Mosaics of Combinatorial Designs
Oliver W. Gnilke, Marcus Greferath, Mario Osvin Pav\v{c}evi\'c

TL;DR
This paper introduces the concept of mosaics of combinatorial designs, representing matrices as decompositions into multiple incidence matrices, with applications in experiment design and extensions to q-analogues.
Contribution
It defines and constructs infinite series of mosaics of designs, extending the concept to q-analogues and posing open problems.
Findings
Constructed infinite series of design mosaics
Extended the notion to q-analogues of designs
Identified open problems in the theory
Abstract
Looking at incidence matrices of - designs as matrices with possible entries, each of which indicates incidences of a -design, we introduce the notion of a -mosaic of designs, having the same number of points and blocks, as a matrix with different entries, such that each entry defines incidences of a design. In fact, a matrix is decomposed in 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 -analogues of designs in a…
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