the performance of polar codes for multi-level flash memories

The Performance of Polar Codes for Multi-level Flash

Memories Yue Li

joint work with Hakim Alhussien, Erich F. Haratsch,

and Anxiao (Andrew) JiangMarch 10th, 2014

2

NAND Flash Memory

…

…

…… … Blocks

4 pages/WL

The circuit board of a SSD

3

Multi-Level Cells10000111

2 bits/cell• Four different kinds of pages:

• Lower even• Lower odd• Upper even• Upper odd

4

Why Polar Codes?• Desire for optimal ECCs.• Excellent properties

– Capacity-achieving– Theoretical guarantee of error floor

performance– Efficient encoding and decoding

algorithms

5confidential

Encoding

Erdal Arıkan, “Channel Polarization: A Method for Constructing Capacity-Achieving Codes for Symmetric Binary-Input Memoryless Channels," IEEE Transactions on Information Theory, 2009.

Frozen bits

Information Bits

Frozen Channels

Input User BitsPolar Codeword

Noisy Codeword

Flash channels

G

6confidential

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.

7confidential

Frozen Channels

Estimated user bits

Noisy Codeword

Successive Cancellation Decoding

Erdal Arıkan, “Channel Polarization: A Method for Constructing Capacity-Achieving Codes for Symmetric Binary-Input Memoryless Channels," IEEE Transactions on Information Theory, 2009.

8confidential

Frozen Channels

Successive Cancellation Decoding

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.

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 C Noisy CK – K’ N – K’ N – K’K’

11

K – K’ K’

Est. M

(N, K, K’)-Shortened Polar Code

1 0 0 … 01 0 … 0

… 0 …1 0

1

ç

ç

(x1, x2, …, xN-k’+1, …, xN)=(u1, u2, …, uN-k’+1, …, uN) G

(u1, u2, …, uN-k’+1, …, uN) K’

K’

(x1, x2, …, 0, …, 0)=(u1, u2, …, 0, …, 0) G

12

Evaluation with Random Data

(0, 1, 1, 0, …, 1)(1, 0, 1, 0, …, 1)…(1, 0, 1, 0, …, 1)

Pseudo-random Data

Cycling / Retention

(0, 0, 1, 1, …, 1)(1, 0, 0, 0, …, 1)…(0, 0, 1, 1, …, 1)

Not generated by polar encoder

13

Treating Random Data as Codewords

(u1, u2, …, uN) = (x1, x2, …, xN) G-1 Invertib

leInputOutputChannel parameters

Construct codesFrozen Bits

14

Hard and Soft Sensing

Cell Voltage

11 01 00 10

LLR = log___________________P( V | bit = 1 ) P( V | bit = 0 )

15

Reference threshold voltages

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

BER

PEC0 R1pec1 R2

pec2 R2pec3

Correction Capability

Is repetitive code construction needed at rate-switching PECs?

22

Why Code Reconstruction is Not Needed?

23

With and Without Code Reconstruction

24Upper odd page Average

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

Thank You