Permutation-based decoding of Reed-Muller codes in binary erasure - - PowerPoint PPT Presentation

Permutation-based decoding of Reed-Muller codes in binary erasure channel

Kirill Ivanov, R¨ udiger Urbanke ´ Ecole Polytechnique F´ ed´ erale de Lausanne, Switzerland July 12, 2018

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Channel polarization

Consider data transmission via binary erasure channel BEC(ǫ) Polarization transform:

ǫ(0) = 1 − (1 − ǫ)2 ǫ(1) = ǫ2

This can be done recursively up to any length n = 2m

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Polar codes

Polarization transform matrix: Am = 1 1 1 ⊗m Encoding of data vector of size k: Select set of frozen subchannel indices F, |F| = n − k with the highest values ǫ(i) Transmit data via k best subchannels and some predefined values (e.g. zeros) via n − k others Reed-Muller code RM(r, m): take F = {i| wt(i) < m − r} Minimum distance of polar code is determined by low-weight non-frozen indices RM rule maximizes the minimum distance of the code

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List decoding

Path ˆ uφ−1 — vector of estimated symbols For φ = 0, . . . , n − 1, decoder constructs all possible path continuations and puts at most L most likely ones in the list The most likely path ˆ un−1 is returned

Figure: Decoding of (16, 8) code with L = 4

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u0 u1 u2 u3 u4 u5 u6 u8 u7 u9 u10 u11 u12 u13 u14 u15

1 1

Factor graph permutations

The encoding with length-2m polar code can be represented with m-layer factor graph Any permutation P of factor graph layers corresponds to particular estimation order by SC-based algorithms

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Permutation list decoding

Brief outline of the algorithm: Take multiple SCL decoder copies corresponding to different permutations Synchronize them via shared buffer of estimated symbols Decoding fails if no decoder can make another step Reed-Muller codes: near-MAP performance for smaller list size

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