Transversal Designs in Classical Planes and Spaces.pdf

Transversal Designs in Classical Planes and Spaces Aiden A Bruen and Charles J Colb ourn Computer Science University of Vermont Burlington VT USA Abstract Possible emb eddings of transversal designs in the classical pro jective spaces on nite elds are characterized Background A transversal design of order or gr oup size n block size k and index denoted TD k n is a triple V G B where V is a set of k n elements G is a partition of V into k classes called gr oups each of size n B is a collection of ksubsets of V called blocks every unordered pair of elements from V is either contained in exactly one group or is contained in exactly blo cks but not b oth When one writes simply TD k n A transversal design TD k n is equivalent to a set of k mutually orthogonal latin squares MOLS of side n and also to an orthogonal array of strength two having n2 columns k rows and n symb ols see Constructions of MOLS have b een widely studied see for a concise survey Principal among the construction metho ds are Wilsons theorem using transversal designs recursively and the BoseShrikhandeParker theorem using pairwise balanced designs Both employ ingredient designs that arise primarily at least currently from congurations emb edded in pro jective planes See for sp ecic examples for a more complete inventory and for uses in making MOLS Often the conguration is of interest as a consequence of the manner in which it is emb edded in the plane see esp ecially Among those congurations most extensively exploited in this way are pro jective subplanes ane subplanes and blo cking