Mosaics of Combinatorial Designs

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 $t$-$(v,k,\lambda)$ designs as $v \times b$ 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 $v \times b$ 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.