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?