Decoding Reed–Muller Codes Using Redundant Code Constraints Mengke Lian ∗ , Christian H¨ ager † , Henry D. Pfister ∗‡ ∗ Department of Electrical and Computer Engineering, Duke University † Department of Electrical Engineering, Chalmers University of Technology ‡ Department of Mathematics, Duke University IEEE International Symposium on Information Theory June 21-26, 2020
Outline Introduction 1 Factor Graphs for Reed–Muller Codes 2 Recursive Decoding Algorithms 3 Collapsed Decoding Algorithms 4 Simulation Results 5 Conclusions 6 Decoding Reed–Muller Codes Using Redundant Code Constraints | M. Lian, C. H¨ ager, H. D. Pfister 1 / 26
Outline Introduction 1 Factor Graphs for Reed–Muller Codes 2 Recursive Decoding Algorithms 3 Collapsed Decoding Algorithms 4 Simulation Results 5 Conclusions 6 Decoding Reed–Muller Codes Using Redundant Code Constraints | M. Lian, C. H¨ ager, H. D. Pfister 2 / 26
Introduction Since the discovery of polar codes, interest in Reed–Muller (RM) codes has surged because of their excellent ML performance for short block lengths Many existing decoding methods exploit the symmetry of RM codes to achieve improved performance in practice [BH86, HC06, SHP18] The recursive projection–aggregation (RPA) decoding algorithm [YA20] was recently shown to perform well on a variety of low-order RM codes Decoding Reed–Muller Codes Using Redundant Code Constraints | M. Lian, C. H¨ ager, H. D. Pfister 3 / 26
Introduction Since the discovery of polar codes, interest in Reed–Muller (RM) codes has surged because of their excellent ML performance for short block lengths Many existing decoding methods exploit the symmetry of RM codes to achieve improved performance in practice [BH86, HC06, SHP18] The recursive projection–aggregation (RPA) decoding algorithm [YA20] was recently shown to perform well on a variety of low-order RM codes Main contributions of this work We connect RPA to (weighted) belief-propagation (BP) decoding by 1 interpreting it as message-passing on a redundant factor graph We propose a novel decoder, called recursive puncturing-aggregation (RXA), 2 that uses puncturing to extend its idea to high-rate RM codes We investigate non-recursive (i.e., collapsed) versions of RPA/RXA and 3 compare with BP decoding using all minimum-weight parity checks Decoding Reed–Muller Codes Using Redundant Code Constraints | M. Lian, C. H¨ ager, H. D. Pfister 3 / 26
Outline Introduction 1 Factor Graphs for Reed–Muller Codes 2 Recursive Decoding Algorithms 3 Collapsed Decoding Algorithms 4 Simulation Results 5 Conclusions 6 Decoding Reed–Muller Codes Using Redundant Code Constraints | M. Lian, C. H¨ ager, H. D. Pfister 4 / 26
Factor Graphs and LDPC Decoding ℓ 7 v 1 v 2 v 3 v 4 v 5 v 6 v 7 V � [ N ] ∂v 3 ˆ λ 3 → 5 λ 7 → 3 ∂c 1 C � [ M ] c 1 c 2 c 3 Belief-propagation (BP) decoding in the LLR domain: λ ( t − 1) λ ( t ) � ˆ v → c = ℓ v + c ′ → v c ′ ∈ ∂v \ c � � λ ( t ) �� ˆ � λ ( t ) c → v = 2 tanh − 1 v ′ → c tanh , 2 v ′ ∈ ∂c \ v Decoding Reed–Muller Codes Using Redundant Code Constraints | M. Lian, C. H¨ ager, H. D. Pfister 5 / 26
Recursive Construction of Reed–Muller Codes Reed–Muller code RM( r, m ) has length N = 2 m Each message is given by an m -variate binary polynomial of degree at most r The codeword formed by evaluating its message at all ( x 1 , . . . , x m ) ∈ { 0 , 1 } m Decoding Reed–Muller Codes Using Redundant Code Constraints | M. Lian, C. H¨ ager, H. D. Pfister 6 / 26
Recursive Construction of Reed–Muller Codes Reed–Muller code RM( r, m ) has length N = 2 m Each message is given by an m -variate binary polynomial of degree at most r The codeword formed by evaluating its message at all ( x 1 , . . . , x m ) ∈ { 0 , 1 } m Definition: RM( − 1 , 0) � ∅ , RM(0 , 0) � { 0 , 1 } , and � � � RM( r, m ) � ( u , u + v ) � u ∈ RM( r, m − 1) , v ∈ RM( r − 1 , m − 1) � Puncturing second half leaves shorter RM( r, m − 1) of same order Adding both halves projects onto shorter RM( r − 1 , m − 1) of lower order Decoding Reed–Muller Codes Using Redundant Code Constraints | M. Lian, C. H¨ ager, H. D. Pfister 6 / 26
b b b b b b b b b b b b Factor Graphs for Reed–Muller Codes (1) RM ( r, m − 1) RM ( r, m − 1) RM ( r − 1 , m − 1) Generalized check node � represents code constraints of labeled RM code RM( r, m ) � { ( u , u + v ) | u ∈ RM( r, m − 1) , v ∈ RM( r − 1 , m − 1) } Puncturing second half leaves shorter RM( r, m − 1) of same order Adding both halves projects onto shorter RM( r − 1 , m − 1) of lower order Decoding Reed–Muller Codes Using Redundant Code Constraints | M. Lian, C. H¨ ager, H. D. Pfister 7 / 26
b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b Factor Graphs for Reed–Muller Codes (2) degree 2 m − 1 π 2 m − 1 subcodes RM ( r − 1 , m − 1) RM ( r − 1 , m − 1) Automorphism group of length- 2 m RM code Affine general linear group over F 2 : AGL ( m, 2) For invertible A ∈ { 0 , 1 } m × m and any b ∈ { 0 , 1 } m , contains permutations σ : { 0 , 1 } m → { 0 , 1 } m x �→ A x + b AGL ( m, 2) generates 2 m − 1 distinct RM( r − 1 , m − 1) constraints Decoding Reed–Muller Codes Using Redundant Code Constraints | M. Lian, C. H¨ ager, H. D. Pfister 8 / 26
Outline Introduction 1 Factor Graphs for Reed–Muller Codes 2 Recursive Decoding Algorithms 3 Collapsed Decoding Algorithms 4 Simulation Results 5 Conclusions 6 Decoding Reed–Muller Codes Using Redundant Code Constraints | M. Lian, C. H¨ ager, H. D. Pfister 9 / 26
b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b Recursive Projection–Aggregation (RPA) [YA20] degree 4 2 m − 1 π 3 1 2 m − 1 subcodes 2 RM ( r − 1 , m − 1) RM ( r − 1 , m − 1) Yellow VN messages initialized to channel LLRs. Then iterate T max times: 1 Projection: Standard message-passing rule from code bits to subcode bits 2 Subcode decoding: Recurse or soft-in/hard-out RM(1 , m − 1) decoding 3 Backward update: Hard output from subcode allows soft code-bit combining 4 VN update: Average of all input LLRs update due to redundant constraints Decoding Reed–Muller Codes Using Redundant Code Constraints | M. Lian, C. H¨ ager, H. D. Pfister 10 / 26
Decoding Path in the Reed–Muller Tableau a recursive puncturing–aggregation (RXA) SPC codes extended RM(4,5) Hamming RM(3,4) RM(4,6) codes RM(2,3) RM(3,5) a RM(1,2) RM(2,4) RM(3,6) ( u , u + v ) a RM(1,3) RM(2,5) RM(3,7) construction d RM(0,2) RM(1,4) RM(2,6) d RM(0,3) RM(1,5) augmented RM(0,4) RM(1,6) Hadamard RM(0,5) codes repeat codes d recursive projection–aggregation (RPA) Decoding Reed–Muller Codes Using Redundant Code Constraints | M. Lian, C. H¨ ager, H. D. Pfister 11 / 26
b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b Recursive Puncturing–Aggregation (RXA) degree 1 2 m − 1 RM ( r, m − 1) All outgoing VN messages are initialized to the channel LLRs. Then, iterate T max times: RM ( r, m − 1) 1 Subcode decoding: if r<m − 3 , recurse. 2(2 m − 1) total Else, extrinsic soft-in/soft-out decoding of π subcodes RM( m − 3 , m − 1) extended Hamming code 2 VN update: Sum channel LLR and RM ( r, m − 1) extrinsic weighted sum 1 of input LLRs RM ( r, m − 1) 2 1 Many redundant constraints means averaging (e.g., w = α/ (deg − 1) for α ∈ (0 , 1) ) works well Decoding Reed–Muller Codes Using Redundant Code Constraints | M. Lian, C. H¨ ager, H. D. Pfister 12 / 26
Hard vs. Soft Decoding Factor graph description naturally leads to a variety of decoders sum product: defines extrinsic soft-in/soft-out decoding max product / min sum – defines another extrinsic soft-in/soft-out decoding weights and corrections: improved performance vs. complexity intrinsic hard decision – RPA uses soft-in/hard-out ML decoder of RM code possible extensions: guided decimation and neural BP Decoding Reed–Muller Codes Using Redundant Code Constraints | M. Lian, C. H¨ ager, H. D. Pfister 13 / 26
Hard vs. Soft Decoding Factor graph description naturally leads to a variety of decoders sum product: defines extrinsic soft-in/soft-out decoding max product / min sum – defines another extrinsic soft-in/soft-out decoding weights and corrections: improved performance vs. complexity intrinsic hard decision – RPA uses soft-in/hard-out ML decoder of RM code possible extensions: guided decimation and neural BP RPA uses soft-projection and intrinsic hard decision decoding Works quite well with low-complexity and no parameters to tune! RXA uses extrinsic soft-in/soft-out RM decoding (no projection required) Increases subcode decoding complexity and requires some α tuning Decoding Reed–Muller Codes Using Redundant Code Constraints | M. Lian, C. H¨ ager, H. D. Pfister 13 / 26
Outline Introduction 1 Factor Graphs for Reed–Muller Codes 2 Recursive Decoding Algorithms 3 Collapsed Decoding Algorithms 4 Simulation Results 5 Conclusions 6 Decoding Reed–Muller Codes Using Redundant Code Constraints | M. Lian, C. H¨ ager, H. D. Pfister 14 / 26
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