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Finite-Length Optimization of Iterative Coding Systems via Scaling

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					Finite-Length Optimization of Iterative Coding
            Systems via Scaling

Abdelaziz Amraoui1 , Jeremie Ezri1 , Andrea Montanari2 , and
                     Rudiger Urbanke1

                                   1 EPFL,    Lausanne
                                   2 LPT-ENS,    Paris


                USC-2006, September 22th, 2006




      Amraoui, Ezri, Montanari, and Urbanke     Finite-Length Optimization of Iterative Coding Systems via Scaling
Codes on Graphs and Iterative Decoding: LDPC




                                   PB (σ)
 20                  10
 19
 18                   9




                                             ≈ 1.19658
 17
 16                   8
 15
 14                   7
 13
 12                   6
 11
 10


                                                     dB
  9                   5

                                            (Eb /N0 )∗
  8                   4
  7
  6                   3
  5
  4
  3                   2
  2                                                                                           (Eb /N0 )dB
  1                   1




        Amraoui, Ezri, Montanari, and Urbanke             Finite-Length Optimization of Iterative Coding Systems via Scaling
Codes on Graphs and Iterative Decoding: Turbo




  x [s]                                       x [s]
                                                          Pb

                                            x [p1 ]




                                                                    (Eb /N0 )BP ≈ 0.537
 permutation
     [2]
    π                                       x [p2 ]




                                                                             dB
                                                                                             `         ´
                                                                                                 Eb /N0 dB




               Amraoui, Ezri, Montanari, and Urbanke   Finite-Length Optimization of Iterative Coding Systems via Scaling
What we have ... ldpcOpt




        Amraoui, Ezri, Montanari, and Urbanke   Finite-Length Optimization of Iterative Coding Systems via Scaling
What we have ... ldpcOpt




        Amraoui, Ezri, Montanari, and Urbanke   Finite-Length Optimization of Iterative Coding Systems via Scaling
What we would like to have ...




  welcome to FINITELdpcOpt
                          channel=BAWGNC                  length=5000

                          ensemble=multi-edge        decoder=BP, 4bits

                          Ptarget=10*e-5                    OPTIMIZE




         Amraoui, Ezri, Montanari, and Urbanke   Finite-Length Optimization of Iterative Coding Systems via Scaling
A slightly less ambitious program for now ...




                                      BEC :-)




         Amraoui, Ezri, Montanari, and Urbanke   Finite-Length Optimization of Iterative Coding Systems via Scaling
Typical Curve: Waterfall + Error Floor


          λ = 0.0739196x + 0.65789x 2 + 0.2681x 12
          ρ = 0.390753x 4 + 0.361589x 5 + 0.247658x 9

                  PB
                       10-1




                       10-2




                       10-3




                       10-4




                       10-5                                                 blocklength=5000

                       10-6
                           0.25   0.3   0.35   0.4   0.45   0.5     0.55   0.6   0.65   0.7




     exact computation is possible ... but does not generalize

                 treat both contributions separately ...

         Amraoui, Ezri, Montanari, and Urbanke                    Finite-Length Optimization of Iterative Coding Systems via Scaling
Typical Curve: Waterfall + Error Floor


          λ = 0.0739196x + 0.65789x 2 + 0.2681x 12
          ρ = 0.390753x 4 + 0.361589x 5 + 0.247658x 9

                  PB
                       10-1




                       10-2




                       10-3




                       10-4




                       10-5                                                 blocklength=5000

                       10-6
                           0.25   0.3   0.35   0.4   0.45   0.5     0.55   0.6   0.65   0.7




     exact computation is possible ... but does not generalize

                 treat both contributions separately ...

         Amraoui, Ezri, Montanari, and Urbanke                    Finite-Length Optimization of Iterative Coding Systems via Scaling
Typical Curve: Waterfall + Error Floor


          λ = 0.0739196x + 0.65789x 2 + 0.2681x 12
          ρ = 0.390753x 4 + 0.361589x 5 + 0.247658x 9

                  PB
                       10-1




                       10-2




                       10-3




                       10-4




                       10-5                                                 blocklength=5000

                       10-6
                           0.25   0.3   0.35   0.4   0.45   0.5     0.55   0.6   0.65   0.7




     exact computation is possible ... but does not generalize

                 treat both contributions separately ...

         Amraoui, Ezri, Montanari, and Urbanke                    Finite-Length Optimization of Iterative Coding Systems via Scaling
How Do Finite-Length Curves Approach Threshold?


            PBP
             b




                                 scaling approach




        Amraoui, Ezri, Montanari, and Urbanke   Finite-Length Optimization of Iterative Coding Systems via Scaling
How Do Finite-Length Curves Approach Threshold?


            PBP
             b




                                 scaling approach




        Amraoui, Ezri, Montanari, and Urbanke   Finite-Length Optimization of Iterative Coding Systems via Scaling
The Waterfall ...



  Lemma (Amraoui, Montanari, Richardson, U)
  As n tends to infinity with argument of Q(·) kept fixed
                                  √                     2
                                      n(   BP
                                                − βn− 3 − )
     PB (n, λ, ρ, ) = Q                                                 1 + O n−1/3
                                                 α



  where α = α(λ, ρ) and β = β(λ, ρ).




          Amraoui, Ezri, Montanari, and Urbanke    Finite-Length Optimization of Iterative Coding Systems via Scaling
  s
 180

 160

 140

 120
                 successful decoding
 100

 80

 60

 40

 20

 0
       0      100       200      300       400      500      600       700       800       900
                                              unknown variables

(3,6) code, length 2048, fixed erasure = 0.425. PB = 0.47528.


           Amraoui, Ezri, Montanari, and Urbanke    Finite-Length Optimization of Iterative Coding Systems via Scaling
  s
 180

 160

 140

 120
               unsuccessful decoding
 100

 80

 60

 40

 20

 0
       0      100       200      300       400      500      600       700       800       900
                                              unknown variables

(3,6) code, length 2048, fixed erasure = 0.425. PB = 0.47528.


           Amraoui, Ezri, Montanari, and Urbanke    Finite-Length Optimization of Iterative Coding Systems via Scaling
  s
 180

 160

 140
                  density evolution
 120
                    actual realization
 100

 80

 60

 40

 20

 0
       0      100        200       300     400      500      600       700       800       900
                                              unknown variables

(3,6) code, length 2048, fixed erasure = 0.425. PB = 0.47528.


           Amraoui, Ezri, Montanari, and Urbanke    Finite-Length Optimization of Iterative Coding Systems via Scaling
   s
  180

  160

  140

  120
                                                                      0.04

  100
                                                                      0.03
  80
                                                                      0.02
  60

  40                                                                  0.01



  20                                                                  0.0
                                                                             0   10   20   30   40   50   60   70   80   90   100

  0
        0   100   200   300   400   500   600   700   800   900
                                                                   empirical dist. of degree one check nodes
                          unknown variables


(3,6) code, length 2048, fixed erasure = 0.425. PB = 0.47528.




              Amraoui, Ezri, Montanari, and Urbanke               Finite-Length Optimization of Iterative Coding Systems via Scaling
How to Compute α ...



  In order to compute α we need to know the variance of the
  decoding trajectory at the critical value. This can be done using
  Covariance Evolution:

  d z (i)
    ¯
          (τ ) = ˆ(i) (z (τ ))
                 f ¯
   dτ                                                                                                              
                                           d                   ˆ(j)
  dδ (ij)                                       δ (ik ) (τ ) ∂ f                   ∂ˆ(i)
                                                                                     f
            (τ ) = ˆ(ij) (z (τ )) +
                   f      ¯                                                       +                      δ (kj) (τ )
   dτ                                                         ∂z (k )               ∂z (k )
                                         k =0                            ¯
                                                                         z (τ )                 ¯
                                                                                                z (τ )

  Problem: nb. of coupled differential equations can be 100 ...




                Amraoui, Ezri, Montanari, and Urbanke    Finite-Length Optimization of Iterative Coding Systems via Scaling
How to Compute α ...



  In order to compute α we need to know the variance of the
  decoding trajectory at the critical value. This can be done using
  Covariance Evolution:

  d z (i)
    ¯
          (τ ) = ˆ(i) (z (τ ))
                 f ¯
   dτ                                                                                                              
                                           d                   ˆ(j)
  dδ (ij)                                       δ (ik ) (τ ) ∂ f                   ∂ˆ(i)
                                                                                     f
            (τ ) = ˆ(ij) (z (τ )) +
                   f      ¯                                                       +                      δ (kj) (τ )
   dτ                                                         ∂z (k )               ∂z (k )
                                         k =0                            ¯
                                                                         z (τ )                 ¯
                                                                                                z (τ )

  Problem: nb. of coupled differential equations can be 100 ...




                Amraoui, Ezri, Montanari, and Urbanke    Finite-Length Optimization of Iterative Coding Systems via Scaling
How to Compute α ...



  In order to compute α we need to know the variance of the
  decoding trajectory at the critical value. This can be done using
  Covariance Evolution:

  d z (i)
    ¯
          (τ ) = ˆ(i) (z (τ ))
                 f ¯
   dτ                                                                                                              
                                           d                   ˆ(j)
  dδ (ij)                                       δ (ik ) (τ ) ∂ f                   ∂ˆ(i)
                                                                                     f
            (τ ) = ˆ(ij) (z (τ )) +
                   f      ¯                                                       +                      δ (kj) (τ )
   dτ                                                         ∂z (k )               ∂z (k )
                                         k =0                            ¯
                                                                         z (τ )                 ¯
                                                                                                z (τ )

  Problem: nb. of coupled differential equations can be 100 ...




                Amraoui, Ezri, Montanari, and Urbanke    Finite-Length Optimization of Iterative Coding Systems via Scaling
Alternative Computation Leads to ...


               ρ(x )2 − ρ(x 2 ) + ρ (x )(1 − 2xρ(x )) − x 2 ρ (x 2 )
                 ¯        ¯          ¯               ¯  ¯      ¯
       α=                                  2 ρ (x )2
                                                                     +
                                L (1)λ(y )      ¯
               2 λ(y )2        2 λ(y 2 )                           1/2
                          −           − y 2 2 λ (y 2 )
                                                                         ,
                              L (1)λ(y )2
                                                                                    1/3
                 4 r 2 ( λ (y )2 r − x(λ (y )r + λ                (y )x))2
                    2              2               2
       β=               2 ρ (x )3 x 10 (2 λ (y )2 r − λ
                                                                                          ,
                L (1)        ¯                     3              (y )r2 x)
                                   j −1           m−1
       ri =           (−1)i+j                          ρm ( λ(y ))j ,
                                   i −1           j −1
              m≥j≥i

  with 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).

          Amraoui, Ezri, Montanari, and Urbanke    Finite-Length Optimization of Iterative Coding Systems via Scaling
Scaling for BEC ... waterfall region




         Amraoui, Ezri, Montanari, and Urbanke   Finite-Length Optimization of Iterative Coding Systems via Scaling
Scaling for BEC ... waterfall region




         Amraoui, Ezri, Montanari, and Urbanke   Finite-Length Optimization of Iterative Coding Systems via Scaling
Scaling for BEC ... waterfall region




         Amraoui, Ezri, Montanari, and Urbanke   Finite-Length Optimization of Iterative Coding Systems via Scaling
Scaling BEC ... error floor ... and the whole curve




         Amraoui, Ezri, Montanari, and Urbanke   Finite-Length Optimization of Iterative Coding Systems via Scaling
Scaling BEC ... error floor ... and the whole curve




         Amraoui, Ezri, Montanari, and Urbanke   Finite-Length Optimization of Iterative Coding Systems via Scaling
Scaling BEC ... error floor ... and the whole curve




         Amraoui, Ezri, Montanari, and Urbanke   Finite-Length Optimization of Iterative Coding Systems via Scaling
Scaling BEC ... error floor ... and the whole curve




         Amraoui, Ezri, Montanari, and Urbanke   Finite-Length Optimization of Iterative Coding Systems via Scaling
Optimization via LP


  Assume we start with any (λ, ρ) which fulfills P(λ, ρ) ≤ Ptarget .
  To find a degree distribution (λ + ∆λ, ρ + ∆ρ) with increased
  rate that still fulfills condition we solve:

   min{ − (1 − r )             ∆λi /i +            ∆ρi /i |
         − min{δ, λi } ≤ ∆λi ≤ δ;
         − min{δ, ρi } ≤ ∆ρi ≤ δ;
              ∆λi = 0;              ∆ρi = 0;
              ∂P(λ, ρ)                         ∂P(λ, ρ)
                       ∆λi +                            ∆ρi ≤ Ptarget − P(λ, ρ)}
                ∂λi                              ∂ρi




           Amraoui, Ezri, Montanari, and Urbanke    Finite-Length Optimization of Iterative Coding Systems via Scaling
Optimization via LP


  Assume we start with any (λ, ρ) which fulfills P(λ, ρ) ≤ Ptarget .
  To find a degree distribution (λ + ∆λ, ρ + ∆ρ) with increased
  rate that still fulfills condition we solve:

   min{ − (1 − r )             ∆λi /i +            ∆ρi /i |
         − min{δ, λi } ≤ ∆λi ≤ δ;
         − min{δ, ρi } ≤ ∆ρi ≤ δ;
              ∆λi = 0;              ∆ρi = 0;
              ∂P(λ, ρ)                         ∂P(λ, ρ)
                       ∆λi +                            ∆ρi ≤ Ptarget − P(λ, ρ)}
                ∂λi                              ∂ρi




           Amraoui, Ezri, Montanari, and Urbanke    Finite-Length Optimization of Iterative Coding Systems via Scaling
Example Run


  Consider the following case
      channel BEC
      length 5000
      type of ensemble LDPC
          Largest variable degree 13
          Largest check degree 10
          Expurgation 6
      decoder BP
      PBtarget 10−4
  We start with a random code and optimize...




          Amraoui, Ezri, Montanari, and Urbanke   Finite-Length Optimization of Iterative Coding Systems via Scaling
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

    -5
  10                                                                                                            -0.06


                                                                                                                -0.07


                                                                                                                -0.08


    -6                                                                                                          -0.09
  10                                                                                                                                                    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 = 0.0739196x + 0.65789x 2 + 0.2681x 12 ,
                                 ρ = 0.390753x 4 + 0.361589x 5 + 0.247658x 9 .

                                                                                   Play it Again Sam!




                               Amraoui, Ezri, Montanari, and Urbanke                            Finite-Length Optimization of Iterative Coding Systems via Scaling
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

    -5
  10                                                                                                            -0.06


                                                                                                                -0.07


                                                                                                                -0.08


    -6                                                                                                          -0.09
  10                                                                                                                                                    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 = 0.0739196x + 0.65789x 2 + 0.2681x 12 ,
                                 ρ = 0.390753x 4 + 0.361589x 5 + 0.247658x 9 .

                                                                                   Play it Again Sam!




                               Amraoui, Ezri, Montanari, and Urbanke                            Finite-Length Optimization of Iterative Coding Systems via Scaling
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

    -5
  10                                                                                                           -0.06


                                                                                                               -0.07


                                                                                                               -0.08


    -6                                                                                                         -0.09
  10                                                                                                                                                   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 = 0.0739196x + 0.65789x 2 + 0.2681x 12 ,
                                ρ = 0.390753x 4 + 0.361589x 5 + 0.247658x 9 .

                                                                                  Play it Again Sam!




                              Amraoui, Ezri, Montanari, and Urbanke                            Finite-Length Optimization of Iterative Coding Systems via Scaling
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

    -5
  10                                                                                                           -0.06


                                                                                                               -0.07


                                                                                                               -0.08


    -6                                                                                                         -0.09
  10                                                                                                                                                   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 = 0.0739196x + 0.65789x 2 + 0.2681x 12 ,
                                ρ = 0.390753x 4 + 0.361589x 5 + 0.247658x 9 .

                                                                                  Play it Again Sam!




                              Amraoui, Ezri, Montanari, and Urbanke                            Finite-Length Optimization of Iterative Coding Systems via Scaling
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

    -5
  10                                                                                                           -0.06


                                                                                                               -0.07


                                                                                                               -0.08


    -6                                                                                                         -0.09
  10                                                                                                                                                   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 = 0.0739196x + 0.65789x 2 + 0.2681x 12 ,
                                ρ = 0.390753x 4 + 0.361589x 5 + 0.247658x 9 .

                                                                                  Play it Again Sam!




                              Amraoui, Ezri, Montanari, and Urbanke                            Finite-Length Optimization of Iterative Coding Systems via Scaling
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

    -5
  10                                                                                                           -0.06


                                                                                                               -0.07


                                                                                                               -0.08


    -6                                                                                                         -0.09
  10                                                                                                                                                   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 = 0.0739196x + 0.65789x 2 + 0.2681x 12 ,
                                ρ = 0.390753x 4 + 0.361589x 5 + 0.247658x 9 .

                                                                                  Play it Again Sam!




                              Amraoui, Ezri, Montanari, and Urbanke                            Finite-Length Optimization of Iterative Coding Systems via Scaling
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

    -5
  10                                                                                                            -0.06


                                                                                                                -0.07


                                                                                                                -0.08


    -6                                                                                                          -0.09
  10                                                                                                                                                    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 = 0.0739196x + 0.65789x 2 + 0.2681x 12 ,
                                 ρ = 0.390753x 4 + 0.361589x 5 + 0.247658x 9 .

                                                                                   Play it Again Sam!




                               Amraoui, Ezri, Montanari, and Urbanke                            Finite-Length Optimization of Iterative Coding Systems via Scaling
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

                          46.01 %
               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

    -5
  10                                                                                                            -0.06


                                                                                                                -0.07


                                                                                                                -0.08


    -6                                                                                                          -0.09
  10                                                                                                                                                    counter:=35

         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 = 0.0739196x + 0.65789x 2 + 0.2681x 12 ,
                                 ρ = 0.390753x 4 + 0.361589x 5 + 0.247658x 9 .

                                                                                   Play it Again Sam!




                               Amraoui, Ezri, Montanari, and Urbanke                            Finite-Length Optimization of Iterative Coding Systems via Scaling
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

                          49.59 %
               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

    -5
  10                                                                                                           -0.06


                                                                                                               -0.07


                                                                                                               -0.08


    -6                                                                                                         -0.09
  10                                                                                                                                                   counter:=40

         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 = 0.0739196x + 0.65789x 2 + 0.2681x 12 ,
                                ρ = 0.390753x 4 + 0.361589x 5 + 0.247658x 9 .

                                                                                  Play it Again Sam!




                              Amraoui, Ezri, Montanari, and Urbanke                            Finite-Length Optimization of Iterative Coding Systems via Scaling
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

                          53.04 %
               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

    -5
  10                                                                                                           -0.06


                                                                                                               -0.07


                                                                                                               -0.08


    -6                                                                                                         -0.09
  10                                                                                                                                                   counter:=45

         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 = 0.0739196x + 0.65789x 2 + 0.2681x 12 ,
                                ρ = 0.390753x 4 + 0.361589x 5 + 0.247658x 9 .

                                                                                  Play it Again Sam!




                              Amraoui, Ezri, Montanari, and Urbanke                            Finite-Length Optimization of Iterative Coding Systems via Scaling
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

                          55.55 %
               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

    -5
  10                                                                                                           -0.06


                                                                                                               -0.07


                                                                                                               -0.08


    -6                                                                                                         -0.09
  10                                                                                                                                                   counter:=50

         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 = 0.0739196x + 0.65789x 2 + 0.2681x 12 ,
                                ρ = 0.390753x 4 + 0.361589x 5 + 0.247658x 9 .

                                                                                  Play it Again Sam!




                              Amraoui, Ezri, Montanari, and Urbanke                            Finite-Length Optimization of Iterative Coding Systems via Scaling
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

                              57.90 %
               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

    -5
  10                                                                                                            -0.06


                                                                                                                -0.07


                                                                                                                -0.08


    -6                                                                                                          -0.09
  10                                                                                                                                                    counter:=55

         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 = 0.0739196x + 0.65789x 2 + 0.2681x 12 ,
                                 ρ = 0.390753x 4 + 0.361589x 5 + 0.247658x 9 .

                                                                                   Play it Again Sam!




                               Amraoui, Ezri, Montanari, and Urbanke                            Finite-Length Optimization of Iterative Coding Systems via Scaling
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

                              60.43 %
               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

    -5
  10                                                                                                           -0.06


                                                                                                               -0.07


                                                                                                               -0.08


    -6                                                                                                         -0.09
  10                                                                                                                                                   counter:=60

         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 = 0.0739196x + 0.65789x 2 + 0.2681x 12 ,
                                ρ = 0.390753x 4 + 0.361589x 5 + 0.247658x 9 .

                                                                                  Play it Again Sam!




                               Amraoui, Ezri, Montanari, and Urbanke                           Finite-Length Optimization of Iterative Coding Systems via Scaling
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

                              63.49 %
               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

    -5
  10                                                                                                           -0.06


                                                                                                               -0.07


                                                                                                               -0.08


    -6                                                                                                         -0.09
  10                                                                                                                                                   counter:=65

         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 = 0.0739196x + 0.65789x 2 + 0.2681x 12 ,
                                ρ = 0.390753x 4 + 0.361589x 5 + 0.247658x 9 .

                                                                                  Play it Again Sam!




                              Amraoui, Ezri, Montanari, and Urbanke                            Finite-Length Optimization of Iterative Coding Systems via Scaling
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

                              65.22 %
               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

    -5
  10                                                                                                           -0.06


                                                                                                               -0.07


                                                                                                               -0.08


    -6                                                                                                         -0.09
  10                                                                                                                                                   counter:=70

         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 = 0.0739196x + 0.65789x 2 + 0.2681x 12 ,
                                ρ = 0.390753x 4 + 0.361589x 5 + 0.247658x 9 .

                                                                                  Play it Again Sam!




                              Amraoui, Ezri, Montanari, and Urbanke                            Finite-Length Optimization of Iterative Coding Systems via Scaling
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

                              66.88 %
               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

    -5
  10                                                                                                           -0.06


                                                                                                               -0.07


                                                                                                               -0.08


    -6                                                                                                         -0.09
  10                                                                                                                                                   counter:=75

         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 = 0.0739196x + 0.65789x 2 + 0.2681x 12 ,
                                ρ = 0.390753x 4 + 0.361589x 5 + 0.247658x 9 .

                                                                                  Play it Again Sam!




                              Amraoui, Ezri, Montanari, and Urbanke                            Finite-Length Optimization of Iterative Coding Systems via Scaling
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

                               67.98 %
               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

    -5
  10                                                                                                           -0.06


                                                                                                               -0.07


                                                                                                               -0.08


    -6                                                                                                         -0.09
  10                                                                                                                                                   counter:=80

         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 = 0.0739196x + 0.65789x 2 + 0.2681x 12 ,
                                ρ = 0.390753x 4 + 0.361589x 5 + 0.247658x 9 .

                                                                                  Play it Again Sam!




                              Amraoui, Ezri, Montanari, and Urbanke                            Finite-Length Optimization of Iterative Coding Systems via Scaling
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

                               70.08 %
               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

    -5
  10                                                                                                           -0.06


                                                                                                               -0.07


                                                                                                               -0.08


    -6                                                                                                         -0.09
  10                                                                                                                                                   counter:=85

         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 = 0.0739196x + 0.65789x 2 + 0.2681x 12 ,
                                ρ = 0.390753x 4 + 0.361589x 5 + 0.247658x 9 .

                                                                                  Play it Again Sam!




                              Amraoui, Ezri, Montanari, and Urbanke                            Finite-Length Optimization of Iterative Coding Systems via Scaling
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

                                72.22 %
               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

    -5
  10                                                                                                           -0.06


                                                                                                               -0.07


                                                                                                               -0.08


    -6                                                                                                         -0.09
  10                                                                                                                                                   counter:=115

         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 = 0.0739196x + 0.65789x 2 + 0.2681x 12 ,
                                ρ = 0.390753x 4 + 0.361589x 5 + 0.247658x 9 .

                                                                                  Play it Again Sam!




                              Amraoui, Ezri, Montanari, and Urbanke                            Finite-Length Optimization of Iterative Coding Systems via Scaling
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

                                   76.77 %
               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

    -5
  10                                                                                                           -0.06


                                                                                                               -0.07


                                                                                                               -0.08


    -6                                                                                                         -0.09
  10                                                                                                                                                   counter:=255

         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 = 0.0739196x + 0.65789x 2 + 0.2681x 12 ,
                                ρ = 0.390753x 4 + 0.361589x 5 + 0.247658x 9 .

                                                                                  Play it Again Sam!




                              Amraoui, Ezri, Montanari, and Urbanke                            Finite-Length Optimization of Iterative Coding Systems via Scaling
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

                                   78.68 %
               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

    -5
  10                                                                                                           -0.06


                                                                                                               -0.07


                                                                                                               -0.08


    -6                                                                                                         -0.09
  10                                                                                                                                                   counter:=355

         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 = 0.0739196x + 0.65789x 2 + 0.2681x 12 ,
                                ρ = 0.390753x 4 + 0.361589x 5 + 0.247658x 9 .

                                                                                  Play it Again Sam!




                              Amraoui, Ezri, Montanari, and Urbanke                            Finite-Length Optimization of Iterative Coding Systems via Scaling
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

                                   79.15 %
               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

    -5
  10                                                                                                           -0.06


                                                                                                               -0.07


                                                                                                               -0.08


    -6                                                                                                         -0.09
  10                                                                                                                                                   counter:=390

         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 = 0.0739196x + 0.65789x 2 + 0.2681x 12 ,
                                ρ = 0.390753x 4 + 0.361589x 5 + 0.247658x 9 .

                                                                                  Play it Again Sam!




                              Amraoui, Ezri, Montanari, and Urbanke                            Finite-Length Optimization of Iterative Coding Systems via Scaling
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

                                   79.72 %
               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

    -5
  10                                                                                                           -0.06


                                                                                                               -0.07


                                                                                                               -0.08


    -6                                                                                                         -0.09
  10                                                                                                                                                   counter:=445

         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 = 0.0739196x + 0.65789x 2 + 0.2681x 12 ,
                                ρ = 0.390753x 4 + 0.361589x 5 + 0.247658x 9 .

                                                                                  Play it Again Sam!




                              Amraoui, Ezri, Montanari, and Urbanke                            Finite-Length Optimization of Iterative Coding Systems via Scaling
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

                                   80.65 %
               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

    -5
  10                                                                                                           -0.06


                                                                                                               -0.07


                                                                                                               -0.08


    -6                                                                                                         -0.09
  10                                                                                                                                                   counter:=500

         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 = 0.0739196x + 0.65789x 2 + 0.2681x 12 ,
                                ρ = 0.390753x 4 + 0.361589x 5 + 0.247658x 9 .

                                                                                  Play it Again Sam!




                              Amraoui, Ezri, Montanari, and Urbanke                            Finite-Length Optimization of Iterative Coding Systems via Scaling
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

                                     81.43 %
               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

    -5
  10                                                                                                            -0.06


                                                                                                                -0.07


                                                                                                                -0.08


    -6                                                                                                          -0.09
  10                                                                                                                                                    counter:=555

         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 = 0.0739196x + 0.65789x 2 + 0.2681x 12 ,
                                ρ = 0.390753x 4 + 0.361589x 5 + 0.247658x 9 .

                                                                                   Play it Again Sam!




                              Amraoui, Ezri, Montanari, and Urbanke                             Finite-Length Optimization of Iterative Coding Systems via Scaling
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

                                     81.66 %
               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

    -5
  10                                                                                                            -0.06


                                                                                                                -0.07


                                                                                                                -0.08


    -6                                                                                                          -0.09
  10                                                                                                                                                    counter:=580

         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 = 0.0739196x + 0.65789x 2 + 0.2681x 12 ,
                                ρ = 0.390753x 4 + 0.361589x 5 + 0.247658x 9 .

                                                                                   Play it Again Sam!




                              Amraoui, Ezri, Montanari, and Urbanke                             Finite-Length Optimization of Iterative Coding Systems via Scaling
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

                                     81.84 %
               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

    -5
  10                                                                                                            -0.06


                                                                                                                -0.07


                                                                                                                -0.08


    -6                                                                                                          -0.09
  10                                                                                                                                                    counter:=595

         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 = 0.0739196x + 0.65789x 2 + 0.2681x 12 ,
                                ρ = 0.390753x 4 + 0.361589x 5 + 0.247658x 9 .

                                                                                   Play it Again Sam!




                              Amraoui, Ezri, Montanari, and Urbanke                             Finite-Length Optimization of Iterative Coding Systems via Scaling
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

                                     82.01 %
               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

    -5
  10                                                                                                            -0.06


                                                                                                                -0.07


                                                                                                                -0.08


    -6                                                                                                          -0.09
  10                                                                                                                                                    counter:=610

         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 = 0.0739196x + 0.65789x 2 + 0.2681x 12 ,
                                ρ = 0.390753x 4 + 0.361589x 5 + 0.247658x 9 .

                                                                                   Play it Again Sam!




                              Amraoui, Ezri, Montanari, and Urbanke                             Finite-Length Optimization of Iterative Coding Systems via Scaling
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

                                     82.13 %
               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

    -5
  10                                                                                                            -0.06


                                                                                                                -0.07


                                                                                                                -0.08


    -6                                                                                                          -0.09
  10                                                                                                                                                    counter:=625

         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 = 0.0739196x + 0.65789x 2 + 0.2681x 12 ,
                                ρ = 0.390753x 4 + 0.361589x 5 + 0.247658x 9 .

                                                                                   Play it Again Sam!




                              Amraoui, Ezri, Montanari, and Urbanke                             Finite-Length Optimization of Iterative Coding Systems via Scaling
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

                                     82.03 %
               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

    -5
  10                                                                                                            -0.06


                                                                                                                -0.07


                                                                                                                -0.08


    -6                                                                                                          -0.09
  10                                                                                                                                                    counter:=660

         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 = 0.0739196x + 0.65789x 2 + 0.2681x 12 ,
                                ρ = 0.390753x 4 + 0.361589x 5 + 0.247658x 9 .

                                                                                   Play it Again Sam!




                              Amraoui, Ezri, Montanari, and Urbanke                             Finite-Length Optimization of Iterative Coding Systems via Scaling
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

                                     82.09 %
               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

    -5
  10                                                                                                            -0.06


                                                                                                                -0.07


                                                                                                                -0.08


    -6                                                                                                          -0.09
  10                                                                                                                                                    counter:=700

         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 = 0.0739196x + 0.65789x 2 + 0.2681x 12 ,
                                ρ = 0.390753x 4 + 0.361589x 5 + 0.247658x 9 .

                                                                                   Play it Again Sam!




                              Amraoui, Ezri, Montanari, and Urbanke                             Finite-Length Optimization of Iterative Coding Systems via Scaling
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

                                     82.13 %
               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

    -5
  10                                                                                                            -0.06


                                                                                                                -0.07


                                                                                                                -0.08


    -6                                                                                                          -0.09
  10                                                                                                                                                    counter:=713

         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 = 0.0739196x + 0.65789x 2 + 0.2681x 12 ,
                                ρ = 0.390753x 4 + 0.361589x 5 + 0.247658x 9 .

                                                                                   Play it Again Sam!




                              Amraoui, Ezri, Montanari, and Urbanke                             Finite-Length Optimization of Iterative Coding Systems via Scaling
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

                                     82.13 %
               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

    -5
  10                                                                                                            -0.06


                                                                                                                -0.07


                                                                                                                -0.08


    -6                                                                                                          -0.09
  10                                                                                                                                                    counter:=713

         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 = 0.0739196x + 0.65789x 2 + 0.2681x 12 ,
                                ρ = 0.390753x 4 + 0.361589x 5 + 0.247658x 9 .

                                                                                   Play it Again Sam!




                              Amraoui, Ezri, Montanari, and Urbanke                             Finite-Length Optimization of Iterative Coding Systems via Scaling
OK – But What About BMS Channels?
  Conjecture – General Channels
  Parameterize channel by entropy h. As n tends to infinity with
  argument of Q(·) kept fixed
                                         √                2
                                             n(hBP − βn− 3 − h)
    PB (n, λ, ρ, h) = Q                                                    1 + O n−1/3
                                                    α
                  PB (σ)
                            ≈ 1.19658
                                    dB
                           (Eb /N0 )∗




                                                                     (Eb /N0 )dB
          Amraoui, Ezri, Montanari, and Urbanke      Finite-Length Optimization of Iterative Coding Systems via Scaling
BSC and Gallager A ... A First Step



        PB
         10-1



         10-2



         10-3



         10-4



         10-5
             0.01             0.02              0.03              0.04




        Amraoui, Ezri, Montanari, and Urbanke     Finite-Length Optimization of Iterative Coding Systems via Scaling
Gallager A: Density Evolution



                                                            l−1
                     1 + (1 − 2xi−1 )r −1                                             1 − (1 − 2xi−1 )r −1
  xi   =   −                                                       + (1 − )
                             2                                                                2


                                         with x0 = .

                                      0.004

                                      0.002

                                                   0.01     0.02    0.03    0.04
                                     -0.002

                                     -0.004

                                     -0.006




           Amraoui, Ezri, Montanari, and Urbanke          Finite-Length Optimization of Iterative Coding Systems via Scaling
Gallager A: Density Evolution




                    Experiments: ((3, 4)-Regular Code)
   (i) single iteration: 13 percent error (α ≈ 1.21)
   (ii) two iterations: 5 percent error (α ≈ 1.12)
   (iii three iterations: 3 percent error (α ≈ 1.07)
   (iv ??
  (∞) α ≈ 1.05




            Amraoui, Ezri, Montanari, and Urbanke   Finite-Length Optimization of Iterative Coding Systems via Scaling
So where do we go from here ...




   the standard LDPC ensemble not really powerful enough ...
      LDPC, turbo, RA, IRA, IRAA, concatenated tree codes,
     multi-edge ensembles, matched liftings, ensembles from
                         protographs, ...
  general channels and decoding algorithms (quantized versions
                             of BP)




         Amraoui, Ezri, Montanari, and Urbanke   Finite-Length Optimization of Iterative Coding Systems via Scaling
So where do we go from here ...




   the standard LDPC ensemble not really powerful enough ...
      LDPC, turbo, RA, IRA, IRAA, concatenated tree codes,
     multi-edge ensembles, matched liftings, ensembles from
                         protographs, ...
  general channels and decoding algorithms (quantized versions
                             of BP)




         Amraoui, Ezri, Montanari, and Urbanke   Finite-Length Optimization of Iterative Coding Systems via Scaling
So where do we go from here ...




   the standard LDPC ensemble not really powerful enough ...
      LDPC, turbo, RA, IRA, IRAA, concatenated tree codes,
     multi-edge ensembles, matched liftings, ensembles from
                         protographs, ...
  general channels and decoding algorithms (quantized versions
                             of BP)




         Amraoui, Ezri, Montanari, and Urbanke   Finite-Length Optimization of Iterative Coding Systems via Scaling
So maybe in a couple of years ...




  welcome to FINITELdpcOpt
                          channel=BAWGNC                  length=5000

                          ensemble=multi-edge        decoder=BP, 4bits

                          Ptarget=10*e-5                    OPTIMIZE




         Amraoui, Ezri, Montanari, and Urbanke   Finite-Length Optimization of Iterative Coding Systems via Scaling
The Error Floor ...


  Lemma (Gallager, Burshtein et. al., Litsyn et. al., Di et. al., ...)
  Let A(G, w) denote the number of stopping sets of weight w of a
  code . Then
                   E[A(G, w)] =    η(ω, e)γ(e),
                                                     e

  where

                η(w, e) = coef                     (1 + xy i )Λi , x w y e ,
                                             i
                                  coef             i ((1   + z)i − iz)Ri , z e
                    γ(e) =                                 Λ (1)
                                                                                         .
                                                             e




           Amraoui, Ezri, Montanari, and Urbanke         Finite-Length Optimization of Iterative Coding Systems via Scaling
A First Approximation of Error Floor ...




                                                              w
                                      PB ≈             Aw
                                                   w



   This bound is not very good: (i) we do not know where to stop,
   (ii) there is clear overcounting, and (iii) “typical codes” can be
                         behave very differently




           Amraoui, Ezri, Montanari, and Urbanke       Finite-Length Optimization of Iterative Coding Systems via Scaling
A First Approximation of Error Floor ...




                                                              w
                                      PB ≈             Aw
                                                   w



   This bound is not very good: (i) we do not know where to stop,
   (ii) there is clear overcounting, and (iii) “typical codes” can be
                         behave very differently




           Amraoui, Ezri, Montanari, and Urbanke       Finite-Length Optimization of Iterative Coding Systems via Scaling
Better: Minimal Stopping Sets and Expurgation ...




                                 A(x) =                  Aw x w
                                                    w
                                 ˜
                                 A(x) = log A(x)


                                P        ˜      w
          PB ≈ 1 − e−              w≥s   Aw
                                                                 expurgation parameter s
                      −
                          P       ˜
       Pgood ≈ e            0<w<s Aw                                  fraction of good codes




        Amraoui, Ezri, Montanari, and Urbanke           Finite-Length Optimization of Iterative Coding Systems via Scaling
Better: Minimal Stopping Sets and Expurgation ...




                                 A(x) =                  Aw x w
                                                    w
                                 ˜
                                 A(x) = log A(x)


                                P        ˜      w
          PB ≈ 1 − e−              w≥s   Aw
                                                                 expurgation parameter s
                      −
                          P       ˜
       Pgood ≈ e            0<w<s Aw                                  fraction of good codes




        Amraoui, Ezri, Montanari, and Urbanke           Finite-Length Optimization of Iterative Coding Systems via Scaling
Better: Minimal Stopping Sets and Expurgation ...




                                 A(x) =                  Aw x w
                                                    w
                                 ˜
                                 A(x) = log A(x)


                                P        ˜      w
          PB ≈ 1 − e−              w≥s   Aw
                                                                 expurgation parameter s
                      −
                          P       ˜
       Pgood ≈ e            0<w<s Aw                                  fraction of good codes




        Amraoui, Ezri, Montanari, and Urbanke           Finite-Length Optimization of Iterative Coding Systems via Scaling
Where does this formula come from ...

  ˜ ˜
  Z1 , Z2 , · · · are jointly independent Poisson random variables.

                    P{        ˜
                              Zw > 0} = 1 − P{                   ˜
                                                                 Zw = 0}
                                                         P˜
                                                        − Zw
                                              =1−e


             ˜
             A(x) =                ˜
                                 E[Zw ]x w                     “minimal stopping sets”
                             ˜
             A(x) = eA(x) ,
                        ˜
                        A(x)i
                  =                                             “regular stopping sets”
                            i!




           Amraoui, Ezri, Montanari, and Urbanke   Finite-Length Optimization of Iterative Coding Systems via Scaling
Justification for independence ...


  Lemma (Orlitsky et. al., Di, Richardson, U)
  Consider the ensemble LDPC(n, λ, ρ) and define
  µ λ (0)ρ (1). Then the expected asymptotic number of
  minimal stopping sets of size w is equal to µw /(2w). Let ν BP
  denote the smallest non-zero critical point of the ensemble and
  let 0 < ν < ν BP . Then for µ < 1
                                                                                  P        µw w
                         MAP/BP
      lim ELDPC(n,λ,ρ) [PB      (G, w ≤ νn, )] = 1 − e−                              w≥1    2w
     n→∞
                                                                 =1−            1−µ




           Amraoui, Ezri, Montanari, and Urbanke   Finite-Length Optimization of Iterative Coding Systems via Scaling

				
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