A *double Youden rectangle* is a set of size *kn* with two partitions into *k* sets of *n* and two into *n* sets of *k* satisfying the appropriate orthogonality and balance conditions as defined here: thus, two partitions of different sizes are orthogonal, and two partitions of the same size form a SBIBD. (In the case of the partitions with smaller number of parts, it is first necessary to delete a fixed number, the floor of *n*/*k*, of incidences between each pair.) The parameters of the SBIBDs are (*n,k*,λ) and (*k,l*,μ), where *l* is the remainder on dividing *n* by *k*. [Here and subsequently we assume that *k* < *n*.]

As I mentioned there, all known examples have *k* dividing *n*±1, and the SBIBD formed by the two partitions with *k* parts is trivial (each part of one partition is incident with exactly one part of the other if *k* divides *n*−1, or incident with all but one part of the other if *k* divides *n*+1). Is this necessarily so?

The smallest set of parameters where something more interesting could happen is *n* = 40, *k* = 27. Now SBIBDs with parameters (40,27,18) and (27,13,6) exist: an example of the first is the complementary design of projective 3-space over the field of three elements, and an example of the second is the Paley design derived from the field with 27 elements.

Is there a 27×40 double Youden rectangle?

### Like this:

Like Loading...

*Related*

## About Peter Cameron

I count all the things that need to be counted.