  For a set
  $\cM=\{-\mu,-\mu+1,\ldots, \lambda\}\setminus\{0\}$ with non-negative
integers $\lambda,\mu<q$ not both 0, a subset $\cS$ of the residue class ring
$\Z_q$ modulo an integer $q\ge 1$ is called a $(\lambda,\mu;q)$-\emph{covering
set} if $$ \cM \cS=\{ms \bmod q : m\in \cM,\ s\in \cS\}=\Z_q. $$ Small covering
sets play an important role in codes correcting limited-magnitude errors. We
give an explicit construction of a $(\lambda,\mu;q)$-covering set $\cS$ which
is of the size $q^{1 + o(1)}\max\{\lambda,\mu\}^{-1/2}$ for almost all integers
$q\ge 1$ and of optimal size $p\max\{\lambda,\mu\}^{-1}$ if $q=p$ is prime.
Furthermore, using a bound on the fourth moment of character sums of Cochrane
and Shi we prove the bound $$\omega_{\lambda,\mu}(q)\le
q^{1+o(1)}\max\{\lambda,\mu\}^{-1/2},$$ for any integer $q\ge 1$, however the
proof of this bound is not constructive.
