Doubly resolvable designs

A problem which has recently been of interest to several authors is to arrange the blocks of a Kirkman triple system or a nearly Kirkman triple system into a square array such that each cell of the array is either empty or contains a block of the design and with the additional property that each element of the system is contained in exactly one cell of each row and column of the array. P. Smith [5] has shown that this is possible for a nearly Kirkman triple system with 24 elements and, in the same paper, indicates that no other such arrays were known. It has recently been shown [2] that a Kirkman triple system with 27 elements can be arranged into a square array of this type. In this paper, we give a number of recursive constructions which allow us to produce infinitely many of these arrays. In fact, we consider a more general structure and establish recursive methods for these.