Unbalanced Hadamard Matrices and Finite Projective Planes of Even Order

By an unbalanced Hadamard matrix we mean a matrix H,, = (hii) such that (i) hii = l/fi or -I/fi, (ii) H, is orthogonal, and (iii) the number of positive entries exceeds the number of negative entries in each row. In particular it is well-known that the dimension n must be an even perfect square if the number of positive entries is the same in each row. It is easy to show that the number of positive entries is 2t’ + t in this case. It is this case which is of interest to us, and we accordingly agree that n = 4t2 and omit future reference to the dimensionality of II. An elementary proof of these statements appears in [7]. By a finite projective plane of order 2t we mean a collection of 4t2 + 2t + 1 objects called points divided into subclasses called lines such that each subclass contains 2t + 1 points, and any two subclasses have precisely one point in common. Thus two lines define a point, and the dual, two points determine a line, also holds. An extensive literature has grown up on this subject since the topic was first explored in [8]. The leading theorems about finite projective planes concern the question of existence of such planes. We have the classic result: if the order is a prime or a power of prime, the geometry exists established in [I]. We also have the now celebrated Bruck-Ryser theorem [5] stating that, if the order s = 1 or 2 (4) and if the decomposition of s contains a prime of the form 4k + 3 to an odd power, then the finite projective plane fails to exist. Despite strenuous efforts and much computer time, no further progress has been made with this fundamental question. As a result, two schools have arisen, one faction believing that finite projective planes exist in all cases not excluded by the Bruck-Ryser conditions while others conjecture that such planes exist only if s is a power of a prime or a prime. The former position was somewhat strengthened by the remarkable discovery