This work was supported in part by National Natural Science Foundation of China (NSFC) (Grant Nos. 61501116, 61571105), Jiangsu Provincial NSF for Excellent Young Scholars (Grant No. BK20140636), Huawei HIRP Flagship (Grant No. YB201504), Fundamental Research Funds for the Central Universities, SRTP of Southeast University, State Key Laboratory of ASIC System (Grant No. 2016KF007), ICRI for MNC, and Project Sponsored by the SRF for the Returned Overseas Chinese Scholars of MoE.
Appendix
Proof for Gray code mapping scheme
proof As mentioned in Subsection $$A = \left( \begin{array}{l}0\\0\end{array} \right), B = \left( \begin{array}{l}1\\0\end{array} \right), C = \left( \begin{array}{l}1\\1\end{array} \right), D = \left( \begin{array} {l} 0\\1\end{array} \right).$$
Each scheme has $2$ rows and if $1$ bit is different from its neighbouring bits in a row, we will call it a change. For example, the combination $ABCD$ indicates the mapping scheme shown in Table
We have already known that raw errors usually happen in overlapped regions. For a $2$-bit cell, there remains $3$ overlapped regions. Assume the raw error probability for each region is $P_1$, $P_2$ and $P_3$, respectively. Therefore, the expectation of raw errors $N_G$ of mapping schemes using Gray code is \begin{equation} N_G=P_1+P_2+P_3, \tag{23}\end{equation}
\begin{equation} \begin{aligned} N_A = \alpha {P_1} + \beta {P_2} + \gamma {P_3} (\alpha + \beta + \gamma = 4\text{ or }5, \alpha \beta \gamma\neq 0). \end{aligned} \tag{24}\end{equation}
$N_A$ is absolutely bigger than $N_G$. In other words, we can tell that Gray code is the best choice for mapping schemes.
Calculation of boundaries in quantized-soft decoder
In Subsection
Since $p^{k}(x)$ is a Gaussian distribution, $p^{(k)}(B_l^{(k)})$ and $p^{(k+1)}(B_l^{(k)})$ are \begin{equation}\left\{ \begin{aligned} & {p^{(k)}}\left(B_l^{(k)}\right)= \dfrac{1}{{{\sigma _k}\sqrt {2\pi } }}\exp \left( - \dfrac{{{{(B_l^{(k)} - {\mu _k})}^2}}}{{2\sigma _k^2}}\right), \\ & {p^{(k + 1)}}\left(B_l^{(k)}\right) =\dfrac{1}{{{\sigma _{k + 1}}\sqrt {2\pi } }}\exp \left( - \dfrac{{{{(B_l^{(k)} - {\mu _{k + 1}})}^2}}}{{2\sigma _{k + 1}^2}}\right). \end{aligned} \right. \end{equation} \begin{equation}\frac{{{\sigma _k}}}{{{\sigma _{k + 1}}}}R = \exp \left( - \frac{{{{(B_l^{(k)} - {\mu _k})}^2}}}{{2\sigma _k^2}} + \frac{{{{(B_l^{(k)} - {\mu _{k + 1}})}^2}}}{{2\sigma _{k + 1}^2}}\right).\end{equation}
Take the log of both sides of the equation, we get \begin{equation}\log\left(\frac{{{\sigma _k}}}{{{\sigma _{k + 1}}}}R\right) = - \frac{{{{(B_l^{(k)} - {\mu _k})}^2}}}{{2\sigma _k^2}} + \frac{{{{(B_l^{(k)} - {\mu _{k + 1}})}^2}}}{{2\sigma _{k + 1}^2}}.\end{equation} \begin{equation*}2{\sigma _k^2\sigma _{k + 1}^2}\log \left(\frac{{{\sigma _k}}}{{{\sigma _{k + 1}}}}R\right) = - \sigma _{k + 1}^2{({B_l}^{(k)} - {\mu _k})^2}+ \sigma _k^2{({B_l}^{(k)} - {\mu _{k + 1}})^2},\end{equation*}
The other fraction can be derived in the same way.
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Figure 1
Modeling and Gray mapping scheme.
Figure 2
(Color online) BER results of a (1024,512) hard-decision polar decoder.
Figure 3
Error correction module in SSD controller.
Figure 4
Flow chart of pre-check scheme.
Figure 5
(Color online) An example for soft LLR calculation based on a specific division of $O_i$ and $Z_i$.
Figure 6
Non-uniform sensing operations
Figure 7
(Color online) Different references for the LSB and MSB.
Figure 8
(Color online) Quantization boundaries for the LSB in MLC.
Figure 9
Channel model for the LSB.
Figure 10
$4$-input, $7$-output MLC model for MMI scheme.
Figure 11
(Color online) $9$ reference in practical scheme.
Figure 12
Channel models for the LSB in practical scheme.
Figure 13
Channel models for the MSB in practical scheme.
Figure 14
(Color online) Detection in binary-input decoder.
Figure 15
Proposed architecture of binary Type I PE.
Figure 16
Proposed architecture of binary Type II PE.
Figure 17
(Color online) FER performance of an $(8192,$ $7168)$ polar code and an $(8192,7168)$ QC-LDPC code.
Figure 18
(Color online) Comparison of decoding complexity between different algorithms.
Figure 1
Modeling and Gray mapping scheme.
Figure 2
(Color online) BER results of a (1024,512) hard-decision polar decoder.
Figure 3
Error correction module in SSD controller.
Figure 4
Flow chart of pre-check scheme.
Figure 5
(Color online) An example for soft LLR calculation based on a specific division of $O_i$ and $Z_i$.
Figure 6
Non-uniform sensing operations
Figure 7
(Color online) Different references for the LSB and MSB.
Figure 8
(Color online) Quantization boundaries for the LSB in MLC.
Figure 9
Channel model for the LSB.
Figure 10
$4$-input, $7$-output MLC model for MMI scheme.
Figure 12
Channel models for the LSB in practical scheme.
Figure 13
Channel models for the MSB in practical scheme.
Figure 15
Proposed architecture of binary Type I PE.
Figure 16
Proposed architecture of binary Type II PE.
Figure 17
(Color online) FER performance of an $(8192,$ $7168)$ polar code and an $(8192,7168)$ QC-LDPC code.
Figure 18
(Color online) Comparison of decoding complexity between different algorithms.