Incidence Structures and Incidence Matrices

The most basic notion in (finite) geometry is that of an incidence structure. It contains nothing more than the idea that two objects from distinct classes of things (say points and lines) may be “incident” with each other. The only requirement will be that the classes do not overlap. We now make this more precise.

Definitions.

An incidence structure is a triple D = (V, B, I) where V and B are any two disjoint sets and I is a binary relation between V and B, i.e. I ⊆ V × B. The elements of V will be called points, those of B blocks and those of I flags. Instead of (p, B) ∈ I, we will simply write pI B and use such geometric language as “the point p lies on the block B”, “B passes through p”, “p and B are incident”, etc.

For reasons of convenience, we will usually not state whether a given object is a point or a block; this will be clear from the context and we will always use lower case letters (e.g. p, q, r, …) to denote points and upper case letters (e.g. B, C, …) to denote blocks. Now let us look at some examples! Of course, familiar (euclidean) geometry provides examples, e.g. taking points and lines (as blocks) in the euclidean plane or points and planes (as blocks) in 3-space. One reason for choosing the term “block” instead of “line” is that we will very often consider planes or hyperplanes as blocks.

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) GDλ(TD*(k), TDλ(k)) ⊆ TDλ(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.)

The subject of this chapter is a detailed study of the Steiner systems S(4, 5; 11), S(5, 6; 12), S(4, 7; 23) and S(5, 8; 24) which were constructed independently by Carmichael (1937) andWitt (1938a).Witt (1938b) also sketched a proof for the uniqueness of these Steiner systems (up to isomorphism); a detailed uniqueness proof was given by Lüneburg (1969). These Steiner systems are now usually called the Witt designs. Their automorphism groups are the Mathieu groups discovered by Mathieu (1861, 1873), which are the only finite t-transitive permutation groups with t ≥4, except for the symmetric and alternating groups. The (binary respectively ternary) codes of theWitt designs are the Golay codes constructed by Golay (1949), the only perfect t-error correcting codes with t ≥2.

Nowadays there are quite a few existence and uniqueness proofs for these Steiner systems, and many books and papers discuss the relationship between the Witt designs, the Mathieu groups and the Golay codes. We have tried to keep our proofs of the existence and uniqueness of the Witt designs as free from using the methods and results of coding theory as possible; for the opposite (and quite effective) approach, see, for instance, MacWilliams and Sloane (1977).

The Existence of the Witt Designs

Introduction. We have already provided existence proofs for the Witt designs in §III.8, using the Kramer–Mesner approach for the construction of t-designs; see Examples III.8.8 and III.8.9.