Let D = (P,L) be a pair of sets such that
A finite projective plane is a point-line pair D such that
A finite projective plane is a symmetric 2–(n2 + n + 1, n + 1, 1) design where the lines of the plane become the blocks of the design. A finite affine plane is a point-line pair D such that
A finite affine plane can also be regarded as a 2–(n2, n, 1) design.
The basic method of defining a finite plane is to specify its point set and the line set. A classical finite projective plane can be defined by specifying a 3-dimensional vector space V over a finite field. The points are the 1-dimensional subspaces while the lines are the 2-dimensional subspaces. A classical finite affine plane is defined by specifying a 2-dimensional vector space V over a finite field. The points are the 1-dimensional subspaces while the lines are the cosets of the 1-dimensional subspaces.
A constructor is provided to create subplanes of projective or affine planes that contain a specified set of points. In the case of classical planes, subfield subplanes may be created. Given a specified line l in a projective plane π, the affine plane obtained from π by removing l can be constructed. Conversely, given an affine plane π, the projective completion of π can be constructed.
A test for a plane being Desarguesian is provided. In addition, in the case of projective planes, a function for testing whether the plane is self-dual is available.
A k-arc in a projective or affine plane P is a set of k points of P, no three of which are collinear. A k-arc is complete if it cannot be extended to a (k + 1)-arc by the addition of another point. A tangent to an arc A is a line which meets A exactly once; a secant is a line which meets A exactly twice; and a passant, or external line, is a line which does not meet A at all. A unital in the classical projective plane PG2(q2) is a set of q3 + 1 points such that every line meeting two of these points meets exactly q + 1 of them. Functions are provided to find arcs, conics, parallelisms, and unitals.
A particularly important calculation for finite planes is the determination of its automorphism group. Magma uses a very efficient backtrack search algorithm developed by J. Leon which employs his partition refinement technique. A variation of this algorithm is used to test pairs of planes for isomorphism.
The automorphism group of an incidence structure D is returned as a permutation group G acting on a set which is determined by properties of D. The Magma G-set machinery may be used to obtain the action of G on different sets associated with D. The two most important actions are those acting on the point set and line set. As the group G does not directly act on D, elements of G that are required to act on D are first cast as mappings of D into itself.
Let p be a point and l a line of a projective plane P. A (p,l)-central collineation is a collineation α of P which fixes l pointwise and p linewise. The line l is called the axis of α and the point p is called the centre of α. A function is provided to find the group G of (p,l)-central collineations of a projective plane. In addition, functions are provided to find the central collineation group with specified axis or centre.
A small group of functions is provided for constructing translation planes by derivation. Let q be a power of a prime. Given the projective plane PG2(q2), a function is provided for finding a Baer subplane of order q. An affine plane may be constructed from PG2(q2) by derivation with respect to a Baer subplane. Finally, given the projective plane PG2(q), where q is a power of 2, a function constructs a translation plane by derivation with respect to an oval.
Given a finite near-field N, there is an affine plane A with point set N×N and lines given by the equations y = xm + b, x = c. Let P be the corresponding projective plane, obtained from A by adjoining a line L∞ called the line at infinity. By carefully labelling the points of A and the line L∞ with triples of elements of N, every collineation of A extends to a collineation of P. A function is provided to construct this plane.
In 1957, Hughes discovered a class of finite projective planes constructed from the Dickson near-fields which have rank 2 over their kernel. Neither these planes nor their duals are translation planes and therefore they cannot be obtained by the coordinatisation method of the previous paragraph. Hughes' methods required the kernel to be central but in 1960 the construction was generalised by Rosati to include the Zassenhaus near-fields. For simplicity, the term ‘Hughes plane’ will include both Hughes planes and generalised Hughes planes. A function is provided which constructs the Hughes plane given a near-field having rank 2 over its kernel.
As an example, we can construct four distinct projective planes of order 49. These are the Desarguesian plane PG2(49), the non-Desarguesian near-field plane over the Dickson near-field of order 49, the non-Desarguesian near-field plane over the Zassenhaus near-field of order 49, and finally the Hughes plane over the Dickson near-field of order 49. The orders of their respective collineation groups are 66437613849600, 22127616, 33191424, and 11261376. Thus, the four planes are distinct from one another. The computation took 5 seconds.
Finite geometry is an active area of research. Many of the major centres for finite geometry use Magma. R. Baker and G. Ebert have studied questions arising in PGn(q) for n ≥ 3 as well as in planes. E. Assmus and J. Key did many computations with finite planes in the course of their work applying coding theory to the analysis and construction of designs. Many workers have used Magma in their study of blocking sets and spreads in finite planes.