Indicator Sets in an Affine Space of any Dimension

Abstract

It is well known that a translation plane can be represented in a vector space over a field F where F is a subfield of the kernel of a quasifield which coordinatizes the plane [1; 2; 4, p.220; 10]. If II is a finite translation plane of order qr (q = pn, p any prime), then II may be represented in V2r(q), the vector space of dimension 2r over GF(q), as follows:(i) The points of II are the vectors in V = V2r(q)(ii) The lines of II are(a) A set of qr + 1 mutually disjoint r-dimensional subspaces of V.(b) All translates of in V.(iii) Incidence is inclusion.