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 kirill.ivanov@epfl.ch (EPFL) Decoding of RM codes in BEC July 12, 2018 1 / 6
Channel polarization Consider data transmission via binary erasure channel BEC( ǫ ) Polarization transform: ǫ (0) = 1 − (1 − ǫ ) 2 ǫ (1) = ǫ 2 This can be done recursively up to any length n = 2 m kirill.ivanov@epfl.ch (EPFL) Decoding of RM codes in BEC July 12, 2018 2 / 6
Polar codes � ⊗ m � 1 0 Polarization transform matrix: A m = 1 1 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 kirill.ivanov@epfl.ch (EPFL) Decoding of RM codes in BEC July 12, 2018 3 / 6
List decoding u φ − 1 Path ˆ — vector of estimated symbols 0 For φ = 0 , . . . , n − 1, decoder constructs all possible path continuations and puts at most L most likely ones in the list u n − 1 The most likely path ˆ is returned 0 u 0 u 1 u 2 u 3 u 4 u 5 u 6 u 7 u 8 u 9 u 10 u 11 u 12 u 13 u 14 u 15 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 Figure: Decoding of (16 , 8) code with L = 4 kirill.ivanov@epfl.ch (EPFL) Decoding of RM codes in BEC July 12, 2018 4 / 6
Factor graph permutations The encoding with length-2 m 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 kirill.ivanov@epfl.ch (EPFL) Decoding of RM codes in BEC July 12, 2018 5 / 6
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 kirill.ivanov@epfl.ch (EPFL) Decoding of RM codes in BEC July 12, 2018 6 / 6
Recommend
More recommend
