Correcting Errors in MLCs with Bit-fixing Coding
Yue Lijoint work with
Anxiao (Andrew) Jiang and Jehoshua Bruck
Flash memories have excellent performance.
Cells are programmed by injecting electrons.◦ Programming is fast.◦ Erasure is expensive.
The growth of storage density brings reliability issues.◦ Cells are more closely aligned ◦ More levels are programmed into each cell.
We propose a new coding scheme for correcting asymmetric errors in multi-level cells (MLCs).
Introduction
Number of Pulses
Voltage
76543210
Reprogram
Correct with ECC
Motivation: Cell Overprogramming
012
012345
0 0123
012345
01234567891011
12345678
8 2 5 0 3 5 110
Asymmetric errors appear…
012345
012345
01
0123
012345678
01234567891011
12345678910
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10 5 1 8
How to correct asymmetric errors ?
How to minimize redundancy in ECC?
0123456789101112131415
000100100011010001010110011110001001101010111100110111101111
0000
Binary Codes
000100110010011001110101010011001101111111101010101110011000
0000
Gray Codes Our Codes
The number of bit errors equals the Hamming weight of the binary representation of a cell error.
Cell error # of bit errors1 12 13 24 1
The number of bit errors depends only on the size of the cell error.
The number of bit errors depends on the size of the error and the cell level
5 26 27 38 1
This work generalizes the code constructions in:
Codes Correcting Asymmetric Errors
[1] R. Ahlswede, H. Aydinian and L. Khachatrian, "Unidirectional error control codes and related combinational problems," in Proc. Eighth Int. Workshop Algebr. Combin. Coding Theory, pp. 6-9, 2002. [2] Y. Cassuto, M. Schwartz, V. Bohossian and J. Bruck, "Codes for asymmetric limited-magnitude errors with application to multilevel flash memories," in IEEE Trans. Information Theory, vol. 56, no. 4, pp. 1582-95, 2010.[3] E. Yaakobi, P. H. Siegel, A. Vardy, and J. K. Wolf, “On Codes that Correct Asymmetric Errors with Graded Magnitude Distribution,” in Proc. IEEE International Symposium on Information Theory, pp. 1021-1025, 2011.
Example of Bit-fixing Coding
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Bit-wise ECC
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Assume asymmetric errors appear…
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0 01234
012345678
01234567891011
12345678
7+1 2+2 5 0 3+1 5+3 110
Binary Codes: there are 12 bit errors.
Bit-fixing code: there are only 5 bit errors! It alternatively corrects bits and cell levels.
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Gray Codes: there are 7 bit errors.
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Decoding round 1: Correct bits
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corrected 3 bit errors€
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Decoding round 1: Correct cell levels
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(7+1) – 2^0 = 7
(3+1) – 2^0 = 3
(5+3) – 2^0 = 7
A B C D
Updating rule: new level = current level – e * 2^i
Value of error Bit index An error in the LSB = a magnitude-1 error on level
1st bit = a magnitude-2 error 2nd bit = a magnitude-4 error
MSB = a magnitude-8 error corrected 3 bit errors
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Decoding round 1: Correct cell levels
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corrected 3 bit errors
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(7+1) – 2^0 = 7
(3+1) – 2^0 = 3
(5+3) – 2^0 = 7
A B C D
Updating rule: new level = current level – e * 2^i
Value of error Bit index
7
Initial Data
Receive Errors
Correct Bits
Correct Cells
7 7+1 7+1(7+1) – 2^0
= 7
Corrected Data
Decoding round 2: Correct bits
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Decoding round 2: Correct cell levels
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Updating rule: new level = current level – e * 2^i
(2+2) – 2^1 = 2
(5+2) – 2^1 = 5
A B C D
Let the total number of cell levels be . Assume the magnitudes of n errors are , where for
◦ The total number of errors need to be corrected is
◦ Let , the total number of iterations is
Correcting asymmetric errors
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log2 e⎡ ⎤
Computes Hamming weight
Computes binary representation
0123456789101112131415
A downward error with magnitude e (less than 0)
An upward error with magnitude (e mod q)
Let the total number of cell levels be . Assume the magnitudes of errors are , where
for .◦ The total number of errors need to be corrected is
◦ Let , the total # of iterations is
Correcting bidirectional errors
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log2 e⎡ ⎤Updating rule: new level = (current level – e * 2^i)
mod q
Suppose errors are bidirectional
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012345
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Decoding round 1: Correct bits
corrected 3 bit errors
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corrected 3 bit errors
(7+1) – 2^0 = 7
(5+3) – 2^0 = 7
(11-1) – 2^0 = 9
Updating rule: new level = (current level – e * 2^i) mod q
A B C D
Total number of cell levels
Decoding round 2: Correct bits
corrected 3 + 3 bit errors
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Decoding round 2: Correct cell levels
7 2 0 3 5
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corrected 3 + 3 bit errors
5-2-2 11-2-2
(5-2) – 2^1 = 1
(11-2) – 2^1 = 7
A B C D
(5+2) – 2^0 = 5
Updating rule: new level = (current level – e * 2^i) mod q
Decoding round 3: Correct bits
corrected 3 + 3 +2 bit errors
7 2 0 3 5
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A B C D
corrected 3 + 3 +2 bit errors
7 2 0 3 5
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Decoding round 3: Correct cell levels
5-2-2-4 11-2-2-4
((5-2-2) – 2^2) mod 16 = 13 (11-2-2) – 2^2 = 3
B C DA
Updating rule: new level = (current level – e * 2^i) mod q
Decoding round 4: Correct bits
7 2 5-2-2-4 0 3 5 11-2-2-4
€
0111
⎛
⎝
⎜ ⎜ ⎜ ⎜
⎞
⎠
⎟ ⎟ ⎟ ⎟
€
0010
⎛
⎝
⎜ ⎜ ⎜ ⎜
⎞
⎠
⎟ ⎟ ⎟ ⎟
€
0101
⎛
⎝
⎜ ⎜ ⎜ ⎜
⎞
⎠
⎟ ⎟ ⎟ ⎟
€
0000
⎛
⎝
⎜ ⎜ ⎜ ⎜
⎞
⎠
⎟ ⎟ ⎟ ⎟
€
0011
⎛
⎝
⎜ ⎜ ⎜ ⎜
⎞
⎠
⎟ ⎟ ⎟ ⎟
€
0101
⎛
⎝
⎜ ⎜ ⎜ ⎜
⎞
⎠
⎟ ⎟ ⎟ ⎟
€
1011
⎛
⎝
⎜ ⎜ ⎜ ⎜
⎞
⎠
⎟ ⎟ ⎟ ⎟
corrected 3 + 3 + 2 + 2 bit errors
B C DA
corrected 3 + 3 + 2 + 2 bit errors
7 2 5-2-2-4-8 0 3 5 11-2-2-4-8
€
0111
⎛
⎝
⎜ ⎜ ⎜ ⎜
⎞
⎠
⎟ ⎟ ⎟ ⎟
€
0010
⎛
⎝
⎜ ⎜ ⎜ ⎜
⎞
⎠
⎟ ⎟ ⎟ ⎟
€
0101
⎛
⎝
⎜ ⎜ ⎜ ⎜
⎞
⎠
⎟ ⎟ ⎟ ⎟
€
0000
⎛
⎝
⎜ ⎜ ⎜ ⎜
⎞
⎠
⎟ ⎟ ⎟ ⎟
€
0011
⎛
⎝
⎜ ⎜ ⎜ ⎜
⎞
⎠
⎟ ⎟ ⎟ ⎟
€
0101
⎛
⎝
⎜ ⎜ ⎜ ⎜
⎞
⎠
⎟ ⎟ ⎟ ⎟
€
1011
⎛
⎝
⎜ ⎜ ⎜ ⎜
⎞
⎠
⎟ ⎟ ⎟ ⎟
Decoding round 4: Correct cell levels((5-2-2-4) – 2^3) mod 16 = 5
((11-2-2-4) – 2^3) mod 16 = 11
B C DA
Updating rule: new level = (current level – e * 2^i) mod q
...…………
€
km
€
km−1
€
k2
€
k1€
...
€
l1
€
l2
€
l3
€
ln−2
€
ln−1
€
ln−1
€
...
Encoding
Redundancy
AllocationSeparateEncoding
Compute Cell Levels
ProgramCells
ECCECCECCECCECC
Compare the rates with those of◦ Binary codes
◦ Gray codes
◦ ALM-ECC with hard decoding An error of any magnitude is seen as a Hamming error in the
inner code.
◦ ALM-ECC with soft decoding The error distribution is used for optimal decoding. (But the
inner ECC is not binary if the maximum magnitude of errors is more than 1).
Evaluation
Y. Cassuto, M. Schwartz, V. Bohossian and J. Bruck, "Codes for asymmetric limited-magnitude errors with application to multilevel flash memories," in IEEE Trans. Information Theory, vol. 56, no. 4, pp. 1582-95, 2010.
Cell levels: 16
Maximum magnitude of asymmetric errors: 3
Error distribution mimics a Gaussian distribution.◦ For , the
probability of an error of size equals .
Error Model
€
i = 0, 1, 2, 3
€
i
€
Pi
Effective◦ Good for correcting asymmetric errors.◦ Efficient decoding and encoding methods.
Flexible◦ Works with existing ECCs.◦ Naturally handles bidirectional errors.
General◦ Supports arbitrary error distribution.◦ Supports arbitrary numeral systems.
Conclusions
A. Jiang, Y. Li and J. Bruck, ”Bit-fixing codes for multi-level cells," in Proc. IEEE Information Theory Workshop (ITW), 2012.