New flag-transitive affine planes of even order (Q1913993)

Let \(V\) be a vector space of dimension \(2k\) over \(GF(q)\) with a nonsingular quadratic form \(Q\). An orthogonal spread is a family of \(q^{k - 1} + 1\) totally singular \(k\)-spaces such that each non-zero singular vector is in exactly one member of the family. The bilinear form \(Q(u + v) - Q(u) - Q(v)\) defines a symplectic structure. For \(q^m > 8,q\) even, let \(m, m_1, \dots\) be a sequence of distinct divisors of \(m\) such that \(m_i\) is divisible by \(m_{i + 1}\). The authors use a process of flipping back forth between symplectic and orthogonal spreads and together with various subfields of \(GF(q^m)\) to construct flag transitive affine planes.