Plane of PG(2,q^(3*ν)) with subgeometry.

The derivation of the plane is discussed elsewhere. This page is concerned with designs derived from this plane.

(B₂) Plane with no secants and 1 point of the sub-geometry.
		q^2ν+q^ν+1	q^3ν-q^2ν-q^ν	q^3ν	q^3ν(q^3ν-1)
		t₁	t₂	x₁	x₂
1	C₀	1,q^2ν+q^ν+1	1,q^3ν-q^2ν-q^ν	0,0	0,0
q^2ν(q^2ν+q^ν+1)	C₁	q^2ν,1	0,0	q^2ν+q^ν+1,q^ν	q^ν+1,q^3ν-q^ν
(q^3ν-q^2ν)(q^2ν+q^ν+1)	C₂₁	q^3ν-q^2ν,1	0,0	0,0	q^2ν,q^3ν
q^4ν(q^2ν-q^ν-1)	C₂₂	0,0	q^3ν,1	q^3ν-q^2ν-q^ν,1	q^3ν-q^2ν-q^ν,q^3ν-1

The complete design above is a BIBD with λ=1.

Drop points of C₁, giving a PBD with q^6ν-q^4ν-q^2ν+1 points, in blocks of sizes from {q^3ν+1,q^3ν-q^2ν+1,q^3ν-q^2ν-q^ν}.
Drop points of C₂₁, giving a PBD with q^6ν-q^5ν+q^3ν+q^2ν+1 points, in blocks of sizes from {q^2ν+1, q^3ν+1, q^3ν-q^2ν+1}.
Drop points of C₂₂, giving a PBD with q^5ν+q^4ν+q^3ν+1 points, and blocks of sizes from {q^3ν+1, q^2ν+q^ν+1}
In addition, could then consider adding a point back, or even spiking all or part of the deletions from a line. Or one could drop a further point, or a line's worth.

Furthermore, there is the ability to drop points of two classes, or even three.

Last Updated on 06/05/2004
By D.H.Rees