Quasi-Cyclic Non-Binary LDPC Codes for MLC NAND Flash Memory
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1 for MLC NAND Flash Memory Ahmed Hareedy LORIS Lab, UCLA CoDESS, UCLA Joint work with: Clayton Schoeny (UCLA), Behzad Amiri (UCLA), and Lara Dolecek (UCLA) Flash Memory Summit 2015
2 1 MLC NAND Flash Model Mixed Normal-Laplace Distribution Capacity C. Schoeny, B. Amiri, A. Hareedy, and L. Dolecek, Quasi-Cyclic Non-Binary LDPC Codes for MLC NAND Flash Memory, NVMW, March / 21
3 1 MLC NAND Flash Model Mixed Normal-Laplace Distribution Capacity 2 LDPC Codes Overview Absorbing Sets C. Schoeny, B. Amiri, A. Hareedy, and L. Dolecek, Quasi-Cyclic Non-Binary LDPC Codes for MLC NAND Flash Memory, NVMW, March / 21
4 1 MLC NAND Flash Model Mixed Normal-Laplace Distribution Capacity 2 LDPC Codes Overview Absorbing Sets 3 C. Schoeny, B. Amiri, A. Hareedy, and L. Dolecek, Quasi-Cyclic Non-Binary LDPC Codes for MLC NAND Flash Memory, NVMW, March / 21
5 1 MLC NAND Flash Model Mixed Normal-Laplace Distribution Capacity 2 LDPC Codes Overview Absorbing Sets 3 4 C. Schoeny, B. Amiri, A. Hareedy, and L. Dolecek, Quasi-Cyclic Non-Binary LDPC Codes for MLC NAND Flash Memory, NVMW, March / 21
6 Mixed Normal-Laplace Distribution Capacity Flash Model 2 / 21
7 Normal-Laplace Distribution Mixed Normal-Laplace Distribution Capacity Key properties of the Normal-Laplace Distribution: Normal-like curve but wider tails. The two tails can behave differently from one another. Normal T. Parnell, N. Papandreou, T. Mittelholzer, and H. Pozidis, Modelling of the Threshold Voltage Distributions of Sub-20nm NAND Flash Memory, Globecomm, Dec / 21
8 Normal-Laplace Distribution Mixed Normal-Laplace Distribution Capacity Key properties of the Normal-Laplace Distribution: Normal-like curve but wider tails. The two tails can behave differently from one another. Normal T. Parnell, N. Papandreou, T. Mittelholzer, and H. Pozidis, Modelling of the Threshold Voltage Distributions of Sub-20nm NAND Flash Memory, Globecomm, Dec / 21
9 Normal-Laplace Distribution Mixed Normal-Laplace Distribution Capacity Key properties of the Normal-Laplace Distribution: Normal-like curve but wider tails. The two tails can behave differently from one another. Normal Normal-Laplace T. Parnell, N. Papandreou, T. Mittelholzer, and H. Pozidis, Modelling of the Threshold Voltage Distributions of Sub-20nm NAND Flash Memory, Globecomm, Dec / 21
10 Normal-Laplace Distribution Mixed Normal-Laplace Distribution Capacity Key properties of the Normal-Laplace Distribution: Normal-like curve but wider tails. The two tails can behave differently from one another. Normal Normal-Laplace T. Parnell, N. Papandreou, T. Mittelholzer, and H. Pozidis, Modelling of the Threshold Voltage Distributions of Sub-20nm NAND Flash Memory, Globecomm, Dec / 21
11 Programming Errors Mixed Normal-Laplace Distribution Capacity Multi-modality is caused by programming errors, in which we program the wrong level. Erase failure. Error in two step programming algorithm. T. Parnell, N. Papandreou, T. Mittelholzer, and H. Pozidis, Modelling of the Threshold Voltage Distributions of Sub-20nm NAND Flash Memory, Globecomm, Dec / 21
12 Programming Errors Mixed Normal-Laplace Distribution Capacity Multi-modality is caused by programming errors, in which we program the wrong level. Erase failure. Error in two step programming algorithm. The NL-distributions and the programming error rates are parameterized by the P/E cycles. T. Parnell, N. Papandreou, T. Mittelholzer, and H. Pozidis, Modelling of the Threshold Voltage Distributions of Sub-20nm NAND Flash Memory, Globecomm, Dec / 21
13 Mixed Normal-Laplace Distribution Capacity Capacity and Optimal Input Distribution These capacity calculations are for hard-sensing. C. Schoeny, F. Sala, and L. Dolecek, Analysis and Coding Schemes for the Flash Normal-Laplace Mixture Channel, ISIT, June / 21
14 Mixed Normal-Laplace Distribution Capacity Capacity and Optimal Input Distribution These capacity calculations are for hard-sensing. Optimal input distribution are uniform until late in lifetime. C. Schoeny, F. Sala, and L. Dolecek, Analysis and Coding Schemes for the Flash Normal-Laplace Mixture Channel, ISIT, June / 21
15 Mixed Normal-Laplace Distribution Capacity Capacity and Optimal Input Distribution - Longer Lifetime is Possible These capacity calculations are for hard-sensing. Optimal input distribution are uniform until late in lifetime. Codes with rate 8/9 can support up to 28,000 P/E cycles. C. Schoeny, F. Sala, and L. Dolecek, Analysis and Coding Schemes for the Flash Normal-Laplace Mixture Channel, ISIT, June / 21
16 LDPC Codes Overview Absorbing Sets LDPC Codes 6 / 21
17 Why Non-Binary LDPC Codes? LDPC Codes Overview Absorbing Sets LDPC codes outperform commonly used BCH codes. Larger Galois field size results in better performance. 7 / 21
18 Binary LDPC Codes LDPC Codes Overview Absorbing Sets LDPC codes are a class of graph-based channel codes with capacity approaching performance. These codes can be described by a bipartite graph called a Tanner graph. 8 / 21
19 Binary LDPC Codes LDPC Codes Overview Absorbing Sets LDPC codes are a class of graph-based channel codes with capacity approaching performance. These codes can be described by a bipartite graph called a Tanner graph. Binary code H = v 1 v v 2 3 v v v v c 1 c 2 c 3 v 1 v 2 Variable nodes v 3 v v v v c 1 c c 2 3 Check nodes Parity check: c 1 = v 1 + v 2 + v 3 + v 5 over GF (2). 8 / 21
20 Non-Binary LDPC Codes LDPC Codes Overview Absorbing Sets LDPC codes are a class of graph-based channel codes with capacity approaching performance. These codes can be described by a bipartite graph called a Tanner graph. Non-binary code, GF(4) H = v v v v v v v c 1 c 2 c 3 v 1 2 Variable nodes v v 3 v v v v c c c Check nodes 3 Parity check: c 1 = 2v 1 + 3v 2 + v 3 + 2v 5 over GF (4). 9 / 21
21 LDPC Codes Overview Absorbing Sets Non-Binary LDPC Decoding Complexity Commonly used decoding algorithms: Binary: Min-Sum Non-Binary: Min-Max D. Declercq and M. Fossorier, Decoding Algorithms for Nonbinary LDPC Codes Over GF, TCOM, Apr Y. Toriyama, B. Amiri, L. Dolecek, and D. Markovic, Field-Order Based Hardware Cost Analysis of Non-Binary LDPC Decoders, Asilomar, Nov / 21
22 LDPC Codes Overview Absorbing Sets Non-Binary LDPC Decoding Complexity Commonly used decoding algorithms: Binary: Min-Sum Non-Binary: Min-Max In non-binary LDPC, decoding complexity is of order O(q log q), where q is the GF size. D. Declercq and M. Fossorier, Decoding Algorithms for Nonbinary LDPC Codes Over GF, TCOM, Apr Y. Toriyama, B. Amiri, L. Dolecek, and D. Markovic, Field-Order Based Hardware Cost Analysis of Non-Binary LDPC Decoders, Asilomar, Nov / 21
23 LDPC Codes Overview Absorbing Sets Non-Binary LDPC Decoding Complexity Commonly used decoding algorithms: Binary: Min-Sum Non-Binary: Min-Max In non-binary LDPC, decoding complexity is of order O(q log q), where q is the GF size. Current research includes complexity reduction through the use of message pruning. D. Declercq and M. Fossorier, Decoding Algorithms for Nonbinary LDPC Codes Over GF, TCOM, Apr Y. Toriyama, B. Amiri, L. Dolecek, and D. Markovic, Field-Order Based Hardware Cost Analysis of Non-Binary LDPC Decoders, Asilomar, Nov / 21
24 LDPC Codes Overview Absorbing Sets Quasi-Cyclic Non-Binary Advantages Construction is based on lifting and labeling protographs. L. Zeng, et al., Constructions of Nonbinary Quasi-Cyclic LDPC Codes: A Finite Field Approach, TCOM, Apr J. Huang, et al., Large-Girth Nonbinary QC-LDPC Codes of Various Lengths, TCOM, Dec L. Zhang, et al., Quasi-Cyclic LDPC Codes: An Algebraic Construction, Rank Analysis, and Codes on Latin Squares, IEEE Trans. Commun., Nov / 21
25 LDPC Codes Overview Absorbing Sets Quasi-Cyclic Non-Binary Advantages Construction is based on lifting and labeling protographs. Due to their implementation-friendly structure and superior performance, QC-NB codes are well-suited for emerging data storage applications requiring very low error rates. L. Zeng, et al., Constructions of Nonbinary Quasi-Cyclic LDPC Codes: A Finite Field Approach, TCOM, Apr J. Huang, et al., Large-Girth Nonbinary QC-LDPC Codes of Various Lengths, TCOM, Dec L. Zhang, et al., Quasi-Cyclic LDPC Codes: An Algebraic Construction, Rank Analysis, and Codes on Latin Squares, IEEE Trans. Commun., Nov / 21
26 LDPC Codes Overview Absorbing Sets Quasi-Cyclic Non-Binary Advantages Construction is based on lifting and labeling protographs. Due to their implementation-friendly structure and superior performance, QC-NB codes are well-suited for emerging data storage applications requiring very low error rates. We design QC-NB codes with improved performance in the low error-rate region by removing small problematic absorbing sets while maintaining desired code parameters. L. Zeng, et al., Constructions of Nonbinary Quasi-Cyclic LDPC Codes: A Finite Field Approach, TCOM, Apr J. Huang, et al., Large-Girth Nonbinary QC-LDPC Codes of Various Lengths, TCOM, Dec L. Zhang, et al., Quasi-Cyclic LDPC Codes: An Algebraic Construction, Rank Analysis, and Codes on Latin Squares, IEEE Trans. Commun., Nov / 21
27 LDPC Codes Overview Absorbing Sets Error Floor is Caused by Absorbing Sets LDPC codes suffer from an error floor when decoded by message passing algorithms, largely due to subgraphs called absorbing sets. Example: A (4,4) binary absorbing set. 12 / 21
28 LDPC Codes Overview Absorbing Sets Error Floor is Caused by Absorbing Sets LDPC codes suffer from an error floor when decoded by message passing algorithms, largely due to subgraphs called absorbing sets. Example: A (4,4) binary absorbing set. The above absorbing set is a fixed point of the bit flipping decoder, since each variable node is connected to more satisfied check nodes than unsatisfied check nodes. 12 / 21
29 LDPC Codes Overview Absorbing Sets Error Floor is Caused by Absorbing Sets LDPC codes suffer from an error floor when decoded by message passing algorithms, largely due to subgraphs called absorbing sets. Example: A (4,4) binary absorbing set. The above absorbing set is a fixed point of the bit flipping decoder, since each variable node is connected to more satisfied check nodes than unsatisfied check nodes. 12 / 21
30 LDPC Codes Overview Absorbing Sets Error Floor is Caused by Absorbing Sets LDPC codes suffer from an error floor when decoded by message passing algorithms, largely due to subgraphs called absorbing sets. Example: A (4,4) binary absorbing set. The above absorbing set is a fixed point of the bit flipping decoder, since each variable node is connected to more satisfied check nodes than unsatisfied check nodes. 12 / 21
31 LDPC Codes Overview Absorbing Sets Error Floor is Caused by Absorbing Sets LDPC codes suffer from an error floor when decoded by message passing algorithms, largely due to subgraphs called absorbing sets. Example: A (4,4) binary absorbing set. The above absorbing set is a fixed point of the bit flipping decoder, since each variable node is connected to more satisfied check nodes than unsatisfied check nodes. 12 / 21
32 Non-Binary Absorbing Sets LDPC Codes Overview Absorbing Sets Edge weights and variable node values are elements of GF (q). The shown absorbing set is called elementary absorbing set. v 1 w w 8 9 w 7 v 3 w1 w2 w 12 w 6 w 11 w 10 w 5 v 2 w 3 w 4 v 4 w i GF(q)\0 v j GF(q) \ 0 Same topological conditions as in the binary case. 13 / 21
33 Non-Binary Absorbing Sets LDPC Codes Overview Absorbing Sets Edge weights and variable node values are elements of GF (q). The shown absorbing set is called elementary absorbing set. v 1 w w 8 9 w1 w2 w 11 v 2 w 3 w i GF(q)\0 w 1 w 7 w 11 = w 2 w 8 w 12 over GF (q). w 7 v 3 w 12 w 6 w 10 w 5 w 4 v 4 v j GF(q) \ 0 Same topological conditions as in the binary case. The edges in the fundamental cycles must satisfy the weight conditions. 13 / 21
34 Non-Binary Absorbing Sets LDPC Codes Overview Absorbing Sets Edge weights and variable node values are elements of GF (q). The shown absorbing set is called elementary absorbing set. v 1 w w 8 9 w 7 v 3 w1 w2 w 12 w 6 w 11 w 10 w 5 v 2 w 3 w 4 v 4 w i GF(q)\0 v j GF(q) \ 0 w 1 w 7 w 11 = w 2 w 8 w 12 over GF (q). w 3 w 5 w 12 = w 4 w 6 w 11 over GF (q). Same topological conditions as in the binary case. The edges in the fundamental cycles must satisfy the weight conditions. 13 / 21
35 Non-Binary Absorbing Sets LDPC Codes Overview Absorbing Sets Edge weights and variable node values are elements of GF (q). The shown absorbing set is called elementary absorbing set. v 1 w w 8 9 w 7 v 3 w1 w2 w 12 w 6 w 11 w 10 w 5 v 2 w 3 w 4 v 4 w i GF(q)\0 v j GF(q) \ 0 w 1 w 7 w 11 = w 2 w 8 w 12 over GF (q). w 3 w 5 w 12 = w 4 w 6 w 11 over GF (q). w 2 w 4 w 9 = w 1 w 3 w 10 over GF (q). Same topological conditions as in the binary case. The edges in the fundamental cycles must satisfy the weight conditions. 13 / 21
36 Absorbing Set Removal LDPC Codes Overview Absorbing Sets 1 Identify a list of problematic absorbing sets. B. Amiri, J. Kliewer, and L. Dolecek, Analysis and Enumeration of Absorbing Sets for Non-Binary Graph-Based Codes, TCOM, Feb / 21
37 Absorbing Set Removal LDPC Codes Overview Absorbing Sets 1 Identify a list of problematic absorbing sets. 2 Find all binary absorbing sets of interest in the unlabeled bipartite graph. B. Amiri, J. Kliewer, and L. Dolecek, Analysis and Enumeration of Absorbing Sets for Non-Binary Graph-Based Codes, TCOM, Feb / 21
38 Absorbing Set Removal LDPC Codes Overview Absorbing Sets 1 Identify a list of problematic absorbing sets. 2 Find all binary absorbing sets of interest in the unlabeled bipartite graph. 3 For each candidate, check if the weight conditions are satisfied. B. Amiri, J. Kliewer, and L. Dolecek, Analysis and Enumeration of Absorbing Sets for Non-Binary Graph-Based Codes, TCOM, Feb / 21
39 Absorbing Set Removal LDPC Codes Overview Absorbing Sets 1 Identify a list of problematic absorbing sets. 2 Find all binary absorbing sets of interest in the unlabeled bipartite graph. 3 For each candidate, check if the weight conditions are satisfied. 4 The labeling parameters in the process of constructing the QC-NB-LDPC code are modified such that the weight condition of absorbing sets is not satisfied. B. Amiri, J. Kliewer, and L. Dolecek, Analysis and Enumeration of Absorbing Sets for Non-Binary Graph-Based Codes, TCOM, Feb / 21
40 Absorbing Set Removal LDPC Codes Overview Absorbing Sets 1 Identify a list of problematic absorbing sets. 2 Find all binary absorbing sets of interest in the unlabeled bipartite graph. 3 For each candidate, check if the weight conditions are satisfied. 4 The labeling parameters in the process of constructing the QC-NB-LDPC code are modified such that the weight condition of absorbing sets is not satisfied. 5 Continue until no more non-binary absorbing sets can be eliminated. B. Amiri, J. Kliewer, and L. Dolecek, Analysis and Enumeration of Absorbing Sets for Non-Binary Graph-Based Codes, TCOM, Feb / 21
41 15 / 21
42 Shifting Gaussian MLC Channel First let us view the results from a previously used model. Each state is modeled as a Gaussian distribution with shifting mean and variance dependent on the P/E cycles. 16 / 21
43 Shifting Gaussian MLC Channel First let us view the results from a previously used model. Each state is modeled as a Gaussian distribution with shifting mean and variance dependent on the P/E cycles. Binary LDPC code decoded using the min-sum algorithm. Non-binary LDPC code decoded using FFT-sum-product algorithm. Block lengths 2k bits. Code rates 8/9. GF sizes = 4. Column weights = / 21
44 Shifting Gaussian MLC Channel 17 / 21
45 Shifting Gaussian MLC Channel 17 / 21
46 Shifting Gaussian MLC Channel 17 / 21
47 Shifting Gaussian MLC Channel 17 / 21
48 Shifting Gaussian MLC Channel Superior performance for non-binary LDPC codes with AS removal compared to BCH and binary LDPC codes. 17 / 21
49 Normal-Laplace Mixture MLC Channel Soft-Information refers to 6-reads. 18 / 21
50 Normal-Laplace Mixture MLC Channel Block length = 1692 bits. Code rate 8/9. GF size = 4. Column weight = / 21
51 Normal-Laplace Mixture MLC Channel Block length = 1692 bits. Code rate 8/9. GF size = 4. Column weight = / 21
52 Normal-Laplace Mixture MLC Channel Block length = 1692 bits. Code rate 8/9. GF size = 4. Column weight = / 21
53 Normal-Laplace Mixture MLC Channel Block length = 1058 bits. Code rate 8/9. GF size = 4. Column weight = / 21
54 Normal-Laplace Mixture MLC Channel Block length = 1058 bits. Code rate 8/9. GF size = 4. Column weight = / 21
55 MLC NAND Flash Model Non-binary LDPC codes offer excellent error-correcting performance over NAND Flash models. 20 / 21
56 MLC NAND Flash Model Non-binary LDPC codes offer excellent error-correcting performance over NAND Flash models. AS removal provides very promising results over the Gaussian Flash channel model. 20 / 21
57 MLC NAND Flash Model Non-binary LDPC codes offer excellent error-correcting performance over NAND Flash models. AS removal provides very promising results over the Gaussian Flash channel model. Further performance improvement is achievable over the NL model by accurately identifying the objects which dominate the error floor (ongoing research). 20 / 21
58 MLC NAND Flash Model Non-binary LDPC codes offer excellent error-correcting performance over NAND Flash models. AS removal provides very promising results over the Gaussian Flash channel model. Further performance improvement is achievable over the NL model by accurately identifying the objects which dominate the error floor (ongoing research). Concurrent work on partial-response channels demonstrated significant performance gain by properly identifying and removing the right objects. A. Hareedy, B. Amiri, S. Zhao, R. Galbraith, and L. Dolecek, Non-Binary LDPC Code Optimization for Partial-Response Channels, Globecom, Dec / 21
59 Thank You 21 / 21
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