A finite projective plane of
order n ≧ 2 can be considered as a ⟨v,k,λ⟩ design where v = n2+ n + 1, k = n + 1,
and λ = 1. As such, it can be characterized by its point-line 0,1 incidence matrix A
of order v satisfying the incidence equation
where J is the matrix of order v consisting entirely of 1’s. Thus, if a plane of order n
exists then (∗) has an integral solution A. Ryser has shown that if A is a normal
integral solution to (∗) or if A is merely an integral solution to (∗) where n is odd,
then A can be made into an incidence matrix for a plane of order n by suitably
multiplying its columns by −1. Such an integral solution to (∗) we shall call a type
I solution. When A is merely an integral solution to (∗) where n is even,
then A may be a type I solution but may also be not of this type. These
latter integral solutions to (∗) we shall call type II solutions. Ryser has
constructed type II solutions for n = 2 and for all n ≡ 0 (mod4) for which there
exists a Hadamard matrix of order n, and Hall and Ryser have constructed a
type II solution for n = 10. In this paper we construct type II solutions for
some infinite classes of values of n ≡ 2 (mod4). Basic to these constructions
is a special class of ⟨v,k,λ⟩ designs called skew-Hadamard designs whose
incidence matrices form a part of the substructure of our type II solutions. We
exhibit examples for n = 26 and 50 and also derive examples for n = 10 and
18.
