The number of secants in PG(3,q2ν) is (q2ν+1)(q2ν+qν+1); and through each secant there are (q2ν+1) planes. Of these, (qν+1) are the planes of the subgeometry(i.e. of type B0): the remainder are of type B1, containing just the one secant. These account for all the planes of the geometry (the numbers are the same as the numbers of points of type C0 and C1)).
The matrix showing the numbers of points per line, and vice versa, class by class is:
| 1 | q2ν(qν+1) | q3ν(qν-1) | ||
|---|---|---|---|---|
| s | t | x | ||
| (qν+1) | C0 | qν+1,1 | 1,q2ν | 0,0 |
| qν(qν-1) | C11 | q2ν-qν,1 | 0,0 | 1,q2ν |
| q4ν | C12 | 0,0 | q2ν,qν+1 | q2ν,qν(qν-1) |
The complete design above is a BIBD with λ=1. As the symmetry between points and lines is lost in the detailed breakdown, different subdesigns are available according to whether points or lines are taken as treatments.
The structure of the plane is easily ascertained. The secant is given. The points of C1 appearing on the secant cannot appear on the tangents, obviously, so two sub-categories of C1 have been introduced. The structure of the tangents is then given, while the number of tangents is just the number of subgeometry points, (qν+1), times one less than the replication of those points, q2ν. The exterior lines must enable the points of C11 to occur with those of C12: but the C11 points have already appeared with each other (in the secant), so can appear in singles only. The rest of the table then follows by simple calculations.
A table showing the relationship between the points and planes of PG(3,q2ν) is as follows.
| (q4ν-1)/(qν-1) | qν(q2ν+1)(q3ν-1) | ||
|---|---|---|---|
| B0 | B1 | ||
| (q4ν-1)/(qν-1) | C0 | q2ν+qν+1,q2ν+qν+1 | qν+1,qν(q3ν-1) |
| qν(q2ν+1)(q3ν-1) | C1 | qν(q3ν-1),qν+1 | q4ν+q2ν-qν,q4ν+q2ν-qν |
The complete design above is a BIBD with λ=q2ν+1.
Last Updated on 29/04/2004
By D.H.Rees