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

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

(B₁) Plane with one secant and q^ν+1 points of the sub-geometry
		1	q^2ν(q^ν+1)	q^2ν(q^2ν-1)	q^4ν(q^2ν-1)
		s	t₁	t₂	x₂
q^ν+1	C₀	q^ν+1,1	1,q^2ν	1,q^3ν-q^2ν	0,0
q^3ν-q^ν	C₁₁	q^3ν-q^ν,1	0,0	0,0	1,q^3ν
q^4ν	C₁₂	0,0	q^2ν,q^ν+1	0,0	q^ν,q^3ν-q^ν
q^4ν(q^2ν-1)	C₂	0,0	q^3ν-q^2ν,1	q^3ν,q^ν	q^3ν-q^ν,q^3ν-q^ν

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

Drop the points of C₀; this gives a PBD for q^6ν+q^3ν-q^ν points, one block of size q^3ν-q^ν and other blocks of sizes from {q^3ν-q^2ν, q^3ν,q^3ν+1}.
Drop points of C₁₁, giving a PBD with q^6ν+q^ν+1 points, in blocks of sizes from {q^3ν+1,q^3ν}.
Drop points of C₁₂, giving a PBD with q^6ν+q^3ν+1-q^4ν points, in blocks of sizes from {q^3ν+1, q^3ν-q^2ν+1, q^3ν-q^ν+1}.
Drop points of C₂, giving a PBD with q^4ν+q^3ν+1 points, one block of size q^3ν+1 and the other blocks of sizes from {q^2ν+1, 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