Hyperplanes of type D1 from projective geometries with dimension 4

BIBDs

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Hyperplanes of type D₁ from projective geometries PG(4,q^ρν), ρ=2

The matrix showing the numbers of points per line, and vice versa, class by class is:

(D₁) Hyperplane with one plane and (q^2ν+q^ν+1) points of the subgeometry.
		q^2ν+q^ν+1	q^ν(q^3ν-1)	q^4ν(q^2ν+q^ν+1)	q^5ν(q^3ν-1)
		s	t₁	t₂	x
q^2ν+q^ν+1	C₀	q^ν+1,q^ν+1	1,q^2ν-q^ν	1,q^4ν	0,0
q^ν(q^3ν-1)	C₁₁	q^2ν-q^ν,1	q^2ν,q^2ν	0,0	1,q^4ν
q^6ν	C₁₂	0,0	0,0	q^2ν,q^2ν+q^ν+1	q^2ν,q^ν(q^3ν-1)

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

The structure of the hyperplane is easily ascertained. The plane of type B₀ is given in the 2x2 subtable top left.

There must be q^4ν(q^2ν+q^ν+1) more tangents, hence the t₂, which necessarily must use a separate class of C₁ points. Finally, the exterior lines give the concurrences between the C₁₁ and C₁₂ points: since all the concurrences within the C₁₁ points have been obtained in the plane of type B₀ aforementioned, they can appear only singly in the exterior lines, hence the form given.

The rest of the table then follows by simple calculations.

A table showing the relationship between the points and hyperplanes of PG(4,q^2ν) is as follows.

Points and hyperplanes of PG(4,q^2ν)
		(q^5ν-1)/(q^ν-1)	q^ν(q^2ν+1)(q^5ν-1)
		D₀	D₁
(q^5ν-1)/(q^ν-1)	C₀	q^3ν+q^2ν+q^ν+1,q^3ν+q^2ν+q^ν+1	q^2ν+q^ν+1,q^ν(q^2ν+1)(q^3ν+1)
q^ν(q^2ν+1)(q^5ν-1)	C₁	q^ν(q^2ν+1)(q^3ν+1),q^2ν+q^ν+1	q^6ν+q^4ν-q^ν,q^6ν+q^4ν-q^ν

The complete design above is a BIBD with λ=q^4ν+q^2ν+1.

My conjectured table showing the relationship between the points and planes of a hyperplane of type D1 is as below. The planes of types B₁ and B₂ are shown here.

Points and planes of PG(3,q^2ν) of type D1
		1	q^2ν(q^2ν+q^ν+1)	q^3ν(q^3ν-1)
		B₀	B₁	B₂
q^2ν+q^ν+1	C₀	q^2ν+q^ν+1,1	q^ν+1,q^2ν(q^ν+1)	1,q^3ν(q^ν-1)
q^4ν-q^ν	C₁₁	q^4ν-q^ν,1	q^2ν-q^ν,q^2ν	q^2ν,q^4ν
q^6ν	C₁₂	0,0	q^4ν,q^2ν+q^ν+1	q^4ν,q^ν(q^3ν-1)

The complete design above is a BIBD with λ=q^2ν+1.

Last Updated on 04/05/2004
By D.H.Rees

BIBDs

Hyperplanes of type D1 from projective geometries PG(4,qρν), ρ=2

Hyperplanes of type D₁ from projective geometries PG(4,q^ρν), ρ=2