Martin Grüttmüller
On the Number of Indecomposable Block Designs
Preprint series: Preprints aus dem Fachbereich Mathematik, Universität Rostock

MSC:

05B05 Block designs, See also {51E05, 62K10}

Abstract: A $t$-$(v,k,\lambda)$ design $\mathcal{D}$ is a system
(multiset) of $k$-element subsets (called blocks) of a
$v$-element set $V$ such that every $t$-element subsset of
$V$ occurs exactly ${\lambda}$ times in the blocks of
$\mathcal{D}$. A $t$-$(v,k,\lambda)$ design $\mathcal{D}$ is
called indecomposable (or elementary) if and only if there
is no subsystem which is a $t$-$(v,k,\lambda ')$ design
with $0<\lambda '<\lambda$. It is known that the number of
indecomposable designs for given parameters $t,v,k$ is finite.
A block design is a is $t$-$(v,k,\lambda)$ design with $t=2$.
The exact number of non-isomorphic, indecomposable block
designs is only known for $k=3$ and $v\le 7$. We computed
the number of indecomposable designs for $v\le 13$ and
$\lambda\le 6$. The algorithms used will be described.

Keywords: design, indecomposable