On Steiner tableaus

AbstractKarl Byleen and Dale Mesner have suggested the following natural generalization of Room squares: An r-dimensional Steiner tableau of side n is an array (ai1i2…ir) of nr cells where each cell is either empty or contains an unordered r-tuple from (n + r − 1) symbols. Further, (T1) for any fixed ik=h, the (r − 1)-dimensional subarray (ai1…ik−1 hik+1−ir)contains each unordered (r − 1)-tuple of the (n + r − 1)-symbols; and (T2) each unordered r-tuple of the (n + r − 1) symbols appears exactly once in the entire array. A necessary and sufficient condition for the existence of an r-dimensional Steiner tableau of side n is mentioned. The above definition reduces to the well-known Room squares when r = 2. For r = 3, we call such a design a Steiner cube. A description is given of a search (aided by machine) which establishes the existence of three Steiner cubes of side 7. Each is a 7 × 7 × 7 cube with each cell either empty or containing a single unordered triple from a 9-set. Each of the (93)=84 unordered triples from a 9-set occurs once, and any of the 7 + 7 + 7 = 21 square cross-sections (slices) of the cube contains the 12 triples of a Steiner triple system on 9 symbols