On the Decoding of Polar Codeson Permuted Factor Graphs

Nghia Doan1, Seyyed Ali Hashemi1, Marco Mondelli2, andWarren Gross1

1McGill University, Quebec, Canada2Standford University, California, USA

GLOBECOM 2018Abu Dhabi, UAEDec 10, 2018

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Problem Statement

I Polar codes: selected for the eMBB control channel in 5G

I Successive Cancellation (SC) List (SCL): gooderror-correction performance, serial nature

I Belief Propagation (BP): reasonable error-correctionperformance, highly parallel→ enable high decodingthroughput !

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Problem Statement

This paper

I Improve the error-correction performance of polar-codedecoding by exploiting factor-graph permutations

I Provide a hardware friendly representation of thefactor-graph permutations

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Polar codesI Introduced by Arıkan1 in 2009I P(N,K ), N: code length, K : message lengthI Code construction: based on polarization phenomenon

I K most reliable channels: information bitsI (N − K ) least reliable channels: frozen bits

P(8, 5) with u0, u1, and u2 are frozen bits

1E. Arıkan. “Channel Polarization: A Method for Constructing Capacity-Achieving Codes for SymmetricBinary-Input Memoryless Channels”. In: 55.7 (2009), pp. 3051–3073. ISSN: 0018-9448. DOI:10.1109/TIT.2009.2021379.Doan et. al On the Decoding of Polar Codes on Permuted Factor Graphs 3/17

Belief Propagation (BP) Decoding

I An iterative message passing algorithmI Belief messages are updated through Processing

Elements (PEs)I Termination conditions: CRC-based condition2, maximum

number of iterations

BP decoding for P(8, 5) A polar PE

2Y. Ren et al. “Efficient early termination schemes for belief-propagation decoding of polar codes”. In: IEEE 11thInt. Conf. on ASIC. 2015, pp. 1–4. DOI: 10.1109/ASICON.2015.7517046.

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BP Decoding on Permuted Factor Graphs

I Permuting the factor-graph layers preserves the code34

I Running BP decoding on multiple factor-graphpermutations improves the error-correction performance56

Various factor-graph permutations of P(8, 5)

3S. B. Korada. “Polar Codes for Channel and Source Coding”. PhD thesis. Lausanne, Switzerland: EPFL, 2009.4Nadine Hussami, Satish Babu Korada, and Rudiger Urbanke. “Performance of polar codes for channel and

source coding”. In: IEEE Int. Symp. on Inf. Theory. 2009, pp. 1488–1492.5A. Elkelesh et al. “Belief propagation decoding of polar codes on permuted factor graphs”. In: IEEE Wireless

Commun. and Net. Conf. 2018, pp. 1–6. DOI: 10.1109/WCNC.2018.8377158.6Ahmed Elkelesh et al. “Belief Propagation List Decoding of Polar Codes”. In: IEEE Communications Letters 22

(2018), pp. 1536–1539.

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BP Decoding on Permuted Factor Graphs

Permuted factor graph representations for P(8, 5)

I Problem: each factor-graph permutation requires adifferent decoding schedule→ require different decodersin hardware

I Solution: transform factor-graph layer permutation tocodeword-position permutation

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Factor-graph Permutation to Codeword-positionPermutation

I Definitions:

I Permutation π : {0,1, . . . ,n − 1} → {0,1, . . . ,n − 1}where n = log2(N)

I Apply a permutation π to the original graph layerL = {ln−1, . . . , l1, l0}:

L = {ln−1, . . . , l1, l0}π−→ Lπ = {lπ(n−1), . . . , lπ(1), lπ(0)}

I Binary expansion of the integer i

{b(i)n−1, . . . ,b

(i)1 ,b(i)

0 }

where b(i)j ∈ {0,1} (0 ≤ i ≤ N − 1;0 ≤ j ≤ n − 1)

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Factor-graph Permutation to Codeword-positionPermutation

Theorem 1Apply permutation π to the original factor graph of a polar code

L = {ln−1, . . . , l1, l0}π−→ Lπ = {lπ(n−1), . . . , lπ(1), lπ(0)}.

Then, the synthetic channel associated with the binaryexpansion

{b(i)n−1, . . . ,b

(i)1 ,b(i)

0 }

of the original factor graph L is the same as the syntheticchannel associated with the binary expansion

{b(i)π(n−1), . . . ,b

(i)π(1),b

(i)π(0)}

of the permuted factor graph Lπ.

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Factor-graph Permutation to Codeword-positionPermutation

Proof.

I On L, the synthetic channel associated with the binaryexpansion {b(i)

n−1, . . . ,b(i)1 ,b(i)

0 } is

W (i)L = ((((W b(i)

n−1)...)b(i)1 )b(i)

0 )

I On Lπ, the synthetic channel associated with the binaryexpansion {b(i)

π(n−1), . . . ,b(i)π(1),b

(i)π(0)} is

W (i)Lπ = ((((W b(i)

π(π(n−1)))...)b(i)π(π(1)))

b(i)π(π(0))

= ((((W b(i)n−1)...)b(i)

1 )b(i)0 ) = W (i)

L

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Factor-graph Permutation to Codeword-positionPermutation

Apply Theorem 1 to a permuted factor graph Lπ:

Lπ = {lπ(n−1), . . . , lπ(1), lπ(0)}π−→ L = {ln−1, . . . , l1, l0}.

Then, the synthetic channel associated with the binaryexpansion

{b(i)n−1, . . . ,b

(i)1 ,b(i)

0 }

of the permuted factor graph Lπ is the same as the syntheticchannel associated with the binary expansion

{b(i)π(n−1), . . . ,b

(i)π(1),b

(i)π(0)}

of the original factor graph L.

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Factor-graph Permutation to Codeword-positionPermutation

An example:

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Selection of Good Permutations

I Construct a set of permutations P

I Fix (n − k) left-most layers of the original graph

I Construct k ! permutations of the k right-most layers

I |P| = k !

I If the decoding algorithm fails on the original graph

I Run the decoding algorithm on each permutation p of PI Numerically evaluate the probability of successful decoding

for each permutation p

I Select the M best permutations of P

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Selection of Good Permutations

2 2.5 3 3.5

10−4

10−3

10−2

Eb/N0 [dB]

FER

k = 4

k = 5

k = 6

k = 7

k = 8

k = 9

FER performance of BP decoding on 16 best permuted factor graphsof P(1024,512) with 24-bit CRC, and |P| = k !.

I Good permutations: found by permuting the layers on theright-most side of the original factor graph

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5G P(1024,512), no CRC, AWGN channel

2 2.5 3 3.5

10−4

10−3

10−2

10−1

Eb/N0 [dB]

FER

PBP-CS78 PBP-R109

PBP-B10 PSC-B10

SC BP

SCL32

7Korada, “Polar Codes for Channel and Source Coding”.8Hussami, Korada, and Urbanke, “Performance of polar codes for channel and source coding”.9Elkelesh et al., “Belief propagation decoding of polar codes on permuted factor graphs”.

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5G P(1024,512), 24-bit CRC, AWGN channel

2 2.5 3 3.510−6

10−5

10−4

10−3

10−2

10−1

Eb/N0 [dB]

FER

SCL32

SCL32-CRC24

PBP-CS-CRC241011

PBP-R10-CRC2412

PBP-B10-CRC24

PSC-B10-CRC24

PBP-R32-CRC2412

PBP-B32-CRC24

PSC-B32-CRC24

PBP-R128-CRC2412

PBP-B128-CRC24

PSC-B128-CRC24

10Korada, “Polar Codes for Channel and Source Coding”.11Hussami, Korada, and Urbanke, “Performance of polar codes for channel and source coding”.12Elkelesh et al., “Belief propagation decoding of polar codes on permuted factor graphs”.

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Conclusion

I Factor-graph permutations can be mapped tocodeword-position permutations→ require a single decoder for hardware implementation

I Propose a method to construct good permutations fordifferent polar-code decoders

I 5G P(1024,512), no CRC, at FER = 10−4: PSC-B10 is 0.4dB better than PSC-R10, and almost equivalent to SCL32

I 5G P(1024,512), 24-bit CRC, at FER = 10−4: PBP-B128is 0.25 dB better than PBP-R128, but 0.3 dB worse thanSCL32

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Thank you !

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