Small examples of mosaics of combinatorial designs

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.