The Performance of Polar Codes for
Multi-level Flash Memories
Yue Li
joint work with
Hakim Alhussien, Erich F. Haratsch,
and Anxiao (Andrew) Jiang
March 10th, 2014
NAND Flash Memory
…
…
…
The circuit board of a SSD
Blocks
…
4 pages/WL
2
Multi-Level Cells
10
00
01
11
2 bits/cell
• Four different kinds of pages:
• Lower even
• Lower odd
• Upper even
• Upper odd
3
Why Polar Codes?
• Desire for optimal ECCs.
• Excellent properties
– Capacity-achieving
– Theoretical guarantee of error floor performance
– Efficient encoding and decoding algorithms
4
Encoding
Input User Bits
Frozen bits
Polar Codeword
Flash channels
Frozen
Channels
G
Noisy Codeword
Information Bits
Erdal Arıkan, “Channel Polarization: A Method for Constructing Capacity-Achieving Codes for Symmetric
Binary-Input Memoryless Channels," IEEE Transactions on Information Theory, 2009.
confidential
5
Successive Cancellation Decoding
Frozen
Channels
Estimated user bits
Noisy Codeword
Erdal Arıkan, “Channel Polarization: A Method for Constructing Capacity-Achieving Codes for Symmetric
Binary-Input Memoryless Channels," IEEE Transactions on Information Theory, 2009.
confidential
6
Successive Cancellation Decoding
Frozen
Channels
Estimated user bits
Noisy Codeword
Erdal Arıkan, “Channel Polarization: A Method for Constructing Capacity-Achieving Codes for Symmetric
Binary-Input Memoryless Channels," IEEE Transactions on Information Theory, 2009.
confidential
7
Successive Cancellation Decoding
Frozen
Channels
Estimated user bits
Noisy Codeword
Erdal Arıkan, “Channel Polarization: A Method for Constructing Capacity-Achieving Codes for Symmetric
Binary-Input Memoryless Channels," IEEE Transactions on Information Theory, 2009.
confidential
8
Is polar code suitable for
flash memories?
1
• Make polar code work in flash memory
2
• Performance evaluations
3
• Adaptive decoding
Code Length Adaptation
• Polar codes have length N = 2m
• The code lengths in flash memory
need to be flexible.
10
Shortening
M
K – K’
C
K’
N – K’
Noisy C
Est. M
N – K’
K – K’
K’
11
(N, K, K’)-Shortened Polar Code
(x1, x2, …, xN-k’+1, …, xN)=(u1, u2, …, uN-k’+1, …, uN) G
(x1, x2, …, 0, …, 0)=(u1, u2, …, 0, …, 0) G
1
K’
0 … 0
0 … 0
… 0 …
1 0
1
ç
(u1, u2, …, uN-k’+1, …, uN)
0
1
ç
K’
12
Evaluation with Random Data
Pseudo-random Data
(0, 1, 1, 0, …, 1)
(1, 0, 1, 0, …, 1)
…
(1, 0, 1, 0, …, 1)
Cycling / Retention
Not generated by
polar encoder
(0, 0, 1, 1, …, 1)
(1, 0, 0, 0, …, 1)
…
(0, 0, 1, 1, …, 1)
13
Treating Random Data as Codewords
(u1, u2, …, uN) = (x1, x2, …, xN) G-1
Invertible
Input
Output
Channel parameters
Construct codes
Frozen Bits
14
Hard and Soft Sensing
Reference threshold voltages
11
01
00
10
Cell Voltage
P(
V
|
bit
=
0
)
___________________
LLR = log
P( V | bit = 1 )
15
2
•Performance
Evaluation
16
Experimental Setup
• Construct one polar code for each kind of page.
• List successive cancellation decoding [Tal and Vardy 2011]
– List size = 32 with CRC
• Block length
– 7943 bits shortened from 8192 bits
• Code rates
– 0.93, 0.94, 0.95
• Flash data
– obtained by characterizing 2X-nm MLC flash chips
– 6-month retention
17
Hard and Soft Decoding
Hard Decoding
Soft Decoding
18
Different Block Lengths
19
Asymmetric and Symmetric
Errors
20
3
•Adaptive
Decoding
21
Code Rate Switching
Is repetitive code construction needed at rate-switching PECs?
BER
Correction
Capability
0
R1
pec1
R2
pec2
R2
pec3
PEC
22
Why Code Reconstruction is Not Needed?
23
With and Without Code Reconstruction
Upper odd page
Average
24
Summary
• On the flash data
– Polar codes are comparable to LDPC codes using hard and soft
sensing
– Larger block lengths do not improve decoding performance a lot
– More symmetric, better decoding performance
– Repetitive code construction is not necessary for adaptive
decoding
25
Future Directions
• Error floor performance
• Comparing with LDPC decoder with the same
hardware latency
• Efficient hardware implementations
26