# Artificial Neural Network Assisted Error Correction for MLC NAND Flash Memory
## 2 Channel Model
### Why Are Artificial Neural Networks Useful for NAND Flash Memory?  
To simplify the analysis, this subsection first discusses the case that the CCI is only generated by the vertical neighboring cell. In this case, the conditional probability distribution function of the threshold voltage, (8), is simplified to (16):  
\[p(V_{\mu}|k\in\{11,01,00,01\})=\frac{1}{4}[\mathcal{N}(\mu_{k}- \mu_{\
u},\sigma_{k}^{2}+\sigma_{t}^{2}+\sigma_{r}^{2})\] \[+\sum_{\mu_{p}}\mathcal{N}(\mu_{k}+\gamma_{y}(\mu_{p}-\mu_{e})- \mu_{t},{\sigma_{k}}^{2}+\gamma_{y}^{2}(\sigma_{p}^{2}+\sigma_{e}^{2}\] \[+2\sigma_{r}^{2})+\sigma_{t}^{2}+\sigma_{r}^{2})]. \tag{16}\]  
In (16), it is seen that the threshold voltage distribution can be divided into four parts: the distribution of cells with CCI from \"11\"-state, \"01\"-state, \"00\"-state and \"10\"-state, which are also shown in Figure 4. In an overlap region, the bits with different CCI noise levels may have different error rates. For instance, in the overlap region between \"01\"-state and \"00\"-state, the bits of the cells in \"00\"-state with CCI from neighboring cells in \"11\"-state may be wrongly detected as \"1\" in LSB. In general, we want to find the optimal reading reference voltage at the intersecting point of the distributions of two states, such  
Figure 4: Plot of the relative log-likelihood ratio (LLR) versus data retention time at \(PE=1K\), \(\triangle=0.05\) and \(s=1\).  
Figure 3: Illustration of the statistic distribution and mathematical distribution at \(s=1\) and \(PE=1K\).  
as the red dotted line in Figure 5. However, once we know the programmed state or the threshold voltage of the cells that donate the CCI to victim cells, the optimal reading reference voltage may change. For example, the optimal reading reference voltage should be selected by the blue dotted line in Figure 5, when the vertical neighboring cell is in the erased state.  
In this paper, we expand the two-dimensional coordinates to three-dimensional, as shown in Figure 6a. The X-axis is the victim cell's voltage, and the Y-axis is the threshold voltage of vertical neighboring cell. By doing so, one can easily find the incorrectly detected cells, marked with red dots. Moreover, we have two important observations:  
1. The correct cells (the blue dots) and the incorrect cells (the red dots) are not interlaced in the three-dimensional space. It means that the correct cells (or the incorrect cells) have similar features, which may be used for clustering them from the incorrect ones.
2. The hard decision may not be the optimal decision when the surrounding cells have been read. In Figure 6a, the gray plane is the hard-decision plane, but not optimal. Suppose that there is a decision plane, shown as Figure 6b, and then we apply this plane to the same data in Figure 6a. One can see that the decision performance by the plane gets significantly improved compared to the plane in Figure 6a.  
These two observations reveal that the detection of bits in a cell can be transformed into a clustering problem, which is to obtain an optimal classification hyperplane. When more surrounding cells are considered, the clustering problem will become more complex and the  
Figure 5: Illustration of the distribution of NAND flash memory at \(s=1.4\) (the cell-to-cell coupling strength factor), \(PE=1K\) and \(Retention\ time=10^{5}\).  
Figure 6: Illustration of the decision of least significant bit (LSB) in the NAND flash memory. (**a**) The conventional hard-decision plane in the three-dimensional coordinates. (**b**) The optimal plane.  
dimensions of the classification hyperplane will increase beyond three. To address this issue, We propose to use the neural network, which is good at solving various clustering problems.