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How to Compute Scaling Parameters for SparseGraph Codes Under Message-Passing Decoding
with a Finite Message Alphabet
R. Urbanke1
Based on joint work with Jérémie Ezri1 and Andrea Montanari2
1EPFL
2Stanford University
Santa Fe, May 5th 2007
R. Urbanke (EPFL) A Scaling Approach to Coding Santa Fe, May 5th 2007 1 / 27
R. Urbanke (EPFL) A Scaling Approach to Coding Santa Fe, May 5th 2007 2 / 27
R. Urbanke (EPFL) A Scaling Approach to Coding Santa Fe, May 5th 2007 2 / 27
R. Urbanke (EPFL) A Scaling Approach to Coding Santa Fe, May 5th 2007 2 / 27
R. Urbanke (EPFL) A Scaling Approach to Coding Santa Fe, May 5th 2007 2 / 27
R. Urbanke (EPFL) A Scaling Approach to Coding Santa Fe, May 5th 2007 2 / 27
R. Urbanke (EPFL) A Scaling Approach to Coding Santa Fe, May 5th 2007 2 / 27
BP, MinSum, LP, Gallager A, Gallager B, Decoder with Erasures, turbo,time variant, ..., schedules, number of iterations, ...
R. Urbanke (EPFL) A Scaling Approach to Coding Santa Fe, May 5th 2007 2 / 27
0.35 0.36 0.37 0.38 0.39 0.4 0.41 0.42 0.43 0.4410-4
10-3
10-2
10-1
0.35 0.36 0.37 0.38 0.39 0.4 0.41 0.42 0.43 0.4410-4
10-3
10-2
10-1
ε
n = 1024
n = 8192
n = +∞
ε∗
PB
R. Urbanke (EPFL) A Scaling Approach to Coding Santa Fe, May 5th 2007 3 / 27
Our Tool: Scaling Law
see talks of David Tse, Eli Ben-Naim and remarks by ChristopherMoore
R. Urbanke (EPFL) A Scaling Approach to Coding Santa Fe, May 5th 2007 4 / 27
Scaling Around a First Order Phase Transition
εBP10-8
10-7
10-6
10-5
10-4
10-3
10-2
10-1
Play it Again!
R. Urbanke (EPFL) A Scaling Approach to Coding Santa Fe, May 5th 2007 5 / 27
Scaling Around a First Order Phase Transition
εBPε− εBP
10-8
10-7
10-6
10-5
10-4
10-3
10-2
10-1
Play it Again!
R. Urbanke (EPFL) A Scaling Approach to Coding Santa Fe, May 5th 2007 5 / 27
Scaling Around a First Order Phase Transition
εBP
1/ν
0 0.5
n1ν (ε− εBP)
10-8
10-7
10-6
10-5
10-4
10-3
10-2
10-1
Play it Again!
R. Urbanke (EPFL) A Scaling Approach to Coding Santa Fe, May 5th 2007 5 / 27
Scaling Around a First Order Phase Transition
εBP
1/ν
0 0.5
n1ν (ε− εBP)
10-8
10-7
10-6
10-5
10-4
10-3
10-2
10-1
Play it Again!
R. Urbanke (EPFL) A Scaling Approach to Coding Santa Fe, May 5th 2007 5 / 27
Scaling Around a First Order Phase Transition
εBP
1/ν
0 0.5
n1ν (ε− εBP)
10-8
10-7
10-6
10-5
10-4
10-3
10-2
10-1
Play it Again!
R. Urbanke (EPFL) A Scaling Approach to Coding Santa Fe, May 5th 2007 5 / 27
Scaling Around a First Order Phase Transition
εBP
1/ν
0 0.5
n1ν (ε− εBP)
10-8
10-7
10-6
10-5
10-4
10-3
10-2
10-1
Play it Again!
R. Urbanke (EPFL) A Scaling Approach to Coding Santa Fe, May 5th 2007 5 / 27
Scaling Around a First Order Phase Transition
εBP
1/ν
0 0.5
n1ν (ε− εBP)
10-8
10-7
10-6
10-5
10-4
10-3
10-2
10-1
Play it Again!
R. Urbanke (EPFL) A Scaling Approach to Coding Santa Fe, May 5th 2007 5 / 27
Scaling Around a First Order Phase Transition
εBP
1/ν
0 0.5
n1ν (ε− εBP)
10-8
10-7
10-6
10-5
10-4
10-3
10-2
10-1
Play it Again!
R. Urbanke (EPFL) A Scaling Approach to Coding Santa Fe, May 5th 2007 5 / 27
Scaling Around a First Order Phase Transition
εBP
1/ν
0 0.5
n1ν (ε− εBP)
10-8
10-7
10-6
10-5
10-4
10-3
10-2
10-1
Play it Again!
R. Urbanke (EPFL) A Scaling Approach to Coding Santa Fe, May 5th 2007 5 / 27
Scaling Around a First Order Phase Transition
εBP
1/ν
0 0.5
n1ν (ε− εBP)
10-8
10-7
10-6
10-5
10-4
10-3
10-2
10-1
Play it Again!
R. Urbanke (EPFL) A Scaling Approach to Coding Santa Fe, May 5th 2007 5 / 27
Scaling Around a First Order Phase Transition
εBP
1/ν
0 0.5
n1ν (ε− εBP)
10-8
10-7
10-6
10-5
10-4
10-3
10-2
10-1
Play it Again!
R. Urbanke (EPFL) A Scaling Approach to Coding Santa Fe, May 5th 2007 5 / 27
Scaling Around a First Order Phase Transition
εBP
1/ν
0 0.5
n1ν (ε− εBP)
10-8
10-7
10-6
10-5
10-4
10-3
10-2
10-1
Play it Again!
R. Urbanke (EPFL) A Scaling Approach to Coding Santa Fe, May 5th 2007 5 / 27
Scaling Around a First Order Phase Transition
εBP
1/ν
0 0.5
n1ν (ε− εBP)
10-8
10-7
10-6
10-5
10-4
10-3
10-2
10-1
Play it Again!
R. Urbanke (EPFL) A Scaling Approach to Coding Santa Fe, May 5th 2007 5 / 27
Scaling Around a First Order Phase Transition
εBP
1/ν
0 0.5
n1ν (ε− εBP)
10-8
10-7
10-6
10-5
10-4
10-3
10-2
10-1
Play it Again!
R. Urbanke (EPFL) A Scaling Approach to Coding Santa Fe, May 5th 2007 5 / 27
Scaling Around a First Order Phase Transition
εBP
1/ν
0 0.5
n1ν (ε− εBP)
10-8
10-7
10-6
10-5
10-4
10-3
10-2
10-1
Play it Again!
R. Urbanke (EPFL) A Scaling Approach to Coding Santa Fe, May 5th 2007 5 / 27
Scaling Around a First Order Phase Transition
εBP
1/ν
0 0.5
n1ν (ε− εBP)
10-8
10-7
10-6
10-5
10-4
10-3
10-2
10-1
Play it Again!
R. Urbanke (EPFL) A Scaling Approach to Coding Santa Fe, May 5th 2007 5 / 27
Scaling Around a First Order Phase Transition
εBP
1/ν
0 0.5
n1ν (ε− εBP)
10-8
10-7
10-6
10-5
10-4
10-3
10-2
10-1
Play it Again!
R. Urbanke (EPFL) A Scaling Approach to Coding Santa Fe, May 5th 2007 5 / 27
Scaling Around a First Order Phase Transition
εBP
1/ν
0 0.5
n1ν (ε− εBP)
10-8
10-7
10-6
10-5
10-4
10-3
10-2
10-1
Play it Again!
R. Urbanke (EPFL) A Scaling Approach to Coding Santa Fe, May 5th 2007 5 / 27
Scaling Around a First Order Phase Transition
εBP
1/ν
0 0.5
n1ν (ε− εBP)
10-8
10-7
10-6
10-5
10-4
10-3
10-2
10-1
Play it Again!
R. Urbanke (EPFL) A Scaling Approach to Coding Santa Fe, May 5th 2007 5 / 27
Scaling Around a First Order Phase Transition
εBP
1/ν
0 0.5
n1ν (ε− εBP)
10-8
10-7
10-6
10-5
10-4
10-3
10-2
10-1
Play it Again!
R. Urbanke (EPFL) A Scaling Approach to Coding Santa Fe, May 5th 2007 5 / 27
Scaling Around a First Order Phase Transition
εBP
1/ν
0 0.5
n1ν (ε− εBP)
10-8
10-7
10-6
10-5
10-4
10-3
10-2
10-1
Play it Again!
R. Urbanke (EPFL) A Scaling Approach to Coding Santa Fe, May 5th 2007 5 / 27
Scaling Around a First Order Phase Transition
εBP
1/ν
0 0.5
n1ν (ε− εBP)
10-8
10-7
10-6
10-5
10-4
10-3
10-2
10-1
Play it Again!
R. Urbanke (EPFL) A Scaling Approach to Coding Santa Fe, May 5th 2007 5 / 27
Scaling Around a First Order Phase Transition
εBP
1/ν
0 0.5
n1ν (ε− εBP)
10-8
10-7
10-6
10-5
10-4
10-3
10-2
10-1
Play it Again!
R. Urbanke (EPFL) A Scaling Approach to Coding Santa Fe, May 5th 2007 5 / 27
0.00 0.05 0.10
p
0.0
0.2
0.4
0.6
0.8
Blo
ck E
rror
Pro
babi
lity
50100200300500100020005000
−1.0 −0.5 0.0 0.5 1.0
(p−pd)N1/ν
0.0
0.2
0.4
0.6
0.8
1.0
Blo
ck E
rror
Pro
babi
lity
50100200300500100020005000
[A. Montanari 02]
R. Urbanke (EPFL) A Scaling Approach to Coding Santa Fe, May 5th 2007 6 / 27
The Simplest Case ... Transmission over the BEC
Theorem (Basic Scaling Law – Amraoui, Montanari, Richardson,Urbanke)
As n tends to infinity with argument of Q(·) kept fixed
PB(n, λ, ρ, ε) = Q(√
n(εBP − ε)α
)(1 + o (1))
R. Urbanke (EPFL) A Scaling Approach to Coding Santa Fe, May 5th 2007 7 / 27
Waterfall Approximation
0.35 0.36 0.37 0.38 0.39 0.4 0.41 0.42 0.43 0.4410-8
10-7
10-6
10-5
10-4
10-3
10-2
10-1
ε ε
n = 1024
n = 8192
n = +∞
ε?
PB
R. Urbanke (EPFL) A Scaling Approach to Coding Santa Fe, May 5th 2007 8 / 27
Refined Scaling Law For LDPC Codes
Conjecture (Refined Scaling Law – Amraoui, Montanari, Richardson,Urbanke – recently proved by Dembo and Montanari for Poissonensembles.)
As n tends to infinity with argument of Q(·) kept fixed
PB(n, λ, ρ, ε) = Q
(√n(εBP − βn−
23 − ε)
α
)(1 + O
(n−1/3))
where α = α(λ, ρ) and β = β(λ, ρ).
R. Urbanke (EPFL) A Scaling Approach to Coding Santa Fe, May 5th 2007 9 / 27
Refined Waterfall Approximation
0.35 0.36 0.37 0.38 0.39 0.4 0.41 0.42 0.43 0.4410-8
10-7
10-6
10-5
10-4
10-3
10-2
10-1
ε ε
n = 1024
n = 8192
n = +∞
ε?
PB
R. Urbanke (EPFL) A Scaling Approach to Coding Santa Fe, May 5th 2007 10 / 27
Computation Leads to ...
α =(
ρ(x)2 − ρ(x2) + ρ′(x)(1− 2xρ(x))− x2ρ′(x2)L′(1)λ(y)2ρ′(x)2 +
ε2λ(y)2 − ε2λ(y2)− y2ε2λ′(y2)L′(1)λ(y)2
)1/2
,
β =(
ε4r22(ελ
′(y)2r2 − x(λ′′(y)r2 + λ′(y)x))2
L′(1)2ρ′(x)3x10(2ελ′(y)2r3 − λ′′(y)r2x)
)1/3
,
ri =∑
m≥j≥i
(−1)i+j(
j− 1i− 1
)(m− 1j− 1
)ρm(ελ(y))j,
with ε the channel erasure probability at the critical point,x and y the erasures probabilities in the decoder at that point,with x = 1− x and y = 1− ρ(1− x).
R. Urbanke (EPFL) A Scaling Approach to Coding Santa Fe, May 5th 2007 11 / 27
Optimization For BEC
Complete approximation for the BEC (waterfall + error floor)
Fix ε, n and a target error probability Ptarg
→ degree distribution optimization using LP
0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.7510-6
10-5
10-4
10-3
10-2
10-1
40.58 %
0.0 1.0rate/capacity
2 3 4 5 6 7 8 9 10
2 3 4 5 6 7 8 9 10 11 12 13
contribution to error floor
6 8 10 12 14 16 18 20 22 24 26
0.0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45-0.1
-0.09
-0.08
-0.07
-0.06
-0.05
-0.04
-0.03
-0.02
-0.01
counter:=0
λ = 0.0739196x + 0.65789x2 + 0.2681x12,
ρ = 0.390753x4 + 0.361589x5 + 0.247658x9.
Play it Again!
R. Urbanke (EPFL) A Scaling Approach to Coding Santa Fe, May 5th 2007 12 / 27
Optimization For BEC
Complete approximation for the BEC (waterfall + error floor)
Fix ε, n and a target error probability Ptarg
→ degree distribution optimization using LP
0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.7510-6
10-5
10-4
10-3
10-2
10-1
40.97 %
0.0 1.0rate/capacity
2 3 4 5 6 7 8 9 10
2 3 4 5 6 7 8 9 10 11 12 13
contribution to error floor
6 8 10 12 14 16 18 20 22 24 26
0.0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45-0.1
-0.09
-0.08
-0.07
-0.06
-0.05
-0.04
-0.03
-0.02
-0.01
counter:=5
λ = 0.0739196x + 0.65789x2 + 0.2681x12,
ρ = 0.390753x4 + 0.361589x5 + 0.247658x9.
Play it Again!
R. Urbanke (EPFL) A Scaling Approach to Coding Santa Fe, May 5th 2007 12 / 27
Optimization For BEC
Complete approximation for the BEC (waterfall + error floor)
Fix ε, n and a target error probability Ptarg
→ degree distribution optimization using LP
0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.7510-6
10-5
10-4
10-3
10-2
10-1
41.34 %
0.0 1.0rate/capacity
2 3 4 5 6 7 8 9 10
2 3 4 5 6 7 8 9 10 11 12 13
contribution to error floor
6 8 10 12 14 16 18 20 22 24 26
0.0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45-0.1
-0.09
-0.08
-0.07
-0.06
-0.05
-0.04
-0.03
-0.02
-0.01
counter:=10
λ = 0.0739196x + 0.65789x2 + 0.2681x12,
ρ = 0.390753x4 + 0.361589x5 + 0.247658x9.
Play it Again!
R. Urbanke (EPFL) A Scaling Approach to Coding Santa Fe, May 5th 2007 12 / 27
Optimization For BEC
Complete approximation for the BEC (waterfall + error floor)
Fix ε, n and a target error probability Ptarg
→ degree distribution optimization using LP
0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.7510-6
10-5
10-4
10-3
10-2
10-1
41.68 %
0.0 1.0rate/capacity
2 3 4 5 6 7 8 9 10
2 3 4 5 6 7 8 9 10 11 12 13
contribution to error floor
6 8 10 12 14 16 18 20 22 24 26
0.0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45-0.1
-0.09
-0.08
-0.07
-0.06
-0.05
-0.04
-0.03
-0.02
-0.01
counter:=15
λ = 0.0739196x + 0.65789x2 + 0.2681x12,
ρ = 0.390753x4 + 0.361589x5 + 0.247658x9.
Play it Again!
R. Urbanke (EPFL) A Scaling Approach to Coding Santa Fe, May 5th 2007 12 / 27
Optimization For BEC
Complete approximation for the BEC (waterfall + error floor)
Fix ε, n and a target error probability Ptarg
→ degree distribution optimization using LP
0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.7510-6
10-5
10-4
10-3
10-2
10-1
42.01 %
0.0 1.0rate/capacity
2 3 4 5 6 7 8 9 10
2 3 4 5 6 7 8 9 10 11 12 13
contribution to error floor
6 8 10 12 14 16 18 20 22 24 26
0.0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45-0.1
-0.09
-0.08
-0.07
-0.06
-0.05
-0.04
-0.03
-0.02
-0.01
counter:=20
λ = 0.0739196x + 0.65789x2 + 0.2681x12,
ρ = 0.390753x4 + 0.361589x5 + 0.247658x9.
Play it Again!
R. Urbanke (EPFL) A Scaling Approach to Coding Santa Fe, May 5th 2007 12 / 27
Optimization For BEC
Complete approximation for the BEC (waterfall + error floor)
Fix ε, n and a target error probability Ptarg
→ degree distribution optimization using LP
0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.7510-6
10-5
10-4
10-3
10-2
10-1
42.33 %
0.0 1.0rate/capacity
2 3 4 5 6 7 8 9 10
2 3 4 5 6 7 8 9 10 11 12 13
contribution to error floor
6 8 10 12 14 16 18 20 22 24 26
0.0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45-0.1
-0.09
-0.08
-0.07
-0.06
-0.05
-0.04
-0.03
-0.02
-0.01
counter:=25
λ = 0.0739196x + 0.65789x2 + 0.2681x12,
ρ = 0.390753x4 + 0.361589x5 + 0.247658x9.
Play it Again!
R. Urbanke (EPFL) A Scaling Approach to Coding Santa Fe, May 5th 2007 12 / 27
Optimization For BEC
Complete approximation for the BEC (waterfall + error floor)
Fix ε, n and a target error probability Ptarg
→ degree distribution optimization using LP
0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.7510-6
10-5
10-4
10-3
10-2
10-1
43.63 %
0.0 1.0rate/capacity
2 3 4 5 6 7 8 9 10
2 3 4 5 6 7 8 9 10 11 12 13
contribution to error floor
6 8 10 12 14 16 18 20 22 24 26
0.0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45-0.1
-0.09
-0.08
-0.07
-0.06
-0.05
-0.04
-0.03
-0.02
-0.01
counter:=30
λ = 0.0739196x + 0.65789x2 + 0.2681x12,
ρ = 0.390753x4 + 0.361589x5 + 0.247658x9.
Play it Again!
R. Urbanke (EPFL) A Scaling Approach to Coding Santa Fe, May 5th 2007 12 / 27
Optimization For BEC
Complete approximation for the BEC (waterfall + error floor)
Fix ε, n and a target error probability Ptarg
→ degree distribution optimization using LP
0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.7510-6
10-5
10-4
10-3
10-2
10-1
46.01 %
0.0 1.0rate/capacity
2 3 4 5 6 7 8 9 10
2 3 4 5 6 7 8 9 10 11 12 13
contribution to error floor
6 8 10 12 14 16 18 20 22 24 26
0.0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45-0.1
-0.09
-0.08
-0.07
-0.06
-0.05
-0.04
-0.03
-0.02
-0.01
counter:=35
λ = 0.0739196x + 0.65789x2 + 0.2681x12,
ρ = 0.390753x4 + 0.361589x5 + 0.247658x9.
Play it Again!
R. Urbanke (EPFL) A Scaling Approach to Coding Santa Fe, May 5th 2007 12 / 27
Optimization For BEC
Complete approximation for the BEC (waterfall + error floor)
Fix ε, n and a target error probability Ptarg
→ degree distribution optimization using LP
0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.7510-6
10-5
10-4
10-3
10-2
10-1
49.59 %
0.0 1.0rate/capacity
2 3 4 5 6 7 8 9 10
2 3 4 5 6 7 8 9 10 11 12 13
contribution to error floor
6 8 10 12 14 16 18 20 22 24 26
0.0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45-0.1
-0.09
-0.08
-0.07
-0.06
-0.05
-0.04
-0.03
-0.02
-0.01
counter:=40
λ = 0.0739196x + 0.65789x2 + 0.2681x12,
ρ = 0.390753x4 + 0.361589x5 + 0.247658x9.
Play it Again!
R. Urbanke (EPFL) A Scaling Approach to Coding Santa Fe, May 5th 2007 12 / 27
Optimization For BEC
Complete approximation for the BEC (waterfall + error floor)
Fix ε, n and a target error probability Ptarg
→ degree distribution optimization using LP
0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.7510-6
10-5
10-4
10-3
10-2
10-1
53.04 %
0.0 1.0rate/capacity
2 3 4 5 6 7 8 9 10
2 3 4 5 6 7 8 9 10 11 12 13
contribution to error floor
6 8 10 12 14 16 18 20 22 24 26
0.0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45-0.1
-0.09
-0.08
-0.07
-0.06
-0.05
-0.04
-0.03
-0.02
-0.01
counter:=45
λ = 0.0739196x + 0.65789x2 + 0.2681x12,
ρ = 0.390753x4 + 0.361589x5 + 0.247658x9.
Play it Again!
R. Urbanke (EPFL) A Scaling Approach to Coding Santa Fe, May 5th 2007 12 / 27
Optimization For BEC
Complete approximation for the BEC (waterfall + error floor)
Fix ε, n and a target error probability Ptarg
→ degree distribution optimization using LP
0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.7510-6
10-5
10-4
10-3
10-2
10-1
55.55 %
0.0 1.0rate/capacity
2 3 4 5 6 7 8 9 10
2 3 4 5 6 7 8 9 10 11 12 13
contribution to error floor
6 8 10 12 14 16 18 20 22 24 26
0.0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45-0.1
-0.09
-0.08
-0.07
-0.06
-0.05
-0.04
-0.03
-0.02
-0.01
counter:=50
λ = 0.0739196x + 0.65789x2 + 0.2681x12,
ρ = 0.390753x4 + 0.361589x5 + 0.247658x9.
Play it Again!
R. Urbanke (EPFL) A Scaling Approach to Coding Santa Fe, May 5th 2007 12 / 27
Optimization For BEC
Complete approximation for the BEC (waterfall + error floor)
Fix ε, n and a target error probability Ptarg
→ degree distribution optimization using LP
0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.7510-6
10-5
10-4
10-3
10-2
10-1
57.90 %
0.0 1.0rate/capacity
2 3 4 5 6 7 8 9 10
2 3 4 5 6 7 8 9 10 11 12 13
contribution to error floor
6 8 10 12 14 16 18 20 22 24 26
0.0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45-0.1
-0.09
-0.08
-0.07
-0.06
-0.05
-0.04
-0.03
-0.02
-0.01
counter:=55
λ = 0.0739196x + 0.65789x2 + 0.2681x12,
ρ = 0.390753x4 + 0.361589x5 + 0.247658x9.
Play it Again!
R. Urbanke (EPFL) A Scaling Approach to Coding Santa Fe, May 5th 2007 12 / 27
Optimization For BEC
Complete approximation for the BEC (waterfall + error floor)
Fix ε, n and a target error probability Ptarg
→ degree distribution optimization using LP
0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.7510-6
10-5
10-4
10-3
10-2
10-1
60.43 %
0.0 1.0rate/capacity
2 3 4 5 6 7 8 9 10
2 3 4 5 6 7 8 9 10 11 12 13
contribution to error floor
6 8 10 12 14 16 18 20 22 24 26
0.0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45-0.1
-0.09
-0.08
-0.07
-0.06
-0.05
-0.04
-0.03
-0.02
-0.01
counter:=60
λ = 0.0739196x + 0.65789x2 + 0.2681x12,
ρ = 0.390753x4 + 0.361589x5 + 0.247658x9.
Play it Again!
R. Urbanke (EPFL) A Scaling Approach to Coding Santa Fe, May 5th 2007 12 / 27
Optimization For BEC
Complete approximation for the BEC (waterfall + error floor)
Fix ε, n and a target error probability Ptarg
→ degree distribution optimization using LP
0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.7510-6
10-5
10-4
10-3
10-2
10-1
63.49 %
0.0 1.0rate/capacity
2 3 4 5 6 7 8 9 10
2 3 4 5 6 7 8 9 10 11 12 13
contribution to error floor
6 8 10 12 14 16 18 20 22 24 26
0.0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45-0.1
-0.09
-0.08
-0.07
-0.06
-0.05
-0.04
-0.03
-0.02
-0.01
counter:=65
λ = 0.0739196x + 0.65789x2 + 0.2681x12,
ρ = 0.390753x4 + 0.361589x5 + 0.247658x9.
Play it Again!
R. Urbanke (EPFL) A Scaling Approach to Coding Santa Fe, May 5th 2007 12 / 27
Optimization For BEC
Complete approximation for the BEC (waterfall + error floor)
Fix ε, n and a target error probability Ptarg
→ degree distribution optimization using LP
0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.7510-6
10-5
10-4
10-3
10-2
10-1
65.22 %
0.0 1.0rate/capacity
2 3 4 5 6 7 8 9 10
2 3 4 5 6 7 8 9 10 11 12 13
contribution to error floor
6 8 10 12 14 16 18 20 22 24 26
0.0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45-0.1
-0.09
-0.08
-0.07
-0.06
-0.05
-0.04
-0.03
-0.02
-0.01
counter:=70
λ = 0.0739196x + 0.65789x2 + 0.2681x12,
ρ = 0.390753x4 + 0.361589x5 + 0.247658x9.
Play it Again!
R. Urbanke (EPFL) A Scaling Approach to Coding Santa Fe, May 5th 2007 12 / 27
Optimization For BEC
Complete approximation for the BEC (waterfall + error floor)
Fix ε, n and a target error probability Ptarg
→ degree distribution optimization using LP
0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.7510-6
10-5
10-4
10-3
10-2
10-1
66.88 %
0.0 1.0rate/capacity
2 3 4 5 6 7 8 9 10
2 3 4 5 6 7 8 9 10 11 12 13
contribution to error floor
6 8 10 12 14 16 18 20 22 24 26
0.0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45-0.1
-0.09
-0.08
-0.07
-0.06
-0.05
-0.04
-0.03
-0.02
-0.01
counter:=75
λ = 0.0739196x + 0.65789x2 + 0.2681x12,
ρ = 0.390753x4 + 0.361589x5 + 0.247658x9.
Play it Again!
R. Urbanke (EPFL) A Scaling Approach to Coding Santa Fe, May 5th 2007 12 / 27
Optimization For BEC
Complete approximation for the BEC (waterfall + error floor)
Fix ε, n and a target error probability Ptarg
→ degree distribution optimization using LP
0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.7510-6
10-5
10-4
10-3
10-2
10-1
67.98 %
0.0 1.0rate/capacity
2 3 4 5 6 7 8 9 10
2 3 4 5 6 7 8 9 10 11 12 13
contribution to error floor
6 8 10 12 14 16 18 20 22 24 26
0.0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45-0.1
-0.09
-0.08
-0.07
-0.06
-0.05
-0.04
-0.03
-0.02
-0.01
counter:=80
λ = 0.0739196x + 0.65789x2 + 0.2681x12,
ρ = 0.390753x4 + 0.361589x5 + 0.247658x9.
Play it Again!
R. Urbanke (EPFL) A Scaling Approach to Coding Santa Fe, May 5th 2007 12 / 27
Optimization For BEC
Complete approximation for the BEC (waterfall + error floor)
Fix ε, n and a target error probability Ptarg
→ degree distribution optimization using LP
0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.7510-6
10-5
10-4
10-3
10-2
10-1
70.08 %
0.0 1.0rate/capacity
2 3 4 5 6 7 8 9 10
2 3 4 5 6 7 8 9 10 11 12 13
contribution to error floor
6 8 10 12 14 16 18 20 22 24 26
0.0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45-0.1
-0.09
-0.08
-0.07
-0.06
-0.05
-0.04
-0.03
-0.02
-0.01
counter:=85
λ = 0.0739196x + 0.65789x2 + 0.2681x12,
ρ = 0.390753x4 + 0.361589x5 + 0.247658x9.
Play it Again!
R. Urbanke (EPFL) A Scaling Approach to Coding Santa Fe, May 5th 2007 12 / 27
Optimization For BEC
Complete approximation for the BEC (waterfall + error floor)
Fix ε, n and a target error probability Ptarg
→ degree distribution optimization using LP
0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.7510-6
10-5
10-4
10-3
10-2
10-1
72.22 %
0.0 1.0rate/capacity
2 3 4 5 6 7 8 9 10
2 3 4 5 6 7 8 9 10 11 12 13
contribution to error floor
6 8 10 12 14 16 18 20 22 24 26
0.0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45-0.1
-0.09
-0.08
-0.07
-0.06
-0.05
-0.04
-0.03
-0.02
-0.01
counter:=115
λ = 0.0739196x + 0.65789x2 + 0.2681x12,
ρ = 0.390753x4 + 0.361589x5 + 0.247658x9.
Play it Again!
R. Urbanke (EPFL) A Scaling Approach to Coding Santa Fe, May 5th 2007 12 / 27
Optimization For BEC
Complete approximation for the BEC (waterfall + error floor)
Fix ε, n and a target error probability Ptarg
→ degree distribution optimization using LP
0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.7510-6
10-5
10-4
10-3
10-2
10-1
76.77 %
0.0 1.0rate/capacity
2 3 4 5 6 7 8 9 10
2 3 4 5 6 7 8 9 10 11 12 13
contribution to error floor
6 8 10 12 14 16 18 20 22 24 26
0.0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45-0.1
-0.09
-0.08
-0.07
-0.06
-0.05
-0.04
-0.03
-0.02
-0.01
counter:=255
λ = 0.0739196x + 0.65789x2 + 0.2681x12,
ρ = 0.390753x4 + 0.361589x5 + 0.247658x9.
Play it Again!
R. Urbanke (EPFL) A Scaling Approach to Coding Santa Fe, May 5th 2007 12 / 27
Optimization For BEC
Complete approximation for the BEC (waterfall + error floor)
Fix ε, n and a target error probability Ptarg
→ degree distribution optimization using LP
0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.7510-6
10-5
10-4
10-3
10-2
10-1
78.68 %
0.0 1.0rate/capacity
2 3 4 5 6 7 8 9 10
2 3 4 5 6 7 8 9 10 11 12 13
contribution to error floor
6 8 10 12 14 16 18 20 22 24 26
0.0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45-0.1
-0.09
-0.08
-0.07
-0.06
-0.05
-0.04
-0.03
-0.02
-0.01
counter:=355
λ = 0.0739196x + 0.65789x2 + 0.2681x12,
ρ = 0.390753x4 + 0.361589x5 + 0.247658x9.
Play it Again!
R. Urbanke (EPFL) A Scaling Approach to Coding Santa Fe, May 5th 2007 12 / 27
Optimization For BEC
Complete approximation for the BEC (waterfall + error floor)
Fix ε, n and a target error probability Ptarg
→ degree distribution optimization using LP
0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.7510-6
10-5
10-4
10-3
10-2
10-1
79.15 %
0.0 1.0rate/capacity
2 3 4 5 6 7 8 9 10
2 3 4 5 6 7 8 9 10 11 12 13
contribution to error floor
6 8 10 12 14 16 18 20 22 24 26
0.0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45-0.1
-0.09
-0.08
-0.07
-0.06
-0.05
-0.04
-0.03
-0.02
-0.01
counter:=390
λ = 0.0739196x + 0.65789x2 + 0.2681x12,
ρ = 0.390753x4 + 0.361589x5 + 0.247658x9.
Play it Again!
R. Urbanke (EPFL) A Scaling Approach to Coding Santa Fe, May 5th 2007 12 / 27
Optimization For BEC
Complete approximation for the BEC (waterfall + error floor)
Fix ε, n and a target error probability Ptarg
→ degree distribution optimization using LP
0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.7510-6
10-5
10-4
10-3
10-2
10-1
79.72 %
0.0 1.0rate/capacity
2 3 4 5 6 7 8 9 10
2 3 4 5 6 7 8 9 10 11 12 13
contribution to error floor
6 8 10 12 14 16 18 20 22 24 26
0.0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45-0.1
-0.09
-0.08
-0.07
-0.06
-0.05
-0.04
-0.03
-0.02
-0.01
counter:=445
λ = 0.0739196x + 0.65789x2 + 0.2681x12,
ρ = 0.390753x4 + 0.361589x5 + 0.247658x9.
Play it Again!
R. Urbanke (EPFL) A Scaling Approach to Coding Santa Fe, May 5th 2007 12 / 27
Optimization For BEC
Complete approximation for the BEC (waterfall + error floor)
Fix ε, n and a target error probability Ptarg
→ degree distribution optimization using LP
0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.7510-6
10-5
10-4
10-3
10-2
10-1
80.65 %
0.0 1.0rate/capacity
2 3 4 5 6 7 8 9 10
2 3 4 5 6 7 8 9 10 11 12 13
contribution to error floor
6 8 10 12 14 16 18 20 22 24 26
0.0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45-0.1
-0.09
-0.08
-0.07
-0.06
-0.05
-0.04
-0.03
-0.02
-0.01
counter:=500
λ = 0.0739196x + 0.65789x2 + 0.2681x12,
ρ = 0.390753x4 + 0.361589x5 + 0.247658x9.
Play it Again!
R. Urbanke (EPFL) A Scaling Approach to Coding Santa Fe, May 5th 2007 12 / 27
Optimization For BEC
Complete approximation for the BEC (waterfall + error floor)
Fix ε, n and a target error probability Ptarg
→ degree distribution optimization using LP
0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.7510-6
10-5
10-4
10-3
10-2
10-1
81.43 %
0.0 1.0rate/capacity
2 3 4 5 6 7 8 9 10
2 3 4 5 6 7 8 9 10 11 12 13
contribution to error floor
6 8 10 12 14 16 18 20 22 24 26
0.0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45-0.1
-0.09
-0.08
-0.07
-0.06
-0.05
-0.04
-0.03
-0.02
-0.01
counter:=555
λ = 0.0739196x + 0.65789x2 + 0.2681x12,
ρ = 0.390753x4 + 0.361589x5 + 0.247658x9.
Play it Again!
R. Urbanke (EPFL) A Scaling Approach to Coding Santa Fe, May 5th 2007 12 / 27
Optimization For BEC
Complete approximation for the BEC (waterfall + error floor)
Fix ε, n and a target error probability Ptarg
→ degree distribution optimization using LP
0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.7510-6
10-5
10-4
10-3
10-2
10-1
81.66 %
0.0 1.0rate/capacity
2 3 4 5 6 7 8 9 10
2 3 4 5 6 7 8 9 10 11 12 13
contribution to error floor
6 8 10 12 14 16 18 20 22 24 26
0.0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45-0.1
-0.09
-0.08
-0.07
-0.06
-0.05
-0.04
-0.03
-0.02
-0.01
counter:=580
λ = 0.0739196x + 0.65789x2 + 0.2681x12,
ρ = 0.390753x4 + 0.361589x5 + 0.247658x9.
Play it Again!
R. Urbanke (EPFL) A Scaling Approach to Coding Santa Fe, May 5th 2007 12 / 27
Optimization For BEC
Complete approximation for the BEC (waterfall + error floor)
Fix ε, n and a target error probability Ptarg
→ degree distribution optimization using LP
0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.7510-6
10-5
10-4
10-3
10-2
10-1
81.84 %
0.0 1.0rate/capacity
2 3 4 5 6 7 8 9 10
2 3 4 5 6 7 8 9 10 11 12 13
contribution to error floor
6 8 10 12 14 16 18 20 22 24 26
0.0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45-0.1
-0.09
-0.08
-0.07
-0.06
-0.05
-0.04
-0.03
-0.02
-0.01
counter:=595
λ = 0.0739196x + 0.65789x2 + 0.2681x12,
ρ = 0.390753x4 + 0.361589x5 + 0.247658x9.
Play it Again!
R. Urbanke (EPFL) A Scaling Approach to Coding Santa Fe, May 5th 2007 12 / 27
Optimization For BEC
Complete approximation for the BEC (waterfall + error floor)
Fix ε, n and a target error probability Ptarg
→ degree distribution optimization using LP
0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.7510-6
10-5
10-4
10-3
10-2
10-1
82.01 %
0.0 1.0rate/capacity
2 3 4 5 6 7 8 9 10
2 3 4 5 6 7 8 9 10 11 12 13
contribution to error floor
6 8 10 12 14 16 18 20 22 24 26
0.0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45-0.1
-0.09
-0.08
-0.07
-0.06
-0.05
-0.04
-0.03
-0.02
-0.01
counter:=610
λ = 0.0739196x + 0.65789x2 + 0.2681x12,
ρ = 0.390753x4 + 0.361589x5 + 0.247658x9.
Play it Again!
R. Urbanke (EPFL) A Scaling Approach to Coding Santa Fe, May 5th 2007 12 / 27
Optimization For BEC
Complete approximation for the BEC (waterfall + error floor)
Fix ε, n and a target error probability Ptarg
→ degree distribution optimization using LP
0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.7510-6
10-5
10-4
10-3
10-2
10-1
82.13 %
0.0 1.0rate/capacity
2 3 4 5 6 7 8 9 10
2 3 4 5 6 7 8 9 10 11 12 13
contribution to error floor
6 8 10 12 14 16 18 20 22 24 26
0.0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45-0.1
-0.09
-0.08
-0.07
-0.06
-0.05
-0.04
-0.03
-0.02
-0.01
counter:=625
λ = 0.0739196x + 0.65789x2 + 0.2681x12,
ρ = 0.390753x4 + 0.361589x5 + 0.247658x9.
Play it Again!
R. Urbanke (EPFL) A Scaling Approach to Coding Santa Fe, May 5th 2007 12 / 27
Optimization For BEC
Complete approximation for the BEC (waterfall + error floor)
Fix ε, n and a target error probability Ptarg
→ degree distribution optimization using LP
0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.7510-6
10-5
10-4
10-3
10-2
10-1
82.03 %
0.0 1.0rate/capacity
2 3 4 5 6 7 8 9 10
2 3 4 5 6 7 8 9 10 11 12 13
contribution to error floor
6 8 10 12 14 16 18 20 22 24 26
0.0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45-0.1
-0.09
-0.08
-0.07
-0.06
-0.05
-0.04
-0.03
-0.02
-0.01
counter:=660
λ = 0.0739196x + 0.65789x2 + 0.2681x12,
ρ = 0.390753x4 + 0.361589x5 + 0.247658x9.
Play it Again!
R. Urbanke (EPFL) A Scaling Approach to Coding Santa Fe, May 5th 2007 12 / 27
Optimization For BEC
Complete approximation for the BEC (waterfall + error floor)
Fix ε, n and a target error probability Ptarg
→ degree distribution optimization using LP
0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.7510-6
10-5
10-4
10-3
10-2
10-1
82.09 %
0.0 1.0rate/capacity
2 3 4 5 6 7 8 9 10
2 3 4 5 6 7 8 9 10 11 12 13
contribution to error floor
6 8 10 12 14 16 18 20 22 24 26
0.0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45-0.1
-0.09
-0.08
-0.07
-0.06
-0.05
-0.04
-0.03
-0.02
-0.01
counter:=700
λ = 0.0739196x + 0.65789x2 + 0.2681x12,
ρ = 0.390753x4 + 0.361589x5 + 0.247658x9.
Play it Again!
R. Urbanke (EPFL) A Scaling Approach to Coding Santa Fe, May 5th 2007 12 / 27
Optimization For BEC
Complete approximation for the BEC (waterfall + error floor)
Fix ε, n and a target error probability Ptarg
→ degree distribution optimization using LP
0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.7510-6
10-5
10-4
10-3
10-2
10-1
82.13 %
0.0 1.0rate/capacity
2 3 4 5 6 7 8 9 10
2 3 4 5 6 7 8 9 10 11 12 13
contribution to error floor
6 8 10 12 14 16 18 20 22 24 26
0.0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45-0.1
-0.09
-0.08
-0.07
-0.06
-0.05
-0.04
-0.03
-0.02
-0.01
counter:=713
λ = 0.0739196x + 0.65789x2 + 0.2681x12,
ρ = 0.390753x4 + 0.361589x5 + 0.247658x9.
Play it Again!
R. Urbanke (EPFL) A Scaling Approach to Coding Santa Fe, May 5th 2007 12 / 27
Optimization For BEC
Complete approximation for the BEC (waterfall + error floor)
Fix ε, n and a target error probability Ptarg
→ degree distribution optimization using LP
0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.7510-6
10-5
10-4
10-3
10-2
10-1
82.13 %
0.0 1.0rate/capacity
2 3 4 5 6 7 8 9 10
2 3 4 5 6 7 8 9 10 11 12 13
contribution to error floor
6 8 10 12 14 16 18 20 22 24 26
0.0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45-0.1
-0.09
-0.08
-0.07
-0.06
-0.05
-0.04
-0.03
-0.02
-0.01
counter:=713
λ = 0.0739196x + 0.65789x2 + 0.2681x12,
ρ = 0.390753x4 + 0.361589x5 + 0.247658x9.
Play it Again!
R. Urbanke (EPFL) A Scaling Approach to Coding Santa Fe, May 5th 2007 12 / 27
Scaling for General Message-Passing Decoders
computation of scaling parameter
R. Urbanke (EPFL) A Scaling Approach to Coding Santa Fe, May 5th 2007 13 / 27
EXIT LIKE CURVES
0.2 0.4 0.6 0.8
0.2
0.4
0.6
0.0 h
hEP
B
R. Urbanke (EPFL) A Scaling Approach to Coding Santa Fe, May 5th 2007 14 / 27
Do Such Curves Exist?
-0.4 -0.1 0.2 0.5p0.0
0.10.20.30.40.50.6
a bc d
eC′
Cµ
-2.2 -1.4 -0.6 0.2 1.0 1.8p0.0 a b
c
d
eCµ
0.2
0.4
0.6
0.8
[Rathi, U 07]
R. Urbanke (EPFL) A Scaling Approach to Coding Santa Fe, May 5th 2007 15 / 27
Admissible EXIT Curves
Admissible Not Admissible
R. Urbanke (EPFL) A Scaling Approach to Coding Santa Fe, May 5th 2007 16 / 27
0 100 200 300 400 500 600 700 800 9000
20
40
60
80
100
120
140
160
180
s
unknown variables
successful decoding
(3,6) code, length 2048, fixed erasure ε = 0.425. PB = 0.47528.
R. Urbanke (EPFL) A Scaling Approach to Coding Santa Fe, May 5th 2007 17 / 27
0 100 200 300 400 500 600 700 800 9000
20
40
60
80
100
120
140
160
180
s
unknown variables
unsuccessful decoding
(3,6) code, length 2048, fixed erasure ε = 0.425. PB = 0.47528.
R. Urbanke (EPFL) A Scaling Approach to Coding Santa Fe, May 5th 2007 17 / 27
0 100 200 300 400 500 600 700 800 9000
20
40
60
80
100
120
140
160
180
s
unknown variables
density evolution
actual realization
(3,6) code, length 2048, fixed erasure ε = 0.425. PB = 0.47528.
R. Urbanke (EPFL) A Scaling Approach to Coding Santa Fe, May 5th 2007 17 / 27
0 100 200 300 400 500 600 700 800 9000
20
40
60
80
100
120
140
160
180
0 10 20 30 40 50 60 70 80 90 1000.0
0.01
0.02
0.03
0.04
s
unknown variables
empirical dist. of degree one check nodes
(3,6) code, length 2048, fixed erasure ε = 0.425. PB = 0.47528.
R. Urbanke (EPFL) A Scaling Approach to Coding Santa Fe, May 5th 2007 17 / 27
PB ∼ Q
√n(ε− βn−
23 − ε?)
∂2ε(x)∂x2 |? limε→ε?(x− x?)
√V
Λ′(1)
V =
E(Xε − Xε)2
nΛ′(1)
binary erasure channel
R. Urbanke (EPFL) A Scaling Approach to Coding Santa Fe, May 5th 2007 18 / 27
PB ∼ Q
√n(h− βn−
23 − h?)
∂2h(x)∂x2 |? limε→ε?(x− x?)
√V
Λ′(1)
V =
E(Xε − Xε)2
nΛ′(1)
general case
R. Urbanke (EPFL) A Scaling Approach to Coding Santa Fe, May 5th 2007 18 / 27
How to Compute Correlation
R. Urbanke (EPFL) A Scaling Approach to Coding Santa Fe, May 5th 2007 19 / 27
How to Compute Correlation
µ0→0 µ0→1 µ1→1 µi−1→i µi→i µl→ ˆl+1
µ0←0 µ0←1 µ1←1 µi−1←i µi←i µl← ˆl+1
ν0 ν1 ν1 ν1−1 νi νi νl
0 1 1 i − 1 i i l. . . . . .
R. Urbanke (EPFL) A Scaling Approach to Coding Santa Fe, May 5th 2007 20 / 27
How to Compute Correlation
µ0→0 µ0→1 µ1→1 µi−1→i µi→i µl→ ˆl+1
µ0←0 µ0←1 µ1←1 µi−1←i µi←i µl← ˆl+1
ν0 ν1 ν1 ν1−1 νi νi νl
0 1 1 i − 1 i i l. . . . . .
cMl/2K(MT)l/2cT
R. Urbanke (EPFL) A Scaling Approach to Coding Santa Fe, May 5th 2007 20 / 27
How to Compute Correlation
µ0→0 µ0→1 µ1→1 µi−1→i µi→i µl→ ˆl+1
µ0←0 µ0←1 µ1←1 µi−1←i µi←i µl← ˆl+1
ν0 ν1 ν1 ν1−1 νi νi νl
0 1 1 i − 1 i i l. . . . . .
cMl/2K(MT)l/2cT
M : λ1 = 1;λ2degenerated
R. Urbanke (EPFL) A Scaling Approach to Coding Santa Fe, May 5th 2007 20 / 27
How to Compute Correlation
µ0→0 µ0→1 µ1→1 µi−1→i µi→i µl→ ˆl+1
µ0←0 µ0←1 µ1←1 µi−1←i µi←i µl← ˆl+1
ν0 ν1 ν1 ν1−1 νi νi νl
0 1 1 i − 1 i i l. . . . . .
correlation for depth l =2(l− 1)c23
λ2e2KeT
3 l(γλ2)l(1 + O(x− x∗))
M : λ1 = 1;λ2degenerated
V(1− γ2λ22)
2 ,(l− 1)c23
2λ2e2KeT
3
R. Urbanke (EPFL) A Scaling Approach to Coding Santa Fe, May 5th 2007 20 / 27
Flipping Probabilities
Quantized BP Quantized MinSum
R. Urbanke (EPFL) A Scaling Approach to Coding Santa Fe, May 5th 2007 21 / 27
Results - (3, 6), BAWGNC, MinSum, MAXL = 5, m = 10
σ∗ = 0.825, α ≈ 0.842,PMinSum
B
σ
10−1
10−2
10−3
10−4
0.7 0.75 0.8
PMinSumB
σ
10−1
10−2
10−3
10−4
0.7 0.75 0.8
R. Urbanke (EPFL) A Scaling Approach to Coding Santa Fe, May 5th 2007 22 / 27
(3, 6), Sequence of Quantized MinSum Decoders
m = 20
0.95
0.905
0.85
R. Urbanke (EPFL) A Scaling Approach to Coding Santa Fe, May 5th 2007 23 / 27
Results - (3, 6), BAWGNC, MinSum, MAXL = 20,m = ∞
σ∗ = 0.82125, α ≈ 0.905,PMinSum
B
σ
10−1
10−2
10−3
10−4
0.7 0.75 0.8
PMinSumB
σ
10−1
10−2
10−3
10−4
0.7 0.75 0.8
R. Urbanke (EPFL) A Scaling Approach to Coding Santa Fe, May 5th 2007 24 / 27
Results - (3, 6), BAWGNC, BP, MAXL = 5.13625, m = 7
σ∗ = 0.86915, α ≈ 0.900005,PBP
B
σ
10−1
10−2
10−3
10−4
0.7 0.75 0.8
PBPB
σ
10−1
10−2
10−3
10−4
0.7 0.75 0.8
R. Urbanke (EPFL) A Scaling Approach to Coding Santa Fe, May 5th 2007 25 / 27
Results - (3, 6), BAWGNC, BP, MAXL = 20, m = ∞
σ∗ = 0.881, α ≈ 0.97,PBP
B
σ
10−1
10−2
10−3
10−4
0.7 0.75 0.8
PBPB
σ
10−1
10−2
10−3
10−4
0.7 0.75 0.8
R. Urbanke (EPFL) A Scaling Approach to Coding Santa Fe, May 5th 2007 26 / 27
scaling in principle allows joint optimization of code and decodercomputational complexity (m6)irregulargeneral ensembleserror flooroptimizationproof :-)
R. Urbanke (EPFL) A Scaling Approach to Coding Santa Fe, May 5th 2007 27 / 27
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