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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 10 th, 2014

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NAND Flash Memory … … … … … Blocks 4 pages/WL The circuit board of a SSD 2

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Multi-Level Cells bits/cell 3 Four different kinds of pages: Lower even Lower odd Upper even Upper odd

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Why Polar Codes? Desire for optimal ECCs. Excellent properties – Capacity-achieving – Theoretical guarantee of error floor performance – Efficient encoding and decoding algorithms 4

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5confidential Encoding Erdal Arıkan, “Channel Polarization: A Method for Constructing Capacity-Achieving Codes for Symmetric Binary-Input Memoryless Channels," IEEE Transactions on Information Theory, Frozen bits Information Bits Frozen Channels Input User Bits Polar Codeword Noisy Codeword Flash channels G

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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.

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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.

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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.

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Is polar code suitable for flash memories? 1 Make polar code work in flash memory 2 Performance evaluations 3 Adaptive decoding

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Code Length Adaptation Polar codes have length N = 2 m The code lengths in flash memory need to be flexible. 10

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Shortening M M C C Noisy C K – K’ N – K’ K’ 11 K – K’ K’ Est. M

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(N, K, K’)-Shortened Polar Code 100…0 10…0 …0… 10 1 ç ç ç ç (x 1, x 2, …, x N-k’+1, …, x N )=(u 1, u 2, …, u N-k’+1, …, u N ) G (u 1, u 2, …, u N-k’+1, …, u N ) K’ (x 1, x 2, …, 0, …, 0)=(u 1, u 2, …, 0, …, 0) G 12

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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

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Treating Random Data as Codewords (u 1, u 2, …, u N ) = (x 1, x 2, …, x N ) G -1 Invertible Input Output Channel parameters Construct codes Frozen Bits 14

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Hard and Soft Sensing Cell Voltage LLR = log ___________________ P( V | bit = 1 ) P( V | bit = 0 ) 15 Reference threshold voltages

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2 Performance Evaluation 16

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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

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Hard and Soft Decoding Hard Decoding Soft Decoding 18

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Different Block Lengths 19

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Asymmetric and Symmetric Errors 20

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3 Adaptive Decoding 21

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Code Rate Switching BER PEC 0 R1R1 pec 1 R2R2 pec 2 R2R2 pec 3 Correction Capability Is repetitive code construction needed at rate-switching PECs? 22

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Why Code Reconstruction is Not Needed? 23

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With and Without Code Reconstruction 24 Upper odd pageAverage

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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

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Future Directions Error floor performance Comparing with LDPC decoder with the same hardware latency Efficient hardware implementations 26

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