## 3 Problem Statement and Principal Approach
### Mutual Information  
Next, we investigate the information that is obtained by counting the number of errors in the read codewords. Later, we predict the shift of the reference voltages based on the number of observed errors in the read codeword. In addition to the total number of errors, we exploit the asymmetry between 0-errors and 1-errors.  
We use information theoretic measures to investigate the relation between the optimal read references and the statistical information obtained by counting errors. In particular, we consider the mutual information. The mutual information \(I(X;Y)\) of two random variables is a measure for the information in bits gained about one variable when knowing the other variable. Let \(X\) and \(Y\) be two random variables with \(X\in\{x_{1},x_{2},...,x_{N}\}\) and \(Y\in\{y_{1},y_{2},...,y_{M}\}\) with probability distributions \(f_{X}(x_{i})\) and \(f_{Y}(y_{j})\) and joint probability distribution \(f_{XY}(x_{i},y_{j})\). The mutual information between \(X\) and \(Y\) is defined as [43]  
\[I(X;Y)=\sum_{i=1}^{N}\sum_{j=1}^{M}f_{XY}(x_{i},y_{j})\cdot log2\Bigg{(}\frac{f _{X|Y}(x_{i}|y_{j})}{f_{X}(x_{i})}\Bigg{)}. \tag{6}\]  
The mutual information is non-negative and upper bounded by the minimum of the entropies of the two variables [43]. For practical applications, it can have advantages using other measures, e.g., the residual error rate of a given decoder. But those measures are very specific to a given implementation.  
Figure 7 shows the mutual information \(I(E;V_{opt})\) between the optimal reference voltage \(V_{opt}\) and the observed number of errors \(E\). The mutual information is determined over different life-cycle states. As a second observation, we consider a binary random variable \(A\), which indicates the direction of the asymmetry between 0-errors and 1-errors. \(A=1\) indicates more 1-errors, and \(A=0\) indicates more 0-errors. The mutual information \(I(A;V_{opt})\) is also depicted in Figure 7. Note that this figure shows the curves for reference \(r_{3}\).  
Figure 6: Number of errors in meta-data over threshold voltage.  
When using the number of errors in the meta-data, we get the maximum information about the optimum reference when reading at about \(-14\), i.e., \(14\) voltage steps lower than the mean optimum reference. Near the optimum, the mutual information is fairly low, and this is due to the fact that the entropy of the number of errors is low in that region. Reading far from the optimum (\(|\Delta V_{th}|\geq 30\)), we obtain no information about \(V_{opt}\) because the read data is not correctable. In the case of a decoding failure, the exact number of errors remains unknown. Near the optimum \(\Delta V_{th}=0\), we can see that the binary variable \(A\) that indicates the asymmetry provides more information about \(V_{opt}\) than the number of errors \(E\).  
The asymmetry for higher charge values shown in Figure 5 also leads to an asymmetry in the mutual information. This can be seen in Figure 8, where the mutual information versus the read voltage is plotted for reference \(r_{7}\). As we can see, the maximum of the mutual information between the asymmetry \(A\) and the optimum threshold voltage is not at \(\Delta V_{th}=0\). Similarly, the local minimum for \(I(E;V_{opt})\) does not occur at \(\Delta V_{th}=0\).