  For a given subset $A\subseteq \mathbb F_q^*$, we study the problem of
finding a large packing set $B$ of $A$, that is, a set $B \subseteq \mathbb
F_q^*$ such that $|AB|=|A||B|$. We prove the existence of such a $B$ of size
$|B|\ge (q-1)/|A/A|$ and show that this bound is in general optimal.
  The case that $q=p$ is a prime and $A=\{1,2,\ldots,\lambda\}$ for some
positive integer $\lambda$ is particularly interesting in view of the
construction of limited-magnitude error correcting codes. Here we construct a
packing set $B$ of size $|B|\gg p (\lambda \log p)^{-1}$ for any $\lambda \le c
p^{1/2}$ for some explicitly calcuable constant $c$. This result is optimal up
to the logarithmic factor.
