X - Transversal designs and nets

Summary

Das Unzulängliche,

Hier wird's Ereignis

(Goethe)

In this chapter we continue the study of transversal designs (or equivalently, nets, and for λ = 1, sets of mutually orthogonal Latin squares). The emphasis will be on providing more advanced existence and non-existence results.

The standard reference on Latin squares is the book by Dénes and Keedwell (1974); the same authors also edited an important collection of surveys, see Denes and Keedwell (1991). Both these books emphasise the viewpoint of Latin squares as opposed to nets and transversal designs; thus they contain comparatively little material on geometric and group theoretic aspects. For a survey on Latin squares, transversal designs and nets with particular emphasis on their automorphism groups, see Jungnickel (1990a).

A Recursive Construction

The following recursive construction has proved to be fundamental for the recursive existence theory of nets and Latin squares. It is essentially due to Bose and Shrikhande (1960b), see also Bose, Shrikhande and Parker (1960) and Hanani (1974b).

1.1 Theorem. Let k and λ be positive integers. Then

(1.1.a) GD_λ(TD*(k), TD_λ(k)) ⊆ TD_λ(k),

and, in particular,

(1.1.b) B(TD*(k)) = TD*(k).

(Recall that TD* (k) denotes the set of g ∈ ℕ for which a TD[k; g] with a parallel class exists. The special case (1.1.b) is Theorem IX.2.11.)