  For a finite lattice $L$, let Gm($L$) denote the least $n$ such that $L$ can
be generated by $n$ elements. For integers $r>2$ and $k>1$, denote by FD$(r)^k$
the $k$-th direct power of the free distributive lattice FD($r$) on $r$
generators. We determine Gm(FD$(r)^k$) for many pairs $(r,k)$ either exactly or
with good accuracy by giving a lower estimate that becomes an upper estimate if
we increase it by 1. For example, for $(r,k)=(5,25\,000)$ and $(r,k)=(20,\
1.489\cdot 10^{1789})$, Gm(FD$(r)^k$) is $300$ and $6000$, respectively. To
reach our goal, we give estimates for the maximum number of pairwise unrelated
copies of some specific posets (called full segment posets) in the subset
lattice of an $n$-element set. In addition to analogous earlier results in
lattice theory, a connection with cryptology is also mentioned among the
motivations.
