CONSTRUCTION OF ASSOCIATION SCHEMES FROM FINITE GROUPS
by
I.M. Chakravarti
University of North Carolina
and
S. Ikeda
Nihon University
Institute of Statistics Mimeo Series No. 579
May 1968
This research was supported in part by the
National Science Foundation Grant No. GP-5790 and
the U.S. Arrrry Research Office Grant No.
DA-ARO-D-3l-124-G9l0
DEPARTMENT OF STATISTICS
University of North Carolina
Chapel Hill, N. C.
CONTENTS
1.
Introduction
2.
Collections of group elements
3.
Association relation of group elements and an extension of the module
theorem
4.
Construction of association schemes from finite groups (1)
5.
Construction of association schemes from finite groups (2)
6.
Construction of association schemes from finite groups (3)
7.
Construction of association schemes from finite groups (4)
References
•
•
1.
Introduction
A basic idea of constructing association schemes from permutation
matrices has been presented by I.M. Chakravarti and W.C. Blackwelder [lJ,
where they have proved an interesting theorem which is an analogue of H.B.
Mann's theorem for symmetrical BIB designs [2J.
We shall first reproduce
their theorem with a modified proof.
Let us divide a given set of v distinct permutation matrices of order v,
(lol)
into m + 1 non-empty subsets
(lo2)
~,
••• , ][m'
and let us put
Ai
= r...t.Pa '
i
= 0,1, ••• ,m,
~
where the sunnnation is taken over all Pa belonging to )[i.
Then, m + 1 matrices
AO = Iv, Al , ••• jAm are all vxv matrices.
Lennna. 2.1 of [3] states that
LEMMA 1.1. A necessary and sufficient condition for a set of m + 1
matrices, (A = Iv, Al, ••• ,Am}, of order v to be the set of association
O
matrices of an m-class association scheme with parameters
(lo4)
i
Pjk'
v,
i,j,k
= O,l, ••• ,m,
is given by a set of conditions:
each Ai is a symmetric {O,l)-matrix, i = O,l, ••• ,m,
m
(lo5)
E A. = J (the vxv matrix of lIs), and
~
v
m '
'=0
~-
~~~)
( .........
AA.'-K
J
= i=O
~
'to
~A .,
P'k
J' ~
j,k
= 0,1, .•• ,m.
2
By (1.3), the conditions given by (1.5) can be rewritten as
(i)
(ii) ~ Pa
(1.6)
(iii)
IttiP~ , i = O,l, ••. ,m,
EriPa •
= J v ' and
Ex PA E. P
j'"
~ 7
• . ~ Pjik
_ l.=0
~ Pa
j,k = O,l, ••• ,m,
i'
where the summation of the right-hand side of (i) is taken over all Pa belonging to J[i.
We shall say that two matrices, R
if the matrix R·S
= (rijsij )
= (r ij )
and S
is the null-matrix.
= (sij)'
are disjoint
Then, the condition (ii)
of (1.6) shows that H given by (1.1) is a set of v mutually disjoint permutation matrices, from which the following lemma follows:
LEMMA 1.2.
For Pats satisfying the condition (ii) of (1.6) and for any
given real numbers ca t s and da t s, the condition
v-l
(1.7)
crla =
1:
a=o
implies that cm = dc,
a
v-l
1:
a--o
dela
= O,l, ••• ,v-l.
The proof of this lemma. is easy and is ami tted.
Let
Hi = (p;l IPa € J[i)' i = 0,1, ••• ,m.
The following theorem due to Chakravarti and Blackwelder [1] is now easy
to prove.
. THEOREM 1.1.
Suppose that the set I of v permutation matrices of order
v, given by (1.1), forms a group of order v.
Then, the set of conditions
given by (1.6) is equivalent to the following set of conditions:
3
-1
][i =][i
(i)
(l.8)
i = 0,1, ... , m.
'
(ii)
(iii) for any given P belonging to ][i' the equation
P~
P7
= Pa
i
has exactly Pjk solutions (P~,P7) such that P~
P
7
PROOF.
€
Uk,
€ ][j
and
for any j and k, j,k = O,l, ••• ,m.
It is evident that the conditions (1.8) are sufficient for (1.6).
11
=.
11'1S a group, ][-1
•
S1nce
For any given i, let us put
v-l
'iPa
= a-~ c-ula
Then, the condition (i) of
v-l
J:
a=<>
(1.6)
cula
and
r:i
-1
Pa
v-l
J:
a=0
d-ula'
implies that
v-l
=
=
dula' i
1:
a--o
Hence it follows from Lemma 1.2 that Cia
= d ia,
= 0,1, ••• ,m.
a
= O,l, ••• ,v-l;
i
= O,l, ..• ,m,
and therefore we have (i) of (l.8).
Let Pjk(Pa ) be the number of solutions, (P~,P7) with P~
of the equation in (iii) of (1.8).
Since I is a group,
Hence
and therefore, by the condition (iii) of (1.6), we have
v-l
1:
a=0
=
Pjk(Pa ) Pa
m
J:
i=O
i
Pjk J: i Pa .
This implies, by Lemma 1.2, that
for any P belonging to][i'
a
i
= 0,
1, ••• , m.
€
nj
and P
7
€
~,
P~7 is again in ••
4
Thus, we have proved that the conditions (1.8) are necessary for those
of (1.6).
This completes the proof of the theorem.
When the set
n of
permutation matrices given by (1.1) forms a group, it
can be regarded as a permutation group, whose elements being permutations on
v objects:
n is isomorphic to a subgroup of the symmetric group of order v!.
Since the order of I is v, H is isomorphic to a regular permutation group.
Group representation theory says that any finite group is represented
as a regular permutation group, from Which one can say that the construction
problem of association schemes from permutation matrices, provided they form
a group, is essentially equivalent to that from finite groups.
The present paper seeks for those association schemes which are constructed from finite groups.
In the following section, we shall introduce some operations on collections of group elements and state some of their properties.
In section
3, a pair-wise relation on group elements of a given finite
group, which we intend to call association relation, is introduced, and the
conditions under which the association relation results the usual association
schemes are discussed.
It comes out that the usual module theorem can be
extended to the case where the group operation is not necessarily commutative.
Section 4 and subsequent sections are devoted to study some types of
group structures which generate association schemes.
2.
Collections of group elements
Let G be a group of finite order.
By
a collection of elements of G we
mean a set of elements of G allowing the multiplicity of each element:
H
= Ca, ••• , a, b, ••• , b, .•• • },
a, b, ••• E G,
5
for which the multiplicities of the elements, a, b, ••• are denoted by f (H),
a
fb(H), ••• , respectively.
Given a collection H, we shall define the
multiplicity of any given element a of G relative to H in the following way:
if a belongs to H, then fa(H) is defined to be the number of times a appears
in H, and, if a does not belong to H, then fa(H)
= O.
Thus, for any given
collection H, there corresponds a set of multiplicities of the group elements
relative to H, [fa (H)la
€
G).
This correspondence is one-to-one.
tion with the corresponding multiplicities [fa(H) = ola
€
A collec-
GJ is the empty
collection.
According to the above definition, a subset H of G in the usual sense is
a special type of collection for which f (H) takes the value 0 or I for each
a
a E G.
It should be noted that the same kind of sets of group elements has been
introduced, together with some operations which are stated below, and used
extensively by M. Masuyama (see, for example, [4).
Now, we shall define some operations on collections of group elements.
It is convenient to denote the collection H with the corresponding
multiplicities [fa (H)la
€
G) by H = H(G)[fa (H»), where H(G} designates the
subset of G which consists of all distinct elements belonging to H.
two collections H and K, H~ K means that fa(H) ~ fa(K} for each a
For any
E
G, with
equality if and only if f a (H) = f a (K) for all a E G. Note that H c:_ K implies
that H(G} ~ K(G}, While He K does not necessarily imply H(G} c K(G}.
We shall introduce some rules of calculation on collections:
(a)
K(G}
Inversion:
H- l
= K if
and only if fa(K}
= H(G}-l, where for any subset A of G,
A-1
(b)
=
(x -1
Scalar multiplication:
IX
E
=f
_1(H}, and hence
a
A).
For any non-negative integer A, K
= AH
if
6
and only if f (K) = )f (H).
a
a
(c) Multiplication by group element:
K = xH if and only if fa(K)
=f
-1 (H), and hence K(G)
x
xA
= xH(G), where
a
= (xa I a
for any subset A of G and any x
For any given element x of G,
€
€
A),
Right-multiplication, Hx, is also de-
G.
fined similarly.
H + K = L if and only if f a (L)
hence L(G) = H(G) U K(G).
(d)
Addition:
(e)
Subtraction:
if f a (L)
(f)
= f a (H)
Union
----
= f a (H)
For any H and K such that H ~ K, H-K
- f (K) for each a
a
€
€
a
= L if and only
G.
HU K = L if and only if f a (L) = max (fa (H), f a (K)).
(g)
Intersection:
_.
(h)
Procuct:
HnK
= L if and only if f a (L)
HK = L if and only if f a (L) = E
xy=a
a
+ f (K), and
= min(f a (H),
f a (K))
f x (H)fy (K), for each
G.
Some of the main properties of the operations defined above are listed
in the following
LEMWI. 2.1.
Let H, K, L and M be any given collections of elements of a
given finite group G, and A, 11,
AH + 11K and
(i)
~L
~
and'T1 be any given integers such that both
+ 11M are certain collections.
(NI+JJK) (~L+1')M)
(ii)
(AH+JJK)-l
(iii)
(HK)-l
Then, it holds that
= AtHL+AT}HM+J1~KL+"t}KM,
= AlI-1 + JJK-1,
= K-~-l.
The proof of this lemma is easy and is omitted.
Let us define the cardinality of a collection of group elements, H, by
(2.2)
/H/
=
1:
aEG
f
a
(H).
7
Then, we can see the following
!EMMA 2.2.
For any given subsets, H and K, of G, the cardinality of
H n K is equal to the multiplicity of the unit element 1 in the product
HK
-1
, i.e.,
l
H n K I = fl(HK- ).
I
(2.3)
The product HK- l can be replaced by anyone of the products, H-~, KH-l and
PROOF.
By
(2.2) and the definition of the intersection, we have
I
H n K I = E min(f (H), fa(K»,
aEG
a
while, by the definition of the product,
f (HK- l ) = E f (H) f (K- l ).
l
xy=l x
y
Since f y (K- l )
= f y -l(K),
and f a (H) and f a (K) take the value
l
fl(HK- ) = E -1 fx(H) f _l(K)
x=y
y
= as}
E
f
a
(H)
f
a
1 or 0, we get
(K) = I Hn K I,
which proves (2.3).
It is easy to see that HK- l can be replaced by one of H-~, KH- l and
This
co~letes
the proof of the lemma..
It is straightforward to extend this lemma to get
(2.4)
that is, the cardinality of the subset aH
n bK
is equal to the multiplicity
of the element a-~ in HK- l , for any subsets H and K and any elements a and
b
of G.
8
Let us denote the subset of G consisting of the unit element only by
LEMMA. 2.3.
(i) For any given subgroup H of G, and any given subset of
H, it holds that
KH = HK = IK/H.
(2.5)
(ii)
A necessary and sufficient condition for any given subset H of G
to be a subgroup of G is that
J?
(2.6)
=
IHI
H.
(iii) A subset S = GO + T of G is a subgroup of G if and only if
rI = ITI
(2.7)
PROOF.
(i) :
GO + (jT/-l)T.
By definition,
f
a
(KH)
=
1: f (K) f (H)
xy=a x
y
=
1:_1
y=x a
1
!KI, if a € H,
{
= 0, otherwise.
y€H,x€K
This proves
(ii):
(2.5).
Necessity follows from (i).
Suppose a subset H of G satisfies the condition (2.6).
Since
I If I
=
I
nH
I
where n =
IHI,
H must be in H, i.e., if x, y E H, then xy
Consider any element x in H.
If
€
H.
Then, there exist y, z in H such that
xy = z, or equivalently, x-lz = y.
(G-H)H=H(G-H) = En -
the product of any two elements of
From (2.6) it follows that
= nG -
which implies that the equality x-lx
=y
nH = n(G-H),
given above is impossible unless
9
x -1
€
H.
Hence, if x
€
H then x -1
€
H.
This completes the proof of (ii).
(iii):
Put /T/ = m, then /S/ = m + 1-
Using (2.7) we have
S2 = (G +T)2
o
= Go+2T#
= Go+ 2T
+ rnGo + (m-l)T
= (m+l)(GO+T) = (m+l)S,
and, conversely, if s2
= (m+l)S,
then we have (2.7).
Thus, (iii) follows from (ii), and the proof of the lemma is completed.
3.
Association relation of group elements and an extension of the module
theorem
Let G be any given finite group of order v, and let
be a partition of G into (m+l) non-empty sUbsets, where, as before, Go = {l},
1 being the unit element of G.
Let cp(x,y) be a mapping from the direct product
G X G
= {(x, y) I x,
Y E G} ,
onto the set of (m+l) integers, {O,l, ••• ,m}, defined by
(3.2)
cp(x,y) = i if and only if x-1y
G , i = 0, 1, ... , m.
i
Note that cp(x,y) is not necessarily symnetric with respect to x and y.
€
We
shall say that the element x is of the ith relation to y with respect to the
partition (3.1), if x and y satisfy the condition (3.2).
The mapping defined above has the following properties:
(a) cp(x,y) is defined for all (x,y) of G X G,
(b) ep(x,y) = 0 if and only if x = y,
10
~(ax,ay)
= ~(x,y),
(c)
for any element a of G,
(d)
if G is abelian, then~(x,y) =~(y-l,x-l), and
(e)
for any given a
€
G, the number of elements y such that
is equal to the cardinality of Gi , /G i /,
~(a,y)
=i
i = 0, 1, .•. , m.
We shall say that ~(x,y) is symmetric, if and only if ~(x,y) = ~(y,x)
for all (x,y) of G X G.
We then have the following
3.1. A necessary and sufficient condition for the mapping
LEMMA
(3.2), to be symmetric is that
defined by
-1
(3.3)
Gi
PROCIF.
of G •
i
~(x,y),
= Gi , i = 0, 1, ••• , m.
Suppose that
~(x,y)
is symmetric, and let c be any given element
Then, there exists at least one (x,y) of G XG such that x-ly = c,
and, of course, ~(x,y) = i.
Since, by assumption, ~(x,y) =~(y,x) = i, it holds that y-lx = (x-ly)-l=
c- l belongs to G.•
~
-1
This means that G. c G..
~
=
~
Then, comparing the cardinali-
-1
ties of Gi and Gi ' we get (3.3), which proves the necessity.
It is quite easy to prove the sufficiency.
If every element of G is such that a2 = 1, then for any partition of G
in the form
(3.1) satisfies the condition (3.3), and therefore
~(x,y)
is
symmetric.
DEFINITION
3.1. The mapping cp (x, y), given by (3.2), is said to define
an m-class association scheme on the v elements of G, if it holds that
(i)
(ii)
~(x,y)
is symmetric, and
i
for any given (x,y) of G X G such that ~(x,y) = i, the number Pjk'
of such elements z that cp(x,z)
=j
and at the same time ~(y,z)
=k
is
11
independent of the initial (x,y), for any i, j and k; i,j,k : O,l, .•• ,m.
Once we got an m-c1ass association scheme on the v elements of G, it is
considered as an m-class association scheme in the usual sense defined on v
objects, regardless of the group operation among them.
i
It is well-known that the parameters, Pjk' satisfy the following conditions:
(a)
= Poii ,
Putting n i
i
= O,l, ••. ,m,
m
1:
'=0
J.-
(3.4)
o
(b)
Pjk
= nj
(c)
P~k
=
(d)
P~k
ni
m
(e)
1:
j=O
n.
= v.
J.
i
8 jk and P jO = 6 ij , i,j,k
P~j'
= O,l, ••• ,m.
i,j,k, = O,l, ••• ,m.
= prk
nj
= P~j ~,
i,j,k
= O,l, ••• ,m.
P~k =~, independently of i; i,k = O,l, ••• ,m.
The following is the main theorem of this section.
THEOREM 3.1.
~(x,y)
A necessary and sufficient condition for the mapping
given by (3.2) to define an m-c1ass association scheme on the v
elements of G is given by a set of conditions
(i)
(3.5)
{ (11)
-1
Gi = Gi ' i
= 0,
1, ••• , m,
m J.•
GjGk = .1: P jk Gi , j,k = 0, 1, ••• , m,
J.=O
for some set of integers
PROOF. (Necessity):
{P~k} (i,j,k
= 0,1, ••• ,m).
SUpPOse that the two conditions of Definition 3.1
are satisfied.
Then, the parameters Pijk in the definition satisfy the conditions (a)
12
through (e) of
(3.4).
First, the condition (i) of
(3.5) follows from (i)
in the definition by
Lemma 3.1.
To prove (ii) of (3. 5) , let c be any given element of G..
~
Then, there
exists at least one (a, b) of G X G such that
a -~
for which, of course,
= c,
~(a,b) =
i.
By the condition (ii) of Definition 3.1,
there exist exactly P~k such z that ~(a,z) =
j
and at the same time ~(b,z)
=~
that is,
a
-1
z
E
Gj and
Since (a-lz)(z-~) = a-~ = c
x
E
Gj , y
E
E
-~
ZOE
Gk .
G., there exist at least P~k such (x,y) that
J
~
Gk and xy = c, which implies that
m •
Gj Gk ~ i~P~k Gi , (j,k = O,l, ••• ,m).
The cardinalities of the collections on both sides of this implication
relation are
and
where we have used the fact that n.~
= /G./
~
and the conditions (d) and (e) of
(3.4). Hence, it follows from (3.6) that·
which proves (ii) of
(3.5).
This completes the proof of necessity.
(Sufficiency):
The condition (i) of Definition 3.1 follows from the
condition (i) of (3.5) by Lemma 3.1.
13
We shall prove (ii) in the definition.
Let (x,y) be any given element of G X G satisfying the condition
~(x,y) =
i.
Then, it is easy to see that an element z of G satisfies the
conditions ~(x,z)
=j
and qJ(y,z) = k simultaneously if and only if
Z E
xG j
n yGk.
Thus, the number of such z that ~(x,z)
=j
and ~(y,z)
=k
for any given (x,y)
satisfying ~ (x, y) = i is equal to the cardinality of the subset xG j
n yGk'
which is, by Lemna 2.2, identical with the multiplicity of the element x-ly
-1
in GjGk, provided that x y
of
E
Gi • This number is seen, by the condition (ii)
(3.5), to be equal to P~k independently of (x,y)
This proves the sufficiency.
Thus the proof of the theorem is completed,
It should be remarked that the above theorem is a multiplicative group
•
version of the usual module theorem (for example, [5 J), when G is an abelian
group.
Since the group G of the theorem is not necessarily abelian, the
theorem is regarded as an extension of the module theorem.
It is also noted that the condition (ii) of
(3.5) does not require for
the parameters p ~k to satisfy the conditions given by
says that P~k in (ii) or
(3.4). The theorem
(3.5) automatically satisfy the conditions (3.4).
It is not so difficult to check that P~k in (ii) of
(3.5) satisfy (3 4) as
will be shown in the following
LEMMA 3.2.
For the parameters
(a) through (e) of
PROaF.
~~k}
(i,j,k = 0,1, .•• ,m), the conditions
(3.4) are all satisfied.
The proof of this lemma will help to understand the structure of
the partition (3.1) satisfying the conditions
abelian.
(3.5), expecially when G is not
14
(a):
(3.5) means that P~i is the multiplicity of
The condition (ii) of
the unit element 1 in the collection G~.
~
P?i
~
= /G./,
i
~
(b):
= O,l, ••• ,m.
From (ii) of
i
which implies that PjO
Then, (a) of
(3.5), we have
(3.4) follows from (3.1).
(3.5) it follows that
= 8ij ,
Again from (ii) of
Thus, by (i) of
where 8ij designates the Kronecker delta.
(3.5) it follows that P~k is the multiplicity of the
unit element 1 in the collection GjGk, which is equal to the cardinality of
Gj
n Gk •
Since
o =n 8 •
j jk
we have Pjk
(c):
(3.4), it suffices to show that
To prove (c) of
(3.7)
Since Gj
GjGk
= GkG j ,
j,k
= Gj-1
and Gk
= Gk-1'
(GjGk)-l = GkG j
But, from (i) and (ii) of
(GjGk )
= O,l, ••• ,m.
it holds that
•
(3.5) it follows that
m i
-1
=(1: P'kG~)
i~ J .
-1
m
= i~
1:
i
P
-1
G.
jk ~
m
= i~
1:
i
PJ.kG i
= Gj Gk "
Hence we have (3.7).
As we have used above, it holds that
m
1: A.G.
'-J"\
~-v
"'1 ~
=
m
1: IJ,.G.
'-J"\
~=v
~ ~
for any non-negative integers
(d):
(ii) of
iff
~
A.
= I£~,
"'i.
i
= 0,1,". ",m,
and J.Li •
(3.5) shows that the total number of such (x,y) that
•
15
X
€
Gj' Y
€
Gk and xy
€
Gi is equal to the cardinality of the collection
P~k Gi , i.e., to P~k ni · Note that this is equal to the number of elements
(x,y,z) of G X G X G which satisfy the equation xy = z,
G and
k
z € Gi , and for each of such (x,y,z), we havean element (z,y-l,x) of G X G X G
-1
-1
which satisfies the conditions zy = x, Z € Gi , Y € Gk and x € Gj .
X
€
Gj , Y
€
Thus, it holds that
i
n
P·k
J ~. <
=
and this relation is convertible, from which we have
i
Pjkni
j
= Pikn j .
Analogously, we can prove that (d) holds true.
(e):
On
Sunnning up both sides of (ii) of (3.5) with respect to j, we get
m
m
.
GGk = 1: (1: P~jk)Gi'
i=O j=O
the other hand, it follows from Lemma 2.3 that
m
1: G••
i=O
~
Hence, by (3.8), we have (e) of (3.4).
This completes the proof of the lemma.
Theorem 3.1 presents the conditions under which the association relation
given by (3.2) becomes an m-class association in the usual sense:
We can
construct an m-class association scheme by finding out a finite group G with
a partition in the form (3.1) satifying the conditions of
(3.5).
The corresponding association matrices are then obtained by the following
proC.e dure.
Numbering the v elements of G in any way, let us put
G
= (aO = 1,
a l ,···, a v _l ).
16
There are two ways of representing the group G as a regular permutation group,
that is, the left regular representation
... ,
<:~. 9)
... ,
~V-l )
,
v-I
and the right regular permutation
sex)
(3.10)
For anyone of these representations, the left regular representation
say, let p(x) be the vxv permutation matrix corresponding to the permutation
.(x).
Then, the set of v permutation matrices,
n.(p(x) IXEG),
forms a group of order v isomorphic to G.
Suppose that G has a partition in the form (3.1) satisfying the two
conditions of (3.5).
For this partition, let us put
p(x),
i
= 0,1, ••• ,m.
Then, A = Iv, and it is easy to see that the (m+1) matrices satisfy all the
o
conditions of Lemma 1.1, (1.5).
Thus it is the set of association matrices
of an m-class association scheme with the parameters (1.4).
4.
Construction of associa.tion schemes from finite groups (1)
As we have seen in the preceding section, it is first necessary to find
out a suitable partition of a. given finite group, from which we construct an
association scheme.
•
17
In the present section, we shall construct the so-called extended group
divisible association scheme.
Let G be a finite group, and suppose that there exists a chain of m
sUbgroups of G:
(4.1)
where Hi _1 is a proper subgroup of Hi' i = 1, .•• ,m, and H = (1).
O
Let us put
(4.2)
/ Hi / = hi' i = O,1, ••• ,m-1, and / ~ / = /G/ = hm' and there-
fore hO = 1 and hm = v.
J 1 , ••• ,Jm, such that
•
hJ..
Then, there exists a set of positive integers,
= h.J.-1J.,
J.
i = 1, ••• ,
m.
Thus,
(4.4)
v = hm
= J 1 •.. Jm.
Let us denote the complementary set of H.J.-1 relative to H.J. by H.J.- 1,
i = 1, ••• ,m.
Then,
H1 =HO +HO'
~ =
H:t
~-1
= Hm_2 + Hm_ ,
2
G
Putting GO = HO' G1
a partition
= Hm-1
= He'
+
H1,
+ Hm- 1·
G2 =
H:t' .•. ,
Gi = Hi _1,· •• ,Gm = Hm_1, we have
18
(4.6)
It is then clear that
Gi
(4.7)
= Gi-1,
i
= O,l, ••• ,m,
and
It is noted that the relations (4.5) can be rewritten as
(4.9)
Hi
= GO
+ Gl + ••• + Gi ,
i
= O,l, •.• ,m,
and conversely
Gi = Hi - Hi _l , i = 0,1, ••• ,m,
where we have put H_l = fJ (empty).
It follows from (4.10) that, for j < k,
(4.10)
GjGk
(4.11)
= (Hj-Hj_l)(Hk-Hk_l)
= Hj~-Hj~_l-Hj_llit+Hj_l~_l
= hjHk-hj~_1-hj_1Hk+hj_1Hk_l
= (hj-hj_1)(lit-Hk_l) = (hj-h j _l ) Gk ,
and for j = k,
G2j = (Hj-Hj _l ) 2
=
(4.12)
~
-
Hj_1Hj-HjHj_l+~_1
= hjH j - 2h j _1Hj + hj_1Hj _l
= (hj-hj_l)H j - hj_l(Hj-Hj_l)
=(hj-hj_l)(GO+Gl+••• +Gj_l) + (h j -2h j _l )G j ,
for j,k
= l, ••• ,m.
Thus, it holds that
•
19
(4.13)
j -;; k, j ,k
= 0, 1, .•• , m.
where
o
Poo
(4.14)
= 1,
i
POk
= 8ik,
i,k
= 1, ..• ,m,
P~k = ~j = 8ik(hj-hj_l) = 8ikJoJ1···Jj_1(lj-1),
j,k
_ /h j -h j _1
= 10'1••• l j _1 (l j -1),
< k,
= 1, ••• ,m,
0 -;; i < j,
-J h j -2hj _1 = Jo'1···Jj_l(lj-2),
l
j
i = j,
0, otherwise.
The m-class association scheme thus constructed is the so-called extended
group divisible association scheme.
In the case m
divisible type:
= 2,
this gives a 2-c1ass association scheme of the group
Putting hO
=' 0 = 1,
h1
= ' 1 = n,
h2
= mn, ' 2 = m,
as in
the usual notation, it follows from (4.14) that
(4.15)
v = mn, nl = n-l, n2 = n(m-1),
111
1
Pl1 = n-2, P12 = P21 = 0, P22 = n(m-1)
2
2
2
2
Pll = 0, P12 = P21 = n-l, P22 = n(m-2),
Which gives the parameters of a 2-c1ass association scheme of the group
divisible type.
The above argument says that if a partition of a given group G
(4.16)
is such that the m+l subsets of G defined by
(4.17)
form a chain of subgroups of G:
(4.18)
20
then, the partition (4.16) satisfies the two conditions of (3.5) and the resu1ting association scheme is of the extended group divisible type.
if we put, "0
= 1,
M:>reover,
and
(4.19)
then the parameters of the association are given by (4.14).
In the last half of this section, we shall prove that if the Partition
(4.16) gives an m-class association scheme of the extended group divisible
type with the parameters (4.14), assuming that
= hi'
/Hi /
= 0,1, .•• , m,
i
then, the m + 1 subsets given by (4.17) form a chain of subgroups of G.
To prove this, it suffices, by Lemma 2.3 (ii), to show that
(4.20)
From (4.13) and (4.14) it follows that
~=
(Go
~
i
= 1:
j=O
i
= 1:
+ G1 + ••• + Gi )
i
G2j
m
2
+ 2
1:
s
1: p .. G +2
j=O s=O
GjGk
j <k
JJ
S
i
m
t
t
<
j
s=O
k
s
Pjk Gs '
for i = 1, ••• ,m-1.
Here we have
m
1: pS G
j=O s=o
+
jj s
i
o
1:
t pS G
j=l s=O jj s
j-1
= G + 1: (1:
and also
m
i
i
1:
j=l s=O
(h -h
j
j
_
1
)G + (h -2h
S
j
j
_
1
)G j
),
21
=
=
i
k-l
Gk 1:
j=O
k=l
i
1:
hk_lGk
k=l
(nj-n j _l )
1:
Hence, it f'ollows that
i i i
~ = GO + 1: (hj-hj_l)(GO+Gl+•• ·+Gj_l) + 1: (hj -2h j _l )Gj + 2 1: ~_lGk
j=l
j=l
k=l
i
= h. 1: Gj = h.Hi ,
J.j=O
J.
which proves (4.20~ and theref'ore H.J. is a subgroup of' G, i = l, ••. ,m.
5.
Construction of' association schemes f'rom f'inite groups (2)
In the present section, we shall consider a series of' association schemes
which are constructed from direct product of' a f'inite group.
The resulting
schemes are seen to be of' an extended L type.
2
Let H be a group of' order n, and let G be the m-f'old firect product of' H:
(5.1)
G = H X H X ••• X H,
whose elements being denoted by (al, ••• ,am), a i
then clear that
/G/
= n
m
€
H,
i = l, ••• ,m.
,
and the unit element of' G is (l, ••• ,l), 1 being the unit
Now, let us put
It is
element of' H.
22
Go
=
{(l, .•. ,l»),
Gl
=
«aI' 1, ••• ,1), ••• , (I, ... ,l,am) I a i
€
H, a i
f 1 ),
.............
G = the set of all elements of G whose i components are not
i
the unit element and the rest (m-i) components are the
unit element,
............
It is obvious that
(5.4)
and G is partitioned in the form
It is also clear that
-1
G.
(5.6)
J.
= G.,
J.
i
= O,l, ... ,m.
We shall check that the partition (5.5) satisfies the condition (ii) of
(3.5) for some set of integers,
i
P'k'
J'
Let us assume that j $ k and let Z
= (Zl""'Zm) belong to Gi . Let,
for z, N(z) and I(z) be the sets of all positions on which the component of
z takes non-unity element and the unit element of H, respectively.
/N(z)/
=i
and /I(z)/
= m-i,
and hence /N(z)/ +/I(z)/
Then,
= m.
Suppose that the equality
z
holds for a pair (x,y), x
=x y
= (Xl, ••• ,xm)
N(z) can be divided into three parts:
E
Gj and Y
= (Yl"",Ym)
E
Gk .
Then,
23
N(z) = NI(Z)
where
N2(Z)
+
+
N3 (Z),
NI(Z) is the set of positions on which z, x and y take non-unity
elements (not all the same) of H, N (Z) the set of positions on which Z and
2
x take non-unity and y takes the unit element, and finally N (Z) the set of
3
positions on which Z and y take non-unity and x takes the unit element.
I(z) can be divided into two parts:
Likewise,
I(z) = II(Z)
where
and
+
12(z),
II(z) is the set of positions on which both x and y take non-unity
z takes the unit element, and 12(z) is the set of positions on which z,
x and y take the unit element of H.
Let
Then, it follows that
(Z)/ = m-i-t.
2
The relations which must be satisfied by these numbers are
/N
3
(Z)/
= i-s-u,
and /I
o~
s, u, t
s +
U
~
m,
+ t = j,
s + (i - s - u) + t
= k,
from the last t'WO of which we have
(5.8)
{
Since s
~
0 and 0
~
t
=k
s
= (i
s +u
~
- i + u,
+ j - k) - 2u.
i, we have 0
~
i + j - k - u
j-k ~ u ~ (i + j - k)/2.
On the other hand, m - i
~
t
~
0 implies that
~
i, or equivalently
24
m- i
~
k - i +u
Thus, it is seen from
~
0, or i - k
~
u
~
m - k.
(5.9) and (5.10) that the number u should satisfy
the following inequalities
where
a(i,j,k)
{
= max(O,i-k),
A('
j k) = rr.un
. (.~,m-,
k i+j-k)
~ ~"
2
•
Now, we can see that, for any fixed z
with x
€
G, y
j
€
G , such that xy
k
"ei!
(5.13)
1: s. u.
=z
€
G., the number of pairs {x,y},
~
is given by
s
t
),• (m-i
t )(n-2) (n-l) ,
~-s-u
where the summation is taken over all integers, s, u, and t, satisfying the
conditions
(5.7). This can be rewritten as
i _ ts(i,j,k)
Pjk ~=a(i,~,k)
i
where p.,
J.K.
=0
if a(i,j,k)
i!
m-i
i+j-k-2u
~i+j-k-2u)!U!(k-j+U)!(k-i+u)(n-2)
k-i+u
(n-l)
~(i,j,k).
i
This gives us the values of Pjk for j < k.
·
S~nce
Pijk
i , t he above
= Pkj
exhausts all the cases.
Hence, for the group G given by
(5.1), the partition (5.5) gives us an
m-class association scheme whose parameters being v
O,l, ••• ,m, given by
type.
= nm and P~k'
i,j,k
=
(5.13). This is regarded as an extension of the usual L2
In fact, in the special case m
= 2,
this gives us the 2-class associ-
ation scheme of L type, whose parameters being calculated by
2
follows:
(5.13) as
25
o
Poo
0
0
0
= PlO = POl = P20
0
0
= P02 = P12
0
= P2l = 0,
o = 2(n-1), P0 = (n-l) 2, P1 = POl
1
= 1,
22
lO
Pl1
1
1
P102 -_ P20
= 0, P11l = ( n-2,) P112 = P2l
= ( n-l,)
1
222
P22 = (n-1)(n-2), Poo = POl = PlO = 0
22222
P20 = P02 = 1, Pll = 2, P12 = P21 = 2(n-2),
P~2
= (n_2)2.
In this case, the following method of calculation is more convenient:
We
shall write as (H, H) instead of H X H, 1. e., in general, Let
(H,K) = ((x,y)
Ix
E
H, Y
E
K).
Then, in the case m=2, the partition (5.5) becomes
where G
= (H,H),
GO
= (Ho,HO)'
G1
= (~,Ho)+(HO'~)
and G2
= (~,~),
with
HO = {1} and ~ = H - HO'
Bu using the relation
~
we
= (n-1)HO + (n-2)H ,
1
can obtain the following:
2
2
2
Gl = (~,HO) + 2(~,HO)(HO,Hl) + (HO,H1 )
= (~,~)
+ 2(H:tHO,HOHl ) +
(~,Hi)
= «n-1)HO+
(n-2)H1, HO) + 2{Hl,H1)+{Ho,(n-l)Ho+(n-2)Hl)
= 2{n-1) GO + (n-2) G1 + 2 G2,
,Hi)
G1 G2 = (Hi,H1 ) + (H1
= ({n-1)H +{n-2)H ,H ) + (H ,(n-l)H +{n-2)H )
1
o
l
o
1 1
= {n-l)G1 + 2(n-2)G ,
2
and finally
26
G~ = (~,~)
= «n-l)Ho+(n-2)Hl , (n-l)Ho + (n-2)Hl )
= (n-l) 2
2
GO + (n-l)(n-2) Gl + (n-2) G2 •
These equalities give the parameters (5.15).
It should be remarked that, for the general case,
2
Gl
= m(n-l)Go +
(n-2) Gl + 2 G2 '
and if we define the association in such a way that two elements, x, y, of
G are 1st associates if and only if x- l y € G and 2nd associates if and
l
l
only if x-ly € G , then we have a 2-class association shceme on the m(n-l)
2
elements of G , induced from the original association scheme, and this turns
l
out to be a 2-class association scheme of group divisible type.
6.
Construction of association schemes from finite groups (3)
In the present section, we shall construct an association scheme of extended T type, that is, an m-class association scheme from which a 2-class
2
association scheme of triangular type is induced.
Let G be a finite group generated by the m generators
(6.1)
subject to the generating relations
(6.2)
Let us define
. .G
o
Gl
= 1,
= ~, ••• ,am
...........
..........
27
Then, these subsets of G form a partition of G:
(6.4)
for which it is evident that
and
(6.6)
= Gi-1,
Gi
i
= O,l, ••• ,m.
The group thus defined is isomorphic to the additive group V , consisting
m
of all binary m-vectors whose components are from the Galois field GF(2).
Thus, if we denote Vm by G, then Gi given by (6.3) are
Go = ((0, ••• ,0»,
G
l
= ((1,0, ••• ,0), ••• , (0, ••• ,0,1))
.......................
(6.7)
....................
Gm = (1, ••• ,1».
Now, let z be any given element of G .
i
pairs x, y such that x
(6.8)
x +Y
E
Gj' Y
E
We want to get the numbers of
Gk and
= z.
It is seen that this relation holds true when and only when
(i)
among the i positions on which z takes 1, J positions are occupied
by 1 of x and by
y and by
° of x,
° of y, and the rest i
- J positions are occupied by 1 of
and (ii) among the m - i positions on which z takes O,j -J
positions are occupied by 1 of both x and y.
Then, the number J must satisfy the following relations.
28
0 $ J $ i,
{
i + j - 2$
= k,
from which it follows that
(6.10)
Hence, the relation
(6.8) holds for some pairs x, y only when the inequalities
(6.10) are satisfied, and the number of such pairs is given by
(6.11)
Let us put
(6.12)
i
Pjk
=
i+~-kI(J:~~i),
I
if' i+k-j ::: 0, and both
i+~-k and J+~-i
are non-negative integers,
0, otherwise.
Then, we have
(6.13)
This condition, together with (6.6), gives us an m-class association scheme
m
wi th parameters v = 2 and p ~k given by (6.12).
It is easy to see that
(6.14)
[
G = GO + G2 + G4 + ••• + G2i +••• + G2 [m/2]'
m-l
] being the usual Gauss symbol, is a SUbgroup of order 2
• From (6.12)
and (6.13) it follows that
2
s (2i) (m-2i)
G2j = i:O i 2j-i G2i , j = O,l, ••• ,s,
(6.15)
r
G2j G2k
=
j+k
2i
m-2i
~
(i+j-k)(j+k-i) G2i, j,k=O,l, ••• ,s,
i=ma.x(O,j-k)
29
where we have put s = [m/2].
(I, ••• , 1), we have x-ty
= x-ty,
GJ.. Gj
(6.16)
Since, putting
z = 1-z,
1 being the vector
it also follows that
= Gm-J..
Gm-J., i,j
= O,l, ••• ,m.
The relations (6.15) show that the partition (6.14) gives us an s-class
association scheme with the parameters given by (6.15), which is defined on
-G.
In the final place, we shall examine the association relation on G
2
induced from the association schemes given above.
From (6.15) we see that
(6.17)
We shall say that two elements x, y of G2 are 1st associates or 2nd associates
according respectively as x -1y € G or as x -1y € G4 • Then, it is easy to
2
check that the resulting association scheme is of the usual triangular type,
whose parameters are
v
(6.18)
= (~),
n1
= 2(n-2),
n2
= (n;2),
1
1
1
1
(n-3)
P11 = n-2, P12 = P21 = n-3, P22 = 2 '
2
2
2
2
n-4
Pll = 4, P12 = P21 = 2{n-4), P22 = ( 2 ).
7.
Construction of association schemes from finite groups (4)
Let G be an abelian group of order v, and suppose that the partition
satisfies the following conditions:
30
(i)
/G i / = n,
i
= 1, ... ,m,
GO + Gi is a subgroup of G, i = l, ••• ,m, and
(1ii) for any given i and j (iFj), there exists n integers
(ii)
(7.2)
k(i,j) u , u = l, ••• ,n, such that
GiG.
J
=
n
~
u=l
Gk(i j) .
' u
Then, it is clear that v = nm + 1, and (n+l)2 divides v.
We first note that a sufficient condition for the condition (iii) in
(7.2) to be satisfied under the other conditions is that each GO+G i has no
proper subgroup other than GO'
This is shown easily as follows:
Since
(GO+Gi)(GO+G j ) is a subgroup of G, containing Gi+G j , it is true that, for
any k (ri,j), the set
(GO+Gi)(GO+G j ) n (GO+Gk ) = GO + (GiG j n Gk )
is a subgroup of G, from which it follows that
This implies the condition (iii).
Since
it is seen that, under the conditions given by (7.2), the subset of G,
n
(7.3)
H(i,j) = GO + Gi + Gj + ~ Gk(i j)
u=l
'u
forms a subgroup of G, for any i and j, i
r j.
It is also seen easily that
H(i,j) is a minimal subgroup of G, which contains Gi+G j , in the sense that
there is no proper subgroup of H(i,j) which contains Gi+G j •
The above condition is equivalent to the condition (iii) of (7.2), as
will be stated in the following
31
LEMMA 7.1.
Suppose, for the partition (7.1) of G, the conditions (i)
and (ii) of (7.2) are satisfied.
Then, for the condition (iii) of (7.2) it
is necessary and sufficient that for any given i and j,
a set of n integers k(i,j) , u
u
i
f
j,
there exist
= l, ••• ,n, such that the subset H(i,j) defined
by (7.3) forms a subgroup of G.
In fact, if H(i,j) forms a subgroup of G, GiG j should be contained in
H(i,j), but GiG j and GiiG j are disjoint.
Hence, by comparing the cardinali-
ties of G.G. and H(i,j), we have the sufficiency of the lemma.
~
J
We can show also that
LEMMA 7.2.
Under the same situation as in the above lemma, suppose that
for a set of (n-+e)Gi's (G i ), j = 1, ••• ,n+2,
j
Go + G.
~l
forms a subgroup of G.
Then, for any given jl and j2 distinct, it holds that
G.
J.
+ ••• + Gi
note
G.
j
1
~j
=
2
The proof of this lemma is quite similar to that of the preceding lemma
and will be omitted.
It is now evident that, under the conditions of (7.2), for any given i
and
j
distinct there exists a unique set of n Gi's satisfying the equality
given in (iii) of (7.2).
):J
Simplifying the notation of suffix, let them be
(i, j) = (G., G., Gk ' ••• , Gk ) •
J.
J
1
n
Lemmas 7.1 and 7.2 say then that this set of Gi's is determined uniquely by
giving any pair of G.J. 's in the set.
Let G be outside of the set ):J(i,j) given by (7.6).
u
Then, for any Gk
32
in
~(i,j)
it is easy to see that
~(i,j)
n U(k,u)
= Gk ,
and
~(i,j)'s
Hence, the family of all
mutually distinct is a configuration
satisfying the condition that
~(i,j)n ~(s,t)
(7.9)
¢,
=
or = Gk for some k, that is, any pair of such
1f(i,j)'s do not possess more than one Gk's in common.
We shall prove the following
LE~
7.3.
Under the conditions of (7.2), it holds that
(7.10)
i
PROOF.
{
<
=
GiG.
1:
2
J
j
n{m-l)
First it is noted that
G~ = nGO +
(n-1.)Gi' i = 1, ••• , m,
G = vG = (nm+l) G.
Now,
Gi )
2
m
= (GO + ~ G.)
. 1
J.=
m
2
J.
m
= G~ + aGO ~ G. + (1: G.)2
i=l J.
i=l J.
m
m
2
= GO + 2 1: Gi + ~ G. + 2 1: G.Gj
i=l
i=l J.
i<j J.
= GO + 2
m
m
1: G. + ~ (nG +(n-l)G.) + 2
i=l
Hence we have the identity
J.
i=l
o
J.
~
i<j
G.G j •
J.
33
m
vG = GO + 2
~
m
G.
'1~
~=
+ nmGO +(n-1)
~
G. + 2
'1~
~
G.G ,
';<j~ j
~=.
from which it follows that
m
2 ~ G.G j = n(m-1) ~ G.,
i<j ~
i=l ~
or equivalently (7.10).
This proves the lemma.
The identity (7.10) means that, for any fixed k, G is contained in
k
exactly n(m-l)/2 of all (~) U(i,j)'s, provided that n(m-l)/2 is a positive
integer.
~
~(i,j)'s
which contain Gk • Then, since
for each ~(i,j) containing Gk there are (n;l) »(i',j') which coincide with
If(i,j), it should be true that
Let
be the number of distinct
n(m-l) _ . (n+l)
2
-f.1
2
'
from which we have
~=
m-l
n+r '
and this value does not depend on any given Gk "
Thus we have seen that each Gk is contained in exactly (m-l)/(n+l)
distinct »(i,j) 'so
It is also clear that any given pair Gk and Gu are contained in exactly one ~ (i, j) belonging to m, the family of all distinct
U(i,j)'s.
Thus we can state the following
THEOREM 7.1.
Suppose that the conditions (i) through (iii) of (7.2) are
satisfied for a partition (7.1) of an abelian group G of order nm+l, and that
(a){n+1)2 divides nm+1, (b) n(m-1) is even and (c) (n+l)(n+2) divides m(m-l).
34
Then, the family CB is a (v, k, A) -configuration ([5)), where
v
= m,
k
= n+2,
A
= 1,
and the other parameters are given by
b
m(m-1)
= (n+1)(n+2) ,
m-1
n+1
r =-
It should be remarked that the configuration of this theorem is a
balanced incomplete block design with the parameters given above.
It is
also noted that the condition (a) in the theorem implies that n+1 divides
m-l.
COROLLARY 7.1.
If a partition (7.1) satisfying the conditions of (7.2)
actually exists for some abelian group, and the conditions (a), (b) and (c)
of the above theorem, then a BIB design with the parameters given by (7,13)
and (7.14) can be constructed.
THEOREM 7.2.
If there exists an abelian group G with the partition
(7.1) satisfying the conditions of (7.2), then there can be found an m-c1ass
association scheme, whose parameters being given by
v
= nm+1,
n.~
= n,
i
= 1, ••• ,m,
1, if Gi,G j and Gk belong to the same class
in CB, and j
f:
k,
n-1, if i = j = k,
0, otherwise,
PROOF.
By the preceding theorem, we can see that for any given j and k
distinct there exists exactly one class,
H(j,k)
(i,j,k = 1, ••• ,m).
= ):J(t,u).
~(t,u)
say, in CB such that
35
Hence the theorem follows from Lemma 7.2.
In the case n
= s-l
and v
= st,
where s is any prime power and t is any
positive integer greater than 2, and hence m = (st-1)/(S-1), the association
scheme given in the above theoren is the geometrical association scheme
given by Y. Fujii [6], basing upon the structure of finite projective geometty.
We shall give an algebraic derivation of this scheme.
Let s be a power of a prime.
Then, there exists a finite field
isomorphic to the Galois field GF(s).
Let us consider the vector space over GF(s):
(7.16)
the cardinality of which is given by
s
t
t
= • E=0
~-
t
.
(.)(S-l)~ •
~
Thus, we have
(7.18)
t
s -1
s-l
m=-
It is evident that Vet) is an abelian group of order v
the vector addition, whose unit is 0
= st
with respect to
= (0, ••• ,0).
Let us put B = {Q}.
o
The expansion of m in the form (7.18) suggests us the following way of
partitioning Vet).
Let Ui be the set of all vectors in Vet)' i components of which are
occupied by 1 and the rest t-i by o.
It is then clear that
(7.19)
For any given vector
e = (0, ••• ,0,1,0, ••• ,0,1,0, ••• ,0,1,0, ••• ,0,1,0, ••• ,0),
36
let us put
we,:) = (0, ••• ,0,1,0, ••• ,o,~,o, •.• ,0,(12,0, ... ,0,ai _1,0, ... ,0) I
a j E GF(s),
j 1'J, j = 1, ••• ,i-1»),
(l
and
W(Ui )
=
~
w(~),
i =
1, ••• ,t.
~ E U
i
Then it is clear that W(U ) contains (~)(B_1)i-1 distinct vectors, i=l, ..• ,t.
i
Now, put
(7.20)
Then, the cardinality of this set is equal to m.
Let us number just for
convenience the whole vectors in Win any way but once for each:
(7.21)
W
= (a.., ••• ,a-m ).
'~l.
For each vector a
of W, let us define
-u
Vue (~I ~EGF(S), l#O), u=l, ••• ,m.
Writing Vet) simp4r as V, we then have the partition
for which it is seen that
/Vi / = s-l,
i
= 1, ••• ,m,
and for each i, Vo +Vi forms a subgroup of V, i
= 1, ••• , m.
It is also shown that the condition (iii) of (7.2) is satisfied in the
present case as follows:
are linearly independent.
First it is easy to see that any two vectors in W
For any given
j
and k distinct, therefore,
37
has (S_1)2 elements.
Putting 7
=
1:
7e.GF(s)
71:°
= iliA, the above can be rewritten as
(A 5:j+~)
I
A
E
GF(s), A 1: OJ.
For any given 7 (Fe), there exists exactly one Vu ' to which ~j+~ belongs,
and for this Vu ' it holds that
It is also seen that if 7 and 7 1 are distinct non-zero elements of GF(s),
then
~j+~
and
!j+rl~
do not belong to the same Vu '
Thus, we have
for some sets, Vu , ••• , Vu _
s l
l
Thus the partition (7.23) meets the three conditions of (7.2), and
therefore we have an m-class association scheme equivalent to that of geometrical type.
38
REFERENCES
[1]
I.M. Chakravarti and W.C. Blackwelder, "On some composition and extension
methods in the construction of block designs from association matrices,"
(Read at Symposium on Combinatorial Mathematics, 1967, at Chapel Hill).
[2]
H.B. Mann (1964), "Balanced incomplete block designs and Abelian difference sets", nlinois Jour. Math., 8, 252-261.
[3]
W.A. Thompson, Jr. (1958), "A note on PBIB design matrices", Ann. Math.
Statist., 29, 919-922.
[4]
M. Masu:yama, Calculus of blocks, Lecture note, 1965-1966, Dept. Statist.,
U.N.C.
[5]
H.J. Ryser (1963), Combinatorial Mathematics,
The Carus Math. M:>nographs,
14.
[6]
Y. Fujii (1967), "Geometrical association schemes and fractional factorial
designs", Jour. Scie. Hiroshima Univ., Japan, Ser.A-I, Vo1.31, 195-209.