On Designs of Maximal (+1,-1 )-Matrices
of Order w-2 (mod 4). IP By C. H. Yang Abstract. Finding maximal (4-1, —l)-matrices constructible in the standard form
(
A
B)
\-BT
AT)
is reduced to the finding of two polynomials circulant submatrices A, B) satisfying (*)
\Ciw)\2+
M2m of order 2m (with odd m)
C(m>), Diw) (corresponding
to the
\Diw)\2 = \im - 1) ,
where w is any primitive mth root of unity. Thus, all M2m constructible by the standard form (see [4]) can be classified by the formula (*). Some new matrices
M2mfor m = 25, 27, 31, were found by this method. | Let M2m be a maximal (4-1, —1)-matrix of order 2m and let S = ((s,-)) be the circulant matrix of order m with the first row entries s< (0 ^ i g m — 1), all zero but Si = 1. When m is odd, it is known that (for m ^ 27, except m = 11, 17; see [1] —[4]), M2m can be constructed by the following matrix :** (a
R = I
(1)
R
^
\ —.£>
\
m_1
a T) ,
A
/
m_1
where A = £ akSk, B = £ oAS4with fc_o
ak and 6*, 1 or —1, and 21indicates the transposed
*=o
matrix. Then the gramian matrix
of R becomes
rrT = \o
p)'
where P is equal to
(2)
AAT + BBT = 2{ml + X) Sh) , \ *-i /
where / is the identity matrix of order m. By applying to the both sides of (2) the transformation L which transforms S into a diagonal matrix W = [wi, • • •, wm] with w¡, all distinct mth roots of unity, (namely, L(")
where m—1
m—I
A (tu) = X) akWk,BOw) = X) &*w* > and tu is any mth root of unity. Since w and wm~l are conjugate to each other, (3) is also equivalent to
[Aiw)\2 + \Biw)\2 = 2 ( m 4-
(4)
X^w*) ■
Let p and q be respectively the numbers of —l's in each row of A and B. By replacing 1 by 0 and —1 by 1 in A and B and performing the similar process as above, we obtain the following formula corresponding to (4). m—1
(5)
E
\Ciw)\2+\Diw)\2
= p + q + rY.wk, k=l
Table
C(w)
I
D(w)
NA NB N
1
13
1
1
1
1
2
2
4
2
4
4
4
8
8
1 + to + w3 + to5 + w7 + vfi
2
2
2
1 4- to + w3 -f-1/>5+ to8 + to9
1 + to + to3 + to7 + to9 4- to10+ to"
9
9
9
1 + to + to2 + to4 + to7 + to12
1 + w + w3 4 to4 + to8 4- to10 4 wu
G
6 12
9
9
1
1 + w 4 w3
1+ w
1 4 w- 4 wb
1 + w 4 w3 + to9 1 + W 4
1 + wo + u-o3+ u'o9
Wl
1 + V) + Î02 + tO4 + W7 + KJ9
or 1 + w 4 w3 + Vf + w7 4 to8
15
1+
w 4 to4 + to6
1 + w 4 w3 + wl + to8 + to10
or 1 + w 4 w3 4 wr' 4 ws + to9 1 + to 4- to4 4 to10 19
or
or 1 4 to 4 to3 4 to124 to144 W5
1 4 to 4 tt)34 to5 4 to9 4 to!0 4 wls
1 4 to 4 w2 4 to3 4 to6 4 wis
1 4 to 4 to5 4 to7 4 to9 4 w124 w'5
1 4 to 4 to2 4 to3 4 w7 4 u)n
1 4 w + w3 4 ws + to8 4 w12 4 ío1s
or
9 1 4 to 4 to3 4 to7 4 to" 4 w11 4 to16
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9
18 18
ON DESIGN OF MAXIMAL (4-1,
I + I 1+ + + I +I+I+1++ + + + + I I + + +++I1+++ 1I++111+ I ++ I ++ I + ++++111+ I + + + + + + I I + I + 1+ I + + 1 ++ I + + + ++++I+++ + I I + + + + I + + I I + I 1+ + + + + + + I 1 + 1 I + + + + I I 1 I + + + I + + I I l + l I + + 1+ 1+ 1+ 1 + I ++ I + I I + + I I I ++ I 1 ++ I ++ I + I + I + + + + + I 1+ I + + + 1 + + + + I I + + I + + + + I + I + + I ++ I + + I I I I I I I I
3
4-
r-
n
cc
4M
S g S + + a*+
^H
-tf
1-1
§ g 8
+ 4- +
++Í
'S g §
o
©
f-i
++
+
+ 4-4g g g + ++ g g "g 4-4-4-
ri