Generalized Steiner systems of type 3-(v, {4,6}, 1)

Summary

Let S be the set of entry positions of a 4×4 square. To each point of S associate the 6-subset of S consisting of those entry positions in its row and column but not including it. It is well-known that these sixteen 6-subsets form one of the three biplanes of order four - in fact, the “best” of the three. And this fact is easy to verify [3, 7].

What is not so well-known is that this biplane's ovals also have an easy description in terms of the 4×4 square. An oval is either the four corners of one of the square's thirty-six subrectangles or the four entry positions for the l's of any of the twenty-four 4×4 permutation matrices.

Only a few minutes reflection are needed to convince oneself that these sixty 4-subsets are, indeed, the biplane's ovals. Moreover, the sixteen 6-subsets that constitute the biplane and the sixty 4-subsets that are its ovals do, when taken together, form a generalized Steiner system of type 3-(16, {4,6}, 1). That is, every 3-subset of S is contained in a unique member of this collection. Again, a minute's reflection is all that is needed - given this geometric description.

The description does have a disadvantage: it suggests that there are two kinds of ovals when indeed there are not. The automorphism groups of the biplane is the automorphism group of the generalized Steiner system and it acts doubly-transitively on S, [3].