  We study whether an asymmetric limited-magnitude ball may tile
$\mathbb{Z}^n$. This ball generalizes previously studied shapes: crosses,
semi-crosses, and quasi-crosses. Such tilings act as perfect error-correcting
codes in a channel which changes a transmitted integer vector in a bounded
number of entries by limited-magnitude errors.
  A construction of lattice tilings based on perfect codes in the Hamming
metric is given. Several non-existence results are proved, both for general
tilings, and lattice tilings. A complete classification of lattice tilings for
two certain cases is proved.
