Small examples of mosaics of combinatorial designs

TL;DR

This paper presents the first known mosaics of three distinct combinatorial designs, explores non-resolvable mosaics, and provides an example of a projective plane mosaic that cannot be generated by tiling groups with difference sets.

Contribution

It introduces new examples of mosaics of combinatorial designs, including the first of their kind with three designs, and addresses open questions about non-resolvable mosaics and construction methods.

Findings

First example of a mosaic of three combinatorial designs with distinct parameters.

Examples of non-resolvable mosaics of 2-(9,3,2) designs.

A mosaic of projective planes of order 3 not obtainable by tiling groups.

Abstract

We give the first example of a mosaic of three combinatorial designs with distinct parameters $2$-$(13,3,1)$, $2$-$(13,4,2)$, and $2$-$(13,6,5)$. Furthermore, we give examples of mosaics of $2$-$(9,3,2)$ designs that are not resolvable, thereby answering a question posed by M. Wiese and H. Boche. Finally, we give an example of a mosaic of projective planes of order $3$ that cannot be obtained by tiling groups with difference sets.