High-Dimensional Holeyominoes

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