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Low-Latency Polar Codes via Hybrid Decoding

Bin Li*, Hui Shen*, David Tse+, Wen Tong*

*Huawei Technologies, China + Stanford University, USA

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Contents

Excellent Performance of Polar Codes

Polar SC Decoder: Latency Problem

Proposed Hybrid Decoder

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Contents

Excellent Performance of Polar Codes

Polar SC Decoder: Latency Problem

Proposed Hybrid Decoder

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Excellent Performance can be achieved by

1) Concatenation with CRC [1]

2) SC-LIST Decoding [1]

3) Adaptive SC-LIST Decoding [2]

[1] I. Tal, A. Vardy, “List Decoding of Polar Codes”, arXiv.1206.0050.

[2] B. Li, H.Shen, D. Tse,” An Adaptive Successive Cancellation List Decoder of Polar

Codes with Cyclic Redundancy Check”, IEEE Communications Letters, Dec. 2012.

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Adaptive List decoded Polar Code (N=2048, R=1/2), 16-CRC

0.25dB from finite block size limit with Lmax=2^18 Li-Shen-Tse, 2012.

1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.910

-4

10-3

10-2

10-1

Eb/No (dB)

FE

R

Polar Code (2048,1040) concatenated with 16-Bit CRC, R=1/2

FER,Lmax=32

FER,Lmax=2048

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N=256, R=1/2, 16-CRC

Much better than state-of-the-art LDPC code

1.8 1.9 2 2.1 2.2 2.3 2.4 2.510

-4

10-3

10-2

10-1

Eb/No(dB)

FE

R

Polar Code (256,144) Concatenated with 16-Bit CRC, R=1/2

LDPC(256,128),T.Richardson

Polar code, Lmax=32

Polar code ,Lmax=2048

Polar code,Lmax=32768

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Contents

Excellent Performance of Polar Codes

Polar SC Decoder: Latency Problem

Proposed Hybrid Decoder

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SC Decoder: Latency Problem

SC decoder decodes bit-by-it in serial fashion

2N-2 cycles are needed to decode one code of length N

This latency cannot be reduced by hardware parallelization

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ML Decoder: Low Latency

ML decoder decodes all bits simultaneously

Latency can be made arbitrarily small by

increasing hardware parallelization.

Infeasible to implement in practice.

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Contents

Excellent Performance of Polar Codes

SC Decoder: Latency Problem

Our Proposed Hybrid Decoder

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Equivalent Encoder Representation

2 Outer Codes+ 2n-1 Inner Codes

N=2n

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Equivalent SC Decoder Representation

2 Outer SC Decoders+ 2n-1 Inner SC Decoders

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General Encoder Decomposition

2k Outer Codes+ 2n-k Inner Codes

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General Decoder Decomposition

2k Outer SC Decoders+ 2n-k Inner SC Decoders

Inner SC

decoders

operate in

parallel:

2k+1-2 cycles

Outer SC

decoders

operate

serially:

2k[ 2n-k+1-2]

=2n+1-2k+1

cycles

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Our Proposed Hybrid Decoder

Hybrid Decoder, k=n, SC decoder, k=0, ML decoder

Inner SC

decoders

operate in

parallel:

2k+1-2 cycles

Outer ML

decoders

operate

serially:

2k cycles

Replace SC

decoder with

ML decoder

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Latency Reduction by Hybrid Decoder

The latency ratio of hybrid over SC decoder is

3*2n-k+1/2n+1=3/2n-k

If k=n-4, 16 inner decoders working in parallel, the above

ration is 3/16.

The parameter k is used to tradeoff latency and complexity

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ML Decoder of Polar Codes

Polar Encoding:

ML decoding:

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Decomposition of Polar Codes

Decomposition of Polar code:

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Decomposition of ML Search

Decomposition of ML Search:

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Simplified ML Decoding

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Complexity Reduction of Simplified ML Decoding

Complexity Reduction:

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Pictorially Explain of Simplified ML Search

Pictorially Explain the Simplification of ML Search:

)1(

1C )1(

1C

)1(

1D )2(

1D

One code is decomposed:

one repetition and two independent codes

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Recursive ML Decoder

Recursive Simplification of ML Search:

)1(

1C )1(

1C

)1(

2C )1(

2C )2(

2C )2(

2C

)5(

2D )6(

2D )7(

2D )8(

2D)1(

2D )3(

2D)2(

2D )4(

2D

)1(

1C )1(

1C

)1(

2C )1(

2C )2(

2C )2(

2C

)1(

3C )1(

3C )2(

3C )2(

3C )3(

3C )3(

3C )4(

3C )4(

3C

)1(

3D )3(

3D )5(

3D )7(

3D)2(

3D )4(

3D )6(

3D )8(

3D

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THANK YOU