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.
Categories: cs.IT math.IT