  We show that if a semisimple synchronizing automaton with $n$ states has a
minimal reachable non-unary subset of cardinality $r\ge 2$, then there is a
reset word of length at most $(n-1)D(2,r,n)$, where $D(2,r,n)$ is the
$2$-packing number for families of $r$-subsets of $[1,n]$.
