On the Existence of Strong Kirkman Cubes of Order 39 and Block Size 3

In this article we are interested in resolvable (v,k,1)-BIBDs. Let D be a (v,k,1)-BIBD and let R and S be two resolutions of the blocks of D. R and S are said to be orthogonal resolutions if any parallel class from R has at most one block in common with any class of S. A set of t resolutions of D is called a set of t orthogonal resolutions if every pair of these resolutions is orthogonal. For t = 2 the design resulting is called a Kirkman square. For t = 3 it is called a strong Kirkman cube. Previously, the smallest order for which a strong Kirkman cube of block size 3 was known to exist was v = 255. This paper gives an algorithm for searching for a particular type of Kirkman square with block size 3. The algorithm was applied to the case v = 39, k = 3 with the result that several strong Kirkman cubes were found. The designs obtained have automorphism groups which are transitive on parallel classes of all three orthogonal resolutions. In order to find the strong Kirkman cubes of order 39 cited above we enumerate all Kirkman squares of order 39 of a specific type.