Graph ensemble design for channel coding A. Montanari 1 A. Amraoui 2 - PowerPoint PPT Presentation

Graph ensemble design for channel coding A. Montanari 1 A. Amraoui 2 T. Richardson 3 R. Urbanke 2 A. Dembo 4 1 ENS, France → Stanford, USA 2 EPFL, Switzerland, 3 Flarion, USA 4 Stanford, USA October 18, 2006 A. Montanari A. Amraoui, T. Richardson, R. Urbanke A. Dembo Graph ensemble design for channel coding

Outline The optimization problem 1 A probabilistic strategy 2 The approximate formula 2 Future directions 3 A. Montanari A. Amraoui, T. Richardson, R. Urbanke A. Dembo Graph ensemble design for channel coding

The optimization problem A. Montanari A. Amraoui, T. Richardson, R. Urbanke A. Dembo Graph ensemble design for channel coding

The object to be optimized: A code, i.e. a graph 1 2 · · · m 1 2 · · · n A. Montanari A. Amraoui, T. Richardson, R. Urbanke A. Dembo Graph ensemble design for channel coding

The objective function Let S ⊆ [ n ] be random with density ǫ ∈ [0 , 1]... P B ( G ) = P ǫ {S contains a ‘stopping set’ } A. Montanari A. Amraoui, T. Richardson, R. Urbanke A. Dembo Graph ensemble design for channel coding

The objective function up-degree ≥ 2 A. Montanari A. Amraoui, T. Richardson, R. Urbanke A. Dembo Graph ensemble design for channel coding

The objective function up-degree ≥ 2 A. Montanari A. Amraoui, T. Richardson, R. Urbanke A. Dembo Graph ensemble design for channel coding

The objective function up-degree ≥ 2 A. Montanari A. Amraoui, T. Richardson, R. Urbanke A. Dembo Graph ensemble design for channel coding

If you did not get it Bipartite graph G ↔ hypergraph H P B = P ǫ { a random subgraph of H contains a 2-core } A. Montanari A. Amraoui, T. Richardson, R. Urbanke A. Dembo Graph ensemble design for channel coding

A probabilistic strategy A. Montanari A. Amraoui, T. Richardson, R. Urbanke A. Dembo Graph ensemble design for channel coding

General strategy 1 Define a graph ensemble with parameters n , m , λ = ( λ 1 , λ 2 , . . . , λ k ), ρ = ( ρ 1 , ρ 2 , . . . , ρ k ). 2 Prove an pproximate formula E λ,ρ P B ≃ Q ( λ, ρ ). [*] 3 Find ( λ ∗ , ρ ∗ ) = arg min Q ( λ, ρ ). 4 Sample G from the ( λ ∗ , ρ ∗ )-ensemble and check it. [Concentration] A. Montanari A. Amraoui, T. Richardson, R. Urbanke A. Dembo Graph ensemble design for channel coding

General strategy 1 Define a graph ensemble with parameters n , m , λ = ( λ 1 , λ 2 , . . . , λ k ), ρ = ( ρ 1 , ρ 2 , . . . , ρ k ). 2 Prove an pproximate formula E λ,ρ P B ≃ Q ( λ, ρ ). [*] 3 Find ( λ ∗ , ρ ∗ ) = arg min Q ( λ, ρ ). 4 Sample G from the ( λ ∗ , ρ ∗ )-ensemble and check it. [Concentration] A. Montanari A. Amraoui, T. Richardson, R. Urbanke A. Dembo Graph ensemble design for channel coding

General strategy 1 Define a graph ensemble with parameters n , m , λ = ( λ 1 , λ 2 , . . . , λ k ), ρ = ( ρ 1 , ρ 2 , . . . , ρ k ). 2 Prove an pproximate formula E λ,ρ P B ≃ Q ( λ, ρ ). [*] 3 Find ( λ ∗ , ρ ∗ ) = arg min Q ( λ, ρ ). 4 Sample G from the ( λ ∗ , ρ ∗ )-ensemble and check it. [Concentration] A. Montanari A. Amraoui, T. Richardson, R. Urbanke A. Dembo Graph ensemble design for channel coding

General strategy 1 Define a graph ensemble with parameters n , m , λ = ( λ 1 , λ 2 , . . . , λ k ), ρ = ( ρ 1 , ρ 2 , . . . , ρ k ). 2 Prove an pproximate formula E λ,ρ P B ≃ Q ( λ, ρ ). [*] 3 Find ( λ ∗ , ρ ∗ ) = arg min Q ( λ, ρ ). 4 Sample G from the ( λ ∗ , ρ ∗ )-ensemble and check it. [Concentration] A. Montanari A. Amraoui, T. Richardson, R. Urbanke A. Dembo Graph ensemble design for channel coding

General strategy 1 Define a graph ensemble with parameters n , m , λ = ( λ 1 , λ 2 , . . . , λ k ), ρ = ( ρ 1 , ρ 2 , . . . , ρ k ). 2 Prove an pproximate formula E λ,ρ P B ≃ Q ( λ, ρ ). [*] 3 Find ( λ ∗ , ρ ∗ ) = arg min Q ( λ, ρ ). 4 Sample G from the ( λ ∗ , ρ ∗ )-ensemble and check it. [Concentration] A. Montanari A. Amraoui, T. Richardson, R. Urbanke A. Dembo Graph ensemble design for channel coding

A ‘standard’ ensemble G random (configuration model) with → Up-degree distribution: ρ = ( ρ 1 , . . . , ρ k ) → Down-degree distribution: λ = ( λ 1 , . . . , λ k ) Good for n = ∞ ! [Luby et al.] A. Montanari A. Amraoui, T. Richardson, R. Urbanke A. Dembo Graph ensemble design for channel coding

Approximate formula for E λ,ρ P B LATER!!! A. Montanari A. Amraoui, T. Richardson, R. Urbanke A. Dembo Graph ensemble design for channel coding

Implementation [Amraoui/Montanari/Urbanke] Sample Run: Minimize m / n , given P B = 10 − 4 ǫ = 0 . 5 n = 5000 Largest degree 13 Expurgation 6 [I did not explain this] A. Montanari A. Amraoui, T. Richardson, R. Urbanke A. Dembo Graph ensemble design for channel coding

Example Run 2 3 4 5 6 7 8 9 10 11 12 13 10 -1 2 3 4 5 6 7 8 9 10 40.58 % 0.0 rate/capacity 1.0 10 -2 contribution to error floor 10 -3 6 8 10 12 14 16 18 20 22 24 26 -0.01 -0.02 10 -4 -0.03 -0.04 -0.05 10 -5 -0.06 -0.07 -0.08 10 -6 -0.09 counter:=0 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 -0.1 0.0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 λ = 0 . 0739196 x + 0 . 65789 x 2 + 0 . 2681 x 12 , ρ = 0 . 390753 x 4 + 0 . 361589 x 5 + 0 . 247658 x 9 . Play it Again Sam! A. Montanari A. Amraoui, T. Richardson, R. Urbanke A. Dembo Graph ensemble design for channel coding

Example Run 2 3 4 5 6 7 8 9 10 11 12 13 10 -1 2 3 4 5 6 7 8 9 10 40.97 % 0.0 rate/capacity 1.0 10 -2 contribution to error floor 10 -3 6 8 10 12 14 16 18 20 22 24 26 -0.01 -0.02 10 -4 -0.03 -0.04 -0.05 10 -5 -0.06 -0.07 -0.08 10 -6 -0.09 counter:=5 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 -0.1 0.0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 λ = 0 . 0739196 x + 0 . 65789 x 2 + 0 . 2681 x 12 , ρ = 0 . 390753 x 4 + 0 . 361589 x 5 + 0 . 247658 x 9 . Play it Again Sam! A. Montanari A. Amraoui, T. Richardson, R. Urbanke A. Dembo Graph ensemble design for channel coding

Example Run 2 3 4 5 6 7 8 9 10 11 12 13 10 -1 2 3 4 5 6 7 8 9 10 41.34 % 0.0 rate/capacity 1.0 10 -2 contribution to error floor 10 -3 6 8 10 12 14 16 18 20 22 24 26 -0.01 -0.02 10 -4 -0.03 -0.04 -0.05 10 -5 -0.06 -0.07 -0.08 10 -6 -0.09 counter:=10 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 -0.1 0.0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 λ = 0 . 0739196 x + 0 . 65789 x 2 + 0 . 2681 x 12 , ρ = 0 . 390753 x 4 + 0 . 361589 x 5 + 0 . 247658 x 9 . Play it Again Sam! A. Montanari A. Amraoui, T. Richardson, R. Urbanke A. Dembo Graph ensemble design for channel coding

Example Run 2 3 4 5 6 7 8 9 10 11 12 13 10 -1 2 3 4 5 6 7 8 9 10 41.68 % 0.0 rate/capacity 1.0 10 -2 contribution to error floor 10 -3 6 8 10 12 14 16 18 20 22 24 26 -0.01 -0.02 10 -4 -0.03 -0.04 -0.05 10 -5 -0.06 -0.07 -0.08 10 -6 -0.09 counter:=15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 -0.1 0.0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 λ = 0 . 0739196 x + 0 . 65789 x 2 + 0 . 2681 x 12 , ρ = 0 . 390753 x 4 + 0 . 361589 x 5 + 0 . 247658 x 9 . Play it Again Sam! A. Montanari A. Amraoui, T. Richardson, R. Urbanke A. Dembo Graph ensemble design for channel coding

Example Run 2 3 4 5 6 7 8 9 10 11 12 13 10 -1 2 3 4 5 6 7 8 9 10 42.01 % 0.0 rate/capacity 1.0 10 -2 contribution to error floor 10 -3 6 8 10 12 14 16 18 20 22 24 26 -0.01 -0.02 10 -4 -0.03 -0.04 -0.05 10 -5 -0.06 -0.07 -0.08 10 -6 -0.09 counter:=20 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 -0.1 0.0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 λ = 0 . 0739196 x + 0 . 65789 x 2 + 0 . 2681 x 12 , ρ = 0 . 390753 x 4 + 0 . 361589 x 5 + 0 . 247658 x 9 . Play it Again Sam! A. Montanari A. Amraoui, T. Richardson, R. Urbanke A. Dembo Graph ensemble design for channel coding

Example Run 2 3 4 5 6 7 8 9 10 11 12 13 10 -1 2 3 4 5 6 7 8 9 10 42.33 % 0.0 rate/capacity 1.0 10 -2 contribution to error floor 10 -3 6 8 10 12 14 16 18 20 22 24 26 -0.01 -0.02 10 -4 -0.03 -0.04 -0.05 10 -5 -0.06 -0.07 -0.08 10 -6 -0.09 counter:=25 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 -0.1 0.0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 λ = 0 . 0739196 x + 0 . 65789 x 2 + 0 . 2681 x 12 , ρ = 0 . 390753 x 4 + 0 . 361589 x 5 + 0 . 247658 x 9 . Play it Again Sam! A. Montanari A. Amraoui, T. Richardson, R. Urbanke A. Dembo Graph ensemble design for channel coding

Example Run 2 3 4 5 6 7 8 9 10 11 12 13 10 -1 2 3 4 5 6 7 8 9 10 43.63 % 0.0 rate/capacity 1.0 10 -2 contribution to error floor 10 -3 6 8 10 12 14 16 18 20 22 24 26 -0.01 -0.02 10 -4 -0.03 -0.04 -0.05 10 -5 -0.06 -0.07 -0.08 10 -6 -0.09 counter:=30 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 -0.1 0.0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 λ = 0 . 0739196 x + 0 . 65789 x 2 + 0 . 2681 x 12 , ρ = 0 . 390753 x 4 + 0 . 361589 x 5 + 0 . 247658 x 9 . Play it Again Sam! A. Montanari A. Amraoui, T. Richardson, R. Urbanke A. Dembo Graph ensemble design for channel coding