Multilevel LDPC Lattices with Efficient Encoding and Decoding and a - PowerPoint PPT Presentation

Multilevel LDPC Lattices with Efficient Encoding and Decoding and a Generalization of Construction D ′ Danilo Silva Paulo R. B. da Silva Department of Electrical and Electronic Engineering Federal University of Santa Catarina (UFSC), Brazil danilo.silva@ufsc.br Lattice Coding & Crypto Meeting Imperial College London London, January 15, 2018

Outline 1. Introduction (background, motivation) 2. Constructions of low-complexity lattices 3. New results ◮ Efficient encoding and decoding for Construction D ′ ◮ A generalization of Construction D ′ ◮ Design examples and simulation results 4. Conclusions and open problems 2 / 39

Introduction

Motivation 1. Lattice codes provide a structured solution to achieve the capacity of the point-to-point AWGN channel [Erez-Zamir’04] ◮ Goal: achieve capacity with efficient encoding and decoding 3 / 39

Motivation 1. Lattice codes provide a structured solution to achieve the capacity of the point-to-point AWGN channel [Erez-Zamir’04] ◮ Goal: achieve capacity with efficient encoding and decoding ◮ Solved by polar lattices [Yan-Liu-Ling-Wu’14] 3 / 39

Motivation 1. Lattice codes provide a structured solution to achieve the capacity of the point-to-point AWGN channel [Erez-Zamir’04] ◮ Goal: achieve capacity with efficient encoding and decoding ◮ Solved by polar lattices [Yan-Liu-Ling-Wu’14] 2. For many network information theory problems, lattice codes can achieve strictly better performance than existing non-structured codes ◮ Compute-and-forward for relay networks [Nazer-Gastpar’11] ◮ Integer forcing for MIMO systems [Zhan-Nazer-Erez-Gastpar’14] ◮ Distributed source coding [Krithivasan-Pradhan’09] ◮ Physical-layer security [Ling-Luzzi-Belfiore-Stehlé’14] ◮ And more (see Zamir’s book) 3 / 39

Example: The Two-Way Relay Channel 1 Source: [Nazer-Gastpar’13] 4 / 39

Routing 2 Source: [Nazer-Gastpar’13] 5 / 39

Network Coding 3 Source: [Nazer-Gastpar’13] 6 / 39

Physical-Layer Network Coding 4 Source: [Nazer-Gastpar’13] 7 / 39

Compute-and-Forward Physical-Layer Network Coding + Lattices = Compute-and-Forward 5 Source: [Nazer-Gastpar’13] 8 / 39

Nested Lattice Codes ◮ If Λ ′ ⊆ Λ is a sublattice of Λ with a fundamental region R Λ ′ , then C = Λ ∩ R Λ ′ = Λ mod Λ ′ is said to be a nested lattice code ◮ A decoder that finds the nearest lattice point (ignoring the shaping region) is called a lattice decoder ◮ Nested lattice codes with lattice decoding are capacity-achieving for the AWGN channel if Λ is AWGN-good and Λ ′ is quantization-good [EZ’04] 9 / 39

Compute-and-Forward (special case) ◮ The users transmit c 1 , c 2 ∈ C = Λ ∩ R Λ ′ 10 / 39

Compute-and-Forward (special case) ◮ The users transmit c 1 , c 2 ∈ C = Λ ∩ R Λ ′ ◮ The relay receives z ∼ N ( 0 , σ 2 I ) y = c 1 + c 2 + z , 10 / 39

Compute-and-Forward (special case) ◮ The users transmit c 1 , c 2 ∈ C = Λ ∩ R Λ ′ ◮ The relay receives z ∼ N ( 0 , σ 2 I ) y = c 1 + c 2 + z , and wishes to compute c 3 � c 1 + c 2 mod Λ ′ ∈ C 10 / 39

Compute-and-Forward (special case) ◮ The users transmit c 1 , c 2 ∈ C = Λ ∩ R Λ ′ ◮ The relay receives z ∼ N ( 0 , σ 2 I ) y = c 1 + c 2 + z , and wishes to compute c 3 � c 1 + c 2 mod Λ ′ ∈ C ◮ To do so, it computes y mod Λ ′ = c 3 + z mod Λ ′ from which it can then decode c 3 ∈ C . 10 / 39

Constructions of Low-Complexity Lattices

Main Problem How to construct capacity-approaching lattice codes that admit efficient encoding and decoding? efficient � linear or quasi-linear complexity in number of information bits 11 / 39

Background on Low-Density Parity-Check Codes ◮ An LDPC code is a linear code with a sparse parity-check matrix 2 : Hx T = 0 } , H ∈ F ( n − k ) × n C = { x ∈ F n 2 ◮ Equivalently represented by a Tanner graph (a bipartite graph, with n variable nodes and m check nodes, whose incidence matrix is H ) v 1 v 2 v 3 v 4 v 5 v 6 v 7   1 1 1 0 1 0 0 H = 1 1 0 1 0 1 0   1 0 1 1 0 0 1 ◮ Can be decoded in O ( n ) by the belief propagation algorithm ◮ Performance depends largely (but not only) on the degree distribution ◮ Approaches the BI-AWGN capacity (achieves it if spatially coupled) 12 / 39

Main Approaches ◮ Low-Density Construction A (LDA) Lattices [di Pietro et al. ’12] ◮ Requires an LDPC code over Z p with large p ◮ High-complexity decoding: O ( p 2 n ) with belief propagation 13 / 39

Main Approaches ◮ Low-Density Construction A (LDA) Lattices [di Pietro et al. ’12] ◮ Requires an LDPC code over Z p with large p ◮ High-complexity decoding: O ( p 2 n ) with belief propagation ◮ Low-Density Lattice Codes (LDLC) [Sommer-Feder-Shalvi’08] ◮ Designed directly in R n with a sparse parity-check matrix ◮ BP decoder must process probability density functions 13 / 39

Main Approaches ◮ Low-Density Construction A (LDA) Lattices [di Pietro et al. ’12] ◮ Requires an LDPC code over Z p with large p ◮ High-complexity decoding: O ( p 2 n ) with belief propagation ◮ Low-Density Lattice Codes (LDLC) [Sommer-Feder-Shalvi’08] ◮ Designed directly in R n with a sparse parity-check matrix ◮ BP decoder must process probability density functions ◮ Multilevel Lattices [Forney-Trott-Chung’00] ◮ Uses multiple nested binary linear codes ◮ Efficient decoding is possible (in principle) using multistage decoding ◮ AWGN-good if each component code is capacity-achieving 13 / 39

Multilevel Lattices: Construction D ◮ Let C 0 ⊆ C 1 ⊆ · · · ⊆ C L − 1 ⊆ Z n 2 be a family of nested linear codes, where each C ℓ has dimension k ℓ and generator matrix   g 1 .  ∈ { 0 , 1 } k ℓ × n G ℓ = .   .  g k ℓ ◮ Construction D: � L − 1 � � 2 ℓ u ℓ G ℓ : u ℓ ∈ { 0 , 1 } k ℓ , 0 ≤ ℓ < L + 2 L Z n Λ = ℓ =0 (note that u ℓ G ℓ is computed over Z ) ◮ Remark: Should not be confused with the “Code Formula” Γ = C 0 + 2 C 1 + · · · + 2 L − 1 C L − 1 + 2 L Z n which does not generally produce lattices 14 / 39

Encoding and Multistage Decoding of Construction D Encoder Decoder u 0 ˆ u 0 u 0 G 0 G 0 mod 2 D 0 + − u 0 G 0 ˆ G 0 u 1 u 1 G 1 G 1 1 2 2 u 1 ˆ mod 2 D 1 u 2 u 2 G 2 G 2 + − u 1 G 1 ˆ G 1 4 1 2 15 / 39

Multilevel Lattices: Construction D ′ ◮ Let C 0 ⊆ C 1 ⊆ · · · ⊆ C L − 1 ⊆ Z n 2 be a family of nested linear codes, where each C ℓ has dimension n − m ℓ and parity-check matrix   h 1 .  ∈ { 0 , 1 } m ℓ × n H ℓ = .   .  h m ℓ ◮ Construction D ′ : Λ = { x ∈ Z n : h j x T ≡ 0 (mod 2 ℓ +1 ) , m ℓ +1 < j ≤ m ℓ , 0 ≤ ℓ < L } ◮ Matrix description: � x ∈ Z n : H ℓ x T ≡ 0 � (mod 2 ℓ +1 ) , 0 ≤ ℓ < L Λ = 16 / 39

Example of Construction D ′ For nested codes C 0 ⊆ C 1 ⊆ C 2 ⊆ Z 4 2 , let  1 1 1 1  � 1 � 1 1 1 � � H 0 = 1 0 1 0 H 1 = H 2 = 1 1 1 1   1 0 1 0 1 1 0 0 Then x T ≡ 0   � � 1 1 1 1 (mod 8)    x ∈ Z 4 : x T ≡ 0  � � Λ = 1 0 1 0 (mod 4) x T ≡ 0  � �  1 1 0 0 (mod 2)   or equivalently H 2 x T ≡ 0   (mod 8)    x ∈ Z 4 :  H 1 x T ≡ 0 Λ = (mod 4) H 0 x T ≡ 0   (mod 2)   17 / 39

Multilevel Lattices: Previous Work ◮ Polar Lattices [Yan-Liu-Ling-Wu’14] ◮ Based on Construction D ◮ Capacity-achieving under MSD ◮ Encoding and decoding complexity O ( Ln log n ) 18 / 39

Multilevel Lattices: Previous Work ◮ Polar Lattices [Yan-Liu-Ling-Wu’14] ◮ Based on Construction D ◮ Capacity-achieving under MSD ◮ Encoding and decoding complexity O ( Ln log n ) ◮ LDPC Lattices [Sadeghi-Banihashemi-Panario’06] [Baik-Chung’08] ◮ Based on Construction D ′ ◮ Only joint decoding considered—complexity O (2 L n ) ◮ Encoding complexity not addressed 18 / 39

Multilevel Lattices: Previous Work ◮ Polar Lattices [Yan-Liu-Ling-Wu’14] ◮ Based on Construction D ◮ Capacity-achieving under MSD ◮ Encoding and decoding complexity O ( Ln log n ) ◮ LDPC Lattices [Sadeghi-Banihashemi-Panario’06] [Baik-Chung’08] ◮ Based on Construction D ′ ◮ Only joint decoding considered—complexity O (2 L n ) ◮ Encoding complexity not addressed ◮ Spatially-Coupled LDPC Lattices [Vem-Huang-Narayanan-Pfister’14] ◮ AWGN-good under BP MSD ◮ Based on Construction D = ⇒ generally dense generator matrices ◮ High-complexity encoding and MSD cancellation step 18 / 39

Challenges with Construction D ′ ◮ How to encode (efficiently)? ◮ How to cancel past levels (efficiently) in MSD? ◮ Nested parity-check matrices: ◮ are difficult to design (for non-SC LDPC codes) ◮ do not perform well under BP MSD (for non-SC LDPC codes) 19 / 39