How Do Parallel Line Groups in Affine Planes Relate to Block Designs?

[andreass]

I'm trying to understand some things in finite geometries, more specific - affine geometry.
Now I'm reading some notes and wiki:
http://en.wikipedia.org/wiki/Finite_geometry
http://en.wikipedia.org/wiki/Block_design

I would like to know how many groups of parallel lines are in affine plane AG(2,q) and how many lines are in each of those groups?

For example AG(2,2) correspond to (v=4,k=2,lambda=1,r=3,b=6) which means affine plane consists of 4 points, 2 points on each line, 2 points have only 1 common line, each point lies on 3 lines and there are 6 lines together.

We can see in image that there are 3 groups of parallel lines (each color is 1 group) and there are 2 parallel lines in each group. Does it somehow correspond to block design (e.g. r=3 or k=v*r/b=2)?

 

[pat_connell]

Thanks.A:In an affine plane of order $n$, there are $n+1$ groups of parallel lines and each group contains $n$ lines. The arrangement of the lines in an affine plane of order $n$ form a $2-(n^2,n,1)$ design, which is also known as a projective plane.