Hokkaido Mathematical
Journal Vol. 17
(1988)
p.
139\sim 146
On equivalence of Hadamard matrices
Dedicated to Professor Nagayoshi IWAHORI on his 60th birthday
Hiroshi KiMURA
(Received May 20, 1987,
Revised September 19,
1987)
1. Introduction
An n -dimensional Hadamard matrix is an n by n matrix of 1’s and-l’s
with HH^{T}=nI . In such matrix, n is necessarily 2 or a multiple of 4. An
automorphism of H is a signed permutation g on the set of rows and columns
such that H^{g}=H . The set of automorphisms forms a group under composiand
tion called the automorphism group of H . Two Hadamard matrices
are equivalent if there exists a signed permutation g of rows and columns
. An Hadamard matrix is normalized if its first row and column
with
consist entirely of 1’s.
be two Hadamard matrices of order n . It is difficult to
and
Let
know whether they are equivalent or not. But we know some invariants for
are
and
equivalence classes. For instance if automorphism groups of
nonisomorphic as permutation groups, they are not equivalent. For n=28
we defined the K matrix K(H) associated with H depending only on equivalence class containing H([2]) . In the case n=28 it seems that K(H) is
very useful. By computing K -matrices we obtained Hadamard matrices
with trivial automorphism groups. Furthermore we constructed about four
hundred new Hadamard matrices of order 28 ([2], [3] and [4]).
In \S 2 we discuss K -matrices in general cases. In \S 3 we introduce a new
invariant called K -boxes. For n=28 we give examples with same if-matrix
such that they have different K -boxes and therefore they are not equivalent.
H_{1}
H_{2}
H_{1}^{g}=H_{2}
H_{1}
H_{2}
H_{1}
H_{2}
2. On K-matrices
Let H=(h_{ij}) be an Hadamard matrix of order n with 0\leq i, j\leq n-1 . H
is equivalent to H’=(h_{ij}’,) with h_{i0}’,=h_{0j}’,=1,0\leq i, j\leq n-1 . From H’ we
have an incidence matrix D(H) of a symmetric 2-(v, k, ) design associated
ed with H. where v=n-1 , k=(v-1)/2 , \lambda=(k-1)/2 :
\lambda
D(H)=d_{i,j}) ,
where
d_{i,j}=\{
i, j=1 , . n-1
1, if h_{ij}’,=1
0, if h_{ij}’,=-1
\ldots
H. Kimura
140
For any different four rows i.
follows:
a_{ijkm}(r)=\{
1, if
0, if
j,
k
and m of H , we define
a_{ijkm}
as
h_{ir}h_{jr}h_{kr}h_{mr}=1
h_{ir}h_{j\gamma}h_{kr}h_{mr}=-1
Then a_{ijkm}(0)+\ldots+a_{ijkm}(n-1) is divisible by 4 ([2] or [5]). Let x be an
integer with 0\leq x\leq n/4 . For fixed i and j , let
be a number of pairs k
and m of rows such that a_{ijkm}(0)+\ldots+a_{ijkm}(n-1)=4x . For 0\leq x\leq n/8 , put
\kappa_{ij}’(x)
\kappa_{ij}’(x)+\kappa_{ij}’(n/4-x)
\kappa_{ij}(x)=\{
\kappa_{ij}’(x)
,
,
if x\neq n/4-x
if x=n/8 .
does not change by multiplication of rows i or j by -1. By a
Then
permutation of coordinates we assume that
if j<k . Put
\kappa_{ij}(x)
\kappa_{ij}(x)\leq\kappa_{ik}(x)
\kappa_{ij}(x)
K_{ij}(x)=\{
,
\kappa_{ij+1}(x)
if i>j
, if i\leq j.
Furthermore the rows of the n\cross(n-1) matrix (K_{ij}(x)) are ordered lexic0graphically, that is, if i<i ’. then K_{ij}(x)=K_{ij}(x) for j=1 , . n-1 , or there
exists an integer j such that K_{i,j}(x)=K_{ij’}"(x) for j’<j and K_{i,j}(x)<K_{ij}"(x) .
We call the matrix K_{x}(H)=(K_{ij}(x)) an associated x -th K matrix of H .
By the construction of K_{x}(H) we have the following:
\ldots
THEOREM 1.
be Hadamard matrices
K_{x}(H_{1})=K_{x}(H_{2})
are equivalent, then
for all 0\leq x\leq n/8 .
Let
H_{1}
and
H_{2}
of order
n which
REMARK 1. By considering the relation of three rows of D(H) it is
trivial that K_{0}(H) is the zero matrix in the case n\equiv 4(8) .
REMARK 2.
K(H) in
[2] is
K_{1}(H)
for
n=28 .
Next we prove the following theorems.
THEOREM 2.
Assume n\equiv 4(8) . Let a and b be two integers with 1\leq
a, b\leq(n-4)/8 . If we know K_{m}(H) for all m\neq a, b, K_{a}(H) and K_{b}(H)
can be obtained.
THEOREM 3. Assume n\equiv 0(8) . Let a and b be two integers with 0\leq
a, b\leq n/8 . If we know K_{m}(H) for all m\neq a, b, then K_{a}(H) and K_{b}(H)
can be obtained.
(1) Case n\equiv 4(8) . We arrange the first three rows of D(H) in the
following form:
On equivalence of Hadamard matrices
1
|_{1\ldots 1}^{1\ldots 1}1\ldots 1
.
\lambda
|\begin{array}{lll}1 \cdots 11 \cdots 1\end{array}|
1\ldots 11\ldots 1
\ldots 1
\lambda-\lambda\lambda-\lambda-\lambda-\lambda
1
|
|\begin{array}{lll}1 \cdots 11 \cdots 1\end{array}|
.
141
\ldots 1
|
|
|
1
\ldots 1
k+\lambda’-2\lambda k+\lambda’-2\lambda k+\lambda’-2\lambda\lambda-\lambda
is a number of columns j such that d_{i,j}=d_{2,j}=d_{3,j}=1 . Then
(0)+\ldots+\alpha_{)123}(n-1)=4\lambda ’+4 .
For 0\leq i\leq\lambda-1 , let be a number of rows j of D(H) such that
where
. Since D(H)^{T} is also an incidence matrix of
design, we have the following:
=i , where
\lambda)
} 123
\sum_{k=1}\lambda d_{jk}
\alpha_{i}
k,
\alpha
\lambda
’
3\leq j\leq n-1
2-(v,
(1)
(2)
(3)
\alpha_{\lambda-1}=n-3
\alpha_{0}+\alpha_{1}+\alpha_{2}+\ldots+
\alpha_{1}+2\alpha_{2}+\ldots+(\lambda-1)\alpha_{\lambda-1}=\lambda(k-2)
\lambda-1G\alpha_{\lambda-2}=_{\lambda}G(\lambda-2)
\alpha_{2}+\ldots+
Let a be an integer with
we have
\lambda/2\leq a\leq\lambda-2
. From equalities
(1)
and
(2)
(4)
\sum_{i\neq a}(a-i)\alpha_{i}=a(n-3)-\lambda(k-2)
From (2) and (3) we have
(5)
\sum_{i\neq a}i(a-i)\alpha_{i}/2=\lambda(k-2)(a-1)/2-(\lambda-2)_{\lambda}G
From (4) and (5) we eliminate
ity (i\neq a, \lambda-1-a) . Then
\alpha_{\lambda-1-a}
. Let
A_{i}
be coefficient of new equal-
A_{j}=(i-\lambda+1+a)(a-i)/2
if and only if j=i or j=\lambda-1-i .
\leq i\leq\lambda/2-1 .
Then we have the following:
Therefore
A_{j}=A_{i}
Put
\beta_{i}=\alpha_{i}+\alpha_{\lambda-i-1}
(6)
’
\sum_{i\neq a}A_{i}\beta_{i}=B
where B’ is a constant. By
for 0
(1)
(7)
\sum_{i=0}^{\lambda/2-1}\beta_{i}=n-3
By (6) and (7) we can obtain
\beta_{a}
and
\beta_{b}
. Put
,
\beta_{a}=\sum_{i\neq a,b}B_{i}\beta_{i}+B
i+F, where B and F are constants.
PROOF THEOREM 2. Fix 0\leq i<j\leq n-1 . We assume that
k0=1 for all 0\leq k\leq n-1 . By the definition of
\beta_{b}=\sum_{i\neq a,b}F_{i}\beta
h_{ik}=1
and
\kappa_{ij}(x)
\sum_{x=1}^{\lambda/2}\kappa_{ij}(x)=_{n-2}G
(8)
h
142
H. Kimura
j
be a number of rows m such that a_{ijkm}(0)+\ldots+a_{ijkm}
. By the above discussion
(n-1)=4x or n-4x . we consider
For
,
k\neq j
let
\delta_{x}(k)
\kappa_{ij}(a)
\delta_{a}(k)=\sum_{x\neq a,b}B_{x-1}\delta_{x}(k)+B
\kappa_{ij}(x)=(\sum_{k\neq i,j}\delta_{x}(k))/2
Therefore
\kappa_{ij}(a)=(\sum_{k\neq i,j}\delta_{a}(k))/2
=( \sum_{x\neq a,b}B_{x-1}\sum_{k\neq i.j}\delta_{x}(k))/2+_{n-2}C_{1}B/2
(9)
= \sum_{x\neq a,b}B_{x-1}\kappa_{i,j}(x)+(n-2)B/2
Form
(8) and (9) Theorem
(2) Case n\equiv 0(8) . We
2 is proved.
arrange the first three rows of D(H) in the
following form
1
|11.\cdot.\cdot\cdot.111\ldots 1
.
\lambda
where
\lambda-
|\begin{array}{lll}1 \cdots 11 \cdots 1\end{array}|
1\ldots 11\ldots 1
\ldots 1
1
|
|\begin{array}{lll}1 \cdots 11 \cdots 1\end{array}|
\ldots 1
|
|
1
’
\lambda-\lambda
\lambda-\lambda\lambda-\lambda k+\lambda
is a number of columns
j
|
\ldots 1
’-2\lambda k+\lambda ’-2\lambda k+\lambda’-2\lambda\lambda-\lambda-
such that
’+4 .
be as in the case
d_{1,j}=d_{2,j}=d_{3,j}=1
. Then
a_{0123}
(0)+\ldots.+a_{\}123}(n-1)=4\lambda
For
0\leq i\leq\lambda
,
let
\alpha_{i}
n\equiv 4(8)
.
Then we have the
following:
(10)
(11)
(12)
\alpha_{\lambda}=n-3
\alpha_{0}+\alpha_{1}+\alpha_{2}+\ldots+
\alpha_{1}+2\alpha_{2}+\ldots+
\lambda\alpha_{\lambda}=\lambda(k-2)
\alpha_{2}+\ldots+_{\lambda}G\alpha_{\lambda}=_{\lambda}G(\lambda-2)
Let a be an integer with
(11) we have
\sum_{i\neq a}
From
(11)
(\lambda+1)/2\leq a\leq\lambda-2
.
From equalities
(a-i) \alpha_{i}=a(n-3)-\lambda(k-2)
(10)
and
(13)
and (12) we have
\sum_{i\neq a}i(a-i)
\alpha_{i}/2=\lambda(k-2)(a-1)/2-(\lambda-2)_{\lambda}G
From (13) and (14) we eliminate
equality (i\neq a, \lambda-1-a) . Then
A_{i}=(i-\lambda+1+a)(a-i)/2
\alpha_{\lambda-1-a}
.
Let
A_{1}
(14)
be coefficient of new
On equivalence
For
j,
j\neq(\lambda-1)/2
or
(\lambda-1)/2
\lambda
,
or
\lambda
,
A_{j}=A_{i}
of Hadamard matrices
if and only if
\beta_{i}=A_{i}+A_{\lambda-1-i’}\beta_{(\lambda-1)/2}=A_{(\lambda-1)/2}
,
143
j=i or j=\lambda-1-i. For
and
\beta_{-1}=A_{\lambda}
i\neq
. Then we have
the following:
’
\sum_{i\neq a}A_{i}\beta_{i}=B
where B is a constant. By the similar way in the case n\equiv 4(8) , we can
prove Theorem 3.
Case n=28 . By Theorem 2 we consider only K_{1}(H) . K matrices are
seemed to be useful for a classification of Hadamard matrices. Actually we
obtained 476 non-equivalent Hadamard matrices of order 28 ([3], [4]).
’
3. On K-boxes
In \S 2 we defined k -matrices. But we have some examples of Hadamard
matrices of order 28 with same K -matrix. We consider another method of
classification of Hadamard matrices. Let H be an Hadamard matrix of
order n .
,
as
For any different six rows i, j, k, i ’. j and k of Hr we define
follow:
’
’
a_{ijkij’ k},
a_{ijki’j’ k},(r)=\{
1, if
0, if
h_{ir}h_{j\gamma}h_{kr}h_{i’ r}h_{j’ r}h_{kr},=1
h_{ir}h_{jr}h_{kr}h_{i’ r}h_{jr},h_{k’ r}=-1
be a
Let x be an integer with 0\leq x\leq n . For fixed i, j and k , let
number of triples i ’. j and k of rows such that a_{ijkij’ k’},(0)+\ldots.+a_{ijki’j’k}\cdot(n1)=x . For 0\leq x\leq n/2 , put
\kappa_{ijk}’(x)
’
’
\kappa_{ijk}(x)=\kappa_{ijk}’(x)+\kappa_{ijk}’(n-x)
\kappa_{ijk}’(x)
,
,
if x\neq n-x
if x=n/2 .
does not change by multiplication of rows i,
Then
a permutation of coordinates we assume that
\kappa_{ijk}(x)
j or k by -1.
\kappa_{ijk}(x)\leq\kappa_{ijk’}(x)
\kappa_{ijk}(x)
K_{ijk}’(x)=\{
,
\kappa_{ij+1k}(x)
if k<k
’
By
Put
if i>j
, if i\leq j.
Next put
if j, j>k
K_{ijk}(x)=\{ K_{ijh+1}’(x) , if i>k\geq j, or
K_{ijk+2}’(x) , if i, j\leq k
K_{ijk}’(x)
,
i\leq k<j
Then, for 0\leq i\leq n-1 , the matrix KUx(H } =(K_{ijk}(x)) is of type (n-1)\cross
(n-2) . For i we rearrange the matrix K_{i,x}(H) as in the case of K-
H. Kimura
1M
matrices. Furthermore we rearrange the collection of matrices K_{i,x}(H)
with 0\leqq i\leq n-1 in the following: if i<i’ , then matrix K_{i,X}(H) equals the
matrix K_{ix}"(H) , or there exist integers s and t such that if j<s , then
(x)=K_{ijk},(x) for all k , if k<t , then K_{isk}(x)=K_{i’sk}(x) and K_{ist}(x)=K_{i’ st}(x) .
We call this collection KB_{x}(H) of n matrices K_{i,x}(H)K -box of degree x
associated with Ht
By the construction of KB_{x}(H) we have the following:
K_{ijk}
THEOREM 4.
Let
H_{1}
and
H_{2}
be Hadamard matrices of order n which
are equivalent, then KB_{x}(H_{1})=KB_{x}(H_{2}) for all 0\leq x\leq n/2 .
In the following table we give five matrices of order 28 with same K(H)
such that they have different K -boxes of degree 6. All rows of K(H) and
K(H^{T}) are of type (0000000000000000000000000111).
In the following table, for example, the number 32511 becomes
0000000000000111111011111111 in the binary system and the first row of H is
(11111111-1111111-1-1-1-1-1-1-1-1-1-1-1-1). In KB_{6}(H) ,
Symbol ’.’ represents 0, and also A , B,
and Z represent 10, 11, .... and 34,
respectively.
Mul =4 means that KB_{6}(H) contains four matrices as same
as a matrix. Moreover, (6 CCGGGGGG I I I I KKKKO OOOOO
SSSS) expresses that the multiplicity of rows of type ( CCGGGGGG
I I I I KKKKOOO0OOSSSS ) equals 6.
Acknowledgment. In [8] Professor V. D. Tonchev wrote to us that he
knew four nonequivalent Hadamard matrices of order 28 having same
\ldots
’
K matrix as K(H) .
References
[1]
[2]
[3]
[4]
[5]
[6]
[7]
[8]
M. HALL, JR., Combinatorial Theory, Ginn(Blaisdell) 1967, Boston.
H. KIMURA, Hadamard matrices of order 28 with automorphism groups of order two. J.
Combin. Theory (A) 43 (1986), 98-102.
H. KIMURA and H. OHMORI, Construction of Hadamard matrices of order 28, Graphs and
Combinatorics 2 (1986), 247-257.
H. KIMURA and H. OHMORI, Hadamard matrices of order 28, Memoirs of the faculty of
Education, Ehime Univ. 7 (1987), 7-57.
Z. KIYASU, Hadamard matrix and its applications, Denshi-tsushin Gakkai(Japanese)
1980.
V. D. TONCHEV, Hadamard matrices of order 28 with automorphisms of order 13, J.
Combin. Theory (A) 35 (1983), 43-57.
V. D. TONCHEV, Hadamard matrices of order 28 with automorphisms of order 7, J.
Combin. Theory (A) 40 (1985), 62-81.
V. D. TONCHEV, Private letter, 1987.
Department of Mathematics
Faculty of Science
Ehime University
145
On equivalence of Hadamard matrices
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