This research was supported by the Army Research Office, Durham, Grant
No. DA-ARO-D-3l, l24-G9l0, and the United States Air Force Office of Scientific
Research, Office of Aerospace Research under Grant No. AFOSR-68-l406.
PARTIAL DIFFERENCE SETS AND PARTIALLY BALANCED WEIGHING DESINGS.
by
I.M. Chakravarti
and
K.V. Suryanarayana
Department of Statistics
University of North Carolina at Chapel Hill
Institute of Statistics Mimeo Series No. 600.14
OCTOBER 1969
PARTIAL DIFFERENCE SETS AND PARTIALLY BALANCED WEIGHING DESIGNS*.
by
I. M. Chakravarti and K. V. Suryanarayana
Department of Statistics
University of North Carolina
Abstract
Balanced weighing designs investigated by R. C. Bose and J. M. Cameron
(2~3]
are of special importance in some practical types of studies where one
wishes to compare the value of the "unknown" objects in terms of the accepted
standards.
With the aim of further reduction of the number of weighings
required~
some study has been made by introducing "association schemes" to
a weighing situation.
This paper deals with the use of "difference sets and
partial difference sets" in constructing this new class of designs called
"Partially Balanced Weighing Designs".
1.
Intro,· lction and Summary.
There has been much work
(6~8]
applications of difference sets.
on the
construction~
non-existence and
Two of the main applications are in the areas
of Design of Experiments and Error correcting codes.
A new class of difference
sets arose in the construction of association schemes and partially balanced
incomplete block designs
[4~5].
Such type of difference
sets~ called~
"partial
difference sets" are described in section 2.
Calibration designs or
and used by Bose and Cameron
balanced weighing designs have been constructed
(2~3]
in comparing the value of the "unknown"
objects in terms of the accepted standards.
The extension of the balanced weighing designs (BWD) to a situation of
"partial balance" based on "association schemes" leads to a new type of designs
*
This research was supported by the Army Research office~ Durham~ Grant No.
DA-ARO-D-3l, 124-G9l0~ and the United States Air Force Office of Scientific
Research~ Office of Aerospace Research under Grant No. AFOSR-68-l406.
2
called "partially balanced weighing designs (PBWD)" [7,10,11].
These are
described in section 3.
The main contribution in this paper is the construction of partial
difference sets and the application of difference sets and partial
difference sets in constructing partially balanced weighing designs.
The methods of construction are described in section 4.
2.
Perfect and Partial Difference Sets
A perfect difference set [6,8] is a set of
modulo
v
such that every
a
to
(mod v)
k
V={d , d , ... d }
k
l
2
integers
can be expressed in exactly
A ways in the form
d
where
d
- d
i
and
i
d
a (mod v),
-
j
are in
j
V.
It is easily seen that
k(k-l)
v-I
A=
In order to exclude some degenerate configurations, we may impose the
restriction
o
V
is called a
A
A
<
<
k
<
v-I.
(v, k, A) - difference set.
(v, k, A) - difference set is equivalent to a cyclic
symmetric balanced incomplete block design.
A = 1, a
For
(v, k, A)
(v, k, A)
difference set yields a cyclic projective plane.
A generalization of the idea of a difference set was noticed in Bose
and Nair [4] and later in Bose and Shimamoto [5].
use the term "partial difference set".
They did not, however,
In [3], for defining a cyclic asso-
ciation scheme with two associate classes, the authors introduced a set
d , ••• , d }
2
n
l
the
of
n
l
integers modulo
d.'s are all different, and
J
v,
having the properties
0 < d. < v
J
(j = 1, 2, ...
nt,
3
differences
(11)
(mod
v)
each of the integers
d , d ,·.·, d
occurs g times and
l
2
nl
each of the integers {e , e , .•. , e } occurs h times, where the
l
2
n2
set {d , d ,· .. d , e , e , ... e } is exactly the set of integers
n
2
l
2
l
nl
2
{I, 2, ••• , v-I}.
D will be called a partial difference set and denoted by
D(v, n , n , g, h).
2
l
We note that the parameters of
D satisfy the following
relations.
n
l
+ n 2 = v-I
= nl(nl-l).
nl g + n2 h
If
= h,
g
D is a (v, n , g)
l
perfect difference set.
The following is the statement of a well known theorem in difference
sets [8].
Theorem 2.1.
If the fourth powers of a primitive root of a finite
field of order
pn
=1
+ 4y2
p
n
4f + 1
where
also sufficient.
form a perfect difference set, then
is odd.
y
The equation
If
P
n
then this condition is
n = 1,
=
1
Z
+ 4y ,
n >
1, y
=1
(mod 2)
do not have solutions.
Theorem 2.2 .
Let
v
a primitive root of
a
=
n
p ,
n
GF(p ).
where
P
Then i f
(v, k, A) - difference set, then (x
forms a
is an odd prime. Let
dZ
dk.
dl
(x
dl+h
(v, k, A) - difference set, for
Proof:
Corollary Z. 1 .
, x
, x
h
,"', x
d2+h
," ., x
)
dk+h
x
be
forms
)
also
where
h
an integer.
Easy.
If
P
is an odd integer, then
is a prime of the form
2
6
4u-2
(x , x , " ' , x
)
4h
Z
+ 1,
forms a perfect difference
4
set
(p, u, (u-l/4)
where
u
4
From Theorem 2.1
Proof:
difference set by
( x 2 t X6t
••• ,
x
2
8
forms a perfect
(x , x ,
u-l
(p, u, --4-).
difference set
= h2 •
Multiplying every element of this
and using Theorem 2.2
x4u-2)
we get that
u-l
is a difference set
(p, u, --4-).
The following lemma is well known, see for instance [1].
It will
be used to prove Theorem 2.3.
Lemma 2.1:
If
x
prime, then among the elements {(x
s
(x - l _ I)}
if
quadratic residues and
(t-l)
Theorem 2.3 : Let
p
n
= 4u
Let
positive integers.
2u
2
GF(p),
- 1), (x
4
P
an odd
- 1), .•. , (x
s-3
there is one zero, (t-l) quadratic residues and
non quadratic residues
the
n
is a primitive root of
x
s
= pn = 4t
p
is a prime,
be a primitive element of
... ,
x 4u }
t
and one zero, (t-l)
non quadratic residues
+ 1, where
{x2 , x 4 , x 6 ,
elements
+ 1
- 1),
if
s
u
and
=
n
GF(p ).
p
n
are
n
Then
constitute a partial
difference set with the parameters (4u + 1, 2u, u - 1, u) .
The
Proof:
2
4
6
(x , x , x ,
2
x (x
2
... ,
- 1),
2u(2u - 1) differences which can be formed from
x
4u )
2
x (x
can be shown in the form of an array:
4
- 1), ... ,
x 2 (x 4u- 2 _ 1)
1
4t-1.
5
Evidently, the
hth
column of the array in (1) contains all even
powers of the primitive root
(x2h
1)
x
is an even power of
multiplied by (x
x
2h
2h
_ 1)
Hence if
or in other words a quadratic residue,
then that entire column exhausts the even powers.
(x
- 1).
Similarly, if
is a non-Q.R., then that entire h-th column exhausts the set of
3
and {x, x ,
... ,
2
6
So the frequencies of the sets {x , x 4 , x ,
x.
all odd powers of
x 4u-l }
non-Q. R. 's among the set
... ,
x 4u }
are the same as the number of Q. R. 's and
{(x2 - 1) , (x4 - 1),
by the lemma, these are respectively
the partial difference set,
(u-l)
... ,
and
u.
2
4
6
{x , x , x , ••• , x 4u}
difference set with parameters
(x4u-2 - I)}.
But
So by definition of
forms a partial
(4u+l, 2u, 2u, u-l, u).
which proves
the theorem.
Example 2.1
Let
u
= 3.
Hence
know that 2 is a primitive root of
(4u+l) (=13)
GF(13).
is a prime power.
We
It can be easily verified
that among the differences which can be formed from
(4, 3, 12, 9, 10, 1) , the set (4, 3, 12, 9, 10, 1)
(22, 2 4 , 26 , 2 8 , 210 , 212 )
occurs 2 times and
each of the other remaining 6 non-zero integers (mod 13), occurs 3 times.
Example 2.2
Let
u = 4.
Hence
(4u+l»=17)
know that 3 is a primitive element of
(1, 2, 4, 8, 9, 13, 15, 16)
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 the
sets of Q.R.'s and non-Q.R.'s among the differences formed from
(1, 2, 4, 8, 9, 13, 15, 16).
6
Example 2.3.
Let u = 9.
Hence
(4u+1) = 37
that 2 is a primitive element of
GF(37).
is a prime power.
We know
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)
are 8 and
9 respectively"
Example 2.4.
Let u = 7.
Hence
that 2 is a primitive root of
(4u+1) (;29)
GF(29).
is a prime power.
We know
It can be verified that the frequen-
cies of Q.R. 's and non-Q.R. 's among the differences formed from the quadratic residues (4,16,6,24,9,7,28,25,13,23,5,20,22,1), are 6 and 7 respectively.
Association shcemes have been developed [4,5], in connection with
the study of partially balanced incomplete block designs.
As has been
mentioned already, the extensions of the balanced weighing designs going to
be described in section 3, arise by the introduction of association schemes.
All the constructions which are made in section 4 are with reference
to a particular case of a general type of association scheme called cyclic
association scheme [5] about which a result is established in corollary 2.1.
Although a simple proof of corollary 2.2 can be given by the direct
use of Theorem 2.3, it is deduced as a particular case of a Theorem due to
Mesner [9], which is stated below.
(~Mesner):
Theorem 2.4
In a finite field of order V with additive
group G and multiplicative group
(V-I)
of
G'
such that
be a generator of
sub -group
0
G'.
f or der
the coset of
a
1
N
G', let
N=(V-l)/m
Let
a
o
F(v, m)
in which two elements
only if
(y - x)
belongs to
~
let
sern,
generate d b y
which contains
is even if
= (0),
i-I
x, y
.
of
~visor
m be a
a
1
an d 1 et
of the order
V is odd, and let
~
be the multiplicative
a., 1. = 1 , 2 , ... m,
1
be
Define an association relation
G are
i-th associates if and
ai' i = 0, 1, 2, ... m.
Then for
i, j, k, in
7
the range 1, 2, ••. m and interpreted as taken modulo
F(V,m)
i
1
= P j+l ,k+l
equal to the number of elements of
by adding the unit element
Corollary 2.1.
(4u+l)
Let
x
1
is
and
which occur in the set obtained
to each element of
is the power of an odd prime.
tion relation
F(4u+l,2)
Let
A = O.
in which two non-zero elements
(y-x) belongs to
A., i
1.
GF(4u+l),
Let
O
3
5
4u-l
A = (x, x , x , ... , x
).
2
and
associates if and only if
F(4u+l,2)
aj_i+l
Pj-i+l ,k-i+l,
be the primitive root of the Galois field
2
4
4u
Al = (x , x , ... x )
Define an associax
= 0,1,2.
and
yare
i th
Then
is a two-class partially balanced association scheme with the
parameters V = 4u+l, n
Proof:
where necessary,
is an m-class partially balanced association scheme with the
parameters
where
m,
l
= n
2
= 2u
and
The proof follows from the above Theorem, by taking
3.
V
4U+l
and m=2.
Partially Balanced Weighing Designs.
Definition of Partially Balanced Weighing Designs with two association classes:
Association schemes have been defined and have been widely used.
V treatments 1,2, ••• V,a
Given
relation satisfying the following 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
associate of the treatment
ment
8,
then
8
a
is the i-th
is the i-th associate of the treat-
a(i=1,2).
(b) Each treatment has
dent of
a.
n , i-th associates, the number
i
n.
1.
being indepen-
8
(c)
If any two treatments are i-th associates then the number of treat-
ments which are j-th associates of
a
and k-th associates of
P~k and is independent of the pair of i-th associates
a
S
and
is
S.
The
pa~ameters of the association scheme are V,n i , P~k (i,j,k=I,2).
A design is said to be a Partially Balanced Weighing Design (PBWD)
with
two association classes and with the parameters (v,b,r,p,A II 'A21' A12' A22)
if there are
in
r
v
treatments arranged in
b
blocks, each treatment occuring
blocks, such that the blocks are of size 2p
and if each block can
be subdivided into two halves with the following conditions:
(1)
Any two first associates occur together in the same half block
times and in the opposite half blocks
(2)
A
21
All
times.
Any two second associates occur together in the same half block
times and in the opposite half blocks
Such a design is denoted by
A
22
A
12
times.
PBWD(v,b,r,p,A l1 'A2l'A12"A22)'
The combinatorial properties of these designs together with the methods
of construction and the analysis are described elsewhere [10].
A good
account of the methods of construction of the PBWD's with the triangular,
Latin square and group divisible association shcemes is given in a separate
paper [11].
The next section is concerned with a method called "the cyclic
generation of the PBWD's", which consists in generating all the blocks of
the design with the help of one or more initial blocks.
The investigation
is restricted to a particular class of cyclic association schemes.
9
4.The construction of PBWn's from difference sets.
Let
{d l , d Z ' " "
(v, n , n Z ' a, S)
l
dn }
l
and let
be a partial difference set with parameters
n = {d , d ' ••• ,
Z
l
and
define a cyclic association scheme.
Suppose
are
(i=l,Z, ... , t)
{a'l'
a,Z'
... , a.1.p }
1.
1.
Zt -sets of distinct integers, each of size
p,
with the following
properties:
The sets
I.
are
disjoint.
II.
Among the differences
(i=l,Z, ... ,t,
A
IZ
V.
occurs
1.",
j:f. £ = 1,Z, ... ,p), each
d
j
1.J
occurs
A
Zl
+
ZZ
times.
AU
e
(q=l,Z, ... ,n )
Z
q
(a .. - b, )
1. £
1.J
,
each
d.
J
times.
Among the differences mentioned in IV, each
A
0 )
1.",
times.
e
Z
(£=1,Z, ... ,n )
2
times.
We treat the initial block
of the
- a . 0 ) , (b,. - b.
Among the differences of the form
IV.
occurs
ij
Among the differences mentioned in II, each
III.
occurs
(a
i-th
set as
{ail' a iZ ' ""
a ip ; b il , biZ' ... , b ip }
'O'-th block of that set and the block
{ail + j, a iZ + j, ' ' ' ' a ip + j; bi! + 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.
~,en
we consider the
With this notation, we can state
10
THEOREM 4.. I. _
{b
il
, b
i2
, ... , b
exist, then the
{ail' a
i2
, ""
ip
vt
If the
}
2t
sets {ail' a i2 , ... , a.~p } ,
i=l,2, ••. ,t,
satisfying the properties I - V
blocks generated by developing the sets
a ip ; b il , b i2 , " ' , b }
ip
treated as the initial blocks,
constitutes a partially balanced weighing design with the parameters:
and with the association scheme defined by the partial difference set
... ,
with parameters:
n -a-l
1
n -a-l
1
_ _(.3)
n -8
)
n 2-:1+B-!
.
Thus the corresponding association scheme called cyclic association
scheme is specified by the parameters
a, S, n,
and
For some verifications arising in the proof.
Proof:
11,)
'-'Ie
c:nl
~avc :\11
idc<l
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
vt
when we start with
In the development of
u •
occurs in the block number
u
a
ij
if and only if we can find an
+
q
=u
or
b
ij + q
=u
initial
is the half-block size.
p
Next, consider a treatment numbered
q,
t
•
So
a
q
ij
or
b
ij
such that
can be uniquely fixed by
11
=
q
u - a
or
ij
blocks generated from
the blocks in which
u - a
ip
' u - b
i1
as the case may be.
u - b ..
1.J
{ail' a
u
, ... , a
i2
ip
; b
So restricting to the
i1
, b
i2
, "', b
occurs are the ones numbered
As there are
, u - b i21 .. ·u - b
ip
ip
} ,
u - ail' u - a i2 \··.
~nitia1
t
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 if and only if we can find two integers
a
s
in
and
a ..
1.J
such that
1£
a .. + q
1.J
From ( 4) and (5) , a.. - a. n
1.J
1.'"
=U
-
S
=
u
--(4)
=
s
--(5)
•
Let the partial difference set
{d , d ' " ' , d }
Z
n
1
mentioned in the
1
hypothesis be denoted by
If
u
element of
Thus
and
s
D
since
(u - s)
u
and
as
d
h
s
D.
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 evLdent that
i-th
can be represented as a difference between two
set, as many times
a
i
's
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
1.
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 ,
occur together.
1.
But
set
12
this number is
All
by the condition II.
Also this entire argument
•
is valAld as long as the pair constitutes a pair of first associates.
In the same way, i t is easily seen that two treatments which are
first associates will occur in opposite half blocks of the same block
1..
times and two treatments which are second associates will occur in
21
the same half block
block
A
ZZ
A
12
times and in opposite half blocks of the same
times.
THEOREM
a cyclic PBWD described in Theorem 1 is that:
Ztp(p - 1)
(i)
=
(ii)
Ztp
Z
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
Ztp(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
{d , d ' ••• , d } or E = {e , e Z ' •.. , e }' The
l
Z
l
nz
nl
existence of the PBWD guarantees that each element of D occurs as
belong to
D
=
a difference between the elements of such pairs
element of
E
occurs
Since there are
it follows that:
This proves (i).
A
IZ
All
times and that each
times.
elements in
D
nlA ll + nZA lZ
and
elements in
Ztp(p-l)
Similar argument establishes (ii).
E ,
13
Now we proceed to develop some results relevant for the construction
of the partially balanced weighing designs, with the cyclic association
scheme described in Corollary 2.2.
Lemma 4.1.
(i) Let
v = 4u+l
n
be the power of a prime (p ),
in a positive
integer.
(ii) Let
(iii) Let
Then
x
be a primit~Lroot of the Galois field
~ be defined by
(x)qw = x4w+2u+qa
where
(x
a = (u-w).
4w
- 1)
= x~
for
n
GF(p );
w
= 1,2, •.. ,(u-l).
14
Proof:
Since
x
is a primitive element of
x 4u _ 1
= 0 or yU - 1 = 0
where
except for the trivial case
x
Since
x
4
y
x
o
implies
u
- 1
u-l
u-2
Y
+ Y
+ ... + Y + 1
X
y
4u
4
= 1 or
:f 1 ,
=5 ,
v
y
GF(4u + 1) ,
=0
or
4 (u-l) + x 4 (u-2) +
... + x 4 + 1
=
0
_ _ (6)
+ ...
4 1)[ x 4(u-l) +x 4(u-2) + ... +x 4w]
- ( xby the use of (4).
=
Noting that the product of the last two expressions is
x 2u
and using the fact that
x4w+2u+qa.
:
{x
by
sl
and
Then
Proof:
If
rand
n
u
be of the form
, x
s2
(4t
xqw+8, " ' , xqw+4u}
the sets
Qw-(2u+4w)+4
= x4w+2u[x4(u-w)_1)
With the same notation and conditions «i) - (iii»
of the previous lemma, let
w ,
x 4w_l
we have
This proves the lemma.
LEMMA 4.2
a fixed
-1,
[x 4 (u-w)_1]
Qw-(2u+4w)+8
, .,., x Qw -(2u+4w)+4u}
±
3)
and let for
and
be denoted
respectively.
and
are disjoint
and
(1 .:. r, n .:. u)
(i. e. )
(empty set) .
are not disjoint, let there exist two integers
such that
x
Qw+4r
= x Qw-(2u+4w)+4n •
15
This implies that
=0
(4r + 4w - 4n) + 2u
(mod 4u) .
Since 4 must divide the left hand side of the above congruence, 4 divides
2u.
This is a contradiction, since
u
is of the form
(4t + 3)
contradiction is due to the assumption that one member of
identical with an element of
s2
Hence it follows that
s2 .
is
and
are disjoint.
LEMMA 4.3. :
With the same notation as in the above lemmas,
and under the same situation,
n
v = p
where
let us put an additional restriction that
Then the number of
{x
sl
This
4w _l,
u
is of the form
(4u + 1) ,
is of the form
(4t + 3)
Q.R. 's among the set of
w=1,2, ... ,(u-l)}
.
(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.
Also if
case, the number is
then
(u-w) = w or
that
u = (4t
for any fixed
X
+
3)
w ,
qw __ x4w+2u+qa ,
u = 2w
for any
w=1,2, ... ,(u-l)
which is a contradiction to the assumption
Hence it follows that
(w
In that
= 1, 2,
... ,
(u-l».
of the Lemma 4.1
and
The conclusion
can be written as:
~
-
4w + 2u + qa (mod 4u)
or
~
-
4w + 2u + qu-w (mod 4u)
(Le.)
qu-w
-
q
-
qw + 2u + 4 (u-w) (mod 4u)
-
qw + 2u + 4a (mod 4u)
w
are distinct
-
4w
2u (mod 4u)
(8)
16
It is obvious from(7 ) (or (8»
q
that if
is also odd and vice versa.
is odd, then
u-w
Similarly, i f
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
But by a remark given in the beginning of the
£
proof, the common frequency of odd or even powers of
w=1,2, ... ,(u-l)}
or
u = 4£ + 1
is
(u-l)
x
among
u-l
-2- = 2£
Hence
2
which is a contradiction, since
by condition (1) .
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:
theorem,~
The Lemmas
(4.I~-
4.3)
are developed to prove the
4.3.2 which is useful for the construction of PBWD's.
THEOREM 4. 3 ~ _,
(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
8
4u
{x , x , ... x },
and
be an odd prime and let
3
{x , x
(iv)
{x 4w _ 1,
7
x
2
6
10
{x , x , x
11
Among the
x
4u-l
(u-l)
}
v
be an integer of the
= pn
v
(4t + 3)
... x
4u-2
GF
pn
5
9
4(u-l)+1
{x, x , x , ... x
l
},
be denoted by
and let the 4 sets
A , A , A , A
l
2
3
4
respectively.
distinct elements
w = 1,2, ... , (u-l)},
let there be
g
quadratic residues.
1'7
u(u-l)
Then among the
{x
4
, x 8 , x 12 ,
and
A
4
Pro0f:~
are
... ,
x
.B. ,
2
.B. ,
2
4u
}
differences formed from
the frequencie s of the sets
u-l-g
2
u-l-g
2
and
A , A , A ,
l
2
3
respectively.
All the possible differences which can be formed from
4
8
12
{x , x ,x
... , x
4u
}
can be written in the form of an array
as follows:
x
x
4+ql , x 4+q2 ,
x 4+qu-1
8+ql
x 8+qu-1
, x
8+q2
,
x 4u+ql , x 4u+q2 , .... x 4u+qu-1
Evidently, all the
Also the
s-th
4s
and
x
elements in a column of this array are distinct.
row of this array can be written as
... ,
x 4s+ql , x 4s+q2 ,
x qu-w+
u
x 4s+qu-1
qw+4s
(
Among these, the two elements
= x 4w+2u+q u-w+4s) b Y Lemma
are distinct, as it is already noted (Lemma 4.3.)
w
(1 ~ w ~ u-l)
is of the form
"q
If
form
4h + 2
is of the form
u-w
4h
qu-w
I
for any fixed
.
is of the form
If
4",1
since
4p"
qu-w
==
u
=
(4t + 3)
4p + 2
2 (mod 4)
,
then
and
2u
4w + 2u + q
is of the form (4c + 2). "
4w + 2u + qu-w
==
2 (mod 4)
The same type of argument leads us to conclude that
"q
u-w
"~ + 4s
==
+ 4s
I
3 (mod 4)"
u-w
So we conclude that
implies" (~ + 4s)
is of the form
since
then evidently
4p ,
(mod 4)"
implies
and
vice versa .
is of the
18
So for a fixed
s,
( x 4s+qu-w , x 4s+qw)
be of the form
(x
the elements can be paired as
for different
4c+ l , x4b+3)
w's
or
such that this unordered pair will
( x 4 c+ 2, x 4b) .
s (s = 1, 2, ""
This assertion is true for all
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, ""
of the
array(~)
u,
xqj
is taken out as common factor,
lead to the conclusion that the columns
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
_l,
g
and
stand for the Q.R. 's and non-Q.R. 's among
w=1,2, ... ,(u-l)} ,
we conclude the following:
1 - g .
(i)
h
(ii)
g I h
(iii)
The number of elements of the form
u -
those of the array
t
h
(by Lemma 4.3.8) .
«(9')
is
.&
2 '
or
eu-l-g)
2
x
t
among
according as
is even or odd.
(iv)
All the
u
elements in a column of the array ($1) 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
t
where
i (mod 4)
19
So with the help of array t91)" i t follows that among the
8
4u
4
Al = {x , x , ... , x } ,
differences formed from
are
of the sets
respectively.
.B.
the frequencies
u-I-g
.B.
Z '
z '
u( u-l)
u-I-g
and
Z
Z
This proves Theorem 4.3.
COROLLARY 4 .1 •
With the same notation as in Theorem
among the differences formed from
Z
6
4u-Z
x , x , .•• , x
A :
Z
.B.
are
the frequencies of
4.3
Z '
u-I-g
Z
.B.
Z '
and
u-I-g
Z
' respectively.
Proof:
or
x
Since
Z{x 4 ,
8
x ,
A
Z
Z
= {x , x
... ,
as in the theorem
resulting
x
t
x
6
... ,x
4u-Z
Z 4u
4
8
4u-4
}=x{x , x , x , ... ,x
}
4u-4 , x 4u} , we get the same type of differences
(for the case
AI)' except for the fact that each
will be multiplied by
x
Z
So denoting the set of differences for
D
s
Al
and
respectively, the assertion of the Theorem 4.3.
+
u-I-g
Z
A3
A
Z
as
that
+
u-I-g
Z
A4
implies
D
+
s
u-I-g
Z
A3
(by the remark of the above paragraph).
Hence the corollary follows.
D
f
and
20
Let
THEOREM 4.4
form
(4u+l)
where
x
Let
4
x -x
4u-2
be an odd prime and let
p
4
8
4u
Al = {x , x , " ' , x }
Le t
is a primitive element of
(i)
Let
(A
u
- AI)
2
(AI - AZ)
(u-l-g)
Proof:
"', x
(4t+3)
{x • x
6
,"'X
4u-Z 1
J,
4u
Z
4
6
(x -x , x -x ,
2
4u 6
1m
4u-2)
-x, x -x, "', x -x
and
(AI - A )
2
2u
2
(u-l) -elements
-differences formed
each quadratic residue occurs
times and each non Q.R. occurs
The set
4
differences
then among the
(A Z - AI) ,
and
Z
g -quadratic residues among the
h=l, 2, ... , (u-l)
from
Z
be of the
be defined.
be of the form
Let there be
=
A
Z
n
GF(pn) .
(AI - A ) stand for the u
Z
8 2
8 6
8 4u-Z
x -x , x -x , ... , x -x
Similarly, let
and
v = p
(g+l)
times.
of differences can be written as an array given
below:
x 6 (x 2_l), ~6(x6_l), x 6 (x lO _l),
x 4u-Z( x 2_ 1') ,x 4u-2( x 6_ 1 ) ,x 4u-i( x 10_1) ,
So any arbitrary element of
where
xhw+4s+2
(x 4w+ 2_l)
(A
= xhw
l
hw+4s+2
=x
x 4u~2( x 4u-2_ 1 )
- A ) will be of the form
2
n
Let h + 4s + Z = R which
w
w
obviously depends on the arbitrary integer
x
x6 (x 4u- Z_l)
s.
With this notation
Rw
The corresponding element (i.e. just the negative of this arbitrary
element) of
(A
Z
- AI)
can be written as
follows from the definition
0
f
x
that
(-l)x
X 2u
Rw = xl\y+Zu
-- (-1).
Since
since it
21
2u
=
in
2(4t+3)
x
Rw
(mod 4)
=
4(2t+l) + 2 , the comparison of R
and
w
Rw+2u
x
leads us to the conclusion that
and
appearing
R +Zu - 0, 1, 2 , 3
w
according as
R
w
Let us write the
and the
R +2u
w
u
(mod 4)
Z, 3, 0, 1
elements of the
---'.(10)
(w+l) -th column of the above array
u -elements corresponding to their negatives as a partitioned
row vector as follows:
--+(11 )
Now let us s.tucl1 the nature of these elements.
(h +4c+2+Zu)
respectively from the first and second part.
it is obvious that if
4(s-c)
(h +4s+Z)
and
w
be two arbitrary elements of this partitioned row vector taken
w
that
Let
= 2u(mod
h +4s+Z
w
4u)
= hw+4c+Z+Zu
and hence that
Comparing these two elements,
(mod 4u),
4
we have to conclude
divides
Zu
which is
impossible.
Also, it is obvious that the first part of(11)is a set of
distinct elements and that the second part of (11)is a set of
u
u
distinct
elements.
So it follows from( 10) that all the
Zu -distinct elements of (11)
are quadratic residues or non-quadratic residues according as the corresponding
x Rw
(see (10)
So i f the array
is a quadratic residue or a non-residue, respectively.
(A
Z
AI)
is formed directly from
(AI -
AZ)
by
change of sign of each element, we have the conclusion that:
(i)
the
exhausts all the
(w+l) -th column of the combined array
ZU
Q.R. 's or all the non-Q. R. 's.
A2]
( Ar
AZ-A
l
either
22
(ii)
{(x
2
-1),
If
(x
6
c
is the number of quadratic residues among
-1), ... ,
(x
4u-2
-1) f ,
quadratic residues repeats itself
Determination of
Q.R. 's among
of
{x
c :
Z
-1,
c
times in the combined array.
By the Corollary (1.1.1) of [ ] there are
x
4
-1,
in the hypothesis,
g
implies that the set of
(i)
x
c
6
-1, " ' ,
=
x
4u-2
-l}
and hence by the notation
uul-g .
So by (i), each quadratic residue occurs
~'A1-AZ )
A -A
2 l
non Q.R.
(u-l-g)
Since the total number of rows is
( Al -A 2 )
occurs among
(u-1)
g
+
1
u
(
times in
in this array, eacll
u-(u-l-g)) times.
A -A
This proves Theorem 4.4.3. 2 l
Let
THEOREM
(4u + 1)
form
(i)
Let
Let
u
4h _l),
x
be an odd prime and let
g
v
n
GF(p )
be a primitive root of
be of the form
(ii) Let there be
{(x
p
=
p
n
of the
.
(4t + 3) .
quadratic residues among the
(u-l) -elements
h=l,Z, ... ,(u-l)}
Then the initial block
4
{x , x
8
,
... ,
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
(v, b, r, p)
=
(4u+l, 4u+l, 2u, u)
and
(g, u-l-g; u-l-g, g+l) .
Proof:
We follow the notation of
A
l
and
A
Z
of the Theorem 4.4.3
and note that the initial block can be written as:
{A ; A } .
l
2
2)
Step 1:
By the Theorem 4.3,
.£
each quadratic residue occurs
u-I-g
Al '
among the differences formed from
times and each non Q.R. occurs
2
times.
2
Step 2:
among the differences formed from
By the Corollary 4.1
.£
each Q.R. occurs
Step 3:
By the Theorem 4.4
among the differences
which are the opposite differences arising from (A ;
l
occurs
(u-l-g)
u-l-g
times and each non Q.R. occurs
2
times and each non Q.R. occurs
{A
l
2
-
Ai
(g+l)
A
2
,
A
2
-
A
l
}
each Q.R.
times.
Steps 1 and Z 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
= (g, u-l-g) .
(All' A )
1Z
Step 3 implies that
(A
Zl
,A
Z2
)
=
Hence the initial
(u-l-g, g+l)
block under consideration satisfies all the 5 conditions of the Theorem Y.1
Z
4
6
4u
{d , d , ... , d } = {x , x , x , ... , x } and
2
Zu
l
3
5
4u-l
{x, x , x , ... , x
} defining the
{e 1 , e Z ' ... , eZ~
with
association scheme.
The parameter sets (1) and (Z) of the Theorem
4.1 are given by:
(12)
(4u+l, 4u+l, Zu, u; g, u-l-g, u-l-g, g+l)
and
n
l
n
Z
Zu ,
B
1
as noted in the Corollary 2.1.
=
t
:)
and
pZ
(:
u
u-l)
--~- ~,-
- (1 ':
EXAMPLE 1:
u
= 3, v = 4
4
8
12
(2 , 2 , 2 )
GF(13)
x
By Corollary 2.1
2
g
is a primitive root of
(22, 2 6 , 210 )
(3, 9, 1)
= (2, 2 3)
(2 4_1, 2 8_1)
= 13
3 + 1
=0
(4, 3, 12, 9, 10, 1)
association scheme and by Theorem
= (4, 12, 10)
~.5
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(12) and(l).
(6, 6)
P1
G:),
P2
(: :)
--- (12 j
(13,13,6,3;0,2,2,1)
---i(1);
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)
and
(x
4w
_1,
w=1,2, ... ,6) = (15,23,6,24,22,19)
g
By the Corollary 2.1 t
4 •
(4, 16, 6, 24, 9, 7, 28, 25, 13, 23, 5, 20, 22, 1)
defines a cyclic association scheme and by Theorem 4.5
(16, 24, 7, 25, 23, 20, 1; 4, 6, 9, 28, 13, 5, 22)
gives a PBWD with the
25
parameters of the association scheme and the design being given by
(using (<t, 2 )and (1')
(14, 14)
(v, b, r, p; All' A2l , A12 , A22 )
=
(29,29,14,7; 4,2,2,5) ,
This is verified separately by the direct method.
EXAMPLE 3:
GF(61)
u
=
15,
v = 4
15 + 1
{x 4w},
The powers
can be seen to be
~.
=
{x 4w+2 1
61.
2
an d
is a primitive root of
J (4w
)
I
X
-1,
{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
w=1,2,'L.,1 4 1r
g = 8.
respectively.
So the parameter sets of the corresponding
association scheme and the PBWD are given by:
=(14
15
(30, 30)
15)
15
P2
=
~5
~5
15)'
14
(61,61,30,15;8,6,6,9) ,
and
THEOREM 4-
'6-
8
4
If {x , x , " ' , x 4Uf
4
8
{x , x ,
•
'!
•
,
x
••• , d
4u
2u
v = 4u+1
Let
x
2
be the power
p
n
of an odd prime, p ,
forms a perfect difference set, then
, x
6
, x
10
•.. , x
4u-2
1
2
4
6
4u
} = {x , x , x , " ' , x }
generates a PBWD, with
defining the cyclic
association scheme and with the following paramter sets of the association
scheme and the design:
26
2u ,
u-l u-l u-l u+l
(v, b, r, p; All' AZI ' A12 , AZZ ) = (4u+I,4u+I,2u,u;--2-'--2-'--Z-'--Z-)
By Theorell' 2.2
Proof:
both half blocks of the initial block are
difference sets, with
II
u-l
= --and hence
4
2
By Theorem2.3, ' the elements
difference set with
{x , x
4
u-l u-l
(All' A21 ) = (-2-' -2-) .
6
, x
,
... , x
{4u+l, 2u, 2u, u-1, u}
4u
form a partial
f
Hence among the opposite
u-1
u-l
u-l-(-) =
Z
2
differences arising from the initial block, there are
QR
. . ' s an dth
. ere are
theorem (
u_(u-1)=u+1
2
2
non QR'
. . s.
4.1 )and the corollary
2
block generates a PBWD and that
association scheme.
COROLLARY 4
.2
'+
t2.1~,
{x , x
4
,
Hence by the
it follows that the initial
••• , x
4u
}
defines the cyclic
Hence the theorem follows.
If
v = 4u+1 is a prime, where
... ,
8
then {x , z ,
x
4u
2
6
x , x
x
10
,
u
is of the form
... , x 4u-2 } generates
a PBWD
with the same association scheme as in the Theorem (4.6 ).
Proof:
This fo1l~s
,. from the Theorem 4.6
4
8
f x , x , ""
the above circumstances,
EXAMPLE 1:
u
= 9, v
4
x
, using the result that under
4u
9 + 1 = 37.
x
2
1
forms a perfect difference set.
is a primitive root of
37.
4
8
f (x ., x , " ' , x
36
),
2
(x , x
6
,
""
x
34
)}
can be written as
{16,34,26,33,10,l2,7,l; 4,27,25,30,36,21,3,11,28}
=
18
= (:
:)
(: :)
27
(37,37,18,9; 4,4; 4,5)
and
which is true by direct verification also.
Remarks:
4
The Section
can be
concluded as an attempttotonstruct
partially balanced weighing designs, with cyclic association schemes. 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)
is a set of
G
0. 1
Z.
(b)
G - {e}
(c)
E
l
v
... ,
' a Z'
E
l
U
elements
a
0.
0
{= e} ,
v-l
E
Z
forms a partial difference set.
Design:
(a)
D
1
and
D
Z
p
(b)
If
distinct elements from
D.D.
1
are two disjoint sets, each containing
J
-1
G
stands for the set of all differences formed by
and
b.
Jm
E
D. ,
J
then
(c)
Rule I says that if such a pair
D , D
Z
1
can be found,
then
(D ; D )
Z
1
leads to a PBWD, provided (E ,E ) can be found with conditions described in 1.
1 Z
28
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)
IDll
I
D 1
2
Rule II says that if such a pair
D , D
2
l
(D ; D )
l
2
can be found,
leads to a PBWD, with the association scheme given by
(E , E ) .
2
l
The parameter sets of the designs constructed under Rules I and I I
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 I I respectively.
Tables
4.l(A) and 4.2(A) describe the parameter-sets for a few of the PBWn's
of this type, whereas 4.l(B) and 4.2(B) describe the actual plans of
such designs.
29
TABLE 4.1.(A)
S. No.
v
4u+l
b
r
p
2u
u
4u+l
n
1
n
Z
All
An
A
12
A
22
2u
2u
g
u-l-g
u-l-g
g+1
C1
13
6
3
13
6
6
0
2
2
1
C2
29
14
7
29
14
14
4
2
2
5
C3
61
30 15
61
30
30
8
6
6
9
All
2
x -1
2
A21
2
x -1
2
A
12
2
x -1
2
An
2
x +l
2
TABLE 4.2. (A)
S. No.
v
r
2
4x +l
2x
37
2
18
x
n
p
b
2
2
4x +1
2x
9
37
18
1
2
n
2x
18
2
2
4
4
4
5
30
TABLE 4.1. (B)
v = 13,
Design (1
r = 6,
= 3,
p
'\11 = 0,
2,
'\21
.r1
= 13, n 1 = 6,
b
;)
2
'3
= ,
\:
'\12
Initial Block
1st Associate
(3,9,1; 4,12,10)
5,4,13,10,11,2
1
Design (2
v
= 29,
= 14,
r
t\
11
= 7,
p
A
4,
6
= 14, n 2
2, 1\
5
22
b = 29,
n
A
2,
21
1'1
12
1
7
7
7
Variety
Initial Block
1st Associate
16,24,7,25,23,20,1;
4,6,9,28,13,5,22
5,17,7,25,10,8,29,26,14,24,6
21,23,2
1
Cd
v = 61,
r
= 30,
i\ 11
p
= 8,
14
7
6
\2
Association Scheme
Design
6
=
Association Scheme
Variety
2
=
l' 2
;
n
= 15, b
A
2J.
/14
.. 1 = (
6,
=
61,
A12
n
= 6,
l
=
30,
,\
. 22
30
9
15') "
15 .
Association Scheme
Variety
1st Associates
Initial Block
1
17,13,10,23,48,21,16,58,59
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
31
TABLE 4.2. (B)
Design (4
v
=
37,
r
I\ll
=
18,
=
P
1~1
4,
II
b
4,
=
Variety
1st Associate
1
17,35,27,10,34,11,13,
8,2,5,28,26,31,37,22,
37, .n
1
A
'C :}
Association Scheme
4,12,29
9,
4,
12
1: 2
=
~2
~C
n
18,
2
18
5
:)
Initial Block
16.34,26,9,33,10,12,
7,1; 4,27,25,30,36,21
3,11,28
32
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1.
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Combinatorial problems of Experimental Designs.
John Wiley and Sons, New York (to be published)
2.
Bose, R. C. and Cameron, J. M. (1965).
"The Bridge Tournament problem
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Journal
of Research of the National Bureau of Standards, 69B, 323-32.
3.
Bose, R. C. and Cameron, J. M. (1967).
"Calibration designs based on
solutions to the Tournament problem."
Jour. of Research of the
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4.
Bose, R. C. and Nair, K. R. (1939).
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5.
"Partially Balanced Incomplete
Sankhya, 4, 337-72.
Bose, R. C. and Shimamoto, T. (1952).
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classes. "
Jour. Amer. Stat. Assoc.,
6.
Hall, M., Jr. (1968).
7.
Chakravarti, 1. M. (1969):
!iL,
"Combinatorial theory."
151-84.
Blaisdell Publishing Company.
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the International Statistical Institute,
8.
Matnn, H. B. (19~).
London
3-11 September 1969.
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9.
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(1964).
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Institute of Statistics Mimeo Series No. 410.
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