Decoding Reed-Muller and Polar Codes by Successive Factor Graph - PowerPoint PPT Presentation

Decoding Reed-Muller and Polar Codes by Successive Factor Graph Permutations Seyyed Ali Hashemi a , Nghia Doan a , Marco Mondelli b , Warren J. Gross a a McGill University, Canada b Stanford University, USA ISTC 2018 Hong Kong December 4, 2018. ISTC 2018 Decoding Reed-Muller and Polar Codes by Successive Factor Graph Permutations 0/19

Motivation ◮ Polar Codes : adopted in 5G eMBB control channel ◮ Requires low-complexity decoders with good performance ◮ Reed-Muller (RM) Codes : very similar to polar codes In this talk: We propose a new low-complexity decoder for RM and polar codes with good performance! ISTC 2018 Decoding Reed-Muller and Polar Codes by Successive Factor Graph Permutations 1/19

RM and Polar Codes: Encoding RM ( 8 , 4 ) , P ( 8 , 4 ) uG ⊗ 3 = x ISTC 2018 Decoding Reed-Muller and Polar Codes by Successive Factor Graph Permutations 2/19

RM and Polar Codes: Encoding RM ( 8 , 4 ) , P ( 8 , 4 ) uG ⊗ 3 = x T  T u 0 1 0 0 0 0 0 0 0     x 0  u 1 1 1 0 0 0 0 0 0 x 1             u 2 1 0 1 0 0 0 0 0 x 2             u 3 1 1 1 1 0 0 0 0 x 3       =       u 4 1 0 0 0 1 0 0 0 x 4             u 5 1 1 0 0 1 1 0 0 x 5             u 6 1 0 1 0 1 0 1 0 x 6       u 7 1 1 1 1 1 1 1 1 x 7 ISTC 2018 Decoding Reed-Muller and Polar Codes by Successive Factor Graph Permutations 2/19

RM and Polar Codes: Encoding RM ( 8 , 4 ) , P ( 8 , 4 ) uG ⊗ 3 = x T  T 0 1 0 0 0 0 0 0 0 x 0      0 1 1 0 0 0 0 0 0 x 1        0   1 0 1 0 0 0 0 0    x 2             u 3 1 1 1 1 0 0 0 0 x 3       =       0 1 0 0 0 1 0 0 0 x 4             u 5 1 1 0 0 1 1 0 0 x 5             u 6 1 0 1 0 1 0 1 0 x 6       u 7 1 1 1 1 1 1 1 1 x 7 ISTC 2018 Decoding Reed-Muller and Polar Codes by Successive Factor Graph Permutations 2/19

RM and Polar Codes: Encoding RM ( 8 , 4 ) , P ( 8 , 4 ) uG ⊗ 3 = x T  T 0 1 0 0 0 0 0 0 0 x 0      0 1 1 0 0 0 0 0 0 x 1        0   1 0 1 0 0 0 0 0    x 2             u 3 1 1 1 1 0 0 0 0 x 3       =       0 1 0 0 0 1 0 0 0 x 4             u 5 1 1 0 0 1 1 0 0 x 5             u 6 1 0 1 0 1 0 1 0 x 6       u 7 1 1 1 1 1 1 1 1 x 7 RM rule : Remove rows with lowest Hamming weights ISTC 2018 Decoding Reed-Muller and Polar Codes by Successive Factor Graph Permutations 2/19

RM and Polar Codes: Encoding RM ( 8 , 4 ) , P ( 8 , 4 ) uG ⊗ 3 = x T  T 0 1 0 0 0 0 0 0 0 x 0      0 1 1 0 0 0 0 0 0 x 1        0   1 0 1 0 0 0 0 0    x 2             u 3 1 1 1 1 0 0 0 0 x 3       =       0 1 0 0 0 1 0 0 0 x 4             u 5 1 1 0 0 1 1 0 0 x 5             u 6 1 0 1 0 1 0 1 0 x 6       u 7 1 1 1 1 1 1 1 1 x 7 Polar rule : Remove rows with lowest reliabilities ISTC 2018 Decoding Reed-Muller and Polar Codes by Successive Factor Graph Permutations 2/19

RM and Polar Codes: Recursive Construction layer 0 1 u 0 x 0 u 1 x 1 ISTC 2018 Decoding Reed-Muller and Polar Codes by Successive Factor Graph Permutations 3/19

RM and Polar Codes: Recursive Construction layer 0 1 2 u 0 x 0 u 1 x 1 u 2 x 2 u 3 x 3 ISTC 2018 Decoding Reed-Muller and Polar Codes by Successive Factor Graph Permutations 3/19

RM and Polar Codes: Recursive Construction layer 0 1 2 3 u 0 x 0 u 1 x 1 u 2 x 2 u 3 x 3 u 4 x 4 u 5 x 5 u 6 x 6 u 7 x 7 ISTC 2018 Decoding Reed-Muller and Polar Codes by Successive Factor Graph Permutations 3/19

RM and Polar Codes: Decoding Successive-Cancellation (SC) layer l + 1 l α i , l , β i , l α i , l + 1 , β i , l + 1 α i + 2 l , l , β i + 2 l , l α i + 2 l , l + 1 , β i + 2 l , l + 1 � � α i , l = f α i , l + 1 , α i + 2 l , l + 1 � � α i + 2 l , l = g α i , l + 1 , α i + 2 l , l + 1 , β i , l β i , l + 1 = β i , l ⊕ β i + 2 l , l β i + 2 l , l + 1 = β i + 2 l , l ISTC 2018 Decoding Reed-Muller and Polar Codes by Successive Factor Graph Permutations 4/19

RM vs Polar ◮ RM: ◮ Minimizes error probability under MAP decoding ◮ High complexity ◮ Channel-independent ◮ Low complexity ◮ Polar: ◮ Minimizes error probability under SC decoding ◮ Low complexity ◮ Channel-dependent ◮ High complexity ISTC 2018 Decoding Reed-Muller and Polar Codes by Successive Factor Graph Permutations 5/19

SC List (SCL) Decoding: Something in Between ◮ SCL instantiates multiple SC decoders ◮ L codeword candidates survive to limit complexity ◮ A CRC can help SCL find the correct candidate SC ← → SCL ← → MAP ◮ Requires a large L to get close to MAP ISTC 2018 Decoding Reed-Muller and Polar Codes by Successive Factor Graph Permutations 6/19

Factor Graph Permutations 0 1 2 3 0 1 2 3 0 1 2 3 ◮ n = log 2 N layers → n ! factor graph permutations ◮ Decode over multiple factor graphs → Pick the best one ◮ High decoding complexity ISTC 2018 Decoding Reed-Muller and Polar Codes by Successive Factor Graph Permutations 7/19

Bit Index Permutations 0 1 2 3 u 0 x 0 u 1 x 1 u 2 x 2 u 3 x 3 u 4 x 4 u 5 x 5 u 6 x 6 u 7 x 7 0 1 2 0 1 2 3 0 1 2 3 u 0 x 0 u 0 x 0 u 1 x 1 u 4 x 4 u 2 x 2 u 1 x 1 u 3 x 3 u 5 x 5 u 4 x 4 u 2 x 2 u 5 x 5 u 6 x 6 u 6 x 6 u 3 x 3 u 7 x 7 u 7 x 7 1 2 0 0 1 2 ISTC 2018 Decoding Reed-Muller and Polar Codes by Successive Factor Graph Permutations 8/19

Architecture 0 1 2 3 u 0 x 0 u 1 x 1 u 2 x 2 u 3 x 3 π − 1 π u 4 x 4 u 5 x 5 u 6 x 6 u 7 x 7 0 1 2 ISTC 2018 Decoding Reed-Muller and Polar Codes by Successive Factor Graph Permutations 9/19

Permutations on Sub-Graphs 0 1 2 3 u 0 x 0 π 0 1 u 1 x 1 π 0 2 u 2 x 2 π 1 1 u 3 x 3 π 0 3 u 4 x 4 π 2 1 u 5 x 5 π 1 2 u 6 x 6 π 3 1 u 7 x 7 0 1 2 ISTC 2018 Decoding Reed-Muller and Polar Codes by Successive Factor Graph Permutations 10/19

Total Number of Permutations ◮ By cyclic shift permutations at each layer: n − 1 ( n − l )( 2 l ) � l = 0 ◮ Much larger than n ! ◮ Quickly grows with n ◮ 1658880 permutations for length 32 ◮ More than 1 . 9 × 10 27 for length 128 Issue: Decoding over all permutations is impossible! ISTC 2018 Decoding Reed-Muller and Polar Codes by Successive Factor Graph Permutations 11/19

Successive Permutations for SC 0 1 2 3 u 0 x 0 π 0 1 u 1 x 1 π 0 2 u 2 x 2 π 1 1 u 3 x 3 π 0 3 u 4 x 4 π 2 1 u 5 x 5 π 1 2 u 6 x 6 π 3 1 u 7 x 7 0 1 2 ISTC 2018 Decoding Reed-Muller and Polar Codes by Successive Factor Graph Permutations 12/19

Successive Permutations for SC 0 1 2 3 u 0 x 0 π 0 1 u 1 x 1 π 0 2 u 2 x 2 π 1 1 u 3 x 3 π 0 π 0 3 3 u 4 x 4 π 2 1 u 5 x 5 π 1 2 u 6 x 6 π 3 1 u 7 x 7 0 1 2 ISTC 2018 Decoding Reed-Muller and Polar Codes by Successive Factor Graph Permutations 12/19

Successive Permutations for SC 0 1 2 3 u 0 x 0 π 0 1 u 1 x 1 π 0 π 0 2 2 u 2 x 2 π 1 1 u 3 x 3 π 0 π 0 3 3 u 4 x 4 π 2 1 u 5 x 5 π 1 2 u 6 x 6 π 3 1 u 7 x 7 0 1 2 ISTC 2018 Decoding Reed-Muller and Polar Codes by Successive Factor Graph Permutations 12/19

Successive Permutations for SC 0 1 2 3 u 0 x 0 π 0 π 0 1 1 u 1 x 1 π 0 π 0 2 2 u 2 x 2 π 1 1 u 3 x 3 π 0 π 0 3 3 u 4 x 4 π 2 1 u 5 x 5 π 1 2 u 6 x 6 π 3 1 u 7 x 7 0 1 2 ISTC 2018 Decoding Reed-Muller and Polar Codes by Successive Factor Graph Permutations 12/19

Successive Permutations for SC 0 1 2 3 u 0 x 0 π 0 π 0 1 1 u 1 x 1 π 0 π 0 2 2 u 2 x 2 π 1 π 1 1 1 u 3 x 3 π 0 π 0 3 3 u 4 x 4 π 2 1 u 5 x 5 π 1 2 u 6 x 6 π 3 1 u 7 x 7 0 1 2 ISTC 2018 Decoding Reed-Muller and Polar Codes by Successive Factor Graph Permutations 12/19

Successive Permutations for SC 0 1 2 3 u 0 x 0 π 0 π 0 1 1 u 1 x 1 π 0 π 0 2 2 u 2 x 2 π 1 π 1 1 1 u 3 x 3 π 0 π 0 3 3 u 4 x 4 π 2 1 u 5 x 5 π 1 π 1 2 2 u 6 x 6 π 3 1 u 7 x 7 0 1 2 ISTC 2018 Decoding Reed-Muller and Polar Codes by Successive Factor Graph Permutations 12/19

Successive Permutations for SC 0 1 2 3 u 0 x 0 π 0 π 0 1 1 u 1 x 1 π 0 π 0 2 2 u 2 x 2 π 1 π 1 1 1 u 3 x 3 π 0 π 0 3 3 u 4 x 4 π 2 π 2 1 1 u 5 x 5 π 1 π 1 2 2 u 6 x 6 π 3 1 u 7 x 7 0 1 2 ISTC 2018 Decoding Reed-Muller and Polar Codes by Successive Factor Graph Permutations 12/19