ABSTRACT
More generally, replacing each of the variables of an OD(n; s1, . . . , su) with 0,±1 gives a W (n,w) where w is the sum of the sis such that xi is replaced by ±1.
2.10 Below, D1 and D2 are OD(2; 1, 1); D3 and D4 are OD(4; 1, 1, 1); and D5 is an OD(4; 1)
D1 = (
a b −b a
) D2 =
( x y y −x
)
D3 =
0 a b c
−a 0 c −b −b −c 0 a −c b −a 0
D4 =
0 x y z −x 0 −z y −y z 0 −x −z −y x 0
D5 =
d 0 0 0 0 d 0 0 0 0 d 0 0 0 0 d
2.11 Theorem Suppose W (n,w) exists. 1. If n > 1 is odd, then w is a perfect square and n ≥ w + √w + 1 (equivalently,
(1) w ≤ n− 1+ √ 4n−3 2 or (2) (n− w)2 − (n −w) + 1 ≥ n). The case of equality,
W (k2 + k + 1, k2), implies the existence of a projective plane (see §VII.2.1) of order k.
