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} which is 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\geq 3 to the construction of a maximal size B_{1}[{4}](2^{k-3}r) set. Finally, we give an 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\leq 106 , we conjecture that those lower bounds are tight.