  Limited magnitude asymmetric error model is well suited for flash memory. In
this paper, we consider the construction of asymmetric codes correcting single
error over $\mathbb{Z}_{2^{k}r}$ and which are based on so called
$B_{1}[4](2^{k}r)$ set. In fact, we reduce the construction of a maximal size
$B_{1}[4](2^{k}r)$ set for $k\geq3$ to the construction of a maximal size
$B_{1}[4](2^{k-3}r)$ set. Finally, we give a explicit formula of a maximal size
$B_{1}[4](4r)$ set and some lower bounds of a maximal size $B_{1}[4](2r)$ set.
By computer searching up to $q\leq106$, we conjecture that those lower bounds
are tight.
