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

License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use

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