Finite Planes

Although finite planes correspond to particular families of designs, separate categories are provided for both projective and affine planes in order to exploit the rich structure possessed by these objects.

• Creation of classical and non-classical finite projective and affine planes
• Subplanes, dual of a projective plane
• Numerical invariants: order, p-rank
• Properties: Desarguesian, self-dual
• Parallel classes of an affine plane
• k-arcs: testing, complete, tangents, secants, passants
• Conics: through given points, knot, exterior, interior
• Unitals: testing, tangents, feet
• Affine to projective planes and vice versa
• Related structures: design, incidence matrix, incidence graph, linear code
• Collineation group, isomorphism testing (optimized algorithm for projective planes)
• Central collineations: testing, groups
• Group actions on a plane: orbits and stabilizers of points and lines
• Symmetry properties: point transitive, line transitive

Apart from elementary invariants, a reasonably fast method is available for testing whether a plane is desarguesian. Among special configurations of interest, a search procedure for k-arcs is provided. A specialized algorithm developed by Jeff Leon is used to compute the collineation group of a projective plane while the affine case is handled by the incidence structure method. The collineation group (order 2³ 3⁸) of a ``random'' projective plane of order 81 supplied by Gordon Royle was found in $1\,202$ seconds. As with graphs and designs the G-set mechanism gives the action of the collineation group on any appropriate set.