Resolvable designs from projective space

Resolvable designs are block designs with the additional property that the blocks can be partitioned into partitions of the points. It is easy to see that lines in affine space form a resolvable design since parallel lines partition the space, and so we can partition the lines according to direction.

Can lines in projective space form a resolvable design?

(Unlike affine spaces, this will not be the case for all projective spaces.)

• Are you talking about 2-designs here? Or λ-designs? And when you say "can lines in projective space form a resolvable design?" do you mean the design formed by the set of all lines and all points? This is certainly not resolvable when we're in the projective plane... I'd have to think some more about higher dimensions. – Nick Gill Apr 23 '13 at 13:06

The fact that every odd dimensional projective geometry ${\rm PG}(n,2)$ over $\mathbb{F}_2$ admits a line parallelism (i.e., the Steiner $2$-design formed by the points and lines of ${\rm PG}(n,2)$ with $n$ odd is resolvable) is a corollary of the classical result proven here:

R. D. Baker, Partitioning the planes of ${\rm AG}_{2m}(2)$ into $2$-designs, Discrete Math. 15 (1976), 205–211

Edit: Note that Nick's comment about this result is apparently assuming the trivial necessary condition: the corresponding Steiner $2$-design for when $n$ is even can't be resolvable because in this case the number of points is not divisible by the block size.

For ${\rm PG}(n,q)$ with $q \not= 2$, all I know is Beutelspacher's result Nick mentioned (i.e., the case when $n = 2^{i+1}-1$).

If there are newer positive results, I'm not aware of them. In any case, here's a survey of parallelisms of projective spaces written in 2003:

N. L. Johnson, Parallelisms of projective spaces, J. Geometry, 76 (2003), 110-182

• You are very generous about what I was assuming... But thanks for giving me the benefit of the doubt :-) – Nick Gill Apr 23 '13 at 20:10

The answer is yes, as the following paper makes clear:

Beutelspacher, Albrecht. On parallelisms in finite projective spaces. Geometriae Dedicata 3 (1974), 35–40.

According to the MSN review, the author proves that that any finite projective space of dimension $d=2^{i+1}−1$ (with $i=1,2,\dots$) admits a parallelism of lines. In other words, the set of points and lines form a resolvable design.

I also found a positive answer (in a different direction) in a comment in this paper:

Hishida, Takaaki; Jimbo, Masakazu Cyclic resolutions of the BIB design in PG(5,2). Australas. J. Combin. 22 (2000), 73–79.

Specifically, the authors remark that $PG(n,2)$ is resolvable for $n\geq 3$, although I don't know where you would find a proof.

I do not know if there is a full characterization of which projective spaces are resolvable.