High-dimensional holeyominoes

What is the maximum number of holes enclosed by a $d$-dimensional polyomino built of $n$ tiles? Represent this number by $f_d(n)$. Recent results show that $f_2(n)\mathrm{/}n$ converges to $1\mathrm{/}2$. We prove that for all $d\ge 2$ we have $f_d(n)\mathrm{/}n\to (d-1)\mathrm{/}d$ as $n$ goes to infinity. We also construct polyominoes in $d$-dimensional tori with the maximal possible number of holes per tile. In our proofs, we use metaphors from error-correcting codes and dynamical systems.