Asymmetric Single Magnitude Four Error Correcting Codes

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.