, to use this type of
approach in constructing balanced weighing designs.
The studies on latin squares have been there as early as in 1782.
For a brief account of the recent developments on mutually orthogonal latin
*'
April 10-14, 196 7 , Chapter 11, 187-199.
6
squares, see [22, 27].
Use of orthogonal latin squares has been made in
many ways in the constructions like those of resolvable balanced incomplete
block designs.
Recently, orthogonal latin squares have been used by
C1atwortpy [21] in constructing some
designs of latin square type.
ne~
families of partially balanced
In this chapter, an attempt is made to use
orthogonal latin squares to construct balanced weignin& designs.
The actual method is described in Section 2.4.
2.2.
The construction of Balanced Weighing Designs using
orthogonal arrays and Partially Balanced arrays.
Definition of an orthogonal array:
A (k xN)
matrix
A with entries from a set
elements is called an orthogonal array of size
s
levels,
strength
the possible
(t
x
1)
t,
if any
(t x N)
N,
of
k
constraints,
sub-matrix of
s (
A(N, k, s, t)
is called the index of the array.
Clearly
N
A.
and the number
= AS t
2)
A contains all
column vectors with the same frequency
an array is denoted by the symbol
~
L
Such
A
•
Definition of a partially balanced array:
Let
a
ij
A = (a .. ) ,
1J
i=1,2, ... ,m,
j=1,2, ... ,N
of the matrix be taken from the set of symbols
Consider the
st - (1 x t)
matrices
Suppose, associated with each
is a positive integer
permutations of
A(X , X ' ""
1
(Xl' X ' ""
Z
Z
X) .
t
0, 1, 2, ""
X~ = (Xl' X2 , ""
be formed by giving different values to the
i=1,2, .•• , t .
and let the elements
X)
t
X. ' s ,
1
Xt
)
(s-l)
that can
X. = 0,1,2, •.• ,(s-1),
1
(t x 1) -matrix
X,
there
which is invariant under the
If, for every
t -rowed sub-matrix
7
of
A,
the
st - (t x 1)
times, then the matrix
t
in
X occur as columns,
A(X , X2 , •.• , Xt )
l
A is called a "Partially Balanced Array of strength
N assemblies,
and the specified
matrices
m constraints (or factors),
s
symbols (or levels)
A(X , X , .•. , X ) -parameters".
2
l
t
Use of orthogonal arrays when the number
If we start with an orthogonal array
and if we restrict to the case
s = 3,
s
of levels is 3 :
(N, k, s, t)
A,
of index
the elements being
-1, 0, +1
the use of this array can be made to construct balanced weighing designs,
by taking columns as blocks and rows as treatments, provided the number
of
l's
is the same as the number of
(-l)'s
in each column.
In order to apply this method, it is necessary to omit the columns with
all
(-l)'s
with all
EXAMPLE (1):
O's
or with all unities.
Table 1 of I.M. Chakravarti [17, p.1182] gives the
following orthogonal array:
A(18, 7, 3, 2) •
Assemblies
Constraints
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
1
0
1
2
0
1
2
0
1
2
0
1
2
0
1
2
0
1
2
2
0
1
2
0
1
2
1
2
0
2
0
1
1
2
0
2
0
1
3
0
1
2
1
2
0
0
1
2
2
0
1
2
0
1
1
2
0
4
0
1
2
2
0
1
2
0
1
0
1
2
1
2
0
1
2
0
5
0
1
2
1
2 ;0
2
0
1
1
2
0
0
1
2
2
0
1
6
0
1
2
2
0
1
1
2
0
1
2
0
2
0
1
0
1
2
7
0
0
0
0
0
0
1
1
1
1
1
1
2
2
2
2
2
2
Let us omit the first three columns and the last row.
columns as blocks.
Keeping
1
as
1
and changing
2
as
Let us take
(-1) ,
we
8
e
can write down the 15 blocks of the resulting design as follows with unities
(-1) 's
to represent treatments of the first half block and
treatments of the second half block.
weighing design with
Then we get the following balanced
=
(v, b, r, p, AI' A )
2
(6, 15, 10, 2, 2, 4) :
Treatments
2
3
4
5
6
1
0
0
1
-1
1
-1
3,5; 4,6
2
1
1
-1
0
-1
0
1,2; 3,5
3
-1
-1
0
1
0
1
4,6; 1,2
4
0
1
0
-1
-1
1
2,6; 4,5
5
1
-1
1
0
0
...1
1,3; 2,6
6
-1
0
-1
1
1
6
4,5; 1,3
7
0
-1
-1
0
1
1
5,6; 2,3
8
1
0
0
1
-1
-1
1,4; 5,6
9
-1
1
1
-1
0
0
2,3; 1,4
10
0
1
-1
1
0
-1
2,4; 3,6
11
1
-1
0
-1
1
0
1,5; 2,4
12
-1
0
1
0
-1
1
3,6; 1,5
13
0
-1
1
1
-1
0
3,4; 2,5
14
1
0
-1
-1
0
1
1,6; 3,4
15
-1
1
0
0
1
-1
2,5; 1,6
e
EXAMPLE (2) :
Plan
1
Blocks
to represent
Table 2 (page 1182) of the same paper [17] , gives a
partially balanced array
(15, 6, 3, 2)
as follows:
Assemblies
~
e
Constraints
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
1
0
0
0
0
0
1
1
1
1
.1
2
2
2
2
2
2
0
2
1
1
2
0
0
1
2
2
0
0
2
1
1
3
1
0
2
2
1
0
2
0
1
2
1
2
0
0
1
4
1
1
0
2
2
2
0
2
0
1
2
1
0
1
0
5
2
2
1
0
1
1
2
2
0
0
0
1
1
0
2
6
2
1
2
1
0
2
1
0
2
0
1
0
1
2
0
9
Taking columns as blocks, and taking symbol
first half-block, symbol
2
1
to indicate the
to indicate the second half block and rows
as treatments, we get the following weighing design:
(6, 15, 10, 2, 2, 4)
=
3, 4; 5, 6
3, 6; 1, 4
4, 6; 2, 5
2, 5; 3, 6
4, 5; 1,
5, 6·, 1,
2, 4·, 1,
2, 3; 1,
2, 6; 3, 4
3, 5·, 2, 4
1,
1,
1,
1,
1,
e
Note:
3
2
6
5
5; 4, 6
6; 3, 5
2·, 4, 5
3; 2, 6
4; 2, 3
The fact that each column contains two unities and two
2's leads
to this representation in order to construct a balanced weighing design,
from a partially balanced array.
EXAMPLE 3:
Let us take the columns as blocks.
block and
y
x
x
x
x
y
y
y
y
x
y
x
y
Taking
X to represent the first half
to represent the second half block, we get the following
weighing design, with
(v, b, r, p, 1. ' 1. ) =
2
1
1, 2·, 3, 4
1, 4; 2, 3
1, 3; 2, 4
(4, 3, 3, 2, 1, 2) :
10
EXAMPLE 4:
array
Omitting the columns numbered
A[24, 12, 3, 2]
1
and
13,
in the orthogonal
[44, p.153], taking zeros to represent treatments
of the first half block and with the same type of representation for
blocks and treatments, we get the following weighing design with the
parameters
(v, b, r, p, 1.. ' A2)
1
= (12, 22, 22,
6, 10, 12)
3,
7,
8,
9, 11, 12;
1,
2,
4,
5,
6, 10
2,
6,
7,
8, 10, 12;
1,
3,
4,
5,
9, 11
1,
5,
6,
7,
9, 12;
2,
3,
4,
8, 10, 11
4,
5,
6,
8, 11, 12;
1,
2,
3,
7,
9, 10
3~
4~
5~
7, 10, 12;
1,
2,
6,
8,
9, 11
2,
3,
4,
6,
9, 12;
1,
5,
7,
8, 10, 11
1,
2,
3,
5,
8, 12;
4,
6,
7,
9, 10, 11
1,
2,
4,
7, 11, 12;
3,
5,
6,
8,
9, 10
1,
3,
6, 10, 11, 12;
2,
4,
5,
7,
8,
9
2,
5,
9, 10, 11, 12;
1,
3,
4,
6,
7,
8
1,
4,
8,
9, 10, 12;
2,
3,
5,
6,
7, 11
1,
2,
4,
5,
6, 10;
3,
7,
8,
9, 11, 12
1,
3,
4,
5,
9, 11;
2,
6,
7,
8, 10, 12
2,
3,
4,
8, 10, 11;
1,
5,
6,
7,
1,
2,
3,
7,
9, 10;
4,
5,
6,
8, 11, 12
1,
2,
6,
8,
9, 11;
3,
4,
5,
7, 10, 12
1,
5,
7,
8, 10, 11;
2,
3,
4,
6,
9, 12
4,
6,
7,
9, 10, 11;
1,
2,
3,
5,
8, 12
3,
5,
6,
8,
9, 10;
1,
2,
4,
7, 11, 12
2,
4,
5,
7,
8,
9;
1,
3,
6, 10, 11, 12
1,
3,
4,
6,
7,
8;
2,
5,
9, 10, 11, 12
2,
3,
5,
6,
7, 11;
1,
4,
8,
9, 12
9, 10, 12
These four weighing designs, thus constructed are listed in the table
at the end of this section.
11
Use of orthogonal arrays when the number I SIc' of. levels is greater than
Even when
s
by grouRing the s
>
3,
3
in few cases it may be possible, sometimes,
elements into 3 classes, identified by
or into 2 classes, identified by
0, X, Y
X, Y to get a balanced weighing design.
Example 5 illustrates this.
EXAMPLE 5:
Let us start with the orthogonal array
by Bose and Bush [8, p.529].
[32, 9, 4, 2] given
Let us omit the first four columns and
the last row.
We are left with 28 columns and 8 rows.
1
2 and 3
as
X and
as
y,
h=0,1,2, •.• ,13.
0
and
in this resulting array, it is easy to
2h + 1, 2h + 2
notice ,:that we get the columns numbered
for
Taking
as identical
So taking only one column from each identical pair,
we get the following 14 distinct columns.
1
2
3
4
5
6
7
8
9
10
11
12
13
14
X
Y
X
Y
X
Y
X
Y
X
'y
X
Y
X
Y
X
Y
Y
X
Y
X
X
Y
X
Y
Y
X
Y
X
Y
X
X
Y
Y
X
X
Y
Y
X
X
Y
Y
X
Y
X
Y
X
X
Y
X
Y
Y
X
Y
X
X
Y
X
Y
X
Y
X
Y
Y
X
Y
X
Y
X
Y
X
X
Y
y
X
Y
X·
Y
X
Y
X
X
Y
X
Y
Y
X
X
Y
.y
X
X
X
Y
Y
X
X
Y
Y
X
Y
X
X
Y
Y
y
X
X
Y
X
Y
Y
X
With the representation of rows as treatments, columns as blocks,
corresponding to the treatments of the first half block
and with
XIS
and
corresponding to the treatments of the second half block, we get
yls
a balanced weighing design.
noticed that the columns
Before writing out the plan, it is to be
{(2£ + 1), (2£ + 2)}
give rise to the same
12
blocks, with the first and the second half blocks interchanged
(~=O,1,2, •.. 6)
.
So essentially we get the following weighing design with
7 blocks instead of 14 and with
(v, b, r, p, AI' 1- )
2
=
(8, 7, 7, 4, 3, 4)
Plan of the balanced weighing design
N.B:
The
[8, p.522]
1,2,5,6;
3,4,7,8
1,2,7,8;
3,4,5,6
1,3,5,7;
2,4,6,8
1,3,6,8;
2,4,5,7
1,4,5,8;
2,3,6,7
1,4,6,7;
2,3,5,8
1,2,3,4;
5,6,7,8
or~hogonal
array
(16, 8, 2, 3)
given by Bose and Bush
is evidently also an orthogonal array
leads to the same balanced weighing design as above
(16, 8, 2, 2)
and this
(first column and
last column of (9.3) of the array given by Bose and Bush, are to be
omitted).
Use of the structure of signs in a factorial type of experiment:
This method is given under the category of the use of orthogonal
arrays, since one of the research papers basic for the motivation in the
constructions of orthogonal arrays, is the one by "Placket and Burman"
[38].
EXAMPLE 6:
The signs of the
follows, with
f , f , f
2
1
3
2
3
-factorial scheme can be tabulated as
being the factors involved and with the
capital letters denoting main effects and,irtteractions':
13
Taking the columns of the above scheme as treatments, tows as blocks
'+'
and with the representation of
,- ,
e
e"
signs for the first half block and
signs for the second half block, we get the following balanced
weighing design with
(v, b, r, p,
1..
1'
1..
2
)
=
(8, 7, 7, 4, 3, 4)
1, 2, 3, 4;
5, 6, 7, 8
1, 2, 5, 6;
3, 4, 7, 8
1, 3, 5, 7;
2, 4, 6, 8
1, 2, 7, 8·,
3, 4, 5, 6
1, 3, 6, 8;
2, 4, 5, 7
1, 4, 5, 8;
2, 3, 6, 7
1, 4, 6, 7;
2, 3, 5, 8
14
e
These 6 examples can be tabulated as follows:
v
b
r
p
Al
A_
2
1
6
15
10
2
2
4
2
6
15
10
2
2
4
3
4
3
3
2
1
2
4
12
22
22
6
10
12
5
8
7
7
4
3
4
6
8
7
7
4
3
4
2.3.
Remark:
Same parameter set as 1.
Same as 5.
The construction of Balanced Weighing Designs using
association matrices.
THEOREM· 2.1:
A sufficient condition for a balanced weighing design
(v, v, 2p, p, AI' A )
2
scheme involving
and
v
to exist is that there exists an
objects and two sets of integers
(jl' j2' ... , jt) ,
where
are all distinct such that
(a)
£
I:
q=l
£ + t < m,
jl' j2' " ' , jt
with the following properties:
+
.
q
.1
+
I:
£
I:
(b)
q=l
2
t
£
I:
I:
w=l q=l
=
I:
w<w'
w=l
q
is the same for all
(c)
(iI' i 2 , " ' , i£)
t
q
p.
1
i l • i 2 • .... i£.
m -class association
.
g
t
N.1
=
q
Jw
w=l
1
=
Pi q,jw
=
N.
I:
2
2
=
P
t
£
I:
I:
w=l q=l
t
£
I:
I:
m
Pi
q·jw
w=l q=l
2
Pi q,jw
=
A
2
.
15
Consider the
Proof:
association~atrices
to the above association scheme.
...
let
M = (B, + B, +
Z
JZ
J1
and
M are respectively
Z
+ B,
u
(b ,
o.~l
u
u
u
(bo. ' + bo.' + ... + bo.' )
J1
J2
Jt
i -th and i·s -th associate
q
.
The
)
Jt
M
1
Let
(B,
~1
+ B,
...
+ ... + B,)
~z
+ bU
~~
and
~1
and
)
o.i~
corresponding
,
cannot simultaneously be
a.
u ,
of
=
m
(a. , u) -th elements of
U
+ b o.i
+
Z
Since
••• , B
(q~
for the pair
s)
(q :f s) ,
it is evident that
U
b o.i
1
=
for at most one
q
q
~
..•
b
L:
q=l
U
q
t
Similarly
°.
or
1
=
,
o.~
U
b a.'
=
J
w=l
w
L:
1
.
°.
or
Also by the same argument, the following cannot happen:
~
...
are the two
(0, 1)
or
L:
t
L:
q=l
w=1
=
(m
rs
(1»
(0; 1) -matrices such that
=
(1)
rs
(m
=
and
(m
(1, 1) •
rs
m
(Z»
(Z»
=
rs
(1, 0),
(0, 0) .
Also it is evident that the total of each row and each column of
is
~
L:
q=l
N.
~q
and the total of each row and each column of
M
2
is
M
1
16
t
~
w=l
N.
Let
Jw
zeros,
M - M = D .
2
l
(-l)'s.
and
l' s
It is evident that it is a matrix with
In a weighing experiment, a particular object may not appear either
on the left hand pan or on the right hand pan.
it can appear only in one pan.
But if at all it appears,
The essence of using association matrices
for weighing experiments lies in the identification of the two incidence
matrices corresponding to the two pans by
The elements of
M
l
and
M •
2
D characterize the actual situation about the
position of the objects.
Let uS now verify the sufficiency conditions
(a), (b), (c) for the existence of a balanced weighing design.. ,
p -condition:
As in the previous theorem, let the columns of
(or equivalenty
M)
2
represent the blocks and let the rows of
M
l
M
l
represent treatments.
The number of objects-in- the· leftdhand pan is the same as (the common
number of unities in any column_ of
t
~l
Similarly
w=
-condition:
(a, [3)
times
N.J
represents the number of objects on the right
w
Therefore condition (a) guarantees the p -condition.
hand pan.
~1
Let
(a, [3)
a
and
To find
[3
Since
all
a
and
be any pair of treatments.
The number of
occur together either on the left hand pan or on the right
pan is the total number of
Case 1:
M)
l
are
d
ah
h,
h
h's
such that
such that
(d
ah
, d[3h) = (1, 1) or (-1,-1)
(d ah , d[3h) = (1, 1) ,
assuming that
g -th associates:
h
= b .
a~l
h
+b . +
a~2
... + bh .
a~S/,
_ bh .
aJ 1
...
-
h
b .
aJ t
for
we have the following implication relationships:
17
(d
ah
, d
Sh
)
(1, 1) <=>
=
t
£
h
l: b .
a~
q=l
h
b . ,
w=l aJ.w
l:
q
£
l:
t
h
b '
w=l SJ w
h
b .
q=l S~q
l:
(1, 1)
=
h
(c.
<=> {
(~l'
e
h
b . )
=
a~f
(1, 1)
for some e and f
}
(1)
The last implication relation in (1) follows, since if
B -matrix
(a-th row, h-th column) -element of a
(say
B. )
is unit,
~
e
the corresponding element in each of the remaining matrices
BO = Iv'
B1 , B2 , ""
for every fixed pair
Bi -1
e
'
Bi +1'
.e
£2
h'S
a~e
g
••• p.
.
~1~£
m
must be zero.
This is true
under this case, we have to consider
possibilities
h
h
{ (b . , b S. )
Fixing
B
(a, h) .
So to count the number of
all the
... ,
e
=1
,
(1, 1) ,
=
~f
and noting that there are
- possible values of
h
h
(b . , b ' )
S~£
a~l
h
=
for
(1, 1)
h
(be.
a~l
b h . ) = (1, 1)
S~l
h
(b .
a~l
h
b ' )
S~£
=
,
(1, 1)
respectively, we conclude that the number of possible values of
corresponding to
_ _(2)
e, f = 1,2, ... ,£ }
e = 1,
h
is
£
l:
_ _ (3)
q=l
Essentially what we have done is that we have subdivided the
in (2), by fixing
e
and considering all possibilities for
£2 -possibilities
f.
18
The argument similar to that which led us to (3), leads in general to the
eonclusio,u that the number of possible values of
the sub-case
"e = n"
h,
corresponding to
is
Q,
q
r.
for
Pi i
q=l
n g
n = 1,2,000,Q, 0
_ _(4)
From (1) and (4), the final conclusion of case: 1 is that under the
assumption that
a
and
S
'hI , with the property
Q,
are
(d
ah
g
p ..
r.
~l~q
q=l
g -th associates, the number of possible
, d
Sh
Q,
+ r.
q=l
) = (1, 1)
is given by
Q,
g
Pi .
2~q
g
+ 000 + r. p . .
q=l
~Q, ~q
Q,
::
Case 2:
a
and
To find
S
are
h
g
g
Pi i + 2 r.
Pi i
q<q'
q q'
q=l
q q
_ _(5)
r.
such that
(d
ah
, d
Sh
) = (-1, -1) , assuming that
g -th associates.
It is obvious that this is essentially the same as
replacing
Q"
j
replacing
i
and
w replacing
~~
1, with
t
q 0
So for example, corresponding to (4) of case: 1, we get:
for
n = 1,2,000' t
---(4) ,
So just like in (5) we come to the conclusion that the number of ¥a1ues of
h
with the property
a
and
S
are
(d
ah
, d
Sh
) = (-1, ·1)
(under the assumption that
g -th associates) is
t
r.
w=l
+ 2 r.
w<w'
---(5) ,
19
From (5) and (5)' and the hypothesis (b) of the theorem, it follows
that the
pair
l2
Al -condition is satisfied,
(a,S)
Al
being the same whenever the
is taken.
-condition:
We know that two fixed treatments of
in opposite blocks, if and only if
(d
Just like in the verification of the
ah
, d
Sh
a,S
) = (1, -1)
occur together
(-1, 1) .
or
Al -condition, these two possibilities
can be included under the cases (1) and (2).
Case 1:
(d
, d
ah
Sh
) = (1, -1)
and
a,S
are the
g -th associates.
We know the application relations:
(d
Q,
{ ~ bh
q=l ai q
<=>
ah
t
~
, d
h
b
13h
)
(1, -1)
=
t
h
q=l Si q
Q,
w=l aJ w
~ bh . } =
b
~
. ,
<=>
for some
Just as in (5) of case 1 in the
(1, -1) ]
w=l SJ w
e
and
f
.
} •
Al -conditipn, we arrive at the
relation (7) as follows:
{ # of
his
such that
assumption
a
t/
ah
, d
Sh
) = (-1, 1)
, d
13
Sh
) = (1, -1) ,
are
t
+ ... +
~
w=l
and
a,S
~
j
and vice versa.
_ _(7)
w=l
are
g -th associates.
This case is essentially the same as 'case 1, with
the role of
under the
g -th associates}
t
+
~
w=l
Case 2: (d
ah
and
t
=
(d
i
replacing
20
So similar to the one in (7) , we arrive at the relation (8)
as follows:
h's
{ II of
such that
assumption
,
(do.h ' d 8h ) = (-1, 1)
and
0.
are
8
under the
g -th associates}
2
=
2
2
g
g
g
E p. i + E p. i
+ ... + E p. i
q=l J l q
q=l J 2 q
q=l J t q
=
t
g
g
E
+
p. i
Pj i +
w=l J w 1
w=l
w 2
t
E
t
=
p
E
w=l
•
g
i ljw +
t
E
t
+
g
p. i
w=l J", 2
E
~2Jw
+
+
g
Pi J.
w=l
2 w
E
From (7) and (8) and the hypothesis (c), it follows that the
A being the same whatever the pair
2
is satisfied,
As already remarked
M
l
and
M
Z
b = v
and
8)
(0.,
A -condition
Z
is taken.
respectively determine the structure
of the first and second half blocks, for each of the
this identification
(8)
t
g
p ..
w=l
...
1
v
blocks.
For
(p, A , A ) -conditions are already verified.
l
2
Hence the theorem follows.
COROLLARY 3.1:
A sufficient condition for a balanced weighing design
(v, v, 2p, p, A , A )
l
2
to exist is that there exist
association scheme and two specific classes
••
(a)
n.
(b)
Pi.i + Pjj
(c)
2Pij
=
~
1
p
1
1
=
=
=
2
zP ij
n.
J
i
and
(i) an
j
such that
.
Z
2
PH + P jj
m
m
PH + Pjj
=
m
2P i j
=
A
2
m -class
=
A
l
21
where
v
Proof:
..
t = 1
is the number of treatments of the association scheme.
This follows from the above theorem by taking
~
= 1
and
and noting the fact that there will not be any pairs from the
i -set alone and the
j -set alone', with the notation mentioned in that
theorem.
EXAMPLE 1:
Let us start with the pseudo-cyclic association scheme
corresponding to
v = 9,
for which
B , B
1
Z
are included in a single
partitioned matrix by Blackwelder [[5], p.Z7]
=
1
1
0
1
0
0
o
o
o
1
0
0
1
0
010
i
0
010
o
1
o o o
1 1 o
o o 1
1 o 1
o 1 o
o 1 1
1 o o
o
1
1
0
1
o o
1
1
1
001
o
1
o
1
1
o
1
1
011
1
1
101
0
1
0
0
( 1
Z
2 )
o
o
1
o
o
o
o
o
1
o
1
0
1
1
0
1
o
=
2
o o o
o 1 1
o 1 1
1 o o
1 o o
1 o 1
1 1 o
o o 1
o 1 o
0
0
0
1
2
Z
1
1
1
1
0
o
1
1
1
1
0
1
1
0
o
o
o
0
1
1
o
o o
o 1
1 o
o 1
1 o
o o
o o
0
2 )
1
Hence it is evident that
(a)
=
(b)
1
1
P11 + PZ2
(c)
4
=
Z
Z
= P11 + PZ2 =
3
= 4
So the conditions of theorem (1) of this section are satisfied in
this example, with
p = 4, A = 3
1
and
A = 4.
Z
Since n +n =8, r=8 •
1 Z
22
23
e
..
Hence it is evident that
6
(a)
n
(b)
1
1
P11 + P22
(c)
1
2P12
=
1
=
n
=
2
2
2
5;:: P11 + P22
=
6.: 2P
2
12
So the conditions of Theorem (1) of this section are satisfied in this
example, with
P = 6, Al = 5, A = 6
2
and
r = n
1
+ n2
=
12
.
So the corresponding balanced weighing design with
(v, b, r, P, AI' A ) = (13 , 13, 12, 6, 5, 6)
2
e
•
RULE 1:
is given below:
3,
6,
7,
8,
9, 12;
2,
4,
5, 10, 11, 13
4,
7,
8,
9, 10, 13;
1,
3,
5,
6, 11, 12
1,
5,
8,
9, 10, 11;
2,
4,
6,
7, 12, 13
2,
6,
9, 10, 11, 12;
1,
3,
5,
7,
8, 13
3,
7, 10, 11, 12, 13;
1,
2,
4,
6,
8,
1,
4,
8, 11, 12, 13;
2,
3,
5,
7,
9, 10
1,
2,
5,
9, 12, 13;
3,
4,
6,
8, 1O, 11
1,
2,
3,
6, 10, 13;
4,
5,
7,
9, 11, 12
1,
2, . 3,
4,
7, 11;
5,
6,
8, 10, 12, 13
2,
3,
4,
5,
8, 12;
1,
6,
7,
9, 11, 13
3,
4,
5,
6,
9, 13;
1,
2,
7,
8, 1O, 12
1,
4,
5,
6,
7, 10;
2,
3,
8,
9, 11, 13
2,
5,
6,
7,
8, 11;
1,
3,
4,
9, 10, 12
9
.
Noting that these two examples are particular cases of the general
cyclic association scheme given in Corollary 4.3.1 (Ch.4, Section 4.3)
we note that the cyclic association schemes for
(d , d ,
2
1
...
e
... ,
d 2)
=
(x, x
3
,
x
5
,
... ,
x
4u-1
)
v = 4u + 1
and
give a symmetric class
of balanced weighing designs with the parameters:
24
(v, b, r, p, AI' 1. )
2
n
Since
l
=
=t
1
PI
=
n
2.4.
2u ' ,=>
:)
1
1
Pu + P22
and
=
2
(4u+l, 4u+l, 4u, 2u, 2u-l, 2u) •
=
E2
P = 2u
-c j
->
u-l
2
2
Pn + P22 = 2u - 1
1
2P12
2
2P12
=
=
2u.
Use of orthogonal latin squares in the construction of BWD's.
The following theorem gives a method of construction of a series of
balanced weighing designs, when there exists a complete set of mutually
orthogonal latin squares.
THEOREM 2.4.1:
If there exists a complete set of mutually orthogonal
•
latin squares of order
2 s (s2-l)
(s,
2
Proof:
s,
then it is possible to construct a BWD
2
' s -1, s, s-l, s) •
Let us start with a complete set [see for example, [7]]
.. "
L s- 1
of orthogonal latin squares of order
Let us
s •
superimpose these on the array (1) given below:
1
s +"1
2
s
.
+. 2
s.
.,3
.
s
-+,3
·F·:·~.·
: .: •.
28 ~
• • • • • • • . • • • . • • . 2
(s-l)s + 1 (s-l)s + 2 (s~l)s + ,3 ••• s
_ _ (1)
We know the equivalence [see for example Sectionl.4 of [7]] of finite
affine planes and complete sets of mutually orthogonal latin squares.
So the proof can be given in terms of lines and points of an affine
25
plane, rather than in terms of cells, rows and columns of latin squares.
The equivalence metioned above uses the fact that there are
distinct points in an affine plane and that the
plane can be divided into
s
lines.
(s + 1)
(s
2
+
s
2
lines of the
s)
parallel pencils, each containing
Let the parallel pencils corresponding to the row set and
the column set of
be denoted by
(s)
and
Let the
(s - 1)
parallel pencils obtained by the super imposition of
on (1), be denoted by
•.. , L
s-
1
vI' v 2 ' ... , v s - l .
Using the basic property that
v ' v c ' vI' v 2 ' •.. , v s - l
R
are parallel
pencils, we note that any pair of lines coming from a pencil are disjoint.
Let us form a weighing design by forming all possible pairs of lines
(s + 1)
from each of the
pencils and by identifying the two lines of
each pair with the two half blocks of the design.
v = s2, p = sand
Let
p(i, j)
(i - l)s + j
b = (s + 1)(
intersection of the
line of the pencil
is
R
2(s - 1).
It can. be treated as the point of
i -th line of the pencil
v
c .
v
with the
R
Since evidently there are
j -th
(s - 1)
blocks
i -th line as a half-block and since there are
more blocks containing
but ion from v
~:)= S(S~-l)
be the point corresponding to the treatment
of the array (1).
containing this
Then evidently
and
v
j .... th line
C
of
V
c
as a half block, the contri-
to the number of replications of
As there passes a unique line of
through the poing
p(i, j) ,
v
t
(i - l)s
r = (s + l)(s - 1) = (s2 - 1) .
+
j
(t=1,2, ... ,(s-1))
the same type of argument gives
as the contribution from the set
(s - 1)
(VI' v 2 ' .•• , v s _ l ) .
Hence
(s - 1)
2
26
By the axioms of the affine plane, there is exactly one line which is
incident with each of two distinct points or in other words two distinct
treatments of the array (1).
there is only one line.
or
which
v
s-l
(a., (3)
Suppose
.
So corresponding to two arbitrary treatments,
This line might belong to either
In any case, we can form exactly
belong to the same half block.
and
a.
of the array (1).
v R' v c ' vI' v 2
(s - 1)
Hence
Al
blocks in
= (s - 1)
are two points corresponding to two treatments
(3
Since the/are distinct points, by one of the axioms,
it follows that there can be exactly one line incident with both (a.
(3.
Since each pencil exhausts all points of the plane exactly once,
(s + 1) - 1 (
it is evident that there are
which
and
a. and
(3
= s)
are on two different lines.
pencils in each of
As only one block can be
formed with two distinct lines of a pencil as half blocks, there are
blocks of the weighing design in which
half blocks.
EXAMPLE:
Hence
=s
A
2
Let us take
s
=3
a.
and
(3
s
are in the opposite
.
1
and
L
1
o
=
2
constitute the complete set of orthogonal latin squares.
following array we get a balanced weighing design
Using the
BWD(9, 12, 8, 3, 2, 3)
described below:
Array:
1
4
7
2
5
8
3
6
9
Plan of the design
row set
column set
1st latin square 2nd latin square
1,2,3; 4,5,6
1,4,7; 2,5,8
1,6,8; 2,4,9
1,5,9; 2,6,7
1,2,3; 7,8,9
1,4,7; 3,6,9
1,6,8; 3,5,7
1,5,9; 3,4,8
7,8~9
2,5,8; 3,6,9
2,4,9; 3,5,7
2,6,7; 3,4,8
4,5,6;
.
...
CHAPTER III
THE COMBINATORIAL PROPERTIES OF
PARTIALLY BALANCED WEIGHING DESIGNS
3.1.
Introduction.
A wide class of designs called partially balanced incomplete block
(PBIB) designs which include the balanced incomplete block designs as a special
case c'were'inttoducedceby
R. C. Bose and K. R. Nair [14] in 1939.
A clear-cut classification and analysis of such type of designs was given
by R.C. Bose and T. Shimamoto [15] in 1952.
For a good account of the
present status of the combinatorial properties of Partially Balanced
Designs and association schemes, R.C. Bose [6] can be referred.
The present work on PBIB designs goes in two directions.
the study of the existing and new types; of association schemes.
One is
The other
is the investigation of the methods of construction of the corresponding
designs with various combinations of the parameters.
The latter aspect
raises the question of existence and non-existence of such arrangements.
For a definition of association schemes and PBIB designs, one can
refer to [6].
The main types of association schemes are:
(a)
the group divisible (GD) association scheme,
(b)
the triangular
(c)
the singly linked block (SLB) association scheme,
(d)
the Latin Square (L ) association scheme, and
(e)
cyclic association scheme.
assbciationscheme~
r
For exhaustive search of this type of designs constructed until
1956, the tables by Bose, Shrikhande and Clatworthy [12] and
28
W~H~Cli:ltworthy
[20] are of immense value both for research work as well
as for ready reference for a practical worker.
Balanced weighing designs were introduced and studied by R.C. Bose
and J.M. Cameron in [9] and [10].
As the main objective of this dissertation work is to study a class
wider than the class of BWD, this chapter is devoted to the definition
and primary properties (see 3.2) of this extended class of weighing
designs, called partially balanced weighing designs.
These can be defined
with reference to Jone or the other of the association schemes developed so
far, about which there has been already a mention in the last few pages.
The actual methods of construction of such designs (starting with
few types of association schemes) are postponed until Chapters 4 and 5,
whereas their analysis is considered in Chapter 6.
The definition of a partially balanced weighing design for a
(~3)
higher
association scheme is just a straightforward extension
of the case with two associate-classes.
3.2.
Some preliminary results on
Partially Balanced Weighing
Des1.gns
Definition of Partially Balanced Weighing Designs with two association classes:
Association schemes have been defined and have been widely used.
Given
v
treatments
1,2, ••• v, a relation satisfying the Iollowing conditions
is said to be an association scheme with 2 classes:
(a)
Any two treatments are either 1st, or 2nd associates, the
relation of association being symmetrical, i.e., if the treatment
a
29
is the i-th associate of the treatment
of the treatment
~)
a
(c)
i
Pjk
S is the i-th associate
i-th associates, the number
n.
~
a.
If any two treatments are i-th associates then the number of
treatments which are j-th associates of
is
then
(i=1,2) .
Each treatment has
being independent of
S,
and is
~ndependent
a
and
S
k-th associates of
of the pair of i-th associates
The parameters of the association scheme are
a
and
S .
(i,j,k=l,Z) •
A design is said to be a Partially Balanced Weighing Design (PBWD)
with two association classes with parameters
if there are
in
r
v
treatments arranged in
(v, b, r, p, All' A , A , A )
21
12
22
b -blocks, each treatment occuring
blocks, such that the blocks are of size
2p
which can be divided
into 2 halves with the following conditions:
(1)
All
times and in the opposite half blocks
(2)
A
12
Any two first associates occur together in the same half block
A2l
times.
Any two 2nd associates occur together in the same half block
times and in the opposite half blocks
Such a design will be denoted by
A
22
times.
(v, b, r, p, All' A21 , A12 , A22 ) •
The relation of this type of designs with the Partially Balanced Incomplete
Block designs can be seen with the help of the following theorem:
THEOREM 3.1:
of a PBWD
A necessary and sufficient condition for the existence
(v, b, r, p, All' A , A , A )
12
2Z
21
with
(n , n ,
2
1
1
p~,
2
Pij)
as parameters of the association scheme is the existence of a PBIB with
the same association scheme but with
A
1
* = All + A21
and
v* = v, b* = b, r* = r, k* = 2p,
AZ* = A12 + A~2' each of whose blocks can be divided
30
into two half blocks such that the half blocks form a PBIBD with
vo
o
0
= v, b O = 2b, r O = r, k O = p, All = All and A2 = A12 .
Proof:
The proof is obvious.
THEOREM 3.2:
A set of necessary conditions for a PBWD
(v, b, r, p, All' A2l , A12 , A22 )
(i)
2pb
to exist is that:
where
-vr
=
(ii)
B. = A2 · - AI·
~
Proof:
Let us consider the
r
e occurs. There are r(p-l)
e.
Since each of the
e appears
Ali
n.
~
~
blocks in which a particular object
other objects in the half blocks containing
objects which are the
times in the same half block as
_i~ttt
associates
of
e,
_ _(1)
=
The previous theorem guarantees that the existence of a PBWD
(v, b, r, p, All' A , A , A )
2l
12
22
design with
implies the existence of a PBIB
(v, b, r, p, A +A , A +A )
ll 2l
12 22
as parameters.
Hence the type of argument given in the above paragraph can be
applied to this PBIB design to see that
r(2p-l)
=
=
=
using (1) .
_ _(2)
.
~
31
By (2), we have
_ _(3)
From (1) and (3),
_ _(4)
~l
where
= A2l - All and 62 = A22 - A12 · This proves (ii).
(i) is obvious as the total number of objects involved in the PBWD
is given both by
NOTE:
2pb
on one hand and by
vr
on the other hand.
By (i) and (ii) it follows that
b
THEOREM 3.3:
A necessary condition for a PBWD
< v
Proof:
We know that
n
l
+ n 2 = v-I
(v, b, r, p,
<
for a two-class association scheme.
By Theorem 3.2,
r + (13 2 - I3 l )n l
or
(v-I) 13
13 2-13 1
2
=
=
nl +
(v-I) 13
r
13 2-13 1
2
3Z
...
since
(SZ-Sl)nl
~
(v-I)
=
(SZ -Sl)
>
by hypothesis.
...
ZSZ - Sl + r
v
>
_ _(5)
Sz
Since
it follows that
=
or
r
- nZ
(SZ -Sl)
nZ(SZ-Sl)
(v-I)
<
-
Sl
since
r
Sl
(SZ-Sl)
Sl
-n Z 2 -1 .
ZS 1 - SZ + r
_ _(6)
Sl
In this part also
Hence
which proves the Theorem 3.3.
only when
Sz
>
Sl .
33
1
P11
(any positive integer), then
Proof:
Let
6,
and
(p-2) + (r-1)
2
.
be any two treatments each of which is the first
~
associate of the other.
e
~
B(6,~)
Let
be any half block containing both
~.
Let
B (e)
i
and
B(6,~)
other than
B
i
(~),
i=l, 2, " ' , (r-1)
cQntaining
e
and
~
be the
(r-l) -blocks
respectively.
It is obvious that the set of blocks
{B. (e)}
~
is distinct from
{B.(~)}.
the set
J
STEP 1:
The contribution of
There are
associates of
(p-2)
e
STEP 2:
1
P11(6,~)
other treatments in
as well as
B(e,
B(e,
~)
~)
is
(p-2) .
which are the first
~.
The contribution of
Bj(~)
We can prove that any
from
<
(r-l) 2
from
cannot have more than one treatment
a2 (6)
in common with any of the half blocks
B (6) ,
l
Suppose if possible that
are two treatments common with
and
t.
J
and
t.
~
t.
J
, ..• B _ (e) .
r 1
B.Q,(e)
cannot be second associates.
Hence
t.
~
are first
and
associates.
Since
All ~ 2.
L~t
B. ( e)
~
and
t.
~
and
t.
J
occur together in two distinct half blocks,
This is a contradiction to the hypothesis that
Rj(~' i)
B. (~) •
J
All
=1 .
stand for the number of treatments common between
The above argument asserts that
R.
J
(~,
i)
=
0
or
1.
34
B.(~)
So the number of treatments common between the fixed
blocks
B (6), B2 (6), ... , B _ (6)
l
r 1
and the
J
is
r-1
R.(~,
E
J
i=l
This being true for all
j
,
i)
<
(r-1) .
the number of treatments
X which
satisfies the following three properties simultaneously is at most
(r-1)2 :
(i)
(X, 6)
occur together in a half block.
(ii)
(X, 6)
occur together in a half block.
does not belong to
(iii)
B(6,
~)
.
Hence Step 2 follows.
Now in order to look for the first associates of both
6 ,
we have to take into account
B(6,
~),
Bi (6),
Bj(~)
are the only blocks which contain the first associates of
(using
A
21
~
and
as these
6
or
~
= 0 or in other words that the first associates do not occur
together in opposite half blocks).
These comments together with the Steps 1 and 2 give that
1
Pn(6,~):
,is the same for all
6,
ep
and the proof holds good for any
<
(p-2)
arbitrary first associate pair.
Hence
+ (r-1) 2 .
35
THEOREM 3.5:
For a partially balanced weighing design with
>
Proof:
e
Let
n
possible that
<
2
first associate
e
of
<P
e
block in which
.
p
first associates of
(p-n )
2
e
.
Band
e€
Let
belonging to
e
B'
.
But since
e
occurs together with
occurs.
.
be the
Suppose i f
e
which belong to
B'
is false.
For a partially balanced weighing design, if
n
=
l
p-l •
and
and
<p
occurs
<p
can occur together in the same half block at most
occur in opposite half blocks in a particular block.
r
times in the entire design.
But this contradicts the assumption that
Hence
A = 0
12
are first associates and suppose if possible that
<p
e
<p
e
any
in every half
First we can prove that, under this given condition,
e
,
r = All
which proves the theorem .
then
Suppose
(p-n )
2
Hence the conclusion that there are
of the first associates of
THEOREM 3.6:
and
This proves that
We know that
<p
A
12
e
This means that
But
and
(r-l) times.
r = All •
cannot occur together in opposite half blocks.
=0
r(p-l) = nlA
r(p-l) = n A .
l 11
proves the theorem.
that
B
B'
This implies that there are at least
...
Proof:
p •
be an arbitrary treatment and let
two half blocks of a block containing
r = All'
Also
ll
+ n A12 .
2
r = All'
So
Hence
n
l
=
A = 0
12
(p-l) ,
implies
which
36
THEOREM 3.7:
I f there exists a PBWD with
1.
12
= 1. 22
'
then
>
Proof:
We know from the proof of Theorem 3.2, that
...
rand
n
=
implies that
being positive integers,
l
COROLLARY 3.1:
1.
since
It
1.
21
11
~
1.
COROLLARY 3.2:
A + 1.
= 0
22
12
0
1.
then
2
>
r = n (1. -1. ) .::.. n
l 21 11
l
= 0 for the latter, then
Proof:
=
=
12
If a PBWD can be obtained from a PBIB design
1.
2
1.
Hence the corollary follows.
using Theorem 3.1 and if
1.
> All .
is evident from Theorem 3.6 that
-1.
=
If there exists a PBWD with
r
Proof:
21
--r
implies
2
are non-negative,the equation
1.
12
and since
1.
1.
and
12
+ 1. 22 = 0 implies
All and
>
21
1.
1.
22
= 1. 22 = 0
12
Hence the corollary follows using Theorem 3.6 and Corollary 3.3.
THEOREM 3.8:
If there exists a PBWD with
(v, b, r, p,
as parameters and with
as parameters of the association scheme, then n , n
2
l
expressed in terms of
v, r, 8 ,
1
and
8
2
provided
can be
8
1
~
8
2
•
.
37
Solving
Proof:
v-1
n
= n 1 + n 2 , r = n1 B1 + n 2 B2 it is evident that
2 =
COROLLARY 3.3:
r - 13 (v-1)
1
13 2 - 13 1
(v-1) 13
and
n
1 =
2
- r
13 2 - 13 1
If PBWD with any type of two class association
scheme exists and if
r
= All'
then
>
Proof:
hence
n2~
By Theorem 3.5,
p
=
(n l +1).
parameters
n2~
Hence
COROLLARY 3.4:
p.
By Theorem 3.6,
n
2
If there exists a PBWD with
n
1
+ 1
or
or
= p-1
and
r
= All'
then the
of the association scheme satisfy the relation
and
~
1
nl+l •
v
>
Proof:
n
-
>
2
by Corollary 3.1.
n
1
+ 1
Hence
n
l
+ n
2
~
< v
-2
2n
1
+ 1
implies
Hence the corollary follows.
THEOREM 3.9:
13
2
= A22
- A •
12
If a PBWD exists with
then
13
1
and
13
2
13 1
= A2l
- All
and
cannot be riegative simultaneously.
being positive integers.
with
1
2
(n1 , n 2 , Pij' Pij)
as parameters of the association scheme, then the
38
parameters
A
22
n , n
1
2
can be expressed in terms of
r, p, All' A , A12 ,
21
as follows:
n1
=
1
_ {A 22 (-SlP + A21 )r}
A [rp
D.
.]
21
provided
Proof:
A
21
and
~O
and
D. ~ 0 ,
We know that
=
2pb'
vr
_ _(1)
_ _CD
By eliminating
n
from (1) and (3), we have
1
r[A
=
P - A (P-1)]
11
21
A11A22-A21A12
Substituting this in (2),
=
which proves the result.
1
-[rpA
21
=
CHAPTER ·IV
PARTIAL DIFFERENCE SETS AND PARTIALLY BALANCED WEIGHING DESIGNS
4.1.
Introduction.
A perfect difference set with parameters
a
(v, k, A) -difference set, is a
modulo v
or in other words
k-set - {d , d , .•. , d }
k
2
l
a t 0 (mod v)
such that every
v, k, A,
of integers
can be expressed in exactly
A ways in the form
d.
~
d
j
-
a (mod v)
where
d., d. ED.
:lJ
The classical papers on difference sets include Singer [46] and
Hall [26].
A general survey of the subject is given by
[26], and H.B. Mann [32].
~.
Hall, Jr.
A more comprehensive account of the present
status of "Difference Sets" can be found in [27].
The extension
of the idea of difference sets lead to
(v, k,A) -group difference sets.
A perfect difference set with
leads to a projective plane,
called cyclic projective plane.
A= 1
For that reason, this type of difference
sets are called "projective difference sets" or "cyclic difference sets".
It has been conjectured that for a projective difference set,
n
to
= (k-A) = (k-l)
n
= 1600
[30].
must be a prime power and this has been verified up
All cyclic difference sets with
k
<
50
are now
known [26, 31, 40, 46].
There are two directions of research on difference sets:
of new difference sets, and
(ii)
certain parameter combinations.
proo~
(i) construction
of impossibility of solutions for
While the progress in the constructions
was comparatively slow, an impressive number of theorems ruling out many
classes of parameter combinations have been obtained in recent years.
40
It is known that Difference Sets give rise to a class of Balanced
Incomplete Block Designs [7, 27].
with parameters
[4N-1, 2N-1, N-1]
matrices of order
4N.
The incidence matrix of a BIB design
can be used to construct Hadamard
Other Hadamard matrices may be obtained from
2
2
2
(4N , 2N -N, N _N) .
BIB designs with parameters
Hadamard matrices
have been used in the construction of binary codes [37].
The incidence
matrices of finite projective planes, finite Euclidean planes, and balanced
incomplete block designs are also being used in the construction of
error correcting codes.
For a brief account of these applications one
can refer to I.M. Chakravarti [19].
These are Some of the considerations
which have led to the generalization of "perfect difference sets" to
"partial differece sets", which are the main concern of this chapter.
PARTIAL DIFFERENCE SETS:
Definition:
Let
be a set of
nl
(i)
(ii)
jofj')
g
the
v
be a positive integer and let
integers satisfying the conditions:
d's
among the
reduced
{d , d ,
2
l
set of integers
o
are all different, and
nl(nl-l)
(mod v)
differences
... ,
dn
1
{l, 2,
The set of integers
d.
<
v
<
J
d.-d. ,
J
J
,
(j=l, 2,
e , e ,
1
2
... ,
(v-1) }
e , e ,
2
l
... ,
n )
l
... ,
n ,
1
occurs
... ,
occurs h times,
2
is a permutation of the
en}
2
and g of h
(dr' d2 , " ' , dn )
... ,
(j , j' = 1, 2,
each of the numbers
times, whereas each of the numbers
where
(d , d , " ' , dn )
2
1
1
en
.
satisfying (i) and (ii)
1
is called a partial difference set and is denoted by
set.
(v,n ,n ,g,h) -difference
1 2
Though the words"Partia1 Difference Sets"we"re not used, they
were applied in the construction of designs as early as in 1939,
41
by Bose and Nair [14], and a more precise formulation of this application
was given by Bose and Shimamoto [15].
The investigations on the methods of construction of weighing designs,
using the partial difference sets, are postponed until Section 4.4,
whereas the general description of the methods of construction of partial
difference sets and the other applications of the same are given
respectively in 4.3 and 4.5.
4.2.
Perfect Difference Sets.
The following theorem concerning perfect difference sets throws some
more light on the existing families of difference sets formed with all the
quadratic residues.
It is very useful in certain types of constructions
of the partially balanced weighing designs.
THEOREM 4. 2 . 1 :
Let
x
4
8
12
2
6
10
p •
{x , x , x
{x , x , x
Proof:
where
= 4u
v
+ 1
t
p
be a primitive root of the Galois field
,
... ,
,
""
= 0,1,2,
forms a
x4u-2 }
(4u+l, u, A)
also forms a
a fixed element of
of an odd prime,
GF(p n ) •
Then i f
difference set, then
(4u+l, u, A)
difference set.
will be of the form
x
t
or 3 (mod 4) •
So starting with the case
{ the
n
be the power
Any non-zero element of
(i)
=
Let
n
GF(p ) •
number of pairs
{the number of pairs
=0
t
(mod 4) ,
let
x
4h
be
Evidently,
(
q, )
w
such that
(q', WI)
=
A
such that
X4h
__ (x 4q+ 2 _ x 4w+ 2 )}
x 4h+ 2
=
(x 4q ' _ x 4w ')}
42
if we assume that the original difference set (by hypothesis) is a
(4u + 1, u, A) -difference set.
(Note:
(q', w') = (q+l, w+l)) .
Similarly starting with the following type of elements, we conclude
as follows:
(ii) {the number of pairs (q, w) such that
{ t h e numb er
=
0f
.
pa~rs
(q,
' w') such t h at
X
x 4h+ 3 = (x 4q '_x 4w ')}
(iii) {the number of pairs (q, w) such that
=
x
{the number of pairs (q' , w') such that
with the same notation of
4h+ l __ (x4q+2_x4w+2)}
4h+Z
x
A•
= (x4q+2_x4w+2)}
4h+4
(q', w')
'(iv) {the number of pairs (q, w) such that
--
4w
4q
= (x '_x ')} = A
,
.
x4h+3 = (x4q+2-x4w+Z)}
~ {the number of pairs (q', w') such that x 4 (h+l)+1= (x4q'~x4w')} = A .
The conclusions (i) - (iv)
elements of
GF(pn)
occurs the same number of times in the set of
2
6
4u-2
{x , x , ... , x
}.
differences formed from
COROLLARY 4.Z.l:
-difference set,
Proof:
~
Hence the theorem follows.
>
••• ,
is a
(4u+1, u, A)
(u-l)
4
u(u-l)
= 4uA gives
A
= (u-l)
4
Z
is an odd prime of the form 4h + 1 ,
2
6
10
4h Z-Z
then {x , x , x , ••. , x
} forms a perfect
If
1 (mod 2) ,
difference set with
Proof:
{x , x ,
A is given by
The obvious relation
h
48
In case
COROLLARY 4.2.2:
where
establish that each of the non-zero
p
hZ-I
(p, hZ, -4-)
as parameters.
By Theorem 8.6 of Mann [31] the 4-th powers form a perfect
difference set, when
p = 4h
Z
+ 1
is a prime and
h ~ 1 (mod Z) •
43
Hence the corollary follows from the Theorem 4.2.1.
Note:
this will be used in the construction of partially balanced weighing
designs (in Section 4.4) under the category "Rule II" •
4.3.
Partial Difference Sets and their construction.
4.3.1.
Partial difference sets:
Although no method of construction was given, few examples of partial
difference sets were given by Bose and Nair [14] and Bose and Shimamoto [15],
in constructing association schemes in partially balanced incomplete
block designs.
A lemma established by Nandi and Adhikary [35] can be restated
as follows:
LEMMA 4.3.1 (Nandi and Adhikary):
difference set if and only if
where the
n
1
(d , d , ""
l
2
dn )
1
is a partial
(d ,d 2 ,···,d ) = (-d ,-4 2 , ... ,-d ),
l
1
nl
nl
-sets are unordered sets.
Next, the following is an obvious necessary condition for the
existence of a (v, n , n , g, h) -difference set.
2
l
LEMMA 4!3!2:
A necessary condition for the existence of a
(v, n , n , g, h) -difference set is that
2
l
=
In the next sub-section, some theorems are developed to construct
new series of partial difference sets, together with those which are
used for the constructions of the partially balanced weighing designs.
44
4.3.2.
Some useful results for constructions of
Partially Balanced Weighing Designs:
THEOREM 4.3.1:
n
Let
is a positive integer.
Then the
= Pn,
4u + 1
Let
x
where
p
be a primitive root of
2
4
6
4u
{x , x , x , .•• , x }
2u -elements
is an odd prime and
n
GF(p ) .
constitute a partial
{4u + 1, 2u, 2u, u-1, u} •
difference set
Remark: It can be seen that this theorem is valid for any
Proof:
The proof of this theorem depends upon a lemma given in Bose
[ [7], Corollary 1.1.1]
LEMMA 4.3.3:
p
and which is stated below.
If
is a primitive element of
x
s 1
(x - _1)
= 4t+1
s
=
there is one zero,
and one zero,
(t-1) Q.R; 's
(t-1) Q.R. 's
n
GF(p) ,
and if
2
4
6
s-3
(x -1) ,(x -1), (x -1), ..• (x
-1) ,
is an odd prime, then among the elements
s
u.
and
and: t :7.non Q. R.
i
if
S
(t-1) -non Q.R. 's if
(4t-1), where QR and non-QR stand for quadratic residue and nonquadratic residul
Proof of the main theorem:
formed from
2
6
4
2u(2u-1)
The
{x , x , x , ••. , x
4u
differences which can be
can be written in the form of an
}
array given below:
x (x -1), x (x -1),
x 2 (x 4u- 2_1)
4 2
4 4
x (x -1), x (x -1),
x\x4u- 2_1)
2
6
2
2
2
6
4
4
x (x -1), x (x -1),
x 6 (x 4u- 2_1)
x 4u( x 2_ 1 ) ,x4u( x 4_ 1 ) , •.. x 4u( x 4u-2_ 1 )
_ _ (1)
45
Evidently, each column of the array in (1) contains all even powers
of the primitive root
x.
Hence if
(x
2h
_1)
is an even power of
x
or
in other words a quadratic residue, then that entire column exhausts
the even powers.
Similarly, if
(x
2h
_1) is a non-Q.R., then that entire
h-th column exhausts the set of all odd powers, 'of
2
4
6
4u
{x , x , x , " ' , x }
of the sets
and
x.
So the frequencies
3
4u-1 }
{x, x , .•. , x
are the
same as the number of Q.R. 's and non-Q.R. 's among the set
2
4
{(x -1), (x -1), "', (x
(u-1)
respectively
set,
4u-2
and
-1)}.
u.
So by definition of the partial difference
2
4
6
4u
{x , x , x , ..• , x }
(4u+1, 2u, 2u, u-1, u) ,
Example 4.3.1:
Let
But by the lemma, these are
forms a partial difference set with parameters
which proves the theorem.
u = 3.
Hence
(4u+1) (=13)
We know that 2 is a primitive root of
is a prime power.
GF(13).
It can be easily
verified that among the differences which can be formed from
(22,24,26,28,21°,212) = (4, 3, 12,9,10,1) ,
(4, 3, 12, 9, 10, 1)
occurs 2 times and each of the other remaining
6 non-zero integers (mod 13) ,
Example 4.3.2:
Let
the set
u = 4
ocurs-3 times.
Hence
know that 3 is a primitive element of
(1, 2, 4, 8, 9, 13, 15, 16)
(4u+1))=17)
GF(17)
is a prime power.
We
and that
is the set of quadratic residues.
It can be easily verified that the frequencies are 3 and 4 respectively
for~.thesets
of Q.R. 's and non-Q.R. 's among the differences formed from
(1, 2, 4, 8, 9, 13, 15, 16) •
46
Example 4.3.3:
Let
u = 9.
Hence
(4u+1) (=37)
We know that 2 is a primitive element of
is a prime power.
GF(37)
It can be verified
that the frequencies of Q.R. 's and non-Q.R. 's among the differences
formed from the Q.R. 's
[1, 3, 4, 7, 9, 10, 11, 12, 16, 21, 25, 26,
27, 28, 30, 33, 34, 36]
Example 4.3.4:
Let
are 8 and 9 respectively.
u = 7
.
Hence
(4u+1) (=29)
We know that 2 is a primitive root of
GF(29)
is a prime power •
can be verified
It
that the frequencies of Q.R. 's and non-Q.R. 's among the differences
fromed from the quadratic residues (4, 16, 6, 24, 9, 7, 28, 25, 13, 23,
5, 20, 22, 1) ,
are
6 and 7
respectively.
The following theorem due to Mesner [ L33], page 16] is going to be
2
4
6
{x , x , x , ""
used to infer that
root
x
x
4u
of
}
GF(4u+1)
with primitive
defines an association scheme.
Mesner's Theorem:
In a finite field of order
group
G and multiplicative group
order
(v-1)
and let
s
of
G'
such that
be a generator of
G' ,
G'
J.
i-th
for
i,
F(v, m)
in the range
modulo m where necessary,
1, ""
F(v, m)
association scheme with parameters
O
= {O}
,
to
is odd,
be the
sm,
which contains
y-x
v
let
in which two elements
associates if and only if
j, k
a
is even if
N generated by
be the co-set of
an association relation
are
Let
•
with additive
m be a divisor of the
N = (v-1)jm
multiplicative sub-group of order
a. , i=1,2, ..• ,m,
let
v
and
let
i-l
S
•
x, y
Define
of
a. , i=O, 1 , 2 , ... ,m •
J.
G
Then
m and interpreted
is an
m -class partially balanced
i
1
= Pj+l,k+l = Pj-i+l,k-i+l'
47
and
is equal to the number of elements of
0'. J-1.
• • +1
which occur
in the set obtained by adding the unit element 1 to each element of
The following is a corollary of the above theorem wli.icn defines
an association scheme, with respect to which partially balanced
weighing designs are constructed in Section 4.4.
COROLLARY 4.3.1:
field
A
O
=
GF(4u+1) ,
Let
{OJ •
Let
be the primitive root of the Galois
4u+1
where
A
l
x
2
is the power of an odd prime.
4
= {x , x , ""
Define an association relation
x
elements
i=O,1,2.
and
Then
are
y
Proof:
}
and
F(4u +1, 2)
A
2
3
5
4u-1 }.
x , x , ••• , x
= {x,
in which two non-zero
=
y -
X E
A.
1.
,
is a 2-class partially balanced association
v = 4u + 1,
F,1
4u
i-th associates if and only if
F(4u + 1, 2)
scheme with parameters
x
Let
n
1
= n
(:-1 :)
2
= 2u
and
(: :-~
P'2
The proof follows from the above theorem, by taking
v = 4u + 1
and
m= 2 .
Now we proceed to develop some results relevant for the construction
of the partially balanced weighing designs, which are dealt in [4.4].
LEMMA 4.3.6:
(i)
Let
v = 4u+1
(ii)
Let
x
(iii)
Let
qw
be of the form of
be a primitive root of
p
n
p
being an odd prime.
GF n
P'
Then
x
qw
x 4w+2u+qa
be defined by
h
were
(x
a = u-w .
4w
q
-1) = x w , for
w = 1,2, ... , (u-l) •
48
Proof:
x
4u
- 1
Since
=a
x
is a primitive element of
yu -
or
1
=a ,
except for the trivial case
where
=5
v
yu y
u-1
u-2
+ Y
y
x
Since
= x 4 :f
= x4
y
4W -
=1
or
1 ,
,
1 =
a
implies
+. .. + Y + 1 = a
or
x 4 (u-1) + x 4 (u-2) + ... + x 4 + 1
(x
4u
GF(4u + 1) ,
= a
_ _ (4)
=
1)
_ (x 4_1) [x 4 (u-1) +x 4(u-2) + ... + x
4w]
by the use of (4).
(x 4w _ 1)
_ x 4w( x 4_ 1 )[ x 4(u-l-w) + x 4(u-2-w) + ... + 1] •
=
Noting that the product of the last two expressions is
x 2u
and using the fact that
=
x
4w+2u+qa •
With the same notation and conditions «i) - (iii»
of the previous lemma, let
{x
w,
Qw-(2u+4w)+4
= x4w+2u[x4(u-w)_1]
This proves the lemma.
LEMMA 4.3.7:
a fixed
= -1, we have x 4w_l
[x 4 (u-w)_1]
the sets
, x
u
be ,of the form
(4t
4
{x qw+ , xqw+8, " ' , xqw+4u}
Qw-(2u+4w)+8
,
.~e,
X
Qw-(2u+4w)+4u}
± 3)
and let for
and
be denoted
respectively.
by
and
Proof:
If
rand
n
are disjoint
(empty set) .
(i.e. )
are not disjoint, let there exist two integers
n
.s.
u)
such that
x
Qw-(2u+4w)+4n
.
49
This implies that
=0
(4r + 4w - 4n) + 2u
(mod 4u) .
Since 4 must divide the left hand side of the above congruence, 4 divides
This is a contradiction, since
2u.
u
is of the form
(4t + 3) .
contradiction is due to the assumption that one member of
identical with an element of
s2
sl
Hence it follows that
s2'
This
is
and
are disjoint.
With the same notation as in the above lemmas,
LElv1MA 4. 3 •8 :
and under the same situation,
n
v = p
where
let us put an additional restriction that
Then the number of
{x4w _l,
u
is of the form
(4u +
is of the form
(4t + 3)
Q.R. 's among the set of
w=1,2, ..• ,(u-l)}
1) ,
.
(u-l) -elements
is different from that of non-Q.R. 's among
the same set.
Proof:
Suppose if possible that the two numbers are the same.
( u-2l) .
case, t h e numb er is
= w or u = 2w
then
(u-w)
that
u = (4t + 3)
for any fixed
w
x qw = x4w+2u+qa
e
,
,
.
Al so 1. f
q
u-w
f or any
= q
w
In that
1)
w= 1 , 2 "'" (u-
which is a contradiction to the assumption
Hence it follows that
(w = 1, 2,
... ,
(u-l» .
qu-w
and
qw
are distinct
The conclusion
of the Lemma 4.3.6 can be written as:
qw
-
4w + 2u + qa (mod 4u)
or
qw
-
4w + 2u + qu-w (mod 4u)
(Le •. )
qu':"w
-
q - 4w
w
2u (mod 4u)
-
qw + 2u
+ 4(u-w) (mod
-
qw
+ 2u + 4a (mod
4u)
(2)
4u)
(3)
50
q
It is obvious from (2) (or (3)) that if
is also odd and vice versa.
is odd. then
'
u-w
Similarly, if
is even, then
is also even and vice versa.
So the odd (even) powers of
{x
4w
_l,
w=1,2, ... ,(u-l)},
of odd (even) powers of
for some suitable
x,
if at all there are any, among
can be paired, and hence the number
x
among this set must be of the form
w=1,2, ..• ,(u-l)}
u =
4~
+ 1
is
(u-1)
x
among
u-l
-Z- = 2~
Hence
2
which is a contradiction, since
by condition (1) .
,
But by a remark given in the beginning of the
~.
proof, the common frequency of odd or even powers of
or
2~
u = (4t + 3)
This contradiction is due to our assumption that
the numbers of
Q.R. 's and non-Q.R. 's are the same among the set under
consideration.
Hence the lemma follows.
Remarks:
The Lemmas 4.3.6 - 4.3.8 are developed to prove a
theorem, viz 4.3.2 which is useful for the construction of PBWD's.
THEOREM 4.3.2:
(i)
form
Let
4u + 1,
p
which can be expressed as
(ii)
Let
u
be of the form
(iii)
Let
x
be a primitive root of
4 x 8 , ..• x 4u} ,
{x ,
~d
be an odd prime and let
3
7
{x , x
(iv)
{x 4w _ 1,
x
2 x 6 , x10
{x,
11
Among the
x
4u-1
(u-1)
v
be an integer of the
= pn
v
(4t + 3) •
GF
pn
.•. x 4u-2 },
} be denoted by
and let the 4 sets
{x, x5 , x9 , ... x 4 (u-l)+l }
A , A , A , A
2
1
3
4
respectively.
distinct elements
w = 1, 2, ... , (u-l)},
let there be
g
quadratic residues.
51
Then among the
12
4
8
{x , x , x ,
... ,
and
.B.
lnrlj.:
{x
4
are
A
4
2
u(u-l)
x 4u }
differences formed from
the frequenceis of the sets
, .B.2 , u-l-g
2
u-l-g
2
and
AI' A2 , A3 ,
respectively.
All the possible differences which can be formed from
,
x
8
,
x
12
,
... ,
x4u }
can be written in the form of an array
as follows:
x
4+ql
, x
4+q2
"
• • . •. x
x
4+qu-l
8+qu-l
_ _ (4)
x 4u+ql , x 4u+q2 , •••• x 4u+qu-l
Evidently, all the
Also the
4s
and
.. . ,
x
elements in a column of this array are distinct.
row of this array can be written as
s-th
4s
x 4s+ql , x +q2 ,
x qu-w+
u
x 4s+qu-l
qw+4s
(
.
Among these, the two elements
= x. 4w+2u+q u-w+4s by Lemma 4.3.6)
are distinct, as it is already noted (Lemma 4.3.8) for any fixed
w
~
(1
w ~ u-l) •
is of the form
If
is of the form
"q
since
4p"
is of the form
u-w
If
form
4h + 2
4h
qu-w
4p ,
q
u-w
u = (4t + 3)
implies "( ~
is of the form
since
then evidently
=2
4p + 2
(mod 4)
then
and
2u
is of the form (4c + 2). "
4w + 2u + qu-w
=2
(mod 4)
The same type of argument leads us to conclude that
"q
il
q
w
u-w
+ 4s
+ 4s
=3
1 (mod 4)"
(mod 4)"
u-w
So we conclude that
+ 4s)
,
4w + 2u + q
and
implies
vice versa .
is of the
52
So for a fixed
s,
( x 4s+qu-w , x 4s+qw)
be of the form
the elements can be paired as
for different
(x 4c+ l , x 4b + 3 )
w's
such that this unordered pair will
or
This assertion is true for all
s (s = 1,2, ""
u).
This together
with the fact that each column exhausts all the possible 4-th powers
(which are
for
u
in number), when
j = 1, 2, ""
u,
xqj
is taken out as common factor,
lead to the conclusion that the columns
of the array (4) can be so paired that
u + u
= 2u individual
elements of each pair of columns exhaust all the Q.R. 's exclusively
or they exhaust all the non-Q.R. 'so
So if
{x
4w
_1,
g
and
h
stand for the Q.R. 's and non-Q.R. 's among
w=1,2, ... ,(u-l)} ,
=u
(i)
h
- 1 - g •
(ii)
g ~
(iii)
The number of elements of the form
h
(by Lemma 4.3.8) .
those of the array (4) is
t
we conclude the following:
£
2 '
or
(u-l-g)
2
x
t
among
according as
is even or odd.
(iv)
All the
u
elements in a column of the array (4) are
distinct and each column contains elements of the form
t
covers exclusively all possible elements of the form
for some
i
(i = 0, 1, 2, or 3) .
x
t
where
t - i (mod 4)
53
So with the help:oLarray (4)" i t follows that among the
differences formed from
of the sets
Al = {x
AI' A2 , A3 , A4
respectively.
4
are
,
x
.82
8
,
,
... ,
x4u} ,
.8.
2
u-l-g
the frequencies
and
2
u-l-g
2
This proves Theorem 4.3.2.
COROLLARY 4.3.2:
With the same notation as in Theorem 4.2.2,
2
6
x , x ,
among the differences formed from
the frequencies of
u-l-g
- 2
,
u(u-l)
.82 '
are
AI' A , A , A
3
2
4
... ,
x
4u-2
u-l-g
.8.
2 '
2
and
.
,respect~ vely.
2
6
4u-2. 2 4u
4
8
4u-4
A = {x , x , "', x
} = x {x , x , x , ... , x
}
2
248
, x 4u-4 ,x4u} , we get the same type of differences
or .x {x , x ,
Proof:
Since
...
as in the theorem
resulting
x
t
(for the case
AI)' except for the fact that each
will be multiplied by
x
2
So denoting the set of differences for
D
s
Al
and
A
Z
as
respectively, the assertion of the Theorem 4.3.2 that
+
u-l-g
2
A3
+
u-l-g
2
A4
implies
+ .8. A + u-l-g A
D
2
s
1
.. 2
4
+
u-l-g
2
A3
(by the remark of the above paragraph).
Hence the corollary follows.
D
f
and
54
4.4.
Let
The construction of PBWD's from difference sets.
{d , d , ""
1
2
~,
(v, n , n ,
2
1
S)
dn }
1
and let
be a partial difference set with parameters
D = {d , d , ""
2
1
and
define a cyclic association scheme.
{a'l'
a'2'
... , a,1p }
1
1
Suppose
are
and {b'l'
b'2'
.•• ,'b,}
1
1
1p
2t -sets of distinct integers, each of size
p ,
(i=1,2, ... ,t)
with the following
properties:
are
I.
disjoint.
Among the differences
II.
(i=1,2, ... ,t,
IV.
occurs
V.
occurs
j:f iI., = 1,2, •.. ,p), each
A
12
dj
1J
- b.
occurs
n)
1",
Among the differences of the form
A
21
+
e
(q=1,2, ... ,n )
2
q
(a., - b, )
1J
1i1.,
,
each
d,
J
times.
Among the differences mentioned in IV, each
A
22
eil.,
(iI.,=1,2, ..• ,n )
2
times.
i-th
{ail + j, a
times.
All
times.
We treat the initial block
of the
1",
1J
Among the differences mentioned in II, each
III.
occurs
(a., - a , n ) , (b"
i2
set as
{a·1 1 , a·1 2 , ""
a.1p ; ~'1'
b'2'
1
1
... , b,
1p
}
'O'-th block of that set and the block
+ j, ... , a ip + J; b i1 + j, ... , b ip + j}
where each
element must be interpreted as belonging to the residue system mod v, is
called
residue
j-th block generated from the initial block.
'0',
it will be taken as
and prove the following theorem:
v.
Wnen we consider the
With this notation, we can state
55
THEOREM 4.4.1:
{b·1. 1 , b.1. 2 , ""
exist, then the
{ail' a i2 , ""
I f the
sets {ail' a i2 , ... , a.~p } ,
i=1,2, ... ,t,
b.lp }
vt
2t
satisfying the properties I - V
blocks generated by developing the sets
a.~p ; b.~ l , b.~ 2 , ""
b.~p }
treated as the initial blocks,
constitutes a partially balanced weighing design with the parameters:
_ _(1)
and with the association scheme defined by the partial difference set
... ,
with parameters:
n1-a-l'j
1\ = (
and
\~l-a-1
2
1
.
.......
/"
_ _ (2)
n -13
13
=
J
n -n +a+1
I
~~-a
n 2-n I +13-1
.
/
Thus the corresponding association scheme called cyclic association
scheme is specified by the parameters
Proof:
and
a, S, n,
n
2
For some verifications arising in the proof, ''Ie can have
,;\11
idea
of the situation by restricting to a single initial block from out of the
set of
t -blocks.
We know that the number of blocks is
blocks.
We also know that
p
q
,
a. ; b
~p
il
, b '
i2
... ,
i f and only if we can find an
a .. + q = u
~J
... ,
or
b .. + q
1.J
=u
when we start with
t
initial
is the half-block size.
Next, consider a treatment numbered
{ail ' a i2 ,
vt
.
So
u.
b. },
l.p
a ..
~J
q
u
or
In the development of
occurs in the block number
b ..
~J
such that
can be uniquely fixed by
56
q
=
u - a
or
ij
u - b
blocks generated from
the blocks in which
u
-
-
a ip ' u
bi!' u
as the case may be.
ij
So restricting to the
{ail' a i2 , ... , a ip ; b il' b i2' ... , b ip} ,
u
-
occurs are the ones numbered
b i2l ... u
-
As there are
b.
l.p
-
u
t
ail' u
a
i2
\ •..
'initial blocks,
.
r = 2pt
u
Let
and
s
be two treatments, which are first associates.
q-th block of the
Then the
i-th
set will contain both
u
and
the first half-block i f and only i f we can find two integers
s
in
and
a ..
1.J
such that
aiQ,
a .. + q
l.J
From (3) and (4)
a ij - aiQ,
=u
- s
=
U
_ _(3)
=
s
_ _(4)
.
Let the partial difference set
If
u
element of
Thus
and
s
D
since
(u - s)
u
and
as
d
h
s
mentioned in the
D.
hypothesis be denoted by
are fixed, then
u
and
s
(u - s)
is also fixed and it is an
are first associates of each other.
can be identified to be
d
h
(say).
occur together in the first half of the
Then it is evident that
i-th
can be represented as a difference between two
set, as many times
a. 's •
l.
Similar argument applies to the other half blocks of the
(b
-
il
, b
i2
, "', b
ip
)'
difference between two
So the number of times
a. 's
l.
d
h
occurs either as
or as difference between two
is the same as the number of times
u
and
s
i-th
b. 's ,
l.
occur together.
But
set
57
this number is
All
by the condition II.
Also this entire argument
is valud as long as the pair constitutes a pair of first associates.
In the same way, it is easily seen that two treatments which are
first associates will occur in opposite half blocks of the same block
A
2l
times and two treatments which are second associates will occur in
the same half block
block
A
22
A
12
times and in opposite half blocks of the same
times.
THEOREM 4.4.2:
A ncessary set of conditions for the existence of
a cyclic PBWD described in Theorem 1 is that:
(i)
=
2tp(p - 1)
(ii)
=
2tp
2
Proof:
(i)
block is
The totality of pairs which can be formed from a single initial
p(p-l)
and hence the total number of all possible pairs both
of whose elements belong to the same half block, is
2tp(p-l) .
Considering
any pair is equivalent to considering the difference between the two integers
representing the pair of treatments.
But difference between any pair must
belong to
D = {d , d , ... , d } or E = {aI' e , ... , e }· The
l
2
2
n2
nl
existence of the PBWD guarantees that each element of D occurs as
a difference between the elements of such pairs
element of
E
occurs
Since there are
it follows that:
This proves (i).
A
12
All
times and that each
times.
elements in
D and
=
elements in
2tp(p-l)
Similar argument establishes (ii).
E ,
58
THEOREM 4.4.3:
(4u+l)
form
where
x
Let
4 4u-2
x -x
p
4
8
Al = {x , x , ""
Let
Let
(A 2 - AI)
u
be of the form
(u-l-g)
Proof:
and
2
}
be of the
6
2
4u-2 }
{x , x , x
=
differences
•.• , x
(4t+3)
4u
4
2
4
6
(x -x , x -x ,
2
6 ..• , x4u-x4u-2)
-x, x 4u-x,
and
g -quadratic residues among the
then among the
(A 2 - AI) ,
(AI - A )
2
2u
2
(u-l) -elements
-differences formed
each quadratic residue occurs
times and each non Q.R. occurs
The set
4u
n
be defined.
h=1,2, ... ,(u-l)
(AI - A2 )
x
v = p
GF(pn) •
(AI - A ) stand for the u
2
8 2
8 6
8 4u-2
x -x , x -x , .•. , x -x
Let there be
from
be an odd prime and let
is a primitive element of
Similarly, let
(i)
Let
(g+l)
times.
of differences can be written as an array given
below:
2
4u-2 -1)
6
4u-2 -1)
......... x (x
......... x (x
x 4u-2( x 2_ 1') , x 4u-2( x 6_ 1 ) , x 4u-2( x 10_1) , •••••.. x 4u~2( x 4u-2_ l ) .
So any arbitrary element of
xhw+4s+2
where
(AI - A ) will be of the form
2
'f f
4w
hw
2
(x + _l) = x
Let h + 48 + 2 = R which
w
w
obviously depends on the arbitrary integer
hw+4s+2
Rw
x
s.
With this notation
=x
The corresponding element (i.e. just the negative of this arbitrary
R
Rw+2u
element) of (A - AI) can be written as (-l)x-~ = x
since it
2
2u
-- (-1). Since
' "
0 f x that x
d f 1n1t10n
f o 11ows f rom the e
59
2u = 2(4t+3) = 4(2t+1) + 2 , the <co.mparisoill of R
and
w
l\v and x Rw+2u
in x
leads us to the conclusion that
(mod 4)
appearing
w
R +2u - 0, 1, 2, 3
w
according as
R
w
Let us write the
and the
R +2u
u
elements of the
_ _(5)
(mod 4)
2, 3, 0, 1
(w+l) -th column of the above array
u -elements corresponding to their negatives as a partitioned
row vector as follows:
w+2u) .
( x 2+hw ,x 6+hw , ... x 4u-2+hwIx 2+hw+2u ,x 6+hw+2ti , ... .4u-2+h
x
Now let us: s.t,uC!W the nature of these elements.
(h +4c+2+2u)
w
it is obvious that if
4(s-c)
(h +4s+2)
and
w
be two arbitrary elements of this partitioned row vector taken
respectively from the first and second part.
that
Let
= 2u(mod
h +4s+2
w
4u)
= hw+4c+2+2u
and hence that
Comparing these two elements,
(mod 4u),
4
we have to conclude
divides
2u,
which is
impossible.
Also, it is obvious that the first part of (6) is a set of
distinct elements and that the second part of (6) is a set of
u
u
distinct
elements.
So it follows from (5) that all the
2u -distinct elements of (6)
are quadratic residues or non-quadratic residues according as the corresponding
xRw
(see (5»
So if the array
e
is a quadratic residue or a non-residue, respectively.
(A
2
A)
l
is formed directly from
(A
l
- A)
2
by
change of sign of each element, we have the conclusion that:
. Al-A2
. either
the (w+l) -th column of the combined array
(i)
A2- Al
exhausts all the
2u
Q. R. 's or all the non-Q.R. 'so
(6)1
60
(ii)
If
c
is the number of quadratic residues among
2
6
4u-2
{(x -1), (x -1), ... , (x
-I)},
quadratic residues repeats itself
Determination of
,
Q.R. s among
of
g
c:
(i) implies that the set of
c
times in the combined array.
By the Corollary (1.1.1) of [3] there are
2
4
6
4u-2
{x -1, x -1, x-I, ... , x
-I}
in the hypothesis,
c
=
Since the total number of rows is
occurs among
(AI -A2 )
g
+
1
A -A
This proves Theorem 4.4.3. 2 l
THEOREM 4.4.4:
(4u + 1) .
(i)
Let· u
Let
Let
4h _l),
p
x
u
times in
( = u-(u-l-g»
v
n
be a primitive root of
g
in this array, each
be an odd prime and let
be of the form
(ii) Let there be
{(x
(1l-1-g)
)
A2-A
l
non Q.R.
form
and hence by the notation
u-l-g .
So by (i), each quadratic residue occurs
l 'A1 '-A2
(u-l)
GF(p )
times.
= p n of the
•
(4t + 3) .
quadratic residues among the
(u-l) -elements
h=1,2, ... ,(u-l)}.
Then the initial block
4
8
{x , x ,
... ,
x
4u
2
6
x , x , •• "
x
4u-2
}
when developed, gives a partially balanced weighing design with the association
scheme defined in the Corollary 4.3.1 and the parameters of the design
being
Proof:
(v, b, r, p) = (4u+l, 4u+1, 2u, u)
We follow the notation of
Al
and
and
A
2
of the Theorem 4.4.3
and note that the initial block can be written as:
{AI; A } .
2
61
Step 1:
By the Theorem 4.3.2, among the differences formed from
each quadratic residue occurs
u-l-g
~
Al '
times and each non Q.R. occurs
2
times.
2
Step 2:
By the Corollary 4.3.2, among the differences formed from
1
each Q.R. occurs
Step 3:
By the Theorem 4.4.3, among the differences
which are the opposite differences arising from (AI;
occurs
(u-l-g)
u-l-g
times and each non Q.R. occurs
times and each non Q.R. occurs
2
{AI - A , A - AI}
2
2
Ai.
(g+l)
each Q.R.
times.
Steps 1 and 2 imply that among the differences formed from the same
half block, each Q.R. occurs
(u-l-g) times.
Hence
g
times and each non Q.R. occurs
(All' A12 )
Step 3 implies that
(A
2l
,A
22
=
(g, u-l-g) .
)
=
(u-l-g, g+l) .
Hence the initial
block under consideration satisfies all the 5 conditions of the Theorem
4.4.1, with
{d , d , ""
2
l
{e , e 2 , "', e 2
l
J
= {x,
d
2u
}
= {x 2 ,
6
4u
4
x , x , ..• , x }
3
5
4u-l
x , x , "', x
}
and
defining the
association scheme.
The parameter sets (1) and (2) of the Theorem 4.4.1 are given by:
=
(4u+l, 4u+l, 2u, u; g, u-l-g, u-l-g, g+l)
u
and
=
=
2u ,
as noted in the Corollary 4.3.1.
and
P 2
=
( u
u "'\
U-l) ,
62
u = 3;
EXAMPLE 1:
v =4
x
3 + 1 = 13
(2 4 , 28 , 212 ) = (3, 9, 1)
GF(13)
(2 4_1, 28_1) = (2, 2 3)
...
.
By Corollary 4.3.1
g =
2
is a primitive root of
(22, 26 , 210) = (4, 12, 10)
°.
(4, 3, 12, 9, 1O, 1)
association scheme and by Theorem 4.4.4,
defines a cyclic
(3, 9, 1; 4, 12, 10)
gives
a PBWD with the parameters of the association scheme and the design
being given by (using (8) and (9»
=
(6, 6)
(13,13,6,3;0,2,2,1)
_ _(9)
This can be seen by the direct verification.
EXAMPLE 2:
GF(29)
u = 7;
v =4
x
7 + 1 = 29.
2
is a primitive root of
and
(16, 24, 7, 25, 23, 20, 1)
g
By the Corollary 4.3.1,
=
4 .
(4, 16, 6, 24, 9, 7, 28, 25, 13, 23, 5, 20, 22, 1)
defines a cyclic association scheme and by Theorem 4.4.4
(16,24,7,25,23,20,1; 4, 6, 9, 28, 13, 5,22)
gives a PBWD with the
63
parameters of the association scheme and the design being given by
(using (8) and (9»
;)
(v, b, r, p; All' A , A , A )
21
12
22
=
(29,29,14,7; 4,2,2,5) .
This is verified separately by the direct method.
EXAMPLE 3:
GF(61)
u = 15,
v = 4
T h e powers
can be seen to be
{x 4w}
15 + 1 = 61.
~
,
2
and
is a primitive root of
{(x
4w _1),
w=1,2, ... ,14}
{16,12,9,22,47,20,15,57,58,13,25,34,56,42,1} ,
{4,3,48,36,27,5,19,.60,45,49,52,39,14,41,46}
and
(228,215,23,255,258,226,25°,252,232,28,29,221,237,254)
So evidently
g
= 8.
respectively.
So the parameter sets of the corresponding
association scheme and the PBWD are given by:
=
=(14
15
(30, 30)
=
15)
15
f5 15)
~5
14
(v, b, r, p; All' A , A , A ) = (61,61,30,15;8,6,6,9) .
22
21
12
and
THEOREM 4.4.5:
Let
v = 4u+1
be the power
p
n
of an odd prime, p .
If Ix 4 , x 8 , "" x 4u} forms a perfect difference set, then
4
8
{x , x ,
, x 4u x2 , x 6 , x 10 , "" x 4u-2 } generates a PBWD, with
2
4
6
4u
•• " d } = {x , x , x , "" x } defining the cyclic
2u
...
association scheme and with the following paramter sets of the association
scheme and the design:
64
Zu ,
=
PZ
u-1 u-1 u-1 u+1
(v, b, r, p; All' AZl ' AlZ ' AZZ ) = (4u+1,4u+l,Zu,u;--Z-'--Z-'--Z-'--Z-)
Proof:
By Theorem 4.Z.1, both half blocks of the initial block are
u-l and hence
A= --4-
difference sets, with
Z 4
6
{x , x , x , ""
By Theorem 4.3.1, the elements
difference set with
(All' A2l ) = (u-l
Z' u-1)
Z
.
{4u+1, Zu, Zu, u-1, u}
x
4u
}
form a. partial
Hence among the opposite
differences arising from the initial block, each Q.R. occurs (u_1)_(u;1)=
u-l) .
u-1
u+l
(--Zt1mes and each non Q.R. occurs u-(--Z-)=(--Z-) times.
Hence by the
theorem (4.4.1) and the corollary (4.3.1), it follows that the initial
Z 4
{x, x , ""
block generates a PBWD and that
association scheme.
4
4u
}
defines the cyclic
Hence the theorem follows.
COROLLARY 4.4.1
(Z9-+1) Z
x
If
8
then {x , z ,
v = 4u+1 is a prime, where
... ,
x
4u
Z
x , x
6
x
10
,
... ,
u
x
is of the form
4u-Z
}
generates
a PBWD
with the same association scheme as in the Theorem (4.5.),
Proof:
This fo11ws from the Theorem 4.4.5, using the result that under
the above circumstances,
4
8
{x , x , ""
EXAMPLE 1:
x
u = 9,
v = 4
x
9 + 1 = 37.
4u
}
forms a perfect difference set.
Z is a primitive root of
37.
4
{(x, x 8 ,
... ,
x
36
Z
6
), (x , x , • '"
x
34
)}
can be written as
{16,34,Z6,33,10,lZ,7,1; 4,Z7,Z5,30,36,Zl,3,ll,Z8}
=
18
-(:
:)
65
and
(v, b, r, p; All' A , A12 , A22 ) = (37,37,18,9; 4,4; 4,5)
2l
which is true by direct verification also.
Remarks:
The Section 4.4 can be :concluded",asan ·attetnpt-:to:construct
partially balanced weighing designs, w:i.tPL:.cycliccassociatd:On;sc1;!.emes:, 'under
two different methods which form particular cases of two rules, mentioned
below as Rule I and Rule II.
Rule I:
1.
Association scheme:
(a)
G
... ,
c/'l' c/'2'
e
2.
v elements
is a set of
c/'v_l
c/,O {= e} ,
.
(b)
G -
{e}
(c)
E
l
forms a partial difference set.
=
E
l
u E
2
Design:
(a)
D
l
and
D
2
.. :p
(b)
If
distinct elements from
D.D.~
are two disjoint sets, each containing
J
l
G
stands for the set of all differences formed by
then
and
(c)
Rule I says that if such a pair
=
D , D
l
2
can be found,
then
(D ; D )
2
l
leads to a PBWD, provided (E ,E ) can be found with conditions described in 1.
l 2
66
Rule II:
1.
Association scheme:
(a), (b), (c) are the same as in Rule I.
2.
Design
(a)
D
l
and
D
2
are disjoint.
(b)
D
l
and
D
2
are perfect difference sets separately.
(c)
IDII
(d)
{(](,l)E } u {(](,2)E }
l
2
=
ID2 1
.
Rule II says that if such a pair
=
D u D '
2
l
D , D
l
2
with
](,1 :/: ](,2
can be found,
leads to a PBWD, with the association scheme given by
(D ; D )
l
2
(E , E ) .
2
l
The parameter sets of the designs constructed under Rules I and II
can be described in Tables 4.1 and 4.2.
Both of the Tables 4.1 and 4.2 describe Some examples of partially
balanced weighing designs with cyclic association scheme.
But they
describe constructions based on Rules I and II respectively.
Tables
4.l(A) and 4.2(A) describe the parameter-sets for a few of the PBWD's
of this type, whereas 4.l(B) and 4.2(B) describe the actual plans of
such designs.
67
..
TABLE 4.1. (A)
v
S. No.
4u+1
b
r
p
2u
u
4u+1
2u
2u
g
.. n 1 .
u
2
}\ll
A12
A
22
u-1-g
u-l-g
g+1
A21 ,
1
13
6
3
13
6
6
0
2
2
1
2
29
14
7
29
14
14
4
2
2
5
3
61
30 15
61
30
30
8
6
6
9
e
TABLE 4. 2. (A)
S. No.
v
r
2
4x +1
2x
2
x
p
b
n
2
2
4x +1
2x
1
2
n
2x
2
2
All
2
x -1
2
A21
2
A12
A22
2
2
x -1 x -1 x +1
2 -22
1
37
18
9
37
18
18
4
4
4
5
2
101
50
25
101
50
iO
12
12
12
13
68
TABLE 4.1. (B)
Design (1
v
==
13,
r
6,
==
3,
A
21
0,
==
All
p ==
2,
==
;)
/
PI
b == 13,
=~
A12
n
E2
;
2,
==
1st' Associate
1
5,4,13,10,11,2
r
==
A11
22
p ==
7,
4,
A21
==
b == 29,
A
2,
n
2,
=
12
;\
7
1
5,17,7,25,10,8,29,26,14,24,6
21,~23, 2
A11
3O,
==
p
8,
5
7
Initial Block
1st Associate
==
14
2
6
Variety
r
==
22
Association Scheme
v == 61,
1
14, n
==
l
7
7
6
Design (3
==
(3,9,1; 4,12,10)
14,
==
A
6
2
Initial Block
Variety
v == 29,
n
~ ~)
==
Association Scheme
Design (2
6,
==
l
==
15,
A21
==
l'1 . (14
b
6,
==
1\
lS)
15
16,24,7,25,23,20,1;
4,6,9,28,13,5,22
61,
n
==
6,
12
'
£2
1
==
;\
==
30,
n
2
==
30
9
22
"
C )
S 15
14
Association Scheme
Variety
1
1st Associates
Initial Block
17,13,10,2~,48,21,16,58,59
16,12,9,22,47,20,15
14,26,35,57,43,2,5,4,49,37
57,58,13,25,34,56,42,1;
4,3,48,36,27,5,19,60,
28,6,20,61,46,50,53,40,15
42,47
45,49,52,39,14,41,46
69
TABLE :4 ~2 • (BY
Design (4
v
=
37,
r
1\1
=
18,
= 4,
p =
1~1 =
Association Scheme
Variety
1st Associate
1
17,35,27,10,34,11,13,
8,2,5,28,26,31,37,22,
4,12,29
9,
b = 37,
4,
n
""12 = 4,
l
=
18,
n
2
= 18
~2 = 5
Initial Block
16.34,26,9,33,10,12,
7,1; 4,27,25,30,36,21
3,11,28
CHAPrER
.V
CONSTRUcr ION OF PBWD WITH TWO ASSOCIATE CLASSES
5.1.
Introduction
Partially Balanced Incomplete Block designs have been introduced by
Bose and Nair [14].
The classification of association schemes has been
explained in a paper by Bose and Shimamoto [15].
This classification
has been followed by Bose, Clatworthy and Shrikhande [12] and Clatworthy
[20] in preparing the tables of partially balanced designs with two
associate classes.
As the cyclic type of association scheme has already
been defined and since the corresponding PBWD's have been constructed
in the previous chapter, this chapter deals with the construction of
PBWD's with an association scheme other than the cyclic one.
Before considering different association schemes, they are first
described below.
(a)
Group divisible
(GD)
association scheme:
In this case there
\
are
mn
each.
treatments, which are divided into
m groups of
n
treatments
Two treatments belonging to the same group are first associates,
and two treatments belonging to different groups are second associates.
The association scheme can be exhibited by writing down the
mn
treatments
in the form of a rectangular array, the treatments of the same group
occupying the same row.
association scheme
So
It is readily seen that the parameters of the
obtained are
v
= ron,
n
1
= n-1,
Pl c:z n(:-l~ and Pz = Cn:l) n::~::)
n
2
= n(m-l),
=
It has been shown by Bose and Connor [13] that for a
GD
design,
Hence they have been divided into three classes:
71
Singular
(i)
(8)
(SR)
(ii) Semi-regular
(iii) Regular
(R)
= Al ,
r
if
if
r > Al
r > Al
if
and
rk - A v
2
(rk - A. v) > 0
2
and
The partially balanced weighing designs of the
GD
0
,
.
type are discussed
in Section 5.5.
(b)
Triangular association scheme:
m(m-l)
fill in the
2
We take an
mx n
square, and
positions above the leading diagonal by different
treatments, taken in any prder.
The positions in the leading diagonal
are left blank, while the positions below;this diagonal are filled so
that the scheme is symmetrical with respect to the diagonal.
in the same row (or same column) are first associates.
Two treatments
Two treatments
which do not occur in the same row or same column are second associates.
It is readily verified that the parameters of the association scheme so
obtained are
PI
(-2
m-3
v
0
= l1).(m-l)/2,
m-3
n
=
l
2m-4,
n
2
(m-2)(m-3) /2 ,
=
2m-8
and
(m-3~(m-4)
P
2
=
Gm:a
(m-4) (m-5)
2
This scheme is called the triangular association scheme.
The PBWD's of the triangular type are constructed in Section 5.3.
(c)
L -association scheme:
2
may be set forth in a
s x s
Consider
v
= s 2 treatments which
scheme as:
1
s+1
(s-l)s + 1
2 ••••••••
s
s+2 ....•.. 2s
(s-1)s + 2 .•. s2
72
We define two treatments as first associates if they occur in the same
.. ,
row or column of the square scheme, and second associates otherwise.
association scheme so defined may be called the
L
2
v = s ,
n
The parameters of the
PI
(-2
=
s-l
L
2
8-1
scheme are
)
P
and
(s-l) (s-2)
The PBWD's with the
=
2
2
l
The
-association scheme.
= 2(s-1),
n
2
=
(s-l)
2
:~:::~)
0<8:2)
L -type association scheme are constructed in
2
Section 5.4.
Before discussing the construction methods of PBWD's with different
association schemes, two general theorems (1 and 2) are proved in Section
(5.2) which enable us to obtain PBWD's from some existing PBWD's and
resolvable PBIB designs.
Lastly the parameter sets corresponding to the PBWD's (which are
constructed) with the
(i) triangular association scheme
association scheme and
Tables
5., ·6\
and
(iii) GD -association scheme are tabulated in
(7A -
~")
5.2.
THEOREM 5.1:
we can deduce a
Proof:
(2) L 2
respectively.
Some general theorems.
If there exists a PBWD(v, b, r, 3; All' A ; A12 , A22 ) ,
21
PBWD(v, 9b, 6r, 2; 3A
ll
, 4A
21
, 3A
12
, 4A
22
) .
Taking any specific block, we can form 9 blocks by taking all
possible pairs from each of its half blocks. Hence for the deduced design,
the number of blocks
b*
= 9b.
To find
r,
it is evident that there
are two possible pairs containing a given object, which can be formed
73
for each of these possibilities.
So whenever a fixed treatment occurs once
in the original design, it occurs 6 times in the deduced one.
number of replications
r* = 6r.
Hence the
Whenever a pair is fixed in the deduced
design, it constitutes a half block three times for each original block
which contains the same.
Hence the
A -parameters
Atl' At2
are given by
Similarly considering the blocks in which two treatments occur in
opposite half blocks, we have
= 4A 2l ,
A~l
A~2
= 4A 22 • Hence the
theorem follows.
THEOREM 5.2:
If there exists a resolvable PBIB design
[v, rt, r, k, A , A ]
l
2
with any association scheme and
b = rt
blocks,
then there exists a
Proof:
Let us start with a set of
t -blocks of the resolvable PBIB
design, which constitute a single replication.
(~) pairs of blocks from out of these
design by taking these
t
Let us form all possible
blocks.
Let us form a weighing
(~) pairs as blocks 6f the former, one block of
each pair being the first half block and the other one as second half
block.
In this way, it is evident that each "resolvable set" of the
given PBIB design gives rise to
design.
Hence
(v*, b*, r*, p*; Atl'
(~) blocks of the corresponding weighing
where
A~l;
Ar2'
A~2)
are parameters of the weighing design.
If we fix a particular treatment
set of
t
8
and consider "a resolvable
blocks", it is evident that the corresponding blocks of the
weighing design contains this treatment
(8),
(t-l)
times.
But this
74
e
arbitrary treatment
r
sets.
Hence
r*
occurs
= r(t-l)
we have to verify the
.
r
times, occuring once in each of the
Since evidently
v*
=v
and
p*
=k
,
A -conditions to complete the theorem.
Since each block of the PBIB design appears
(t-l)
times as a half
block of the weighing design, the number of times any two first (or second)
associates occur together in the same half block is
by the corresponding number
Al (or
A)
Z
(t-l)
multiplied
in the original PBIB design
(the half block being treated as a complete block).
Hence the number of times any two treatments occur together is
(t-I)A
or
I
(t-I)A
according as the two treatments are first or second
Z
associates respectively.
Hence
Atl
sets~'
e
and
~
(t-I)A I ' AtZ
A~l
To find the other parameters
(r - AI) -"resolvable
=
and
A~2'
=
(t-I)A Z .
we note that there are
in which two "first associate"-treatments
occur in different blocks rather than in the same block.
By
this construction method of the weighing design, it is thus evident that
A~l
= (r - AI)'
Similar argument applies to two "second-associate"A~Z
treatments, giving
NOte:
=r
- A '
Z
This completes the proof.
Although this theorem applies to any resolvable PBWD to be constructed,
it might not be economic to use the method proposed here, for the construction
of PBWD, in case
t
is very large.
for constructions when
t
= Z.
latter sections are cases where
So it is wise to use this theorem
Most of the examples considered in
t
=Z
.
75
5.3.
THEOREM 5.3:
treatments, with
PBWD with triangular association scheme.
From a triangular association scheme with
n
even,
v =
n(n-l)
2
we can construct a Partially Balanced Weighing
Design with the same association scheme and with parameters
=
l · n(n-l)
2
2(n-2)!
n-l
2
'
(n-3) !
(n-3)!
0,
Proof:
We know that there are
(n-l)
diagonal elements will be omitted.
all possible blocks formed by taking
elements in each row, since
Suppose we construct from each row
(n-l)
objects in each half block.
2
If we consider all possible partitions of the
there are evidently
1:. [
2
As there are
n
~
(n-l)!
]
(n;l) ! (n;l) !
(n-l)
elements of a row,
distinct partitions.
rows in the array for a traingular association scheme,
the number of distinct blocks which can be formed by this procedure is
Since by definition of this scheme, the array is symmetrical
about the principal diagonal, any element
row in which
i
i
appears twice, once in the
is present and secondly in the row corresponding
to the column in which
i
appears.
By the procedure of constructing
the blocks, it is evident that for every row in which
." 1 ,
is present,
76
we can form
distinct possible blocks in which
Ii'
occurs in either half block.
Hence
r
= 2
[
(n-2)!
_ (n/ ) ! (n
To determine
All'
-J
/)~
we have to fix two first associates'1i:n'fo t'Ul:ing".;
all possible blocks, with these two lying in the same half block.
out of
(n - 1 - 2)
( =
n;5)
=
(n - 3)
So
remaining objects, we have to put
in the half block with this fixed pair and remaining
in the second half block.
As we can assume without loss of generality
that the two fixed treatments occur in the first half block, the number is
=
(n-3)!
As there is only one row with a given fixed pair of numbers,
All
is
given by the above number.
The same type of argument applied for two first associates to
occur together in opposite half blocks gives the following:
=
By the method of construction of the half blocks and the blocks, any
two second associates can occur together neither in the same half block
nor in the opposite half blocks.
the theorem follows.
Hence
A
12
=0 =
A
22
Hence
77
EXAMPLE 1:
n = 5 ,
From the association scheme given on page 229 [12] with
we get a
PBWD(10, 15, 6, 2; 1, 2; 0, 0)
given as follows:
1,2; 3,4
2,5; 8,9
4,7; 9,10
1,3; 2,4
2,8; 5,9
4,9; 7,10
1,4; 2,3
2,9; 5,8
4,10; 7,9
1,5; 6,7
3,6; 8,10
1,6; 5,7
3,8; 6,10
1,7; 5,6
3,10; 6,8
.
The parameters of this design can be compared with those of the
BWD given by Bose and Cameron [see Design 13, page 158 of [10]].
The parameters of this BWD are given by
=
EXAMPLE 2:
(la, 45, 18, 2; 2, 4) .
The association scheme given in T31 of page 238 [12]
corresponds to
n = 7
and by substitution of this value we get a
PBWD(21, 70, 20, 3; 4, 6; 0, 0)
with the first 10 blocks (using the
first row of the scheme) given below:
1,2,3; 4,5,6
1,2,4; 3,5,6
1,2,6; 3,4,5
1,3,4; 2,5,6
1,3,5; 2,4,6
1,3,6; 2,4,5
1,4,5;
2~3,6
1,4,6; 2,3,5
1,5,6; 2,3,4
78
These parameters can be compared with those given by Bose and
Cameron in 2.3.1 of page 151 [10] by substituting
those given by
t = 5
and hence with
.
(v, b, r, p; AI' A ) = (21, 70, 20, 3; 2, 3)
2
Examples 3,4, 5 which are given below are obtained from T-3, T-16
and T -22 [12], by the subdivision of the blocks.
EXAMPLE 3:
The plan of the
PBWD[10, 15, 6, 2; 1, 2; 0, 0]
obtained
from T-3 of page 230 [12], is given below:
Association Scheme
Plan
e
1,2; 3,4
3,8; 8,10
x
1
2
3
4
5,6; 7,1
4,9; 7,10
1
x
5
6
7
8,9; 2,5
1,4; 2,3
2
5
x
8
9
10,3; 6,8
1,6; 5,7
3
6
8
x 10
4,7; 9,10
2,9; 5,8
4
7
9 10 x
1,3; 2,4
3,8; 6,10
1,5; 6,7
4,10; 7,9
2,8; 5,9
EXAMPLE 4:
The plan of the
PBWD[10, 10, 6, 3; 2, 2; 0, 2]
obtained
from T-16 of page 233 [12], is given below:
Association Scheme
Plan
5,6,7; 10,9,8
1,3,6; 8,5,2
2,5,1; 4,7,9
x
1
2
3
4
10,9,4; 2,3,8
9,5,8; 6,7,10
8,6,3; 1,2,5
1
x
5
6
7
4,3,1; 6,7,10
9,2,4; 3,8,10
2
5
x
8
9
7,9,5; 2,1,4
3,4,10; 1,6,7
3
6
8
x 10
4
7
9 10 x
79
EXAMPLE 5:
The plan of the PBWD(15, 10, 4, 3; 1, 0; 0, 2)
obtained
from T-22 of page 235 [12], is given below:
Plan
Association Scheme
4,8,1; 10,12,14
6,9,12; 3,4,13
x
1
2
3
4
5
4,5,15; 6,7,10
1,3,7; 11,12,15
1
x
6
7
8
9
8,9,15; 2,3,10
3,5,14; 6,8,11
2
6
x 10 11 12
1,2,6; 13,14,15
7,9,14; 2,4,11
4
8 11 13 x 15
2,5,12; 7,8,13
1,5,9; 10,11,13
5
9 12 14 15 x
Examples 6, 7
given below are obtained from examples
4, 5 ,
using Theorem 5.1.
EXAMPLE 6:
From the design given in Example 4, we get a
PBWD[10, 90, 36, 2; 6, 8; 0, 8] ,
with a triangular association scheme,
for which the blocks can be easily formed.
This can be compared with
J3WD 13 of page 158 [10] with parameters, (v, b, r, p; A , A ) =
2
1
(10, 45, 18, 2, 2, 4) .
EXAMPLE 7:
From the design given in Example 5, we get a PBWD with
triangular association scheme, with parameters
(15,90,24,2;3,0;0,8)
for which 9 of the blocks obtained by taking
(12,14)
(10,12), (10,14) and
as second half-blocks are givetlhalow:"
';
4,8; 10,12
4,8; 10,14
4,8; 12,14
4,1; 10,12
4,1; 10,14
4,1; 12,14
8,1; 10,12
8,1; 10,14
8,1; 12,14 •
80
This can be compared with the minimal BWD given by Series A [page 324,
[9]], with
t
= 3.
Its parameters are
(v, b, r, p; AI' A2 ) = (15, 105, 28, 2; 2, 4) .
5.4.
THEOREM 5.4:
of a
n
2
L
2
PBWD with latin square association scheme.
A PBWD[s2, s(s-l), 2(s-1), s, (s-l), 1, 0, 2]
-association scheme on
s -symbols, with
n
l
= 2(s-1)
and
= (s-1)2 can be constructed directly from the associatd:6nscheme.
Proof:
Let us take each row or each column as a half block and: let
us form all the possible pairs of rows and all possible pairs of columns
to form the blocks.
As blocks of this design can be formed in
from columns and since the two sets of blocks thus formed are distinct,
it is clear that
If we fix a treatment
i,
it must belong to a specific row and
a specific column of the latin square formed below for the association
scheme:
/'
........
1
2
(s+l)
(s+2)
(2s+l)
(2s+2)
• • •-
•
(I
•
s
2s
(1)
0
(s-l)s + 1
But there are
(s-l)
.
s(s-:t.)s
(I
•••••
+ 2
...
3s
.
s
2
blocks of the weighing design, containing any
specific row as a half block.
Similarly the number is
(s-l)
for blocks
81
formed from columns.
it is clear that
p
Hence
= s.
r
= 2(s-1)
•
By the method of construction,
e
If we fix two treatments
and
¢ which
are first associates, they must belong to a single row or to a single
column of (1).
e
But as
in either case, there are
to the same half block.
and
(s-l)
Hence
¢ determine the half block uniquely,
blocks with
All
=
(s-l) .
e
¢ belonging
and
For the same
e
and
¢
(mentioned above) to belong to opposite half blocks, we have to count the
number of possible ways the blocks can be formed with the row (column)
e
containing
¢ as
as one half block and the row (column) containing
the other half block.
This is possible only in one way.
1
A~l
Hence
By the method of construction of half blocks and blocks, it is evident
that
A
12
are the
= O.
e
Lastly consider two second associates
(i, j) -th and
and
¢ which
(iI' jl) -th elements of the array (1), the
first and second suffices representing the row number
number respectively. Ihey being second associates
i
~nd
~
i
l
the column
and j
~
Hence evidently there are two blocks
(i-th row; il-th row) and
(j-th column; jl-th column) in which
¢ and X occur together in
opposite half blocks.
Note:
A22
=
2 •
The type of argument can be extended to construct in general
a PBWD with
p
=r
symbols, provided
sets of
Therefore
jl .
2r
x s
2r
from a latin square association scheme with
<
s .
This can be done by taking all possible
rows which are partiuioned into two sets of
in all possible ways.
The
rs
elements belonging to the
of any such partitioning are the elements of a half block.
illustrated
for· s
= 4,
s
by the Example 3.
r -rows each
r -rows
This will be
82
The examples 1 and 2 are the direct illustrations of the Theorem 5.4.
EXAMPLE 1:
scheme with
Let
s = 3
0
2
5
8
U
P
with
1
By applying the Theorem 5.4, with
PBWD[9, 6, 4, 3; 2, 1; 0, 2]
represent the
;)
(1
=
L -association
2
and P 2 = (2
i) .
= 3, we get
s
given below.
We know that
n
l
= 4,
n
Plan
EXAMPLE 2:
1,2,3; 4,5,6
1,4,7; 2,5,8
1,2:,3; 7,8,9
1,4,7; 3,6,9
4,5,6; 7,8,9
2,5,8; 3,6,9
By applying the Theorem 5.4 for
s = 4,
we have the
L -scheme given by
2
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
\
\
with
P,
1
=
and
P
The plan of the
is given below.
2
=
PBWD[16, 12, 6, 4, 3, 1, 0, 2]
We know that
n
1
=9
and
n
2
=6
thus obtained
2
4.
83
EXAMPLE 3:
1,2,3,4; 5,6,7,8
1,5,9,13; 2,6,10,14
1,2,3,4; 9,10,11,12
1,5,9,13; 3,7,11,15
1,2,3,4; 13,14,15,16
1,5,9,13; 4,8,12,16
5,6,7,8; 9,10,11,12
2,6,10,14; 3,7,11,15
5,6,7,8; 13,14,15,16
2,6,10,14; 3,8,12,16
9,10,11,12; 13,14,15,16
3,7,11,15; 4,8,12,16
As pointed out in the note under Theorem 5.4, we can construct
blocks, by considering more than one row (column) as a half block.
Thus starting with
s
=4
and the following association scheme,
we get a PBWD[16, 6, 6, 8; 3, 3; 2, 4] .
of
L -association scheme with
2
Also
p
2
We know that for the case
and
=
Association
Scheme
1 2 3 4
Plan
1,2,3,4,5,6,7,8; 9,10,11,12,13,14,15,16
1,2,3,4,9,10,11,12; 5,6,7,8,13,14,15,16
5
1,2,3,4,13,14,15,16; 5,6,7,8,9,10,11,12
9 10 11 12
6
7
8
13 14 15 16
1,5,9,13,2,6,10,14; 3,7,11,4,8,12,16
1,5,9,13,3,7,11,15; 2,6,10,14,4,8,12,16
1,5,9,13,4,8,12,16; 2,6,10,14,3,7,11,15
Examples 4,5, 6
given below are the PBWD's of the
L
2
~type,
are obtained by the subdivision of blocks of the designs
and the dual of
LSI [12].
which
L82, L88
84
EXAMPLE 5:
from
L82
The plan of the PBWD[9, 18, 8, 2; 0, 2; 2, 2]
of page 241 [12] is given below:
Association
Scheme
Plan
EXAMPLE 6:
:LS8
obtained
1,6; 9,2
7,3; 6,8
4,3; 9,5
1
2
3
6,8; 2,4
3,5; 8,1
5,1; 7,6
4
5
6
8,1; 4,9
1,9; 6,2
7,6; 3,8
7
8
9
9,2; 5,7
6,2; 8,4
3,8; 1,5
2,4; 7,3
8,4; 1,9
4,9; 3,5
9,5; 2,7
5,7; 1,6
2,7; 4,3
The plan of the
PBWD[9, 6, 4, 3; 2, 1; 0, 2]
obtained from
of page 243 [12] is given below:
Plan
Association
Scheme
4,5,6; 7,8,9
1
2
3
9,8,7; 2,1,3
4
5
6
3,1,2; 4,6,5
7
8
9
5,8,2; 6,3,9
7,4,1; 3,9,6
1,4,7; 2,8,5
This can be compared with BWD 11, on page 158 [10] which has parameters
[9, 12, 8, 3; 2, 3].
85
EXAMPLE 7: The plan of the dual of
PBWD[9, 9, 4, 2; 0, 2; 1, 0]
Dual of
LS1, of page 241 [12], and the
obtained from this dual are given below:
Ls1
Plan of PBWD
Association
Scheme ';u"
1,3,7,9
1,9; 3,7
1
2
3
1,2,4,5
1,5; 2,4
4
5
6
5,6,8,9
5,9; 6,8
7
8
9
2,3,5,6
3,5; 2,6
4,6,7,9
4,9; 6,7
1,2,7,8
1,8; 2,7
4,5,7,8
4;8; 5,7
2,3,8,9
3,8; 2,9
1,3,4,6
1,6; 3,4
This can be compared with BWD-10 given on page 158 [:10] which has
parameters (9, 18, 8, 2; 1, 2) .
5.5.
5.5.1.
PBWD with group divisible association scheme.
Singular group divisible weighing designs.
There are four main methods which can be used to construct this type
of design.
One of them consists in using Theorem 5.1 or its generalization
provided we start from a PBWD of large block size and construct ones with
small block size.
The second method consists in using Theorem 5.2 to get PBWD from
a resolvable grpup divisible design of singular type.
The third method
consists in starting from a BWD and using the following theorem.
Lastly
the 4th method consists insubdiving/the,!?l:O:eks1-i.o;£"the .corresponding-,.· :;
PBIB design.
86
THEOREM 5.4:
If there exists a BWD with parameters
(v*, b*, r*, p*, At,
with
n
A~)
,
then by replacing each of the treatments
treatments, we get a PBWD of singular group divisible
type, where the
n -treatments which replace a particular one of the
original design form a group.
Proof:
Obviously
v
= nv*.
= b.
As there is no difference in the number
of blocks,
b*
occurs
times by the definition of groups
r*
Since the
n
Since each treatment of the original design
~ntr6duced,
= r* .
All
treatments constituting a group either all occur
(in case the corresponding treatment occurs in the original BWD) in
a half block or do not occur at all, it follows that
A
Zl
=0
.
Two treatments are second associates if they arise from groups
of two different treatments.
Hence they occur together in the same
half block or opposite half block respectively that many times
as
the original treatments occur together or in opposite half blocks
these two frequencies are
= A*Z
of
r
p*
= r*
.
At
and
A~.
AIZ
= At and
The number of replications is unchanged and each block
treatments is now replaced by
and
Hence
p
= np*.
np*
treatments.
Hence
This proves the theorem.
All singular group divisible designs obtained in this way are given
in Table . 7C: for
v
p < -2
and
v
<
30 .
Lastly all singular group divisible weighing designs which can
be obtained from the corresponding PBIB designs are given in
Tables 7A and 7B.
87
5.5.2.
Regular group divisible designs.
The methods of construction of these designs are the same as for
singular type, except for the fact that these cannot be obtained from
BWD's.
The designs obtained by this method are given in Table 8.
5.5.3.
Semi-regular group divisible weighing designs.
The methods of construction are the same and the corresponding
PBWD's are given in Table 9.
5.6.
PBWD with some miscellaneous association schemes.
The following is an example of a PBWD[7,' 7, 4, 2, 1, 1, 0, 2]
obtained from the Yonden square 13.2 [22], with the association defined
below:
Association Scheme
1st Assoc. ',2nd"Assoc.
Plan
3,5; 6,7
1
2,3,6,7
4,5
4,6; 7,1
2
3,4,7,1
5,6
5,7; 1,2
3
4,5,1,2
6,7
6,1; 2,3
4
5,6,2,3
7,1
7,2; 3,4
5
6,7,3,4
1,2
1,3; 4,5
6
7,1,4,5
2,3
2,4; 5,6
7
1,2,5,6
3,4
The parameters of the association scheme are:
P
1
=
=
CHAPTER VI
ANALYSIS OF BALANCED AND PARTIALLY BALANCED HEIGHING DESIGNS
6.1.
Introduction
In weighing experiments, intrablock estimates and their standard errors
have been studied by Bose, R.C. and Cameron, J .M. [10].
They restricted
to the cases that in a balanced weighing design
(i)
The sum of all objects, i.e.
8
1
+ 8 2 + ... + 8
v
is a known
constant
(ii)
The sum of any arbitrary number
t , is a known constant,
t
i.e. ,
~8.
1 1.
is given in advance.
In this chapter this analysis is extended in two ways
(a)
v
to study the estimates under one or more general restrictions
v
(l:k.e.:h;\;'(lY- l: k .. e.=YYlijJ·=1,2, .•. ,t)
1 1. 1.
i=l J1. 1. -~
(b)
for the same type of designs
to formulate similar investigations for the new class of designs,
viz, PBWb.
Section 6.2 deals with the extensions for BWD, in which it is noticed
at various stages that the estimates and standard errors reduce and tally
with those of Bose and Cameron [10] in the particular cases.
In Section 6.3,
some specific formulae have been derived for the estimates and their
variances for the PBWD's under one linear restraint.
Lemma 6.2.1 of Section 6.2 was used by Bose and Cameron [10],
although the proof was left as it is more or less obvious.
and 6.2.3 are given under excercises in [43].
Lemmas 6.2.2
89
6.2.
LEMMA 6.2.1:
with
v
(i
Some extensions of analysis of Weighing Designs
N = «n. ,»
l.J
If
is the incidence matrix of a design
treatments (objects) and
stands for treatment and
b blocks, with block totals
j
stands for block), then under the condition
v
that
yl' y2' .•• , Yb
L k, ,e, = m, ,
j=1,2, •.. ,t are known restraints on
i=l J l. l.
J
reduced normal equations to estimate e are given by
e's,
the
~
I
k'
0
k
k
where
=
k
e
k
T
represents a vector of
Proof:
e
The model
N1:.
=
t
E(~
k
n
k
l2
k
21
k
22
tl
t2
k
••• k
2
.v
tv
lv
and
unknown variables.
xl) = N'..a
must be considered as the model and
must be estimated subject to the restrictions that
for
v
L k,.
e.
i=l J l. l.
j=I,2, ... ,t .
The least square estimates are therefore given by minimizing
b
2
L (Yl.' - n l l.,8 1 - n 2 l.,8 2 ..• - n ,8 )
i=l
Vl. v
t
+
2
L
ep
{
v
L k
,8
u=l . u i=l Ul. i
- m }
u
w.r.t.
m,
J
90
and the Legrange multipliers
8 , 8 , "', 8
v
2
1
2 is unimportant).
W.RoT 8
q
<PI'
<P
2
, ••• , <P
t
(constant
The partial derivative of the above expression
is obviously
Equating this to zero and simplifying we get
(L:nl,n
1 q1.
L: n .n, • k lq
V1 q1 .
L:n 2 1·n q1.
T
8
q
v
<PI
<P
But row vector on L.R.8. is obviously the
q -th
t
row of
(NN': k) •
The reduced normal equations are given by
(i)
(NN' • k)
(ii)
~'~ =
(~)
m
..
or writing in terms of a partitioned matrix by
NN'
v
~
and
= T
k
v
k'.
t x v
v
x
t
o
txt
which proves the Lemma 6 .2.1.
Note:
In the application of this result to weighing designs we note
(a)
that
N = ((n ij ) )
(b)
that
Yj
involves
(-l)'s
also and
stands for difference between the half block totals
(left and right pans) in the
j -th block (weighing) .
91
NN' = (r + (3) I -
I t has been noted [10] that
weighing design.
Hence we have:
COROLLARY 6.2.1:
with
t
for a balanced
(3J
For a balanced weighing design
~'~ = ~,
restraints
(r+ S) 1- J
k'
t x v
x
-e-v
t
1
T
=
0
t
x
1
the estimates are given by
k
v
BWD(v,b,r,p;A ,A )
..P.t
t
1
m
We mention here two well known results in matrix theory without proof,
which we need later.
LEMMA 6.2.2:
If
A, D
are symmetric and
E
-1
and
-1
A
and
exist, then
(*)-1
LEMMA 6.2.3:
If
Am x m
=
(i)
tAl
= (a_b)n-l [a
+ (n-l)b]
(ii)
If
A-I
A-I = (a ij )
a
ii
exists,
=
+ (n-2)b
[a + (n-l)b] [a-b]
a
when the denominator does not vanish.
b
a
b
b
b
b
then
is given by
a
iJ'
=
-b
[a + (n-l)b] [a-b]
2
92
The following is a known result.
LEMMA 6.2.4:
(a)
If
A is a
~
m
matrix of rank
r,
then
by a suitable interchange of columns and rows of
B
F =
D
A,
it can
F given below:
be brought to the form of a matrix
where
n
r x r
C
(m-r)x r
E (m-t') x (n-r)
;r x (n-:r)
B is non-singular and
o X.m
-r
(b)
o(l1.-r)
x r
P-:.·-
LEMMA 6.2.5:
,_-.. "
'.
(
.. ~'-"
~
.
.....
0·...·...
Then a necessary and sufficient condition for
... , kv )
k' = (k , k ,
2
1
v
this design is that
i~lki
estimation space.
;.
Suppose we have a balanced weighing design
a vector
Necessity:
x
(.
BWD(v, b, r, p, A , A ) .
1
2
Proof:
is a generalized inverse of
=0
Suppose
Since
there exists a vectbr
.
k'
·NN.~
to belong to the estimation space of
=
(k , k , ..• , k )
v
1
2
belongs to the
=,.{r:l:-S).I .:.. SJ.:-UOY " .·this means that
(c , c , ""
2
1
c )
v
such that the following
relations are true:
k
v
= c 1r -
S (E c. -
=
S (E c. -
=
~
~
c r v
S~c.
~
- c )
v
_ _(1)
F.
93
Adding these
v
relations, we have
=
~k.
r~c.
-
(v-l)S~c.
1 1 1
since
=
0
r = (v-l)S .
Hence the condition is necessary.
Sufficiency:
j' [(r+!3)I -
If
j' = (1, 1, •.. ,1) ,
SJJ vxv
is
0'
(the null vector).
to the estimation space of the design.
the
formula
seen that
r = (v-l)S,
it is evident that
the space T
that
of
1.
of dimension
(k , k , ""
l
2
T
is
k)
is orthogonal
which is valid for any BWD, it can be easily
Hence the rank of the estimation space is
'T'
j'
By using the Lemma 6.2.3, and
(r+S)I(v_l)x(v_l) - SJ(v-l)~(v-l)
orthogonal space
Hence
1.
Hence
is non-singular.
(v-I) .
(1, 1, .•. ,1)
The condition that
Hence the rank of the
is a generator of
v
fk
i
is orthogonal to the generator
=
0
guarantees
(1, 1, ... , 1)
and hence the former belongs to the estimation space.
Hence the
sufficiency follows.
Remark:
Lemma 6.2.5 states that in a balanced weighing design the linear
contrasts and<only the linear constrasts are estimable.
Much of our
interest in practical situations is the comparison between the weights
of two objects.
The following is a direct consequence of this lemma.
LEMMA 6.2.6:
With the same notation as in the above lemma, a
necessary and sufficient condition that
k' = (k , k , ""
2
l
not belong to the estimation space is that
v
i~lki ~
0 •
k )
v
does
94
The following is the statement of a well known result in the theory
of determinants.
LEMMA 6.2.7:
Let
b 1'2~ ; _:,' ~." •
bi2,-,'" .~ ••
=
B
b
, ~ n~2, '
i. .•• ___
be a ,matrix of order
of
where
~J
.
(n x n) • Then the value
(n-l) (n-l)
b
"
IBI
r(n-l)
of the determinant
B is given by:
IBI
b.,
b
(n-I)2
B
nn
=
b
B
nn nn
n-l n-l
L E b .b, B -i'
i :f j = 1 nJ ~n nn:.; J
is the cofactor of
b
in a matrix obtained from
nn
and
Bnn~ij
is the cofactor of
B by omitting the last row and the
last column.
LEMMA 6.2.8:
The value
of the deterrr-lnant of
L1V+I
-13
-13
-13
r
A
~'\\
l
-13
-13
k
r
k
=
k
v
2
Ov
is given by:
L1V+l
Proof:
=
Thi.s follows from the previous lemma by taking
(r
B
+ 13) I
=
[
k'
-
I3J
~J
95
Obviously, in this case, for every
i
we have
-13
nn::.ij
r
G
=
B
,-13
I (Ml )!
.:=
M
l
is a square matrix of order
B
and the value of
B ... (i
nn:';!.J
~
-13
(say)
.••
nn:'~J
where
j
-13
-13
-13
-13
r
-13
·
-$
r
·
··
-13
..
is the same as above as it can be
=
B
r
It is obvious that
brought to the above form by even num,'?-er of
Then the common value of
-13
·
j)
Let
-~
(v-I)
=
nn~.12
...
...
is
interchang~s;.
-13
-13
-13
r
-13
-13
-
1M
2
I
-13'\
-13
•
Hence
L:>v+l
=
As it can be very easily verified that
IMII
and
IM2 1
we get
L:>v+I
v-2
= (r+S) [r- (v-2)S]
v-2
- S(r +s )
= _ S[vS]v-2
= -{ S(vS)v-2(Ek i )2}
= _ SV-Ivv-2(~k.)2
I
This proves the lemma.
= S[vS] v-2
~
96
The following two theorems help to determine the estimates of
A
8
using the Lemma 6.2.2 and the Lemma 6.2.4 respectively when
(k , " ' , k)
l
is a vector not belonging to
THEOREM_ 6.2.1:
Let the results of the
BWD(v, b, r, p, AI' A2 ),
with
the model be written as:
E(~xl)
OT
belonging to the
e~timation
space.
b -weighings of a
B = A2 - AI'
be written as
where
= N'8
y
N = «n.,))
Let
is given
1J
as follows:
n ..
1J
=
1
if the ith treatment belongs to the first half of the jth block
-1
i f the ith treatment belongs to the second half of the jth bloc
0
i f the ith treatment does not belong to the jth block.
v
= T-(vxl)
-
Let
Under this notation, if
NY
... ,
is a linear restraint such that
l:k.8 =m
i=l 1 i
k )
v
does not belong the estimation space, then the best unbiased linear estimate
@ of
8
is given by:
A
1
8.
vB
1
@
and
where
w
's
ij
v
=
are given by
e
By taking
t
= 1
-
J
v-I
l: z. IT . +
(~+~l 2) T
h2
111
7(a), Zij'S
Z.l
- Z 1. 2m]
1
V
by 6(a)
+ vB m
v
h
and further when
in the Corollary 6.2.1, the solution
is given by
( r+13 ) I - iiI
[
1
[
vB
1J
and
=
Proof:
[l: (l+w .. )T . + T. 1
k'
(lxv)
A
8
of
97
k'
But
does not belong to estimation space implies :Ik.
1.
the matrix
IMI
By Lemma 6.2.7~ the value of the determinant
by Lemma 6.2.6.
~ O~
of
M of the normal equations is given by
IMI =
Bose and Cameron [9J have shown that in
r
~
and
Al
A = gn(p-l)
1
(i)
A
2
of a BWD
can be expressed as
A = gnp
2
and
r = gn(v-l)
,
and
= gnv, where g,n,v are positive integers.
r + 13
So for any BWD,
the parameters
Hence it follows that in general
13 = A2 - Al = gn
(ii)
generaL~
13
~
a and r+13
Ik.
Nbw by hypothesis,
1.
~
~
a •
O.
Hence it follows that
and that the unique inverse of the first matrix on the left had side of
2(a) exists.
The remaining part of the proof
consis~of
@ of e.
explicit expression of the estimate
finding the
For that, let us
partition the above mentioned matrix in such a way that we get a
(2 x 2) -matrix
M
22
on the lower right
Let this partitioned matrix
be
M
By the Lemma 6.2.3, it follows that
since
r
=
(v-l)13
=
(r+l3)(v-l)-l [r + {(v-l)-l}(-I3)J
=
(r+l3)
v-2
[r - (v-2)I3J
Since by hypothesis
for any design to exist,
IMlll ~ a .
r+13
~
=
a and since 13
~
a
98
Also by the same lemma
is given by:
r + [(v-1)-Z](-S)
ii
m
= [r + (v-Z) (-S) ] [r+s]
ij
Similarly
=
= vS
D
G
~
(vf
112 1
112
vS
In the Lemma (.,t.,;, , taking
B'
A
= Mi2 =
e
=
= M22 =
D
given by
-1
M
=
ell
1
~:
M
Z2
_(4)
MIl
k
Ml~)
Mi2
'--
c
...
= vS
= S(vS)
1
§.
= S(VS)
III
Z
Z~
r - (v-3)S
= [r - (v-Z)S][r+S]
-1
-S
-s )
kZ
k _
v 1
k~)
and
we have
e~ + FE-IF'
=
-E-1F
-1
M
-FE-j
E-1
1
(5)
where
' -1
E = M2Z - M12M11M12
F'
=
1
V
= ...!.
v~
and
(~~
["VS
R,
1
-1
F = M11M12
,
... -s)
-S
k ..• k 2
v 1
-vs
R,2
:vSJ
y-1
1
2
1
1
1
Z
99
where
v-I
£. = k. + E k. ,
1
1
j=1 J
for
i=I,2, ... ,(v-l) .
l
= .J:.. FV(3
v(3 l-£I
-
I
v(3
(v-l)v(3
v-I 2
l: k.
i=1 1
= QI
-(3
.
k
-(3
k
1
2
v-I
E£.k.
1
1
v-I
E k.
i=1 1
.k
v-I
-vB E k
i=1 i
2
-(3E£ .
Let
-(3
-v(3 .•. -v(3
£2 ... £v-li
1
= Q •
2
It is obvious that,
v-I
v-I
r 2.
I
v-I
E (k.
I
1
=
1
+
v
E k.)
I
1
vEk.
=
I
and
1
(v-I)
E Q..k .
. i=1 1 1
;-
=
v(3 : -vSQI
( :v
E
k
v
0
)
!
Q2+QIJ
-Q I "l
Q2+Qy
vS
-Q 1
...
-v(3Ql
-I
[ (v-1) B
:::
=
. 2
I ,v(v-I:(3
i
J
M' M- 1M
12 11 12
M
22
(V-1)B
_Q
1
-Q1
Q2+Q1 2
v(3
)
100
where
h =
lEI = _h 2
Zk.
1 ].
under the hypothesis that
:f 0 •
(k , k , "', k )
2
v
1
does not belong to the
estimation space.
-J-h
Hence
o
_ _(6)
hvS
o
-vS
..•
1
·-1
FE
hvS
-vS
o
-vS
1
-hvS
-hvS
=
1
vS
z·
v-l,l
z·
v-l,2
101
=
-,vs
for every
h
•
1
FE-1
=
i=I,2, ••• ,(v-l)
1 [
vS ";1 ":'2
_ _ (6a)
]
_ _(7)
-vS
-vS ••• -vS
.Q,1
.Q,Z···.Q,v-l
.Q,1zl2- vSz U
\1;;'. ::,:....'.14 2Z,12:"'~~.1i' ~~~._ •••..•• .Q,v-1z12-vS z 11
.Q,lz22-vSz21
.Q, 2z22 .".~:S:z.21· ;:••: •••••• x'v_1z22 -vS z21
= v 2S 2
w
1,v-1
=
1
vS
w
2,v-1
w(v-1),1
...
M~i +
w(v-1),2 ••• wv - 1 ,v-1
FE-IF' :'.
=
v~
_ _(7a)
[J + I + w] •
_ _(8)
t.
.
1
Substitut~g
(5), (6), (7) and (8) in
~v-I
'%v
t¢
2(a), we have
T
v
m
102
-i [v~1
=
VI-'
\.
+
J
v-I
L: w..T. - zi1T
+
T.
J.
1
... ,
i = 1, 2,
@
and
or
v
-
vS
t
L:
J.
1
= vS
1
L:
z'lT.
J. J.
It is evident that when
(k1~
v
J.
V~l
1
@
(v-I)
--v-I z'IT. +
1
=
- zJ.' 2m
v
J.J J
-
which proves the result.
Remark:
~
k 2 , .•.
k ) =
v
v-1
= k. +
J.
II."
J.
L:
= v
=
(v-1)
J
1
Q2
1
=
zil
v-1
k.
hQ, .
J.
h
~
2
L:
1
@
v
Similar substitution in
=
A
~i
= - 1
T
z·1
J. J..
V
v-I
L: T
1 i
v~ ~v + v~mJ
gives~
A
e.J.
...
~
k2
=
(v-1)
1 i·
1)
~
= v
2
v -(v-1)v
=
2
v
2
v-1
Hence
L:
h
and
2
(Q2+Q1)
-
=
QZ
(1~
T
=
=
l.
v
v
v
T
--Y+~
=
vS
Ti
m
vS
V
= -- + -
for
v
i=1~2~ ... ~v-1
.
Hence the estimates tally' with those obtained by Bose and Cameron [10]
when
(k , k , •••
1
2
~
THEOREM 6.2.2:
if
(k1~
•..
~
k )
v
k) =
(1~ 1~
.•• ~ 1) •
With the same notation as in the Theorem
be1ongs~to
6.2.2~
the estimation space, then the estimate
103
of any estimable linear parametric function
~'e
is
Ji,'
A
e where
is given by:
@.
=
1\
f
@
or
j
Proof:
=
1:.
vl3
T
T
2
vl3
1
vl3
1
vl3
1
vl3
0
0
1
vl3
2
vl3
1
vl3
1
vl3
0
0
1
vl3
1
vl3
1
vl3
2
vl3
0
0
0
0
0
0
0
0
0
0
0
0
0
~
Just as in Theorem 6.2.1, the estimate
k
T
v
m
1 x V
k'
@ of
e is obtained by solving
v x
o
k'
Under the condition that
=0
:2
v
(
1
1
vl3
j
r+l3)I - I3J
L:k.
T
belongs to the estimation space, we know that
Using that the rank of
[(r+I3)I
is
f?J]
(v-I)
and
k'
belongs to the estimation space, it can be easily proved that the rank of
k
v
is
k'
(v-I) .
0
1 x v
Also it can be seen that the above matrix can be partitioned as
(
where
:H
:H
,
H
21 :
H
ll
12
22
.
= (r+l3) I (v-1)~(v-1)
)
I3J
(v-1)x(v-1)
104
By the Lemma 6.2.3, it can be very easily seen that
(v-I)
and
H
il
is of rank
vIS [J (v-I) x(v-l) + I( v- 1) x ( v- 1)] .
=:
Hence by Lemma 6.2.4,
it follows that
o(v-l)x2
)
is a generalized inverse of
(r+S) I - SJ
(
k'
1 x v
Hence it is known that the estimate
=:
v
Hence using, L:T i
=:
0
,
(
v~
~,~
of
£'8
(J+I) (v-I) x (v-I)
is given by substituting
o(v-l)x2
°2x(v-l)
it follows that
1
1\
8.
=:
J
-1T
vS j
)
T
v
-:vS
which proves the result.
In particular the estimate of
v~ (T j
- T£)
for
j
(8
j
- 8£)
is given by
# ~. This tallies with the estimate of
(8 j -
8~)
v
obtained by Bose and Cameron [10] in the two special cases when' (1) L:8.
1
t
and
(ii)
L:8.
1
1
=:
m (t
<
v) .
1
=:
m
105
6.3.
Analysis of Partially Balanced Weighing Design under a linear restraint
THEOREM 6.3.1:
If there exists a PBWD(v, b, r, p, All' A , A , A )
22
12
2l
involving the objects with unknown weights
(8
1
8
1
, 8 , "', 8
2
v
and if
+ e2 + ... + e) = m (a known constant:), then the estimate
~'i of an estimable paramteric function ~'~ is given by substituting
given below, provided
where
d
l
and
d
are given by (21).
2
= m + (hI1 + h 2l )f ol m + (h 12 + h 22 )f o2 m
~fI.,
_ (h il .+ h 21 )s I (Q fI., ) _ (h I2 -+ h 22 )s2(Qn)
IV
In particular,
A
8f1.,
A
- 8
q
=
(hI 1 + h
+ (h
12
2l
+ h
) [sl(Qq) - sl(QfI.,)]
22
) [s2 (Qq) - s2 (QfI.,) ]
where
i=1,2, •.. ,v ,
h
ij
Proof:
"
-
being given in l3(a) - ~6(a)
Let
N = «n
ij
»
s2(Q)
-
= B2Q and
be the incidence matrix of the partially balanced
weighing design under consideration, i.e. the matrix with the
and j-th column -element given by:
i-th row
106
n ..
J.J
=
1
if the i-th treatment belongs to 1st half of j-th block
0
if the i-th treatment does not belong to j-th block
-1
if the i-th treatment belongs to 2nd half of j-th block
By the definition of the partially balanced weighing design
PBWD(v, b, r, p, All' A ,
21
, A ) ,
22
we have
+ n.
0
A
12
n1j + n 2j +
=
vJ
for all
or
naln Sl + n a2 n S2 +
according as
a
and
S
_ _(1)
j=1,2, ... ,b
are first or second associates.
and
Hence using the
and noting that by
the definition of association matrices,
and
j
bil
j
b
i2
=
1
iff
i
and
j
are 1st associates
=
1
iff
i
and
j
are 2nd associates,
naln Sl + n a2n S2 + ... + nabn Sb
for all possible treatments
Also,
=
-S b S 1 cd
(a, S)
=
r
_ _(2)
_ _(3)
Since the model is
E(~
x 1)
=
N' e
=
the normal equations reduce to
NN'e
=
NY
_ _(4)
107
N ~ = Q . I t is evident from the relations (Z), (3) given
Let
ahove that
NN'
B. = (b S .)
where
en
~
=
is the association matrix corresponding to the ith class.
So (4) can. be written as:
_ _(5)
Qi = nilYl + niZY Z + ... + nibYb .
where
Let· s. (8)
~
a
denote the sum of the weights of all the
which are i-th associates of
So
(8 ) = 8
a
for which
a
j
Similarly, let
is the
for
a
s. (Q )
and
(8)
a
BO' Bl
a
and
~
We note that
denote the sum of all
i-th associate of object
system (5) of normal equations by
get (6) , (7)
~
i=O,l,Z.
n. -objects
.
Q: s
J
Pre-multiplying the
B
respectively we
Z
.
(say)
(r I
_ _(6)
Sl(Ql)
~l(QZ)
.
51 (Q)
(7)
..
sZ(Ql)
~2(Q2)
.
5Z(Q)
_ _(8)
108
Using the known relations:
B B
1 2
=
2
B
1
=
(P~lBO
+ Pn B1 + pi1B~
B B
2 1
=
~~2BO
+ P12B1 +
P~2B0
BO! = I! = SO(8) ,
B1i
1
1
=
(n1 l
+ pi1 Bl +" pi1B
-&ih
+
P~2B0
and
(6), (7)
and
(8)
reduce to
Using the relations,
B! =
2
where
and
s2(~)
=
sl(8)
and
and writing the three sets of normal equations, we have
faa
= r;
0
109
Since
8 1 + 8 + ... + 8
2
v
So (8 .Q,)
=
m (a known constant) ,
=
+sl(8.Q,) + s2(8.Q,)
8.Q,
Using the relations:
=
=
SO(8.Q,)
m
.Q,
for all
.
.
m - sl(8.Q,) - s2(8.Q,)
r = n1f\ + n2132
1
1
1
POI + PH + P 21
=
n
1
1
1
P02 + P12 + P22
=
n
2
2
2
POI +P n + P 21
= n1
2
2
2
P02 + P12 + P22
=
n
1
2
2
or
1
1
Pn + P21
=
n:l - 1
or
1
1
P12 + P22
=
n
or
2
2
Pn + P21
or
2
2
P12 + P22
n
=
n
2
1
2
- 1
v
Using (1), it can be shown that
1;
Q.
=0
or
j=l J
Hence equation (9) is a linear combination of equations (10) and (11).
So the solution
(8 , 8 , "', 8 )
1
2
v
by solving (10) and (11) only
of
(9), (10) and (11)
can be obtained
(if at all the system is solvable).
no
the latter reduces to
...
_ _(12)
=
(say)
=
_ _(14)
=
_ _(15).
III
From (13) - (16) ,
h
ll
+ h
Z1
=
(£11 - f Ol ) + (f 1Z - f OZ )
=
(f
=
13 1 - (-n l I3 1 - n Z13 2)
=
13
1
11
+ f
+ r
)
lZ
-
(£01 + f 02 )
.
_ _(17)
Similarly it can be seen that
_ _(18)
NOw coming back to (12) and writing
gi(2)
= si(Q£)
- fOim,
Under the hypothesis we can show that the determinant of
H
is different from
O.
we have
112
12
ll
-1
Denoting
H
h
by
(
h
h
2l
h
)
we can evaluate
22
Before this evaluation, we note that
=
for all
£
= 1,
Summing
A
8£
A
==
SO(8£)
2, " ' , v .
sl(8£)
and using the relation
and
A
= m - sl(8£)
=
we have
m + (h
_ (h
ll
ll
+ h
+h
2l
2l
)f
ol
m + (h
)sl(Q£) _ (h
12
12
+ h
+h
22
22
)f
02
m
)s2(Q£)
--
(9)
By (19),
=
sl(Q£)]
_ _ (20)
- s2(QQ,)]
(19) and (20) give the expressions for
A
Var(8£) ,
h
ij
and
A
1\
Var(8£ - 8 ) '
can be found out.
q
@£
and
@£ - ~q
it can be ver1fied that
Before evaluating
IHI.
Further, (18) can be used to express
~ 0
and
h
as
22
J
ell
IHI
h
Z1
h (11~21 hdh2~j
1J
zz
h
h
B2
. (1 + 1
r
1\ (SZ-Sl)PIZ
= (r + Sl)
h
Z1
+r
=
(
~r
'-1 '
h
ZZ
J
21
11
B:~:) I
r+Sz+piz (S z-f\)
Z
r + Sz + P1Z (SZ - Sl) ]
z
Z
1
= r + r[(Sl + Sz) + (Sz - Sl)(PIZ - P1Z )]
Z
1
+ SlSZ + (Sz - Sl)[SlP 1Z - SzP 1Z ]
Since by hypothesis
- d1
±
hi Z
IH I
:f
o.
4d Z
_ _(21)
114
h
h
h
h
h
ll
ZZ
lHT
12
Z1
ZZ
-h
=
==
Z1
THT
-h
1Z
= lHT =
h
=
ll
lHT
Z
Z
. ( Z
P1l8 1 + P ZZ + ZP 1Z + 2)8 2
Z
r + d r + d
1
Z
1
(Sl - SZ)P 1Z
2
r + d r + d
1
Z
(15a)
Z
(S2 - Sl)P 1Z
2
r + cl r + d
1
Z
(Z
+
1
2P 12
(l3a)
(14a)
1
1
+ P ll )Sl + P 22 BZ
Z
r + d r + d
2
(l6a)
1
where
and
THEOREM 6.3.2:
as in Theorem 6.4,
and
Under the same conditions and with the same notation
for
~
1 q
are given by:
115
1\
1\
2
An
var(e Q, - 8 )
q
[2n (r - 213 IP 1
n
l
+2d'(I3B + 13 B ) e
1 1
2 2
213 2pi2)
] 02
Q,
13 (q 1
+ 13 b
2A2 [r + 13 b
22
2 q2 - 1 buP n
1 ql
+
Q
2
+ b t2 P ll)
1
1
+ nl(r - 213 1P ll - 213 2P 1 2) + r.~' (13 1Bl + 13 2B2) ~]] 0 2
where
(h ij) ) S
are given on page 114 and
where
e
and
All
Proof:
=
(b
~'
=
(b .Q,1
ll
21
= (h + h ) ,
1
b
,
b
2
ql
2
U
b
-,
= (h 12 + h 22 )
A22
...
b
V
ql
(a) Var(Q,Q,)'
(b) Cov(QQ,' Qq)'
)
v )
U
.
var(~Q,) and Var(~Q, - @q)'
In order to find
evaluate
l
ql
d"
first we
(c) Var[sl (Q,Q,)]
,
and
(a)
But
Var(Q)
(b)
Var(Q,Q,)
=
= { ,Q,-th diagonal element of Var(Q)} .
Var(N!.)
=
NN' 0
2
= {,Q,-th
=
=
(rI - 13 1B1 - 132B2)0
2
.'. Var(QQ,)
row and q-th column element of
,Q,
,Q,
2
- 13 2b )0
( - 13 b
q2
1 q1
= ro
2
Var(Q)f
116
(c)
= Var[
v
=
v
L:
i
b 1 Q. ]
i=l i J.
v
i
.
b
Var(Q.) + 2
L: bJ. b~l Cov(Q., Q )
J.=l NOl
J.
i#u=l Q, l N
J.
U
.L:
i
whether
b Q,1
= 0 or 1.
2
°
2
-28 1 0
2
2
u
..•
u
Q,-th
row of
B .
l
( Q,-th diagonal element of
°2
- 28 2
( Q,-th diagonal element of
But
The
v
P\lx,x -28 0
L: b'2x,x
.
1 1. J. U . 2 J.,u=
.
1 J. J. U
J.,u=
or the
where
v
L:
Q,-th diagonal element
= nlPil
for all
Q,.
117
Hence the
~-th
...
1
= nlP 1Z
diagonal element
Var[sl(Q~)]
1
n [r - Zf;\P
1
n
=
Z
1
ZSZP1Z]a
n
(d)
Cov[Q~,
slCQ~)]
=
l
covC·L:1Q£..., Qg)
~=
when
~i
stands for any of the
n
l
nl
~.
i:l (13 1bQ.~
=
(e)
v
=
L:
covCQC' Q~)
~
-first associates of
= -
=
=
~
~.
+
~-th
SZb Q.~) a
Z
treatment.
118
=
_0
2
[(:31
E bUt1dteu + f3 2 E b lid e'
t2, t-u
:t, u
for the particular values of
i
q
and
£.
by using part (d).
(g)
=
=
if
£
and
q
are the first associates and
if
£
and
q are
second associates.
119
(h)
Nbw we can find out
A
Var(8Q,) .
A
eQ,
By using the formula for
112
+ 2 [n l (S1 - r + 2S l Pll + 2S2P12)]AllA220
_ _ (17)*
by (c), (h), and (f).
Next by the formula for
Var[Ae
Q,
A
A
eQ, - eq
- AS]
= (hI1 + h 2l )2 Var[s1(Qq) - sl(Qn)]
q
N
+ (h
+ 2(h
12
+ h
22.2
) Var[s2(Qq) - s2(QQ,)]
11 21
12 22
+h )(h +h )Cov[sl(Qq)-sl(QQ,),s2(Qq)-s2(QQ,)]
But
- - (18)
*
120
Hence
+ 2[Cov(Qt' sl(Qt) + Cov(Qq' sl(Qq»
- Cov(Qq' sl (Qt»
- Cov(Q~, sl(Qq»]
_ _(19)*
The expression in the last parenthesis is
by (d) and (g) •
_ _(20)
*
121
Substituting (18)~ (19)*and (20)*in
•
we have
A
var[e9, -
lie]
q
var[&9, - &q] , i.e. in
(l7)~
122
t)
BIBLIOGRAPHY
[1]
Adhikary, Basudeb (1965).
"On the properties and the construction
of Balanced Block Designs with variable replications."
Calcutta Stat. Assoc. Bull., 14, 36-64.
[2 ]
Adhikary, Basudeb (1965).
"A difference theorem for the con-
struction of Balanced Block Designs with variable replications."
Calcutta Stat. Assoc. Bull., 14, 167-70.
[3]
Adhikary, Basudeb (1967).
"A new type of Higher Associate
Cyclical Association Schemes."
Calcutta Stat ".. Assoc. Bull.,
19, 40-44.
[4]
Barra, J. R. (1965).
"Carres Latins Et. Eu1eriens."
the International Statistical Institute,
[5]
Blackwelder, William C. (1966).
11,
Review of
16-23.
"Construction of Balanced
Incomplete Block Designs from Association Matrices."
Institute of Statistics Mimeo Series No. 481.
[6]
Bose, R. C. (1963).
"Combinatorial properties of partially
balanced designs and association schemes,"
[7]
Bose, R. C. (1969).
Designs.
[8]
Bose, R. C.
Sankhya)£, 109-36.
'Combinatoria1 problems of Experimental
John Wiley and Sons, New York (to be pilblished)
and
B:ush~,K.
A.>
(19~2)~
. "Orth'bg.orial arrays of ('
st'rerigth 2 and 3. "Ann, Math • .sfiiUst. , . 23, 508"·24.
[9]
Bose ,:R.:Ci:'"'an.d Game ron , ;S. M; (1965)., :~B1heBtidge-TOurnament
probleman.dcafibration designs for comparing pairs of
objects."
Journal of Research of the National Bureau
of Standards, 69B, 323-32.
123
•
[10 ]
Bose, R.C. and Cameron, J .M. (1967).
"Calibration designs
based on solutions to the Tournament problem."
2.!... the
Research
[~l]
Jour. of
National Bureau.£!. Standards, 7lB, 149-60.
Bose, R.C. and Shrikhande, S.S. and Bhattacharya, K.N. (1953).
"On the construction of group divisible incomplete block
Ann. Math. ·Stat., ~, 167-95.
designs."
[12]
Bose, R.C. and Clatworthy, W.H. and Shrikhande, S.S. (1954).
"Tables of partially balanced designs with two associate
classes."
North Carolina
~
Expt. Sta. Tech. Bull.,
,No. 107.
[13]
•
Bose, R.C. and Connor, W.S. (1952). "Combinatorial properties
of group divisible incomplete block designs."
Ann. Math.
---
Stat., 23, 367-83.
[14]
Bose, R.C. and Nair, K.R. (1939).
"Partially Balanced Incomplete
Block designs." Sankhya, 4, 337-72.
[15]
Bose, R.C. and Shimamoto, T. (1952).
"Classification and analysis
of partially balanced incomplete block designs with two
associate classes."
[16]
Jour. Amer. Stat. Assoc.,
Bose, R.C. and Shrikhande, S.S. (1960).
1:2,
151-84.
"On the composition
of balanced incomplete block designs."
~L
Math.,
Q,
177-88.
[17]
Chakravarti, LM. (1961).
"On some methods of construction of
partially balanced arrays."
[18]
•
Chakravarti, LM. (1963).
Ann. Math. ~.,
E,
1181-85.
"Orthogonal and partially balanced
arrays and their application in Design of Experiments."
Metrika,
I,
231-43.
124
[19]
Chakravarti, LM. (1965).
"On the construction of difference
sets and their use in the search for othogonal Latin
Squares and error correcting codes. 1I
Bulletin of the
International Statistical Institute, 10, 957-58.
[20]
Clatworthy, W.H. (1956).
"Contributions on Partially Balanced
Incomplete Block Designs with two associate classes."
National Bureau of Standards Applied Mathematics
Series,
[21]
!!1..
Clatworthy, W.H. (1956),
"Contributions on Partially Balanced
Block Designs with two associate classes."
of Standards Applied Mathematics Series,
[22]
CPilZl1X a n,
W.G. and Cox, G.M. (1950).
National Bureau
!!1..
Experimental designs. '
Wiley Mathematical Statistics Series.
[23]
Ehrenfeld, Sylvian (1955).
designs. 1I
[24]
Ann. Math.
"On the efficiencies of experimental
Stat.,~,
Fisher, R.A. and Yates, F. (1953).
247-55.
Statistical tables for
biological, agricultural and medical research.
Oliver and Boyd.
[25]
Hall, M., Jr. (1947).
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[26]
Hall, M., Jr. (1956).
IIA survey of difference sets. II
Proc. Amer. Math. Soc.,
[27]
Hall, M., Jr. (1968).
1, 975-86.
IICombinatorial theory."
Blaisdell Publishing Company.
125
[28]
Hotelling, H. (1944).
"Some improvements in weighing and other
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[29]
Kishen, K. (1945).
Ann. Math. Stat., 15, 297-306.
"On the design of experiments. for weighing
and making other types of measurements."
Ann. Math. Stat.,
16, 294-300.
[30]
Mann, H.B. and Evans
Sankhya, 11, 357-64.
[31]
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"Addition theor.ems." Tracts in Mathematics,
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[32]
Mann, H.B. (1967).
"Recent advances in difference sets."
Ann. Math. Monthly,
[33]
Mesner, Dale M.
~,
(1964).
229-35.
"Negative Latin Square designs,"
Institute of Statistics Mimeo Series No. 410.
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[36]
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[37]
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[39]
Raghavarao, D. (1959),
"Some optimum weighing designs."
Ann. Math. Stat., 30, 295-303.
[40]
Rankin, R.A. (1964). "Difference sets."
2.,
(41)
Acta Arithmetic.. ,
161-68.
Rao, C.R. (1946).
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d
and a
system of confounding in factorial experiments."
Bull. Calcutta Math, Soc., ~, 67- 7 8.
[42]
Rao, C.R. (1947).
"Factorial arrangements derivable from
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[43]
Rao, C.R. (1965).
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Seiden, Esther (1954).
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[45]
128-39.
Linear Statistical inference and its
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[44]
l,
Jour. Roy.
Shrikhande, S.S. (1963).
Ann. Math. Stat., 17,151-56.
"Some recent developments on
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Mathematics Student,
31, 167-77.
(46)
Singer, J. (1938).
"A theorem in finite projective geometry
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Trans. Amer.
Math. Soc., 43, 377-85.
[47]
Yamamoto, K. (1963).
"Decomposition of difference sets."
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[48]
Yates, F. (1935).
"Complex experiments."
Roy. Stat. Soc., 2, 181-247.
Supp!. Jour.
127
APPENDIX
TABLE 1
Parameters of Balanced Weighing designs from orthogonal arrays
v
r
p
b
A
1
A_
2
6
10
2
15
2
4
6
10
2
15
2
4
4
3
2
3
1
2
12
22
6
22
10
12
8
7
4
7
3
4
8
7
4
7
3
4
TABLE 2
Parameters of Balanced Weighing designs from association matrices
A_
2
v
r
p
b
A
1
9
8
4
9
3
4
13
12
6
13
5
6
TABLE 3
Parameters of Balanced Weighing designs from orthogonal latin squares
~v_--=r:-.-_p~_-=b:....-_;.;.A1·_~A2_
9
f
8
3
12
2
3
128
TABLE 4
Parameters of Partially Balanced Weighing designs with cyclic association scheme
,<l
4.1.
S. No.
nl_ _
n2_ _
An----2.21_ _
A12--A22-
v
r
p
b
1
13
6
3
13
6
6
o
2
2
1
2
29
14
7
29
14
14
4
2
2
5
3
61
30
15
61
30
30
8
6
6
9
4.2
S.No.
e
Use of Rule I
Use of Rule II
v
r
p
b
1
37
18
9
37
18
18
4
4
4
5
2
101
50
25
101
50
50
12
12
12
13
A11--A --A --A22n 1---!'!.2_ _
21
12
TABLE 5
Parameters of PBWD with triangular association scheme
S. :tb.
e
A11_ _
A12-A22n 1---!'!.2_ _
A21--
v
r
p
b
1
10
6
2
15
6
3
1
2
0
0
2
21
20
3
70
10
10
4
6
0
0
3
10
6
2
15
6
3
1
2
0
0
4
10
6
3
10
6
3
2
2
0
2
5
15
4
3
10
8
6
1
0
0
2
6
10
36
2
90
6
3
6
8
0
8
7
15
24
2
90
8
6
3
0
0
8
129
c
TABLE 6
Parameters of PBWD with L.S. association scheme
S.No.
e
v
r
p
b
A22A1 2 - n 1_ _
n 2_ _
AU _ _
A21--
1
9
6
4
3
4
4
2
1
0
2
2
16
12
6
4
9
6
3
1
0
2
3
16
6
6
8
9
6
3
3
2
4
4
9
8
2
18
4
4
0
2
2
2
5
9
4
3
6
4
4
2
1
0
2
6
9
4
2
9
4
4
0
2
1
0
7
9
2Lj
2
~4
4
4
6
4
0
8
130
TABLE 7
Parameters of PBWD's with the singular GD association scheme
,~
7(A) .
S. No.
1
2
3
4
5
6
7
8
9
10
11
e
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
..
v
r
p
8
10
12
12
16
16
16
18
18
18
20
20
24
24
24
27
30
30
30
30
32
32
40
40
42
45
50
50
56
63
3
8
3
10
3
6
7
8
10
20
3
36
10
21
20
8
10
28
36
20
15
21
21
36
60
28
24
36
54
60
4
4
6
4
8
8
8
6
6
6
10
4
8
6
8
9
10
6
6
: 10
8
8
10
8
6
9
10
10
8
9
Resolvable method
b
3
10
3
15
3
6
7
12
15
30
3
90
15
42
30
12
15
70
90
30
30
42
42
90
210
70
60
90
189
210
nl----!!.2--All--A21--A12--A221
1
2
1
3
3
1
1
2
2
4
1
3
2
3
2
4.
L
2
4
1
3.
4
3
1
2
1
4
1
2
6
8
9
10
12
12
14
16
15
15
15
18
20
21
20
24
25
28
27
25
31
28
35
36
40
42
48
45
54
60
3
8
3
10
3
6
7
8
10
20
3
36
10
21
20
8
15
28
36
2Gl
15
2i
21
36
60
28
24
36
54
60
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
2
1
2
1
2
3
2
2
4
1
4
2
3
4
2
2
4
4
4
3
3
3
4
6
4
4
4
6
6
2
4
2
4
1
4
4
2
4
8
2
8
4
6
8
3
4
6
8
8
4
6
6
8
9
6
5
8
8
9
131
e
..
""
e
.
e
132
PBWD's of singular group divisible type using Theorem 5.4
7 (C) .
S. No.
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
* Design
v
r
p
26
26
26
26
27
27
30
30
30
30
22
24
24
24
24
24
24
24
24
25
26
26
18
18
20
20
20
20
20
21
21
22
22
22
18
18
20
8
10
12
12
14
14
15
16
16
16
16
18
12
60
12
8
8
8
18
18
36
18
10
7
21
7
'>4
10
12
6
9
12
6
9
12
15
10
6
9
12
4
6
8
10
12
10
4
6
9
4
10
8
4
9
8
6
9
4
6
8
6
8
10
4
4
6
4
4
6
6
8
4
6
8
6
11
11
22
55
11
4
12
6
10
8
3
4
18
18
36
12
6
20
30
40
8
8
18
3
4
3
10
12
6
4
3
7
21
7
10
b
39 '
78
13
18
12
9
45
30
45
18
11
14
28
7
33
22
33
66
11
5
39
26
10
18
3
5
45
30
45
21
.7
55
55
55
12
9
18
3
5
3
15
21
7
5
3
14
28
7
15
A12-A22n 2_ _
All_ _
A21-n 1_ _
1
1
1
2
2
2
2
2
2
2
1
2
2
2
1
1
1
1
1
4
1
1
2
1
4
3
1
1
1
2
2
1
1
1
1
1
1
1
1
2
1
1
1
2
3
1
1
1
2
24
24
24
24
24
24
27
27
27
27
20
21
21
21
22
22
22
22
22
20
24
24
15
16
15
16
18
18
18
18
18
20
20
20
16
16
18
6
8
9
10
12
12
12
12
14
14
14
15
12
60
12
8
8
8
18
18
36
18
10
7
21
7
11
11
22
55
11
4
12
6
10
·8
3
4
18
18
36
12
6
20
30
40
8
8
18
3
4
3
10
12
6
4
3
7
21
7
10
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
number of the table on page 158 [10].
1
20
5
1
2
3
2
2
12
8
4
1
6
3
1
2
6
20
5
1
4
2
4
1
1
1
2
2
12
2
2
2
6
12
2
3
8
1
1
1
2
2
2
1
1
1
6
3
2
2
25
6
2
3
4
4
3
16
10
5
.2
9
4
2
.3
8
25
6
2
6
.3
6
2
2
2
4
3
16
4
3
4
9
16
3
4
10
2
2
2
4
4
3
2
2
2
.9
4
4
*
(26)
(29)
(30) ,
(10)
(11)
(12)
(13)
(14)
(15)
(16)
(20)
(7)
(8)
(9)
(21)
(22)
(23)
(24)
(25)
(2)
(26)
(27)
(4)
(10)
(1)
(2)
(13)
(14)
(15)
(5)
(6)
(17)
(18)
(19)
(11)
(12)
(16)
(1)
(2)
(1)
(3)
(5)
(6)
(2)
(1)
(7)
(8)
(9)
(3)
133
TABLE 8
Parameters of PBWD's with regular GD association scheme
S.No.
v
•
v
r
p
b
1
6
8
2
12
1
4
0
4
2
3
2
10
8
2
20
1
8
0
0
1
2
3
16
6
2
24
3
12
2
0
0
1
e
Al 2 - A22nl __
n 2_ _
An--,\21--
TABLE 9
Parameters of PBWD's with semi-regular GD association scheme
1
12
10
3
20
1
10
o
o
2
3
134
TABLE 9
Parameters of PBWD' s with semi-regular GD association scheme
•
S. No.
v
r
b
m
n
'\11--'\21--'\12--'\22-
2
3
2
6
2
6
2
9
2 12
2 15
2
6
3
4
2
8
8
3
2 18
3 12
3 15
2 10
5
4
3 10
2 20
2 20
4 10
4
3
6
3
6
4
2 15
3 10
5
6
3 12
4
9
4 12
3 18
2 30
2 30
3 20
4 15
3
3
3
3
3
3
4
4
4
4
4
3
3
5
5
5
5
5
5
3
3
3
3
3
3
3
3
3
3
3
3
3
3
2
2
2
2
2
2
2
2
2
2
2
3
3
2
2
2
2
2
2
4
4
4
4
4
4
4
4
4
4
4
4
4
2
0
4
6
0
10
3
0
6
0
9
8
10
4
4
4
0
8
0
2
3
4
5
0
0
6
6
8
9
10
0
0
10
p
• .f)
£
'e
)
."
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
2
6
4
6
4
6
6
6
8
6
6 10
3
8
3
8
6
8
6
8
8
9
9
8
9 10
10
4
10
4
10
6
8
10
10
8
10
8
2
12
12
3
12
4
5
12
12
5
12
5
6
12
6
12
8
12
12
9
12 10
12 10
12 10
12 10
~.
0
4
0
0
8
0
0
3
0
6
0
0
0
0
0
2
8
0
8
0
0
0
0
5
5
0
0
0
0
0
10
10
0
0
1
0
0
2
0
0
1
0
2
0
0
0
0
1
1
1
0
3
0
0
0
0
1
2
0
0
0
0
0
1
2
0
1
1
2
3
2
5
1
1
2
2
3
4
5
1
2
2
1
2
3
1
1
2
1
1
2
2
3
4
3
2
1
2
5