Spatially-Coupled Codes: Recent Trends and Applications in Data - - PowerPoint PPT Presentation

SLIDE 1

Spatially-Coupled Codes: Recent Trends and Applications in Data Storage and Memories

• Prof. Lara Dolecek

ECE Department, UCLA NVMW, UC San Diego, March 2019

SLIDE 2

w As data generation continues to grow…

Data-driven Demand for Dense and Reliable Memories and Storage

1

[Source: HortonWorks]

SLIDE 3

Data-driven Demand for Dense and Reliable Memories and Storage

1

[Source: HortonWorks] [Source: Western Digital]

w As data generation continues to grow…so does the need for

reliable storage

SLIDE 4

ECC methods have had a long and successful career in data storage Hamming and parity codes BCH and Reed-Solomon Codes Constrained Codes LDPC Codes Convolutional Codes

1950 1960 1970 1980 1990 2000 2010 2020

2

SLIDE 5

ECC methods have had a long and successful career in data storage

w ECC methods have been the trusted go-to workhorse for

improving reliability in data storage and memories. Hamming and parity codes BCH and Reed-Solomon Codes Constrained Codes LDPC Codes Convolutional Codes

1950 1960 1970 1980 1990 2000 2010 2020

2

SLIDE 6

Data-driven Demand for Dense and Reliable Memories and Storage

w Transformative, data-driven technological revolution is already

underway.

w Storage systems must support this modern data revolution.

3

SLIDE 7

Data-driven Demand for Dense and Reliable Memories and Storage

w Transformative, data-driven technological revolution is already

underway.

w Storage systems must support this modern data revolution. w Reliability requirements are associated with additional sources of

errors in modern memory and storage devices.

3

[Parnell ’14] [Garani ’18]

SLIDE 8

ECC methods have had a long and successful career in data storage Hamming and parity codes BCH and Reed-Solomon Codes Constrained Codes LDPC Codes Convolutional Codes

1950 1960 1970 1980 1990 2000 2010 2020

Spatially Coupled Codes -- a Solution ?

[Parnell ’14] [Garani ’18]

4

SLIDE 9

Today’s Talk: Spatially-Coupled (SC) Codes

w SC codes: complexity/latency advantages, and capacity-

approaching performance. – Many recent exciting results [Urbanke, Iyengar and Siegel, Pfister, Costello

et al., Lentmeier et al., Mitchell et al., Kasai, and many others]

– Still, large room for improving their finite-length performance for both canonical and non-canonical channels.

w We provide a combinatorial framework for design and

• ptimization of spatially-coupled (SC) codes.

– One dimensional and multi-dimensional constructs

• Theoretical properties
• Experimental results

Our goal is to design and develop advanced ECC methods that exploit properties of the underlying physical channels.

5

SLIDE 10

Laboratory for Robust Information Systems at UCLA

w UCLA students and PI w Our collaborators

Ah Ahmed Ha Hareedy (now Post stdoc c at Duke ke) Siyi yi (Prisca sca) Ya Yang Ruiyi yi (Jo John) Wu Wu Lev Lev Ta Tauz Ho Homa Esf sfahaniza zadeh An Andrew Ta Tan Ze Zehui (Alex) x) Ch Chen Dr

• Dr. Dariush

sh Divsa vsalar Pr

• Prof. Laura Conde-Canenci

cia Pr

• Prof. Pu

Puneet Gu Gupta ta

• Prof. Yuva

val Cassu ssuto

Lar Lara a Dol

• lec

ecek ek

6

SLIDE 11

Laboratory for Robust Information Systems at UCLA

w UCLA students and PI w Our collaborators

Ah Ahmed Ha Hareedy (now Post stdoc c at Duke ke) Siyi yi (Prisca sca) Ya Yang Ruiyi yi (Jo John) Wu Wu Lev Lev Ta Tauz Ho Homa Esf sfahaniza zadeh An Andrew Ta Tan Dr

• Dr. Dariush

sh Divsa vsalar Pr

• Prof. Laura Conde-Canenci

cia Pr

• Prof. Pu

Puneet Gu Gupta ta

• Prof. Yuva

val Cassu ssuto

Spatially y co coupled co codes s rese search ch team

Ze Zehui (Alex) x) Ch Chen

Lar Lara a Dol

• lec

ecek ek

6

SLIDE 12

Spatially Coupled Codes Research Team at UCLA

w UCLA students and PI w Our collaborators

Ah Ahmed Ha Hareedy (now Post stdoc c at Duke ke) Siyi yi (Prisca sca) Ya Yang Ruiyi yi (Jo John) Wu Wu Lev Lev Ta Tauz Ho Homa Esf sfahaniza zadeh An Andrew Ta Tan Dr

• Dr. Dariush

sh Divsa vsalar Pr

• Prof. Laura Conde-Canenci

cia Pr

• Prof. Pu

Puneet Gu Gupta ta

• Prof. Yuva

val Cassu ssuto

Givi ving talks ks at NV NVMW 2019

Ze Zehui (Alex) x) Ch Chen

Lar Lara a Dol

• lec

ecek ek

6

SLIDE 13

Talk Outline

w Mathematical Preliminaries w Combinatorial Framework for SC Codes w Concluding Remarks and Future Directions

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SLIDE 14

Talk Outline

w Mathematical Preliminaries

– Graphical and parity check matrix description

• f SC codes

– Decoding errors under iterative decoding

w Combinatorial Framework for SC Codes w Concluding Remarks and Future Directions

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SLIDE 15

w Parity Check Matrix

Parity-Check Matrix and Tanner Graph

9

circles: variable nodes squares: check nodes

𝐼 =

columns: variable nodes rows: check nodes

SLIDE 16

w Parity Check Matrix w Tanner Graph

Parity-Check Matrix and Tanner Graph

10

circles: variable nodes squares: check nodes

𝐼 =

columns: variable nodes rows: check nodes

SLIDE 17

w Parity Check Matrix w Tanner Graph

Parity-Check Matrix and Tanner Graph

10

columns: variable nodes rows: check nodes circles: variable nodes squares: check nodes

𝐼 =

𝝉𝒈𝒋,𝒌 is a circulant matrix with size p×p and power 𝑔

+,,.

𝐽 𝐽 𝐽 𝐽 𝐽 𝐽 𝜏 𝜏0 𝜏1 𝜏2 𝐽 𝜏0 𝜏2 𝜏 𝜏1 𝐽 𝜏1 𝜏 𝜏2 𝜏0

SLIDE 18

Spatially-Coupled Code as a Chain of Block Codes

w Uncoupled Block Codes: w 1D-SC Code:

11

𝝉𝒈𝟐,𝟐 𝝉𝒈𝟐,𝟑 𝝉𝒈𝟑,𝟐 𝝉𝒈𝟑,𝟑 𝝉𝒈𝟐,𝟒 𝝉𝒈𝟑,𝟒 𝝉𝒈𝟐,𝟐 𝝉𝒈𝟐,𝟑 𝝉𝒈𝟑,𝟐 𝝉𝒈𝟑,𝟑 𝝉𝒈𝟐,𝟒 𝝉𝒈𝟑,𝟒 𝝉𝒈𝟐,𝟐 𝝉𝒈𝟐,𝟑 𝝉𝒈𝟑,𝟐 𝝉𝒈𝟑,𝟑 𝝉𝒈𝟐,𝟒 𝝉𝒈𝟑,𝟒 𝝉𝒈𝟐,𝟐 𝝉𝒈𝟐,𝟐 𝝉𝒈𝟐,𝟐 𝝉𝒈𝟑,𝟐 𝝉𝒈𝟑,𝟐 𝝉𝒈𝟑,𝟐 𝝉𝒈𝟑,𝟑 𝝉𝒈𝟑,𝟑 𝝉𝒈𝟑,𝟑 𝝉𝒈𝟐,𝟑 𝝉𝒈𝟐,𝟑 𝝉𝒈𝟐,𝟑 𝝉𝒈𝟐,𝟒 𝝉𝒈𝟐,𝟒 𝝉𝒈𝟐,𝟒 𝝉𝒈𝟑,𝟒 𝝉𝒈𝟑,𝟒 𝝉𝒈𝟑,𝟒

SLIDE 19

w Spatially-coupled (SC) code with memory 𝒏 and coupling length

𝑴 is constructed by: a) Partitioning a parity check matrix 𝐼 into a number of component matrices: 𝐼9, 𝐼:, … , 𝐼< such that 𝐼 = ∑ 𝐼+

> +?9

b) Coupling 𝑀 copies of the component matrices together to make a chain of coupled block codes.

Spatially-Coupled Code as a Chain of Block Codes

12

SLIDE 20

Message Passing Decoding Algorithm -- Example

𝐽 𝐽 𝐽 𝐽 𝐽 𝐽 𝜏 𝜏0 𝜏1 𝜏2 𝐽 𝜏0 𝜏2 𝜏 𝜏1 𝐽 𝜏1 𝜏 𝜏2 𝜏0

𝐼 =

columns: variable nodes rows: check nodes Each 𝜏k is a 5x5 circulant

13

SLIDE 21

Message Passing Decoding Algorithm – Example

𝐽 𝐽 𝐽 𝐽 𝐽 𝐽 𝜏 𝜏0 𝜏1 𝜏2 𝐽 𝜏0 𝜏2 𝜏 𝜏1 𝐽 𝜏1 𝜏 𝜏2 𝜏0

𝐼 =

columns: variable nodes rows: check nodes circles: variable nodes squares: check nodes Each 𝜏k is a 5x5 circulant

14

SLIDE 22

Message Passing Decoding Algorithm – Example 1

……….… 0 0 … 0 circles: variable nodes squares: check nodes Assume all-zeros codeword was transmitted. Received sequence:

15

SLIDE 23

Message Passing Decoding Algorithm – Example 1

……….… 0 0 … 0 circles: variable nodes squares: check nodes Assume all-zeros codeword was transmitted. Variable to check messages: Variable value is 1 Variable value is 0

16

SLIDE 24

Message Passing Decoding Algorithm – Example 1

……….… 0 0 … 0 circles: variable nodes squares: check nodes Assume all-zeros codeword was transmitted. Check processing: Check is not satisfied Check is satisfied

17

SLIDE 25

Message Passing Decoding Algorithm – Example 1

……….… 0 0 … 0 circles: variable nodes squares: check nodes Assume all-zeros codeword was transmitted. Check to variable node messages: Flip your current value (F) Keep your current value (S)

18

SLIDE 26

Message Passing Decoding Algorithm – Example 1

……….… 0 0 … 0 circles: variable nodes squares: check nodes Assume all-zeros codeword was transmitted. Variable node processing – majority vote: F, F, F, F F, S, S, S S, S, S, S F, F, F, S

19

SLIDE 27

Message Passing Decoding Algorithm – Example 1

……….… 0 0 … 0 circles: variable nodes squares: check nodes Assume all-zeros codeword was transmitted. Variable node processing – majority vote: F, F, F, F F, F, F, S F, S, S, S S, S, S, S

20

SLIDE 28

Message Passing Decoding Algorithm – Example 1

0 0 ……….… 0 0 0 … 0 circles: variable nodes squares: check nodes Assume all-zeros codeword was transmitted. Variable node processing – majority vote:

21

SLIDE 29

Message Passing Decoding Algorithm – Example 1

0 0 ……….… 0 0 0 … 0 circles: variable nodes squares: check nodes Assume all-zeros codeword was transmitted. Variable to check messages: Variable node is 1 Variable node is 0

22

SLIDE 30

Message Passing Decoding Algorithm – Example 1

0 0 ……….… 0 0 0 … 0 circles: variable nodes squares: check nodes Assume all-zeros codeword was transmitted. Check processing: Check is not satisfied Check is satisfied

23

SLIDE 31

Message Passing Decoding Algorithm – Example 1

0 0 ……….… 0 0 0 … 0 circles: variable nodes squares: check nodes Assume all-zeros codeword was transmitted. Check processing: Check is not satisfied Check is satisfied End of decoding: all checks are satisfied DECODING IS SUCCESSFUL

24

SLIDE 32

Message Passing Decoding Algorithm – Example 2

0 ……….… ……….… 0 0 circles: variable nodes squares: check nodes Assume all-zeros codeword was transmitted. Received sequence:

25

SLIDE 33

Message Passing Decoding Algorithm – Example 2

0 ……….… ……….… 0 0 circles: variable nodes squares: check nodes Assume all-zeros codeword was transmitted. Variable to check messages: Variable value is 1 Variable value is 0

26

SLIDE 34

Message Passing Decoding Algorithm – Example 2

0 ……….… ……….… 0 0 circles: variable nodes squares: check nodes Assume all-zeros codeword was transmitted. Check processing: Check is not satisfied Check is satisfied Checks that receive two 1’s so they are satisfied

27

SLIDE 35

Message Passing Decoding Algorithm – Example 2

0 ……….… ……….… 0 0 circles: variable nodes squares: check nodes Assume all-zeros codeword was transmitted. Check to variable node messages: Flip your current value (F) Keep your current value (S)

28

SLIDE 36

Message Passing Decoding Algorithm – Example 2

0 ……….… ……….… 0 0 circles: variable nodes squares: check nodes Assume all-zeros codeword was transmitted. Variable node processing – majority vote: S, S, S, S S, S, F, S S, F, F, S S, F, S, S S, S, F, S S, F, S, S

29

SLIDE 37

Message Passing Decoding Algorithm – Example 2

0 ……….… ……….… 0 0 circles: variable nodes squares: check nodes Assume all-zeros codeword was transmitted. Variable node processing – majority vote: S, S, S, S S, S, F, S S, F, F, S S, F, S, S S, S, F, S S, F, S, S No updates are made despite the presence of unsatisfied checks.

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SLIDE 38

Message Passing Decoding Algorithm – Example 2

0 ……….… ……….… 0 0 circles: variable nodes squares: check nodes Assume all-zeros codeword was transmitted. Variable node processing – majority vote:

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SLIDE 39

Message Passing Decoding Algorithm – Example 2

0 ……….… ……….… 0 0 circles: variable nodes squares: check nodes Assume all-zeros codeword was transmitted. Variable to check messages: Variable value is 1 Variable value is 0

32

SLIDE 40

Message Passing Decoding Algorithm – Example 2

0 ……….… ……….… 0 0 circles: variable nodes squares: check nodes Assume all-zeros codeword was transmitted. Check processing: Check is not satisfied Check is satisfied

33

SLIDE 41

Message Passing Decoding Algorithm – Example 2

0 ……….… ……….… 0 0 circles: variable nodes squares: check nodes Assume all-zeros codeword was transmitted. Check processing: Check is not satisfied Check is satisfied Continue ad infinitum with the same unsatisfied check nodes End of decoding: reach maximum number of iterations. DECODING IS UNSUCCESSFUL

34

SLIDE 42

Decoding Errors of Iteratively-Decoded Graph Codes

An (a, b) absorbing set (AS): – is a subgraph of the Tanner graph; – parameter a is the number of variable nodes in the configuration; – parameter b is the number of unsatisfied check nodes connected to the configuration; – each variable node is connected to more satisfied than unsatisfied check nodes.

w Example: (3, 3) absorbing set.

– Assume all-zeros codeword is transmitted – Assume only these 3 variable nodes are in error.

w Other notions used in the literature: Trapping sets, Stopping

Sets, Pseudocodewords [Richardson, Vasic, Milenkovic, Banihashemi,…].

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S S S S S F F F

SLIDE 43

w Elementary AS:

– is an AS wherein each satisfied check node is of degree 2, and each unsatisfied check node is of degree 1.

Elementary Absorbing Sets

(4,4) elementary AS

36

(3,3) elementary AS

SLIDE 44

w Elementary AS:

– is an AS wherein each satisfied check node is of degree 2, and each unsatisfied check node is of degree 1.

w Elementary AS is commonly used to capture AWGN performance.

Elementary Absorbing Sets

(4,4) elementary AS

36

(3,3) elementary AS

SLIDE 45

w Binary absorbing sets are described in terms of topological

conditions only.

w Non-binary (NB) absorbing sets:

– Both the topological conditions and the edge weight conditions are required.

• Choice of w’s determines satisfied/unsatisfied checks.
• Conditions are congruential equations over GF(q).

Binary and Non-binary (NB) Absorbing Sets

37

SLIDE 46

w Binary absorbing sets are described in terms of topological

conditions only.

w Non-binary (NB) absorbing sets:

– Both the topological conditions and the edge weight conditions are required.

• Choice of w’s determines satisfied/unsatisfied checks.
• Conditions are congruential equations over GF(q).

Binary and Non-binary (NB) Absorbing Sets

𝑥: 𝑥0 𝑥1 𝑥2 𝑥B 𝑥C 𝑥D 𝑥E 𝑥F 𝑥:9 𝑥:0 𝑥:: 𝑥:1 𝑥:2 𝑥:B 𝑥:C

(4, 4) NB AS; w’s are in GF(q)\{0}

37

SLIDE 47

w Binary absorbing sets are described in terms of topological

conditions only.

w Non-binary (NB) absorbing sets:

– Both the topological conditions and the edge weight conditions are required.

• Choice of w’s determines satisfied/unsatisfied checks.
• Conditions are congruential equations over GF(q).

Binary and Non-binary (NB) Absorbing Sets

𝑥: 𝑥0 𝑥1 𝑥2 𝑥B 𝑥C 𝑥D 𝑥E 𝑥F 𝑥:9 𝑥:0 𝑥:: 𝑥:1 𝑥:2 𝑥:B 𝑥:C

37

(4, 4) NB AS; w’s are in GF(q)\{0}

SLIDE 48

w Binary absorbing sets are described in terms of topological

conditions only.

w Non-binary (NB) absorbing sets:

– Both the topological conditions and the edge weight conditions are required.

• Choice of w’s determines satisfied/unsatisfied checks.
• Conditions are congruential equations over GF(q).

Binary and Non-binary (NB) Absorbing Sets

𝑥: 𝑥0 𝑥1 𝑥2 𝑥B 𝑥C 𝑥D 𝑥E 𝑥F 𝑥:9 𝑥:0 𝑥:: 𝑥:1 𝑥:2 𝑥:B 𝑥:C

37

(4, 4) NB AS; w’s are in GF(q)\{0}

SLIDE 49

Effect of Asymmetry on Absorbing Sets (ASs)

w Asymmetry in the channel (e.g., in Flash) results in:

– NB ASs with unsatisfied check nodes having degree > 1. – NB ASs with satisfied check nodes having degree > 2.

• Primarily due to high VN error magnitudes.

w Such dominant objects are non-elementary.

38

SLIDE 50

Effect of Asymmetry on Absorbing Sets (ASs)

w Asymmetry in the channel (e.g., in Flash) results in:

– NB ASs with unsatisfied check nodes having degree > 1. – NB ASs with satisfied check nodes having degree > 2.

• Primarily due to high VN error magnitudes.

w Such dominant objects are non-elementary. w Example: (6, 4) non-elementary NB AS (γ = 3).

38

Th Threshold voltage distribution fo for MLC Flash Memories

SLIDE 51

Effect of Asymmetry on Absorbing Sets (ASs)

w Asymmetry in the channel (e.g., in Flash) results in:

– NB ASs with unsatisfied check nodes having degree > 1. – NB ASs with satisfied check nodes having degree > 2.

• Primarily due to high VN error magnitudes.

w Such dominant objects are non-elementary. w Example: (6, 4) non-elementary NB AS (γ = 3). w (6, 2), (6, 3), and (6, 4) are all problematic because of asymmetry.

38

Th Threshold voltage distribution fo for MLC Flash Memories

SLIDE 52

Detrimental Objects in Case of Storage Channels

w These objects are general absorbing sets of type two (GASTs). w Define an (a, b, d1, d2, d3) GAST over GF(q) [Hareedy ‘16]:

– a is the number of variable nodes in the configuration (its size). – b is the number of unsatisfied check nodes (degree 1 or 2). – d1 (resp., d2 and d3) is the number of degree 1 (resp., 2 and > 2) check nodes. – Each variable node is connected to strictly more satisfied than unsatisfied check nodes (for some VN values in GF(q)\{0}).

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SLIDE 53

Detrimental Objects in Case of Storage Channels

w These objects are general absorbing sets of type two (GASTs). w Define an (a, b, d1, d2, d3) GAST over GF(q) [Hareedy ‘16]:

– a is the number of variable nodes in the configuration (its size). – b is the number of unsatisfied check nodes (degree 1 or 2). – d1 (resp., d2 and d3) is the number of degree 1 (resp., 2 and > 2) check nodes. – Each variable node is connected to strictly more satisfied than unsatisfied check nodes (for some VN values in GF(q)\{0}).

(4, 2, 2, 5, 0) GAST (6, 2, 0, 9, 0) GAST

39

SLIDE 54

w Cycles 6 and cycles 8 are common denominators of problematic

absorbing sets over AWGN, Flash, and PR channels.

Common Denominator of Problematic Objects

40

common denominator (4, 2) (5, 3)

SLIDE 55

w Cycles 6 and cycles 8 are common denominators of problematic

absorbing sets over AWGN, Flash, and PR channels.

w By minimizing the population of common denominator instances,

we can simultaneously target a number of problematic objects and minimize their cumulative population.

Common Denominator of Problematic Objects

40

common denominator (4, 2) (5, 3)

SLIDE 56

w Cycles 6 and cycles 8 are common denominators of problematic

absorbing sets over AWGN, Flash, and PR channels.

w By minimizing the population of common denominator instances,

we can simultaneously target a number of problematic objects and minimize their cumulative population.

w The common denominator structure has a simpler combinatorial

description than the relevant super-structures.

Common Denominator of Problematic Objects

40

common denominator (4, 2) (5, 3)

SLIDE 57

Talk Outline

w Mathematical Preliminaries w Combinatorial Framework for SC Codes

– Analytical description – Applications and simulation results – Recent extensions: multidimensional SC

w Concluding Remarks and Future Directions

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SLIDE 58

Framework Outline

w First we optimize the unlabeled graph.

– Stage 1: We derive the optimal partitioning (OO) corresponding to the minimum number of detrimental objects (common denominator instances) in the protograph. – Stage 2: We devise a circulant power optimizer (CPO) to further reduce the number of problematic subgraphs in the unlabeled graph.

w We use the OO-CPO technique to design binary codes. w Stage 3: For the non-binary codes, we then apply the weight

consistency matrix (WCM) framework.

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SLIDE 59

Framework Outline

w First we optimize the unlabeled graph.

– Stage 1: We derive the optimal partitioning (OO) corresponding to the minimum number of detrimental objects (common denominator instances) in the protograph. – Stage 2: We devise a circulant power optimizer (CPO) to further reduce the number of problematic subgraphs in the unlabeled graph.

w We use the OO-CPO technique to design binary codes. w Stage 3: For the non-binary codes, we then apply the weight

consistency matrix (WCM) framework.

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SLIDE 60

w Partitioning a CB block code into 𝒏 + 𝟐 component matrices,

not necessarily contiguous: For memory 𝑛 = 1: 𝐼 = Green circulants belong to 𝐼9 Blue circulants belong to 𝐼: For memory 𝑛 = 2: 𝐼 = Green circulants belong to 𝐼9 Blue circulants belong to 𝐼: Gray circulants belong to 𝐼0

Stage 1: A Flexible Partitioning Using Overlap Rules

43

𝜏K

L,L

𝜏K

L,M

𝜏K

L,N

𝜏K

L,O

𝜏K

L,P

𝜏K

L,Q

𝜏K

L,R

𝜏K

M,L

𝜏K

M,M

𝜏K

M,N

𝜏K

M,O

𝜏K

M,P

𝜏K

M,Q

𝜏K

M,R

𝜏K

N,L

𝜏K

N,M

𝜏K

N,N

𝜏K

N,O

𝜏K

N,P

𝜏K

N,Q

𝜏K

N,R

𝜏K

L,L

𝜏K

L,M

𝜏K

L,N

𝜏K

L,O

𝜏K

L,P

𝜏K

L,Q

𝜏K

L,R

𝜏K

M,L

𝜏K

M,M

𝜏K

M,N

𝜏K

M,O

𝜏K

M,P

𝜏K

M,Q

𝜏K

M,R

𝜏K

N,L

𝜏K

N,M

𝜏K

N,N

𝜏K

N,O

𝜏K

N,P

𝜏K

N,Q

𝜏K

N,R

SLIDE 61

w Partitioning a CB block code into 𝒏 + 𝟐 component matrices,

not necessarily contiguous: For memory 𝑛 = 1: 𝐼 = Green circulants belong to 𝐼9 Blue circulants belong to 𝐼: For memory 𝑛 = 2: 𝐼 = Green circulants belong to 𝐼9 Blue circulants belong to 𝐼: Gray circulants belong to 𝐼0

w We developed optimum overlap (OO) partitioning that results in

the minimum number of problematic objects for protograph SC codes [Esfahanizadeh ‘18].

Stage 1: A Flexible Partitioning Using Overlap Rules

43

𝜏K

L,L

𝜏K

L,M

𝜏K

L,N

𝜏K

L,O

𝜏K

L,P

𝜏K

L,Q

𝜏K

L,R

𝜏K

M,L

𝜏K

M,M

𝜏K

M,N

𝜏K

M,O

𝜏K

M,P

𝜏K

M,Q

𝜏K

M,R

𝜏K

N,L

𝜏K

N,M

𝜏K

N,N

𝜏K

N,O

𝜏K

N,P

𝜏K

N,Q

𝜏K

N,R

𝜏K

L,L

𝜏K

L,M

𝜏K

L,N

𝜏K

L,O

𝜏K

L,P

𝜏K

L,Q

𝜏K

L,R

𝜏K

M,L

𝜏K

M,M

𝜏K

M,N

𝜏K

M,O

𝜏K

M,P

𝜏K

M,Q

𝜏K

M,R

𝜏K

N,L

𝜏K

N,M

𝜏K

N,N

𝜏K

N,O

𝜏K

N,P

𝜏K

N,Q

𝜏K

N,R

SLIDE 62

w Overlap parameters describe a property of the mapping from

circulants to component matrices. This in turn determines the total number of common denominator structures in the protograph.

w Consider an example:

– Number of rows γ = 3, number of columns κ = 7, memory m = 1. – There are 7 independent overlap parameters:

44

𝜏K

L,M

𝜏K

L,O

𝜏K

L,Q

𝜏K

L,R

𝜏K

M,M

𝜏K

M,N

𝜏K

M,P

𝜏K

M,R

𝜏K

N,N

𝜏K

N,P

𝜏K

N,Q

What Are the Overlap Parameters?

𝐼9 =

SLIDE 63

w Overlap parameters describe a property of the mapping from

circulants to component matrices. This in turn determines the total number of common denominator structures in the protograph.

w Consider an example:

– Number of rows γ = 3, number of columns κ = 7, memory m = 1. – There are 7 independent overlap parameters: 𝑢9 = 4

44

𝜏K

L,M

𝜏K

L,O

𝜏K

L,Q

𝜏K

L,R

𝜏K

M,M

𝜏K

M,N

𝜏K

M,P

𝜏K

M,R

𝜏K

N,N

𝜏K

N,P

𝜏K

N,Q

What Are the Overlap Parameters?

𝐼9 =

SLIDE 64

w Overlap parameters describe a property of the mapping from

circulants to component matrices. This in turn determines the total number of common denominator structures in the protograph.

w Consider an example:

– Number of rows γ = 3, number of columns κ = 7, memory m = 1. – There are 7 independent overlap parameters: 𝑢9 = 4 𝑢: = 4

44

𝜏K

L,M

𝜏K

L,O

𝜏K

L,Q

𝜏K

L,R

𝜏K

M,M

𝜏K

M,N

𝜏K

M,P

𝜏K

M,R

𝜏K

N,N

𝜏K

N,P

𝜏K

N,Q

What Are the Overlap Parameters?

𝐼9 =

SLIDE 65

w Overlap parameters describe a property of the mapping from

circulants to component matrices. This in turn determines the total number of common denominator structures in the protograph.

w Consider an example:

– Number of rows γ = 3, number of columns κ = 7, memory m = 1. – There are 7 independent overlap parameters: 𝑢9 = 4 𝑢: = 4 𝑢0 = 3

44

𝜏K

L,M

𝜏K

L,O

𝜏K

L,Q

𝜏K

L,R

𝜏K

M,M

𝜏K

M,N

𝜏K

M,P

𝜏K

M,R

𝜏K

N,N

𝜏K

N,P

𝜏K

N,Q

What Are the Overlap Parameters?

𝐼9 =

SLIDE 66

w Overlap parameters describe a property of the mapping from

circulants to component matrices. This in turn determines the total number of common denominator structures in the protograph.

w Consider an example:

– Number of rows γ = 3, number of columns κ = 7, memory m = 1. – There are 7 independent overlap parameters: 𝑢9 = 4 𝑢: = 4 𝑢0 = 3 𝑢{9,:}= 1

44

𝜏K

L,M

𝜏K

L,O

𝜏K

L,Q

𝜏K

L,R

𝜏K

M,M

𝜏K

M,N

𝜏K

M,P

𝜏K

M,R

𝜏K

N,N

𝜏K

N,P

𝜏K

N,Q

What Are the Overlap Parameters?

𝐼9 =

SLIDE 67

w Overlap parameters describe a property of the mapping from

circulants to component matrices. This in turn determines the total number of common denominator structures in the protograph.

w Consider an example:

– Number of rows γ = 3, number of columns κ = 7, memory m = 1. – There are 7 independent overlap parameters: 𝑢9 = 4 𝑢: = 4 𝑢0 = 3 𝑢{9,:}= 1 𝑢{9,0} = 1

44

𝜏K

L,M

𝜏K

L,O

𝜏K

L,Q

𝜏K

L,R

𝜏K

M,M

𝜏K

M,N

𝜏K

M,P

𝜏K

M,R

𝜏K

N,N

𝜏K

N,P

𝜏K

N,Q

What Are the Overlap Parameters?

𝐼9 =

SLIDE 68

w Overlap parameters describe a property of the mapping from

circulants to component matrices. This in turn determines the total number of common denominator structures in the protograph.

w Consider an example:

– Number of rows γ = 3, number of columns κ = 7, memory m = 1. – There are 7 independent overlap parameters: 𝑢9 = 4 𝑢: = 4 𝑢0 = 3 𝑢{9,:}= 1 𝑢{9,0} = 1 𝑢{:,0} = 2

44

𝜏K

L,M

𝜏K

L,O

𝜏K

L,Q

𝜏K

L,R

𝜏K

M,M

𝜏K

M,N

𝜏K

M,P

𝜏K

M,R

𝜏K

N,N

𝜏K

N,P

𝜏K

N,Q

What Are the Overlap Parameters?

𝐼9 =

SLIDE 69

w Overlap parameters describe a property of the mapping from

circulants to component matrices. This in turn determines the total number of common denominator structures in the protograph.

w Consider an example:

– Number of rows γ = 3, number of columns κ = 7, memory m = 1. – There are 7 independent overlap parameters: 𝑢9 = 4 𝑢: = 4 𝑢0 = 3 𝑢{9,:}= 1 𝑢{9,0} = 1 𝑢{:,0} = 2 𝑢{9,:,0} = 0

44

𝜏K

L,M

𝜏K

L,O

𝜏K

L,Q

𝜏K

L,R

𝜏K

M,M

𝜏K

M,N

𝜏K

M,P

𝜏K

M,R

𝜏K

N,N

𝜏K

N,P

𝜏K

N,Q

What Are the Overlap Parameters?

𝐼9 =

SLIDE 70

Computing the OO Parameters

w Discrete optimization problem:

– Here, F is the total number of structures of interest. – The constraints of our optimization problem are the conditions under which the overlap parameters and the subsequent partitioning are valid.

45

SLIDE 71

Computing the OO Parameters

w Discrete optimization problem:

– Here, F is the total number of structures of interest. – The constraints of our optimization problem are the conditions under which the overlap parameters and the subsequent partitioning are valid. – Brute force search quickly becomes unwieldy. – We developed a fast enumeration approach that takes the advantage of the repetitive nature of the graph and so-called replicas [Esfahanizadeh ’17, Esfahanizadeh ‘18].

45

SLIDE 72

Computing the OO Parameters

w Discrete optimization problem:

– Here, F is the total number of structures of interest. – The constraints of our optimization problem are the conditions under which the overlap parameters and the subsequent partitioning are valid. – Brute force search quickly becomes unwieldy. – We developed a fast enumeration approach that takes the advantage of the repetitive nature of the graph and so-called replicas [Esfahanizadeh ’17, Esfahanizadeh ‘18].

w Example: For the case of γ = 3, and m = 1, we denote the solution

to the problem as 𝐮∗ = [ 𝑢9

∗ 𝑢: ∗ 𝑢0 ∗ 𝑢{9,:} ∗

𝑢{9,0}

∗

𝑢{:,0}

∗

𝑢{9,:,0}

∗

]. We call 𝐮∗an optimal vector. All optimal vectors perform the same.

45

SLIDE 73

w Lemma: The VNs of a cycle of length 6 in an SC

code span at most 𝑛 + 1 consecutive replicas Ri.

w Example: cycles-6 on 𝐼[\

]

with parameters γ = 3, κ = 5, m = 2, L = 3.

• Cycles-6 in:

– solid lines span one replicas, – dash lines span two replicas, – dash-dot lines span three replicas.

• Component matrices are illustrated in gray.

Case Study: Cycle-6 as the Common Denominator Structure

46

R1 R2 …. RL R1 R2 R3

SLIDE 74

w Theorem: The number of cycles of length 6 (cycles-6) in the binary

protograph (p = 1) of an SC code with γ, κ, m, and L is:

– 𝐺

: _ is the number of cycles that start in first replica and span k replicas.

w Example: 𝐼[\

] with parameter κ = 5, γ = 3, m = 2, L = 3.

Case Study: Cycle-6 as the Common Denominator Structure

47

𝐺 = 3𝐺

: :

+𝐺

: 1

+ 2𝐺

:

Expressions for 𝐆𝟐

𝐥 are also derived explicitly.

SLIDE 75

Stage 2: Circulant Power Optimizer (CPO)

w After 𝐈𝐓𝐃

𝐪 is designed using t*, we apply the CPO to further

reduce the number of the common denominators structures in the graph of 𝐈𝐓𝐃.

w Example for cycle-6:

Protograph Unlabeled graph

48

SLIDE 76

Framework Outline

w First we optimize the unlabeled graph.

– Stage 1: We derive the optimal partitioning (OO) corresponding to the minimum number of detrimental objects (common denominator instances) in the protograph. – Stage 2: We devise a circulant power optimizer (CPO) to further reduce the number of problematic subgraphs in the unlabeled graph.

w We use the OO-CPO technique to design binary codes. w For the non-binary codes, we then apply the weight consistency

matrix (WCM) framework (Stage 3).

49

SLIDE 77

Framework Outline

w First we optimize the unlabeled graph.

– Stage 1: We derive the optimal partitioning (OO) corresponding to the minimum number of detrimental objects (common denominator instances) in the protograph. – Stage 2: We devise a circulant power optimizer (CPO) to further reduce the number of problematic subgraphs in the unlabeled graph.

w We use the OO-CPO technique to design binary codes. w For the non-binary codes, we then apply the weight consistency

matrix (WCM) framework (Stage 3).

49

SLIDE 78

w Recall:

– For a non-binary AS, the values of the variable nodes matter, and weight conditions have to be included. – We use notation (a, b, d1, d2, d3) for a GAST over GF(q).

w Example: (6, 2, 0, 9, 0) GAST (γ = 3):

– W: Satisfied. D: Unsatisfied. W D

Stage 3 ingredients: GASTs and their matrix descriptions

50

SLIDE 79

How to Remove a GAST

w Objectives of the removal:

– The code structure and properties are preserved. – Manipulating the edge weights such that the GAST is completely removed (not converted into another GAST).

w For the shown (6, 0, 0, 9, 0) GAST,

the resulting object after removal: – is not a (6, 0, 0, 9, 0) GAST. – is not a (6, 1, 0, 9, 0) GAST. – is not a (6, 2, 0, 9, 0) GAST. – is not a (6, 3, 0, 9, 0) GAST. bmax = 3

w We need a compact set of submatrices such that when we break

their weight conditions, the GAST is removed. The weight consistency matrices (WCMs)

51

SLIDE 80

Stage 3 ingredients: GASTs and their matrix descriptions

Weight Conditions: – There exists a submatrix W in the matrix of check nodes A, such that the null space of W contains at least one vector with all its entries ≠ 0.

52

𝐄

≠ 0 ≠ 0

. . .

SLIDE 81

Stage 3 ingredients: GASTs and their matrix descriptions

Weight Conditions: – There exists a submatrix W in the matrix of check nodes A, such that the null space of W contains at least one vector with all its entries ≠ 0. – The matrix W is the matrix

• f satisfied check nodes,

while D is the matrix of unsatisfied check nodes. – Each column in A has more non-zero entries in W than in D.

52

𝐄

≠ 0 ≠ 0

. . .

SLIDE 82

WCM Extraction Example

53

w Example: (6, 0, 0, 9, 0) GAST (γ = 3).

We construct the set of WCMs, by excluding c1 first: – Select c1 (mark c1 as unsatisfied). – c2, c7, c6, and c8 cannot be selected concurrently with c1. – Options: c3, c4, c5, and c9. – Select c3. – c4 and c9 cannot be selected. – Select c5 (the remaining one). – Finally, we get 2 WCMs for c1 obtained by removing: (c1, c3, c5) and (c1, c4, c9).

SLIDE 83

Stage 3: WCM Optimization Framework

w Input: Tanner graph. Output: Optimized Tanner graph.

• 1. Identify the set (G) of problematic GASTs.
• 2. For each candidate GAST, extract its subgraph from the Tanner

graph of the code.

• 3. Determine the set of WCMs of that GAST.
• 4. For each WCM in that set:
• a. Find the null space of the WCM.
• b. Break the weight conditions of that WCM via the edge weights.
• 5. If the GAST removal is successful, reflect the edge weight

changes in the Tanner graph of the code.

• 6. This process continues until all GASTs in G are eliminated or no

more GASTs can be eliminated.

w Theoretical analysis in [Hareedy ’16, Hareedy ’19].

54

SLIDE 84

Talk Outline

w Mathematical Preliminaries w Combinatorial Framework for SC Codes

– Analytical description – Applications and simulation results – Recent extensions: multidimensional SC

w Concluding Remarks and Future Directions

55

SLIDE 85

Simulation Over AWGN Channel: Code Parameters

56

Code Name 𝛿 Code Spec Length and Rate Uncoupled Code 2 4 Uncoupled (𝑛 = 0) AB circulant powers length = 8,670, rate = 0.765 SC Code 4 SC (𝑛 = 1) CV-AB construction length = 8,670, rate = 0.757 SC Code 5 SC (𝑛 = 1) OO-AB construction SC Code 6 SC (𝑛 = 1) OO-CPO construction Code Name 𝛿 Code Spec Length and Rate Uncoupled Code 1 3

Uncoupled (𝑛 = 0) AB circulant powers

length = 8,670, rate = 0.824 SC Code 1

SC (𝑛 = 1) CV-AB construction

length = 8,670, rate = 0.818 SC Code 2

SC (𝑛 = 1) OO-AB construction

SC Code 3

SC (𝑛 = 1) OO-CPO construction

SLIDE 86

Our well-designed SC codes compared to uncoupled block codes and SC codes constructed by previous method of cutting vectors. Column weight 𝛿 = 3

57

Simulation Over AWGN Channel: BER Analysis

SNR (dB)

4 4.5 5 5.5 6 6.5 7

BER

10-12 10-11 10-10 10-9 10-8 10-7 10-6 10-5 10-4 10-3 10-2 Uncoupled Code 1 SC Code 1, CV SC Code 2, OO SC Code 3, OO-CPO

SLIDE 87

Our well-designed SC codes compared to uncoupled block codes and SC codes constructed by previous method of cutting vectors. Column weight 𝛿 = 3 Column weight 𝛿 = 4

w

SNR (dB)

3 3.5 4 4.5 5 5.5

BER

10-10 10-9 10-8 10-7 10-6 10-5 10-4 10-3 10-2 10-1 Uncoupled Code 2 SC Code 4, CV SC Code 5, OO SC Code 6, OO-CPO

57

Simulation Over AWGN Channel: BER Analysis

SNR (dB)

4 4.5 5 5.5 6 6.5 7

BER

10-12 10-11 10-10 10-9 10-8 10-7 10-6 10-5 10-4 10-3 10-2 Uncoupled Code 1 SC Code 1, CV SC Code 2, OO SC Code 3, OO-CPO

Orders of magnitude improvement

SLIDE 88

Simulation over Flash Channel, Non-binary SC Codes

w Channel: normal-Laplace mixture (NLM) Flash channel.

– MLC channel with 3 reads and the sector size is 512 bytes. – RBER is raw BER. UBER is uncorrectable BER (FER/(512×8)).

w All the codes have 𝜹 = 𝟒, length 14,440 bits, and rate 0.834.

(𝜆 = 𝑞 = 19, 𝑛 = 1, 𝑀 = 20, and 𝑟 = 4)

w The OO-CPO-WCM approach

• utperforms existing methods:

– Code 6 outperforms Code 2 by 2.5 orders of magnitude. – Code 6 achieves 200% RBER gain compared to Code 2. – Code 6 achieves 500% RBER gain compared to Code 1.

58

SLIDE 89

SC codes for Channels with SNR variation

w Channels with SNR Variation

– Each section has a Δ𝑇𝑂𝑆+ that modulates a nominal 𝑇𝑂𝑆 which is common among all sections.

Section 1 Section 2 Section 𝑗 Section 𝑂

…

w The intrinsic structure of SC codes

makes them robust to the SNR variation.

– Example: CN rreceives information from (𝑛+1) consecutive sections.

𝐷𝑂

59

SLIDE 90

Simulation over Channel with SNR Variation, SC Codes Outperform Uncoupled Block Codes

w Channel: It has 30 sections. Each section is an AWGN channel

with a different SNR value.

w All the codes have 𝜹 = 𝟒 and length 8,670 bits.

(𝜆 = 𝑞 = 17, and 𝑀 = 30)

w Uncoupled Code 1:

𝑛 = 0 and rate 0.824 SC Code 1: 𝑛 = 1 and rate 0.818 SC Code 2: 𝑛 = 2 and rate 0.812

w Outperforms block codes even

with bit-level interleaving and more improvement by increasing the memory. [Esfahanizadeh ‘18b]

60

SNR (dB)

4.5 5 5.5 6 6.5 7

BER

10-12 10-11 10-10 10-9 10-8 10-7 10-6 10-5 10-4 10-3 Uncoupled Code 1 Uncoupled Code 1, Reg. Intrvg SC Code 1 SC Code 1, Eff. Intrvg SC Code 2

SLIDE 91

Talk Outline

w Mathematical Preliminaries w Combinatorial Framework for SC Codes

– Analytical description – Applications and simulation results – Recent extensions: multidimensional SC

w Concluding Remarks and Future Directions

61

SLIDE 92

Multi-Dimensional Spatially-Coupled (MD-SC) Codes

w In 1D-SC code, a number of block codes are connected together to

form a chain of coupled block codes.

62

Talk given in AM session by Homa Esfahanizadeh

SLIDE 93

w In 1D-SC code, a number of block codes are connected together to

form a chain of coupled block codes.

w The idea of a MD-SC code is to connect several SC codes to obtain

an MD coupling among the underlying SC codes.

62

Multi-Dimensional Spatially-Coupled (MD-SC) Codes

Talk given in AM session by Homa Esfahanizadeh

SLIDE 94

• 1. Identify circulants that are the most problematic (contribute to

the most cycles-𝑙 and 𝑙 is girth).

• 2. We relocate each chosen circulant to the same position in one of

the auxiliary matrices 𝐵+, i.e., 𝑗 ∈ {1,2, … , 𝑂 − 1}. Three 1D-SC codes MD-SC code with 𝑂 = 3 𝐼†‡ = 𝐼†‡

ˆ + A: + A0

MD-SC Code Design

63

𝐼†‡ 𝐼†‡ 𝐼†‡ 𝐼†‡

ˆ

𝐵0 𝐵: 𝐵: 𝐼†‡

ˆ

𝐵0 𝐵0 𝐵: 𝐼†‡

ˆ

SLIDE 95

Optimized Relocation of Circulants to Minimize the Number of Problematic Cycles

w Relocation: Moving a problematic circulant 𝐷+,, in 𝐈Š‹ to an

auxiliary matrix 𝐵+, i.e., 𝑗 ∈ {1,2, … , 𝑂 − 1}.

w MD density (𝑈): The maximum number of relocations per replica. w MD Mapping: 𝑁: 𝐷+,, → 0,1, 𝑂 − 1

a) If 𝐷+,, → 𝐵+, 𝑁 𝐷+,, = 𝑗, b) If 𝐷+,, is not relocated (kept in 𝐈Š‹

ˆ ), 𝑁 𝐷+,, = 0.

64

SLIDE 96

Optimized Relocation of Circulants to Minimize the Number of Problematic Cycles

w Relocation: Moving a problematic circulant 𝐷+,, in 𝐈Š‹ to an

auxiliary matrix 𝐵+, i.e., 𝑗 ∈ {1,2, … , 𝑂 − 1}.

w MD density (𝑈): The maximum number of relocations per replica. w MD Mapping: 𝑁: 𝐷+,, → 0,1, 𝑂 − 1

a) If 𝐷+,, → 𝐵+, 𝑁 𝐷+,, = 𝑗, b) If 𝐷+,, is not relocated (kept in 𝐈Š‹

ˆ ), 𝑁 𝐷+,, = 0.

w Theorem: Let 𝐷•‘ = 𝐷+M,,M, 𝐷+N,,N, … , 𝐷+‘,,‘ be the circulants

visited by a cycle-𝑙 , or 𝑃_, in a clock wise order. If the following equation holds, the cycle-𝑙 is preserved in 𝐈Š‹

“” (ineffective),

• −1 –𝑁 𝐷+—,,— = 0

_ –?:

mod 𝑂 ,

• therwise the relocations are effective.

64

SLIDE 97

SNR (dB)

2 2.5 3 3.5 4 4.5 5

BER

10-12 10-10 10-8 10-6 10-4 10-2 100

SC-Code-2 MD-SC-Code

w SC Code 1:

length 2.89 K bits and rate 0.74. (𝛿=4,𝜆=17,𝑞=17,𝑛=1,𝑀=10, and OO-CPO)

w SC Code 2:

length 14.45 K bits and rate 0.76. (𝛿=4,𝜆=17,𝑞=17,𝑛=1,𝑀=50, and OO-CPO)

w MD-SC Code with 𝑶 = 𝟔:

Five SC Code 1 as constituent SC codes, length 14.45 K bits, and rate 0.74. Here: MD density 𝑈=23 or 33.82%

A Promising Result on M-D SC Codes over AWGN Channel

65

Code SC Code 2 MD-SC Code # cycles-6 153,714 1,700

More than 5 orders of magnitude

SLIDE 98

Talk Outline

w Mathematical Preliminaries w Combinatorial Framework for SC Codes w Concluding Remarks and Future Directions

66

SLIDE 99

w Summary:

– SC codes are a promising class of ECCs. – We developed a combinatorial framework for design and

• ptimization of finite-length SC codes for data storage

applications. – Theoretical Innovations: analysis of graphical substructures;

• ptimal partitioning scheme; circulant power optimizer; WCM

for edge weight selection; multi-dimensional structures. – Experimental results confirm excellent performance.

67

Our goal is to design and develop advanced ECC methods that exploit properties of the underlying physical channels.

SLIDE 100

w On-going Research:

– Framework extension for irregular SC codes. – Theoretical development of multidimensional SC codes (𝑂 > 3) and testing on representative channels. – Design and implementation of accompanying low-latency decoding algorithms.

w Future Extensions and Connections:

– Graph coding with locality and with random access [see also: Ram and Cassuto, NVMW 2019]. – Graph coding in ML applications and with natural redundancy [see also: Jiang at al. NVMW 2017; NVMW 2018].

68

Our goal is to design and develop advanced ECC methods that exploit properties of the underlying physical channels.

SLIDE 101

References

w OUR RECENT RESEARCH RESULTS

–

• A. Hareedy, C. Lanka, N. Guo, and L. Dolecek, “A combinatorial methodology for optimizing

non-binary graph-based codes: theoretical analysis and applications,” IEEE Trans. On Information Theory, 2019, to appear. –

• H. Esfahanizadeh, A. Hareedy, and L. Dolecek, “Finite-length construction of high performance

spatially-coupled codes via optimized partitioning and lifting,” IEEE Trans. On Communications, Jan. 2019. –

• A. Hareedy, H. Esfahanizadeh, A. Tan, and L. Dolecek, “Spatially-coupled code design for partial

response channels: optimal object-minimization approach,” in Proc. IEEE Globecom, 2018. –

• H. Esfahanizadeh, A. Hareedy, and L. Dolecek , “Multi-dimensional spatially-coupled code

design through informed relocation of circulants,” in Proc. IEEE Allerton, 2018. –

• H. Esfahanizadeh, A. Hareedy, R. Wu, R. Galbraith, and L. Dolecek, “Coding for channels with

SNR variation: spatial coupling and efficient interleaving,” IEEE Trans. Magnetics, Aug. 2018. –

• H. Esfahanizadeh, A. Hareedy, and L. Dolecek, “A novel combinatorial framework to construct

spatially-coupled codes: minimum overlap partitioning,” in Proc. IEEE ISIT, 2017. –

• A. Hareedy, H. Esfahanizadeh, and L. Dolecek, “High performance non-binary spatially–

coupled codes for Flash memories,” in Proc. IEEE ITW, 2017.

–

• A. Hareedy, C. Lanka, and L. Dolecek, “A general non-binary LDPC code optimization framework

suitable for dense Flash memory and magnetic storage,” IEEE J. Sel. Areas Communications, Sept. 2016.

69

SLIDE 102

Acknowledgment

w Research supported in part by NSF, ASTC, and WDC. We

gratefully acknowledge this support.

70

SLIDE 103

Thank You