Possible embeddings of transversal designs in the classical projective spaces on finite fields are characterized.
2000 Academic Press
1. BACKGROUND
A transversal design of order or group size n, block size k, and index *, denoted TD * (k, n), is a triple (V, G, B), where
(1) V is a set of kn elements;
(2) G is a partition of V into k classes (called groups), each of size n;
(3) B is a collection of k-subsets of V (called blocks);
(4) every unordered pair of elements from V is either contained in exactly one group, or is contained in exactly * blocks, but not both.
When *=1, one writes simply TD(k, n). A transversal design TD(k, n) is equivalent to a set of k&2 mutually orthogonal latin squares (MOLS) of side n, and also to an orthogonal array of strength two having n 2 columns, k rows, and n symbols; see [1,2].
Constructions of MOLS have been widely studied; see [6] for a concise survey. Principal among the construction methods are Wilson's theorem using transversal designs recursively, and the Bose Shrikhande Parker theorem using pairwise balanced designs. Both employ ingredient designs that arise primarily (at least currently) from configurations embedded in projective planes. See [7,8,10] for specific examples, [6] for a more complete inventory, and [1,5] for uses in making MOLS. Often the configuration is of interest as a consequence of the manner in which it is embedded