  We consider lattice tilings of $\R^n$ by a shape we call a
$(\kp,\km,n)$-quasi-cross. Such lattices form perfect error-correcting codes
which correct a single limited-magnitude error with prescribed
maximal-magnitudes of positive error and negative error (the ratio of which is
called the balance ratio). These codes can be used to correct both disturb and
retention errors in flash memories, which are characterized by having limited
magnitudes and different signs.
  We construct infinite families of perfect codes for any rational balance
ratio, and provide a specific construction for $(2,1,n)$-quasi-cross lattice
tiling. The constructions are related to group splitting and modular $B_1$
sequences. We also study bounds on the parameters of lattice-tilings by
quasi-crosses, connecting the arm lengths of the quasi-crosses and the
dimension. We also prove constraints on group splitting, a specific case of
which shows that the parameters of the lattice tiling of
$(2,1,n)$-quasi-crosses is the only ones possible.
