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 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 ≥ 3 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 ≤ 106, we conjecture that those lower bounds are tight.Index Terms−Asymmetric error, single error, flash memories, limited magnitude error.
I IntroductionFlash memory is a kind of non-volatile memory which has higher transfer speed, longer life span and less sensitive of vibration than hard disks. But the material of flash memory is expensive and has fixed blocks, which makes it necessary to increase the density of flash memory. At the same time, it faces many challenges such as how to implement codes correcting asymmetric errors into the flash memories. In [1], the asymmetric channel with limited magnitude errors was introduced and the further results were given in [2,3]. An error model with asymmetric errors of limited magnitude is a good model for some multilevel flash memories. In the asymmetric error model, a symbol a over an alphabet Z q = {0, 1, · · · , q − 1} may be modified during transmission into b, where b ≥ a, and the probability that a is changed to b is considered to be the same for all b > a. For some applications, the error magnitude b − a is not likely to exceed a certain level λ. In general, the errors are mostly asymmetric and some classes of construction of asystematic codes correcting such errors were studied in [5,6,7]. Also, several constructions of systematic codes correcting single errors are given in [6] and the symmetric case is closely related to equi-difference conflict-avoiding codes see e.g., [10,13]. In addition, splitter sets can be seen as codes correcting single limited magnitude errors in flash memories see e.g., [4,7,8,9,11,12,13].On the other hand, construction of codes correcting t errors can be transformed to B t [λ](q) sets and the construction of a maximal size B 1 [3](2 k r) set, B 1 [3](3 k r) set and B 1 [4](3 k r) set can be found in [7]. In this paper, we consider the construction of a maximal size B 1 [4](2 k r) set. In Section II, we briefly introduce B 1 [λ](q) set and linear codes over the ring Z q . Indeed, we recall some basic results on B 1 [λ](q) sets. In Section III, we reduce the construction of a maximal size B 1 [4](2 k r) set for k ≥ 3 to the construction of a maximal size B 1 [4](2 k−3 r) set. In Section IV, we give an exact formula for calculating a maximal size * The authors are with school
