A weighing matrix is an n x n matrix W = W{n, k) with entries from {0,1, -1} , satisfying WW* = kl . We shall n call k the degree of VI . It has been conjectured that if.
BULL. AUSTRAL. MATH. SOC.
05B2O, I5A36
VOL. 10 (1974), 119-122.
Families of weighing matrices Anthony V. Geramita, Norman J. Pullman, and Jennifer S. Wallis A weighing matrix is an n x n matrix W = W{n, k) with WW* = kl . We shall n call k the degree of VI . It has been conjectured that if n = 0 (mod U) then there exist n x n weighing matrices of every degree k £ n . We prove the conjecture when n is a power of 2 . If n is not a power of two we find an integer t < n for which there are weighing matrices of every degree 5 t . entries from
{ 0 , 1 , -1} , satisfying
Taussky [/] suggested the following generalization of Hadamard matri ces: A weighing matrix is an n * n matrix W = W(n, k) with entries from {0, 1, -1} , satisfying WW* = kln . We shall call k the degree of W . In [3, p. 1*33], it was conjectured that (*) If n = 0 (mod h) then there exist n * n weighing matrices of every degree k 5 n . (Note that an n x n weighing matrix of degree n is an Hadamard matrix and so (•) is a generalization of the conjecture on the existence of Hadamard matrices of order n for every n = 0 (mod h) .) In [2] the validity of (») was established for n € {U, 8, 12, 16, 20, 2k, 28, 32, Uo} and partial results were obtained Received 9 October 1973. The work of the first two authors was supported in part by the National Research Council of Canada. I 19
120
Anthony V. Geramita, Norman J . Pullman, and Jennifer S. Wai I is
for n € {36, kk, 52, 56} in that sets of values of which W{n, k) e x i s t s . For a l l n l e t g{n) weighing matrices W(n, k) i s equivalent t o : (*)
k were obtained for
be the maximum degree q for which there exist for all degrees k 5 q . Thus, conjecture (•)
g(n) = n for a l l
n = 0 (mod h) .
The methods of [2] can be used to show that
g[2n) > 3U for a l l
n > 5 • We show [Corollary 2 to our theorem] that in fact g[2n) = 2n for a l l n and hence establish (•) for a l l powers of 2 . As another i kn)\ > 2k for a l l odd n and all corollary to the theorem we show that g[2 k > 1 . This is better, asymptotically, than results obtained by the methods of C 2]. Call
{M , M , ...,
M}
an M-family of order
n i f for each
i ,
1 5 i 2 m: (l)
M. is a weighing matrix of order n and degree
Let \i(n) be the largest Evidently g(n) 2 u(n) . THEOREM. If Proof. A =
1 0
m for which an M-family of order
n
exists.
y(«) > m then \i(2n) > 2m .
Suppose '
i , and
H =
1
{M , M , ..., 1
and
I
M}
is an M-family of order
is the
? *p
n ,
identity matrix.
Define (a)
AT = I2 ® M^ for each i ,
(b)
M . =~M. + A ® M m+i
(c)
^
i = ^®\
for each
i ,
1 < i S m-X , and
tn •
I t i s easily verified that 2n .
1 5 i l .
Each matrix J ® M. i s a weighing matrix of order if
M. i s a weighing matrix of order
n
and degree
nm and i .
Lemma 1 ( i ) , 2 ( i ) and ( i i i ) of [2] imply immediately that (t)
I f («) holds f o r n
then
g{2tn)
> n + 2 t for all integers
But Corollary 1* gives far better estimates of a l l sufficiently large
t .
g\2 n)
t > 0
.
than does ( t ) for
For example, the r e s u l t s of [Z] and ( t ) give
us ^(2*21*) > 2U + 2t but Corollary h gives us b e t t e r estimate for a l l t > 2 .
g{2 2U) > 2 t + 3
which i s a
References
[I]
Olga Taussky, " ( l , 2 , h, 8)-sums of squaresxand Hadamard matrices", Combinatorics, 229-233 (Proc. Symposia Pure Math., 19. Amer. Math. S o c , Providence, Rhode Island, 1971) •
[2]
Jennifer Wai I i s , "Orthogonal (0, 1, - l ) - m a t r i c e s " , Proa. First Austral. Conf. Combinatorial Math., Newcastle, 1972, 6l-81* (TUHRA, Newcastle, 1972).
122 [3]
Anthony V. Geramita, Norman J . Pullman, and Jennifer S. Wai I i s W.D. Wai I i s , Anne Penfold Street, Jennifer Seberry Wai I i s ,
Combinatorics: Room squares, sum-free sets, Hadamard matrices (Lecture Notes in Mathematics, 292. Springer-Verlag, Berlin, Heidelberg, New York, 1972).
Department of Mathematics, Queen's University, Ki ngston, Ontario, Canada.
