Mosaics of Combinatorial Designs
Abstract
Looking at incidence matrices of t-(v,k,λ) designs as v × b matrices with 2 possible entries, each of which indicates incidences of a t-design, we introduce the notion of a c-mosaic of designs, having the same number of points and blocks, as a matrix with c different entries, such that each entry defines incidences of a design. In fact, a v × b matrix is decomposed in c incidence matrices of designs, each denoted by a different colour, hence this decomposition might be seen as a tiling of a matrix with incidence matrices of designs as well. These mosaics have applications in experiment design when considering a simultaneous run of several different experiments. We have constructed infinite series of examples of mosaics and state some probably non-trivial open problems. Furthermore we extend our definition to the case of q-analogues of designs in a meaningful way.