Transversal system

transversal design, transversal scheme, -system

A system of sets defined for a given collection of pairwise-disjoint finite sets , each of which has cardinality . Namely: a transversal system is a system of sets (blocks or transversals), containing elements each and such that:

1) ; ; ;

2) for .

In a transversal design, any two elements and , , occur together in exactly one block. The existence of a transversal design is equivalent to the existence of an orthogonal array .

Transversal designs are used in recursive constructions of block designs (cf. Block design).

A set of transversals in is called parallel if no two of them intersect. If a transversal design contains (or more) parallel classes, then it is denoted by .

Some of the basic properties of transversal systems are:

a) if and exist, then so does ;

b) exists if and only if exists.

References

[1]	M. Hall, "Combinatorial theory" , Wiley (1986)
[2]	H. Hanani, "The existence and construction of balanced incomplete block designs" Ann. Math. Stat. , 32 (1961) pp. 361–386

Comments

The finite sets making up the design are called point classes or point groups.

The existence of a transversal design is equivalent to the existence of mutually orthogonal Latin squares of order . If a exists, and , then .

See also [a1]–[a3] for the recursive construction of and existence results for transversal designs (and of their generalization "transversal design with holes" , [a2]).

One of the most important results on recursive construction of transversal designs (due to R.M. Wilson [a4], see also [a1]) is as follows:

Let be a with point classes and , let be a -subset of and put for , and for every block of . Assume the existence of for and of for each block of . Then also exists.

Transversals designs (and their dual structures, nets) are also of considerable geometric and algebraic interest. For instance, a is equivalent to an affine or projective plane of order , and a is basically the same as a quasi-group of order . Thus, the geometric and algebraic properties of transversal designs (including the study of their automorphism groups) have found considerable interest, cf. [a4].

References

[a1]	T. Beth, D. Jungnickel, H. Lenz, "Design theory" , Cambridge Univ. Press (1986)
[a2]	A.E. Brouwer, "Recursive constructions of mutually orthogonal Latin squares" Ann. Discr. Math. , 46 (1991) pp. 149–168
[a3]	D. Jungnickel, "Latin squares, their geometries and their groups: a survey" D. Ray-Chaudhuri (ed.) , Coding Theory and Design Theory , IMA Vol. Math. Appl. , 21 , Springer (1990) pp. 166–225
[a4]	R.M. Wilson, "Concerning the number of mutually orthogonal Latin squares" Discr. Math. , 9 (1974) pp. 181–198
[a5]	H. Hanani, "Balanced incomplete block designs and related designs" Discrete Math. , 11 (1975) pp. 255–369