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COMBINATORIAL
MATHEMATICS
YEA R
FebmaPy 1969 - June 19'10
ANTISYMMETRIC HADAMARD DIFFERENCE SETS
by
Paul Caution
Uni.vem ty of Tou louse
TouZoU8e~ PRance
Department of Statistics
University of North Carolina at Chapel Hill
Institute of Statistics M:baeo Series No. 600.30
June, 1970
ANTISYMMIn'RIC HADAMARD DIFFERENCE SETS
Paul Camien
UniveZOBi ty of TouZouse
Toulouse" Frtance
1. IN'1'RlDlrn ~,
In [1], we have studied binary relations
(X, U)
complete and anti-
symmetric t that is
V
i,j ~ X:
4U
(i.j) ~ U <a:> (j,t)
for which there exist a group of automorphisms verifying
1.
g
For every pair of ares,
~
G with
{i)g
(itj), (i',j')
The identity alone fixes two points.
3.
Let
A be the set of elements of
a
Ut
there exist a
it {j)g. j'.
c
2.
there exists an
~
~
A and two points
the unique element of
A with
G moving all the points,
i
and
j
such that
a
is
(i)a. j.
We proved the following theorem.
The group of automorphisms of
(X, U) verifies 1, 2, 3 iff
set of elements of a near-field K where there exists a subset
X is the
P c: K
verifying
This research ",as done ",hiZe the author ",as visiting the Department of
Statistics" University of North Carolina at Chapel. Bin" in May" 19'10" and
",as paPtiatly supported by the Altmy Research Office" Duztham" under Grant No.
DA-ARO-D-31-124-G910 and by the u.s. Ail' Force Office of Scientific Research
under Contract No. AFOSR-68-1406.
2
P n (-P)
V
and where
m
E
•
P\ {O, l} ,
{oJ
P u (-P)
•
the mapping defined by
x
morphism of the additive group
with
B-a
~
P.
and
K,
and
K.
1+
x-xm
is an auto-
U is the set of couples
Moreover, in the case where
(a,B)
X is finite, condition 3 may
be dropped, the near-field has an order congruent to 3 mod 4 and
P
is the
set of its squares.
In the finite case, such a group of automorphisms may be defined as a
sharply two homogeneous group (*), and a short proof of that special case may
be found in [2], where the theorem of Feit and Thompson is used.
We consider here a generalization of the finite case.
graph
(X,U)
has an abelian group of automorphism
which its vertices are labele.
G,
That is, the
with the elements of
and the set
{(o,x)lco,x)
E U}
is a difference set.
That condition is necessarily fulfilled when 1 and 2 are verified but
the question of knowing if it is sufficient has not been cleared out in this
paper except when
G is cyclic.
However, we prove that
G 18 a p-group, and our argument leads to a
proof of the fact that condition 1 added to the difference set condition implies
G 18 elementary abelian.
The case where
G is cyclic is thoroughly
cleared out.
(*) We are indebted to Professor Michel Jean of the College Royal de Quebec
who brought that fact to our attention.
3
2.
MIS'MaRIC
HArJAr.MD DIFFERENCE SETS.
2. 1. Defi n1 t1 on •
Let
G be an abelian group and
subset of
G such that the integer
=
Aa
(1)
does not
dep~nd
1{(x,Y)lx,y
on
E
feztence set whenever k
Now
(2)
D, X-Y
= all
aEG\{O}.
The common value of the
A
a
is denoted
= (v-l)/2,
D
will be called
V
XEG\{O},
XEO
A difference set
as a
D be a diffeztence set [3], that is a
where
k
A
and
= IDI
0
and
is a HadaJnar>d
&if-
v:: IGJ
antiaymmetric if
=>
-x(D.
D verifying all those conditions will be referred to
A.H.D.S.
2.2. Statement of the result.
THEOREM 2.1.
An
abel-ian g:roup fJJhich contains an antisymmetric Badt:znmtd
diffe:rence set is a p-gl'OUP
~th ONett
ctmgl'USnt to three modulo four.
it is aycZic, the difference set is the set of the quadNtic residues
When
Ott
set of the quadNtic non-residues of a pz-ime.
2.3. General properties of A.H.D.S.
2.3.1.
Properties of
101 •
jective,
(v-I) /2
I-DI
and
A.
and as the mapping
= (v-1)/2.
But by (2)
(3)
v, k
-D n D = 0
x l~ -x of
G into itself is in-
the
4
and since
-DuDc:G\{O}, I-DUDI ... v-I
(4)
-D
U D
implies
G\{O}.
...
On the other hand, the identity
(5)
k(k-I)"
A(v-l)
is verified for any difference set which implies in the present case
(6)
A ...
(k-I)/2
...
Hence the order of
3 mod 4.
G is congruent to
Properties of the incidence matrix of a A.H. configuration.
2.3.2.
Let
y€G,
(v-)/4.
be the incidence matrix of
A
that is
A'" (a
x,y
),
a
x,y
.. 1
<=I>
G
versus the subsets
x€y+D.
{y}+D,
(3) and (4) are equivalent
to
A
(7)
+
T
A
1
J,
..
D 1s a difference set,
and since
T
(8)
+
AA
nl + AJ~
...
where, as usually,
n" k-A.
We now have
PRoPERTY 2.1.
A be a
Let
v
by
v
matf1'i3: with entnes
foZl<Ming conditiOl18 are pairwise equivalent.
T
I.
AA
II.
A
III.
•
nl + U
+ AT +
2
A + A
1
..
...
J
n(J-l).
0
and 1.
The
5
I implies
(9)
=
JA
kJ,
and
II I follows immediately from I and II and II follows from I, I II, (9)
and (10), hence
(11)
(I & II)
I follows from II and III by simple calculation.
On the other hand,
PRoPERlY 2.2.
(I & III).
<~
For any difference set D,
condition III on its incidence
matPiz is equivalent to
V
V
(12)
I ...
ZE:D,
I {xl xeD, z-xE:D}
Z4D,
Z ~ OI{xlxE:D, Z-XE:D}/
n-l.
...
n
-D n D·".
Actually if
(13)
V
ZE:G:
CoROLLARY 2.1.
A" (a
L
XE:G
a
x,y
)
and
a
0
Z,x x,
2
A • (b
x,y
),
III is equivalent to
+ a Z, 0 .. n(l-6 z, 0)
..
b
Z,
O'
Condition (121 implies, for any difference set D,
-DuD = G\ {O}.
Remark.
n-1
(14)
.<
= k->'-l
If
= >..
V
D verifies conditions (12) it is a A.H.D.S. and
Also
k-l-(n-l)'" A and
ZE:D, l{xlxE:D, Z-X4D}I
=
A.
6
2.3.3.
Equations in the convolution algebra
over the ring
The matrix
(15)
a
x,y
and the transpose
(16)
T
ay,x =
G
11. of integers.
A· (a
..
llG of the abelian group
x,y
is a sum of permutation matrices, 1.e.
)
26
,
x-g,y
g~D
AT .. (aT )
y,x
is given by
L 6y x-g"
geD'
L 6y+g x
gED'
=
2
gE-D
6
•
y-g,x
Consequently, conditions I, II, II I may be considered as identities in the
~-elgebra generated by the set
{(oX-g,y)}gEG.
Now let us represent as in [3] by a polynomial
(17)
an element of ~G.
g
In the canonical basis
the linear mapping
multiplication by
{X }
8E
g
L
G of the lZ-mdule llG,
r
xM'
S ,:
G a X 1+
G a
8
gE
g
gE
g
8
X
is precisely
,
corresponding to the
We thus have the isomorphisms
(19)
lG ::::
ZZ({S}
8 ge
G)
::
11. [{ 6
x-g,y
}) •
Now let us use the following notations.
(20)
D(X- l )
..
L
gE-D
X8 ,
the matrix of
T(X)
7
Then the conditions I, II, III become by (19)
II.
III.
D(X)D(X- 1)
=
+ AT(X)
D(X) + D(X- l ) + 1 •
D2 (X) + D(X)
111 1 •
n
o
Here
X
•
T(X)
n(T(X)-l).
is denoted by
1,
and as usual,
n = k-A.
2.4. Proof of the main lemma leading to Theorem 2.1.
The group
G
may be considered as a module over
G by integers.
use multiplication of elements of
l..aft\ 2.4.
I f DcG is a A.H.D.S. oJ then for everry integer
tD •
lJhere
is the Jacobi
(-)
syTTiJo~
~
2,
(22)
9-1
(-1) 2
,
hence
(23)
and if
q •
lAJith
v
[4]
q a prime
(;H;> •
t
(t)D,
By the quadratic reciprocity law, since
q
7J. so that we shall
2,
(24)
From (23), (24) and from the property
v:: 3 mod 4,
one has for
8
(25)
where
t
a prime,
i
V
i,
all we have to prove is
q-l
(26)
for
qD"
(-1) 2 (;)D,
q • 2
odd, and for
q
2
v -1
qD
(27)
(-1) 8
..
pztime
Fop every
D.
q~
we shall consider the group
G*
of characters of
e
[a]" F*, where F
is the splitting field of X-l
q'"
q'"
being the exponent of G. (i.e., '" is the smallest integer
G into the group
over
F,
q
such that
e
q"'-l; 0 mod e.)
We shall consider, as in [5], the isomorphism
For brevity, we write
D
for the image of
X
D(X)
1.l
-1
of
F G onto
q
G*
F •
q'"
\D1der the character
X£G*.
For every non principal character
(28)
2
D
X
+
D
X
+ n ..
0,
X,
one has
Xf:G*\{O}.
So, for every non principal character
X,
D is one of the two roots of
X
the first member of (28).
Let us first suppose that the polynomial
does split over
(29)
1 -
F ,
q
4(~+1)
is a quadratic residue
that is
-v
..
mod 9,
9-1
(30)
(-1) 2
(v)
q
2
y +Y+n,
..
1,
or in other words
(n
is taken
mod
9)
9
if
q
is odd, and if
(31)
=
n
A+ 1
q. 2
=0
In that case, since
mod
2,
Dy ",
= k€F,
q
'v
or
v _ -1
mod
8.
every character sends
D(X)
in
F
q
and we know that this is possible if and only if (see for example [5] for
a proof)
since
lJ
-1
is an isomorphism, which means that
qD· D.
(33)
Now, on the contrary, if for q ~ 2, (_1)(q-l/2)(7!) = -1, and, for
q
2
2
q • 2, when (_1)(v -1/8) • 1, y +y+u is irreducible over F.
q
2
In that case, y +y+u divides XqV-l_l, v is even, and the Galois
group of
v
q -1;
over
F
qV
is given by all the even powers of
F 2
q
every odd power of
b
But since
(35)
q
q
D
X
F
q
automorphism of
modulo
F
qV
which
and, in particular,
c.
•
= -1,
b+c
defines an
2
Y +Y+n • (Y-b)(Y-c),
permutes the roots of
(34)
q
q
=
one has for every non principal character
X,
-1 - D
X
or, by II I ,
q
(36)
D •
X
But also
D
xO
(-D) •
X
= (-D)Xo
the Fourier transfonn
for the principal character
lJ -
XO'
G*
hence,
since
1 is an isomorphism onto F v'
q
(37)
C.Q.F.D.
10
2.5.
Consequences of LeIIIIB 2.4. for the group G.
PRoPERTY 2.5.1.
to the
origin~
If G' is a subset of G~
symmetric with respect
one has for D' = DnG'
-D' u D'
a
G'\{O}
-D' n D'
.. fJ.
-D' u D'
•
-(DnG') u (DnG')
..
(-DuD) n G'
..
(-DnG') n (DnG')
(38)
-D' n D'
•
(-DnG') u (DnG')
a
G'\{O}.
.. fJ.
.. -D n D n G'
In the hypothesis of Letrmrl 2. 4. ~
CoR:u..ARv 2.5.1.
v cannot have
any prime factor congruent to one modulo four.
We shall show that if
v
has a factor
r
four, there exists a non trivial subgroup
Gt
empty, which is impossible for a A.B.D.S.
D.
We split
G
a subgroup G"
G isomorphic with
r.
tG '
We write
integer
(41)
of
Now, if
( 40)
t
of G such that
=
Gt
l
is the exponent
(t,v)
c:
1,
tG n
e
of G.
l
It
divides
G'
e,
and
of
and
•
c
v = v,r ,
= 1 mod
r
contains a subgroup
G
=G
So
and we choose, as it is possible, an
verifying
t
is
DnG'
G into the canonical direct sum
where the order of
order
congruent to one modulo
VI'
t - -1 mod r,
and
(t,v)
= 1.
11
Now, let
(42)
tD'
D'
III
DnG'.
We have
= -D',
by (41).
But also
tD'
(43)
a
t(DnG')
tD n G'
=
and by Lemma 2.4,
tD'
(44)
...
...
Now, let us compute
Since
t
(.!)
(45)
v
since
-1
:: 1 mod vI'
...
(£.) (DnG')
V
..
(!.) •
v
(£. ) ... 1,
(.! )(.! )
vI r e
and
vI
...
(£. )
r
e
...
is a quadratic residue modulo
(!.)e
r
r.
...
1,
But (42), (44) and (45) im-
ply
(46)
-D'
lIlI
Dr,
which is impossible by (38).
CoRou..ARv 2.5.2.
If G contains a A.H.D.S.
a prime faato1> of v and q
>
v
a prime number.
(42)
By (26), we have to prove that
(48)
D~
~en
let p:: 3 mod 4 be
12
As in the lines following (39), we may find a subgroup
p,
contained in a cyclic subgroup
G"
of order e,
G'
of order
direct factor of
Then
(49)
=
qD'
=
q(DnG')
qD n G'.
Now, suppose (48) is not true.
(!l)
Case 1:
=
p
1-
Then
(50)
But
=
qD'
=
-D n G'
-(DnG')
•
q is a quadratic residue modulo
p,
and (50) implies that it generates in
even order.
-D'.
p:: 3
~(p)
mod 4,
by assumption,
of multiplicative group of
This is impossible.
Case 2:
(!l)
p
•
-1-
Then
qD t
(51)
=
=
D n Gt
Dt •
But this is impossible, since
p-l
(!l)
p
(52)
=
-1
<
:>
q 2
_
-1 mod p
and the last identity means that
p-l
q 2 Dt
(53)
CoRoLLARY
•
2.5.3.
-D'.
If G contains a A.H.D.S.
D,
v oonnot contain
t1J)o distinct prime factors congruent to three moduZo fOur.
Let
prime q
p - r :: 3
>
v,
4
and prIv.
By Corollary 2, one has for every
G.
13
(54)
Now, by the quadratic reciprocity law,
(55)
Hence
than
pr
v.
is a quadratic residue modulo
This implies that
pr
q,
for every prime
is a square.
q
greater
(A proof of that fact may
be found in [4].)
2.6. Proof of Theorem 2.1.
2.6.1.
General discussion.
Let
G· Gl$ •••$G
r
Let
L
where
G
i
has order ii
Be the subgroup of order
pr
of
~
1.
G whose elements are of
the type
api-l ,
(56)
Then the quotient group
GIL
has order
of G
so that a class in
GIL
is a set
p
t-4-r
•
1
S
a
S
p-1.
We may write an element
14
Now, a set Bs
by Property 2.5.1,
ID n Cs I
(59)
::I
CsuC
is symmetric with respect to the origin and
-s
IDnBs I
:I
P
r
IB s 1/2
:0
-
X
ID
s'
•
n
p
r
for
C I
-s
s
o.
~
Let
x.
II:
S
For cOmlting the number of differences giving elements of
count all the differences in the sets
2 •
(60)
B
s
and in
Co\{o},
we may
and subtract
Co
x ,
s
We obtain
1.-r
r r
p
-1
p (p -1)
2
+
(61)
r
r
E...=!
E....:1
_2
2
2
\
r
1.
(p -x )
lSssp -r
s
l.
X
s
and this must equal
r
l
(p -l)(p -3)
(62)
4
The sum of the first two terms of (61) minus (62) gives
r
.e....::!.
(63)
1.
p -p
2
0
2
r
This shows that the
r
::I
n -1
~.
2
x
l-r-1 r
p
2
p.
cannot be all equal if
s
abelian, since in this case we would have
l • r,
x
r
s
:0
p,
G is not elementary
and
l
p -p
r
• 0,
i · l, ••• ,r.
l i · 1,
We arrive to the conclusion that a condition to be fullfilled by the
order of
G and the number
r
of its direct factors is, by (61), (62)
and (63) that there exist integers
(64)
xs s
p
r
x ,
s
s
:0
1, ••• Pl-2 ,
verifying
15
One sees here the reason why if there exists a group of automorphism
of
G
in which the stabilisor of
leaves
{a}
invariant and operates
D
transitively on it, then
G
all the intersections of
Cs
and
C
s
must be entirely contained in
2.6.2.
is elementary abelian.
Indeed, in that case,
D must have the same cardinality.
The cyclic case.
In the case
r == 1,
D or
-D.
First, we have
tC
(65)
s
..
==
by Lemma 2.4.
Now let
We write
i-I
ap
+s
s" s,pj,
and
api-l+s
by
s 'xpi-l + ap£'-l
(66)
Since
xpi-j-l+l
£.-1+s be any two distinct elements of
(s',p). 1.
with
x
Multiplying
a'p
_
Let
.
a'p
we obtain, since
i-l
is a quadratic residue
j < £.-1,
+ s.
mod p,
it follows from the preceding discussion that
the thesis is proved and
i · 1.
As a last remark, we observe that the present argument proves that
x
s
- 0
mod p.
s
(a'-a)/.' mod p.
xp£'-j-l+l,
+s
C •
16
BIBLIOGRAPHY
[1]
P. CAMION,
"Groupes d' automorphisms de graphes complete
antysymetriques et presque-corps"
Un1versite de Toulouse, Rapport de recherche 1967
(unpublished.)
[2]
P. DEMBOWSKI, Finite Geometries ~
Springer-Verlag Berlin, Heidelbelg, New York, 1968.
[ 3]
HENRY B. MANN,
Addition Theorems.J
Intersciences Publishers, 1965.
[ 4]
LANDAU,
Elementary nurrbel'
theo~ 11.1
Chelsea.
[5]
P. CAMION,
"Abelian Codes",
MRC Technical Summary Report # 1059.