Successive Cancellation Inactivation Decoding for Modified - - PowerPoint PPT Presentation
International Symposium on Information Theory Los Angeles, USA June, 2020 Successive Cancellation Inactivation Decoding for Modified Reed-Muller and eBCH Codes Mustafa Cemil Cokun 1,2 Joint work with J. Neu 3 and H. D. Pfister 4 1 German
International Symposium on Information Theory Los Angeles, USA – June, 2020
Mustafa Cemil Coşkun1,2 Joint work with J. Neu3 and H. D. Pfister4
1German Aerospace Center 2Technical University of Munich 3Stanford University 4Duke University
Introduction Preliminaries Successive Cancellation Inactivation Decoding Numerical Results Conclusions
Introduction Preliminaries Successive Cancellation Inactivation Decoding Numerical Results Conclusions
Page 1/19
Successive cancellation (SC) inactivation decoding is introduced as an
efficient implementation of SC list (SCL) decoding over the binary erasure channel (BEC)
decoder with L ≥ 2150, for a (8192, 4096) code
Page 1/19
Successive cancellation (SC) inactivation decoding is introduced as an
efficient implementation of SC list (SCL) decoding over the binary erasure channel (BEC)
decoder with L ≥ 2150, for a (8192, 4096) code
Analysis of the number of inactivations required for the ML decoder
Page 1/19
Successive cancellation (SC) inactivation decoding is introduced as an
efficient implementation of SC list (SCL) decoding over the binary erasure channel (BEC)
decoder with L ≥ 2150, for a (8192, 4096) code
Analysis of the number of inactivations required for the ML decoder Numerical results for various code constructions with dynamic frozen bits,
including a modified Reed–Muller (RM) code (denoted d-RM)
Page 2/19
The BEC, while a special case, can be useful for the design of good codes
for general channels3
good under SCL decoding with small list size on the AWGN channel
Page 2/19
The BEC, while a special case, can be useful for the design of good codes
for general channels3
good under SCL decoding with small list size on the AWGN channel
The number of inactivations quantifies the average complexity for an ML
decoder, implemented via SC inactivation (or SCL) decoding
Introduction Preliminaries Successive Cancellation Inactivation Decoding Numerical Results Conclusions
Page 3/19
xn
1 = un 1 K⊗m 2
where K2
1 1
n = 2m
channels,” T-IT, Jul. 2009.
Page 3/19
xn
1 = un 1 K⊗m 2
where K2
1 1
n = 2m
BEC(0.5) y1 BEC(0.5) y2 BEC(0.5) y3 BEC(0.5) y4 BEC(0.5) y5 BEC(0.5) y6 BEC(0.5) y7 BEC(0.5) y8
⊕
x2 x3 x4 x5 x6 x7 x8 u1 u2 u3 u4 u5 u6 u7 u8
channels,” T-IT, Jul. 2009.
Page 3/19
xn
1 = un 1 K⊗m 2
where K2
1 1
n = 2m
BEC(0.5) y1 BEC(0.5) y2 BEC(0.5) y3 BEC(0.5) y4 BEC(0.5) y5 BEC(0.5) y6 BEC(0.5) y7 BEC(0.5) y8
⊕
x2 x3 x4 x5 x6 x7 x8 ǫ1 = 0.9961 ǫ2 = 0.8789 ǫ3 = 0.8086 ǫ4 = 0.3164 ǫ5 = 0.6836 ǫ6 = 0.1914 ǫ7 = 0.1211 ǫ8 = 0.0039 frozen frozen frozen info frozen info info info F = {1, 2, 3, 5} A = {4, 6, 7, 8}
channels,” T-IT, Jul. 2009.
Page 4/19
The value of a frozen bit can also be set to a linear combination of
previous information bits (rather than 0)
A frozen bit whose value depends on past inputs is called dynamic The SC/SCL decoders are easily modified to decode polar codes with
dynamic frozen bits5
Any binary linear block code can be represented as a polar code with
dynamic frozen bits5 (e.g., PAC code)
Introduction Preliminaries Successive Cancellation Inactivation Decoding Numerical Results Conclusions
Page 5/19
For linear codes on the BEC, any uncertainty in the information bits
always takes the form of an affine subspace
decoding,” T-IT, Dec. 2008.
Commun., Control, and Comput., Sep. 2010.
Page 5/19
For linear codes on the BEC, any uncertainty in the information bits
always takes the form of an affine subspace
The SCL decoder on the BEC lists all valid paths in this subspace The SC inactivation decoder stores a basis instead of listing all valid paths
decoding,” T-IT, Dec. 2008.
Commun., Control, and Comput., Sep. 2010.
Page 5/19
For linear codes on the BEC, any uncertainty in the information bits
always takes the form of an affine subspace
The SCL decoder on the BEC lists all valid paths in this subspace The SC inactivation decoder stores a basis instead of listing all valid paths Similar methods have been proposed for low-density parity-check6 7 8 and
raptor9 codes
For polar codes, a BP decoder with inactivations was proposed,10
improving bit-error rate, but it does not use SC decoding schedule
decoding,” T-IT, Dec. 2008.
Commun., Control, and Comput., Sep. 2010.
Page 6/19
The SC inactivation decoder has the same message passing schedule as
the SC decoder
Whenever an information bit is decoded as erased, it is replaced by a
dummy variable (i.e., inactivated)
Page 6/19
The SC inactivation decoder has the same message passing schedule as
the SC decoder
Whenever an information bit is decoded as erased, it is replaced by a
dummy variable (i.e., inactivated)
It continues decoding using SC decoding for the BEC, where the message
values are allowed to be functions of all inactivated variables
The inactivated bits are resolved, later, using linear equations derived from
decoding frozen bits
Page 7/19
BEC(0.5) BEC(0.5) BEC(0.5) BEC(0.5) BEC(0.5) BEC(0.5) BEC(0.5) BEC(0.5) ? ? ? ?
⊕
u2 = 0 u3 = 0 u4 = info u5 = u4 u6 = info u7 = info u8 = info
Page 7/19
BEC(0.5) BEC(0.5) BEC(0.5) BEC(0.5) BEC(0.5) BEC(0.5) BEC(0.5) BEC(0.5) ? ? ? ?
⊕
u1 = 0 (frozen) ˆ u1 = ? ? ? ? ? ? ? ?
Page 7/19
BEC(0.5) BEC(0.5) BEC(0.5) BEC(0.5) BEC(0.5) BEC(0.5) BEC(0.5) BEC(0.5) ? ? ? ?
⊕
u1 = 0 g = 0 ˆ u2 = 0 (frozen) ˆ u2 = ? ? ? ? ? ? ?
Page 7/19
BEC(0.5) BEC(0.5) BEC(0.5) BEC(0.5) BEC(0.5) BEC(0.5) BEC(0.5) BEC(0.5) ? ? ? ?
⊕
u1 = 0 ˆ u2 = 0 g = 0 ˆ u3 = 0 (frozen) ˆ u3 = ? ? ? ? ? ?
Page 7/19
BEC(0.5) BEC(0.5) BEC(0.5) BEC(0.5) BEC(0.5) BEC(0.5) BEC(0.5) BEC(0.5) ? ? ? ?
⊕
u1 = 0 ˆ u2 = 0 ˆ u3 = 0 g = 0 ˆ u4 = ? ? ? ? ?
Page 7/19
BEC(0.5) BEC(0.5) BEC(0.5) BEC(0.5) BEC(0.5) BEC(0.5) BEC(0.5) BEC(0.5) ? ? ? ?
⊕
u1 = 0 ˆ u2 = 0 ˆ u3 = 0 g = 0 ˆ u4 = x ? ? ? ?
Page 7/19
BEC(0.5) BEC(0.5) BEC(0.5) BEC(0.5) BEC(0.5) BEC(0.5) BEC(0.5) BEC(0.5) ? ? ? ?
⊕
u1 = 0 ˆ u2 = 0 ˆ u3 = 0 ˆ u4 = x g = 1 ˆ u5 = ˆ u4 (frozen) ˆ u5 = 0 ? ? ?
Page 7/19
BEC(0.5) BEC(0.5) BEC(0.5) BEC(0.5) BEC(0.5) BEC(0.5) BEC(0.5) BEC(0.5) ? ? ? ?
⊕
u1 = 0 ˆ u2 = 0 ˆ u3 = 0 ˆ u4 = x ˆ u5 = 0 x = 0 g = 1 ˆ u6 = 0 ? ?
Page 7/19
BEC(0.5) BEC(0.5) BEC(0.5) BEC(0.5) BEC(0.5) BEC(0.5) BEC(0.5) BEC(0.5) ? ? ? ?
⊕
u1 = 0 ˆ u2 = 0 ˆ u3 = 0 ˆ u4 = x ˆ u5 = 0 x = 0 ˆ u6 = 0 g = 1 ˆ u7 = 0 ?
Page 7/19
BEC(0.5) BEC(0.5) BEC(0.5) BEC(0.5) BEC(0.5) BEC(0.5) BEC(0.5) BEC(0.5) ? ? ? ?
⊕
u1 = 0 ˆ u2 = 0 ˆ u3 = 0 ˆ u4 = x ˆ u5 = 0 x = 0 ˆ u6 = 0 ˆ u7 = 0 g = 1 ˆ u8 = 0
Page 7/19
BEC(0.5) BEC(0.5) BEC(0.5) BEC(0.5) BEC(0.5) BEC(0.5) BEC(0.5) BEC(0.5) ? ? ? ?
⊕
u1 = 0 ˆ u2 = 0 ˆ u3 = 0 ˆ u4 = x ˆ u5 = 0 x = 0 ˆ u6 = 0 ˆ u7 = 0 ˆ u8 = 0 g = 1
Page 8/19
Assume that the SC inactivation decoder inactivates g bits in total during
a decoding attempt
The final step of SC inactivation decoding is to solve a system of linear
equations in g unknowns
This will have a unique solution only if the equations obtained from frozen
bits have rank g
Page 9/19
An inactivated bit may be resolved right after decoding a frozen bit
whenever it provides an informative equation
Page 9/19
An inactivated bit may be resolved right after decoding a frozen bit
whenever it provides an informative equation
The SC inactivation decoder with consolidations mimics the path pruning
stage of SCL decoding
Page 10/19
BEC(0.5) BEC(0.5) BEC(0.5) BEC(0.5) BEC(0.5) BEC(0.5) BEC(0.5) BEC(0.5) ? ? ? ?
⊕
u2 = 0 u3 = 0 u4 = info u5 = u4 u6 = info u7 = info u8 = info
Page 10/19
BEC(0.5) BEC(0.5) BEC(0.5) BEC(0.5) BEC(0.5) BEC(0.5) BEC(0.5) BEC(0.5) ? ? ? ?
⊕
u1 = 0 (frozen) ˆ u1 = ? ? ? ? ? ? ? ?
Page 10/19
BEC(0.5) BEC(0.5) BEC(0.5) BEC(0.5) BEC(0.5) BEC(0.5) BEC(0.5) BEC(0.5) ? ? ? ?
⊕
u1 = 0 g1 = 0 ˆ u2 = 0 (frozen) ˆ u2 = ? ? ? ? ? ? ?
Page 10/19
BEC(0.5) BEC(0.5) BEC(0.5) BEC(0.5) BEC(0.5) BEC(0.5) BEC(0.5) BEC(0.5) ? ? ? ?
⊕
u1 = 0 ˆ u2 = 0 g2 = 0 ˆ u3 = 0 (frozen) ˆ u3 = ? ? ? ? ? ?
Page 10/19
BEC(0.5) BEC(0.5) BEC(0.5) BEC(0.5) BEC(0.5) BEC(0.5) BEC(0.5) BEC(0.5) ? ? ? ?
⊕
u1 = 0 ˆ u2 = 0 ˆ u3 = 0 g3 = 0 ˆ u4 =? ? ? ? ?
Page 10/19
BEC(0.5) BEC(0.5) BEC(0.5) BEC(0.5) BEC(0.5) BEC(0.5) BEC(0.5) BEC(0.5) ? ? ? ?
⊕
u1 = 0 ˆ u2 = 0 ˆ u3 = 0 g3 = 0 ˆ u4 = x ? ? ? ?
Page 10/19
BEC(0.5) BEC(0.5) BEC(0.5) BEC(0.5) BEC(0.5) BEC(0.5) BEC(0.5) BEC(0.5) ? ? ? ?
⊕
u1 = 0 ˆ u2 = 0 ˆ u3 = 0 ˆ u4 = x g4 = 1 ˆ u5 = ˆ u4 (frozen) ˆ u5 = 0 ? ? ?
Page 10/19
BEC(0.5) BEC(0.5) BEC(0.5) BEC(0.5) BEC(0.5) BEC(0.5) BEC(0.5) BEC(0.5) ? ? ? ?
⊕
u1 = 0 ˆ u2 = 0 ˆ u3 = 0 ˆ u4 = 0 ˆ u5 = 0 g5 = 0 ˆ u6 = 0 ? ?
Page 10/19
BEC(0.5) BEC(0.5) BEC(0.5) BEC(0.5) BEC(0.5) BEC(0.5) BEC(0.5) BEC(0.5) ? ? ? ?
⊕
u1 = 0 ˆ u2 = 0 ˆ u3 = 0 ˆ u4 = 0 ˆ u5 = 0 ˆ u6 = 0 g6 = 0 ˆ u7 = 0 ?
Page 10/19
BEC(0.5) BEC(0.5) BEC(0.5) BEC(0.5) BEC(0.5) BEC(0.5) BEC(0.5) BEC(0.5) ? ? ? ?
⊕
u1 = 0 ˆ u2 = 0 ˆ u3 = 0 ˆ u4 = 0 ˆ u5 = 0 ˆ u6 = 0 ˆ u7 = 0 g7 = 0 ˆ u8 = 0
Page 10/19
BEC(0.5) BEC(0.5) BEC(0.5) BEC(0.5) BEC(0.5) BEC(0.5) BEC(0.5) BEC(0.5) ? ? ? ?
⊕
u1 = 0 ˆ u2 = 0 ˆ u3 = 0 ˆ u4 = 0 ˆ u5 = 0 ˆ u6 = 0 ˆ u7 = 0 ˆ u8 = 0 g8 = 0
Page 11/19
Corollary Let G be a r.v. equal to the total number of inactivations made by the decoder during a decoding attempt. Then, E[G] =
i∈A ǫi.
performance,” T-COM, Sep. 2014.
Page 11/19
Corollary Let G be a r.v. equal to the total number of inactivations made by the decoder during a decoding attempt. Then, E[G] =
i∈A ǫi.
The performance improvement under ML decoding when interpolating
from polar to RM codes is driven by the weight spectrum improvement, e.g., the minimum distance increases.11 12
The quantity E[G] quantifies this complexity increase
performance,” T-COM, Sep. 2014.
Introduction Preliminaries Successive Cancellation Inactivation Decoding Numerical Results Conclusions
Page 12/19
We first consider the following codes with parameters (n = 128, k = 64):
is a uniform random linear combination of bits ui, i ∈ A
Page 13/19
0.2 0.25 0.3 0.35 0.4 0.45 0.5 10−6 10−5 10−4 10−3 10−2 10−1 100 Erasure Probability ǫ BLER Berlekamp’s Random Coding Bound (BRCB) Singleton Bound (SB)
Page 13/19
0.2 0.25 0.3 0.35 0.4 0.45 0.5 10−6 10−5 10−4 10−3 10−2 10−1 100 Erasure Probability ǫ BLER polar RM eBCH random BRCB SB
Page 14/19
0.2 0.25 0.3 0.35 0.4 0.45 0.5 3 6 9 12 15 Erasure Probability ǫ Inactivations E[G] E[GeBCH] E[GRM] E[Gpolar]
Page 15/19
10 20 30 40 50 60 70 80 90 100 110 120 1 2 3 4 5 6 7 8 9 10 Decoding stage i Unresolved Inactivations E[Gi] (128, 64) eBCH (128, 64) RM (128, 64) polar
Page 16/19
Since the (n = 128, k = 64) RM code provides a good complexity performance trade-off, we propose a variant of it:
d-RM code
information bit(s)
2020.
Page 16/19
Since the (n = 128, k = 64) RM code provides a good complexity performance trade-off, we propose a variant of it:
d-RM code
information bit(s)
Note the similarity of the Arıkan’s PAC code2 and d-RM code:
Toeplitz matrix T
2020.
Page 16/19
Since the (n = 128, k = 64) RM code provides a good complexity performance trade-off, we propose a variant of it:
d-RM code
information bit(s)
Note the similarity of the Arıkan’s PAC code2 and d-RM code:
Toeplitz matrix T
index set is A (Arıkan chooses A of the RM code) and dynamic frozen bits are specified by T 13 14
2020.
Page 17/19
0.2 0.25 0.3 0.35 0.4 0.45 0.5 10−6 10−5 10−4 10−3 10−2 10−1 100 Erasure Probability ǫ BLER RM d-RM BRCB SB
Page 18/19
0.37 0.38 0.39 0.4 0.41 0.42 0.43 0.44 0.45 0.46 0.47 0.48 0.49 0.5 10−7 10−6 10−5 10−4 10−3 10−2 10−1 100 Erasure Probability ǫ BLER RM d-RM BRCB SB
Introduction Preliminaries Successive Cancellation Inactivation Decoding Numerical Results Conclusions
Page 19/19
An inactivation decoder is proposed using the schedule of the SC decoding
for transmisson over the BEC
Using density evolution, the expected number of inactivations is computed
analytically
The expected number of unresolved inactivations quantifies the
performance vs. complexity trade-off for the SC inactivation or SCL decoder to achieve ML decoding performance
Page 19/19
An inactivation decoder is proposed using the schedule of the SC decoding
for transmisson over the BEC
Using density evolution, the expected number of inactivations is computed
analytically
The expected number of unresolved inactivations quantifies the
performance vs. complexity trade-off for the SC inactivation or SCL decoder to achieve ML decoding performance
Future work: Can we generalize the analysis for the SCL decoder over
general BMS channels? Does the analysis also help for code design?15