Minimum-sized generating sets of the direct powers of free distributive lattices

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.