  Consider an independent site percolation model on $\Z^d$, with parameter $p
\in (0,1)$, where all long range connections in the axes directions are
allowed. In this work we show that given any parameter $p$, there exists and
integer $K(p)$ such that all binary sequences (words) $\xi \in \{0,1\}^{\N}$
can be seen simultaneously, almost surely, even if all connections whose length
is bigger than $K(p)$ are suppressed. We also show some results concerning the
question how $K(p)$ should scale with $p$ when $p$ goes to zero. Related
results are also obtained for the question of whether or not almost all words
are seen.
