Iterative Coding for Network Coding · Montanari, Rathi, Urbanke Iterative Coding for Network...

Information is not orangesPractical network coding

Noisy network codingConstruction and decoding algorithm

Achieving capacity

Iterative Coding for Network Coding

Andrea Montanari∗, Vish Rathi† and Rudiger Urbanke†

∗Stanford, †EPFL

March 26, 2008

Montanari, Rathi, Urbanke Iterative Coding for Network Coding

Achieving capacity

Outline

1 Information is not oranges

2 Practical network coding

3 Noisy network coding

4 Construction and decoding algorithm

5 Achieving capacity

Achieving capacity

Outline

Achieving capacity

Outline

Achieving capacity

Outline

Achieving capacity

Outline

Achieving capacity

Information is not oranges

Achieving capacity

Current approach to communication networks

sourcedestination

Achieving capacity

sourcedestination

Achieving capacity

sourcedestination

Achieving capacity

sourcedestination

Achieving capacity

What are we losing? The butterfly example

source

destination 1 destination 2

[Ahlswede,Cai,Li,Yeung, 2000]

Achieving capacity

The butterfly example: Routing

Achieving capacity

1.5 bits per cycle

Achieving capacity

The butterfly example: Network Coding

Achieving capacity

The butterfly example: Netwok Coding

2 bits per cycle

Achieving capacity

Practical network coding

Achieving capacity

Problem

No one knows the network structure.

Source/destination do not control intermediate nodes.

Achieving capacity

Problem

No one knows the network structure.

Source/destination do not control intermediate nodes.

Achieving capacity

α x2 + β x2

Forward random combinations

[Chou, Wu, Jain/ Ho, Kotter, Medard, Karger, Effros 2003]

Achieving capacity

How can that possibly work?

α1 +β1

α2 +β2

Achieving capacity

Wait a minute. . .

How am I supposed to figure out α, β, γ?

Achieving capacity

Wait a minute. . .

How am I supposed to figure out α, β, γ?

Achieving capacity

Idea: Header

Input:

= [1 0| · · · · · · x1 · · · · · · ]

= [0 1| · · · · · · x2 · · · · · · ]

Achieving capacity

Idea: Header

Output:

= [α γ| · · ·αx1 + γx2 · · · ]

= [δ β| · · · δx1 + βx2 · · · ]

Achieving capacity

Noisy network coding

Achieving capacity

Rank Metric Channel (Symmetric Network Channel)(Gabidulin, 1985)

y = x ⊕ z ,

z uniformly random with rank(z) = `ω

Achieving capacity

Asymptotics, Rate, Capacity

` = Nλ, m = N(1− λ), N →∞

R =log2 |Code|

C (λ, ω) =1− λ− ω + λω2

1− λ.

Achieving capacity

` = Nλ, m = N(1− λ), N →∞

R =log2 |Code|

C (λ, ω) =1− λ− ω + λω2

1− λ.

Achieving capacity

` = Nλ, m = N(1− λ), N →∞

R =log2 |Code|

C (λ, ω) =1− λ− ω + λω2

1− λ.

Achieving capacity

Matrix channels

x = xij ∈ Fm×`2

Achieving capacity

Network coding???

(Kotter, Kschischang, 2007)

headers information

1∣∣∣ x

(Chou, Wu, Jain, 2003)

Achieving capacity

Reliable network

∣∣∣ x] [

G∣∣∣ Gx

∣∣∣ x]

NET G−1

Achieving capacity

Faulty network

∣∣∣ x] [

G∣∣∣ Gx + z ′

∣∣∣x + G−1z ′]

NET G−1

Achieving capacity

Binary Erasure Channel

(Elias, 1954)

xij with prob. ε,∗ with prob. 1− ε.

independently

Achieving capacity

` = Nλ, m = N(1− λ), N →∞

R =log2 |Code|

C (ε) = 1− ε .

Achieving capacity

` = Nλ, m = N(1− λ), N →∞

R =log2 |Code|

C (ε) = 1− ε .

Achieving capacity

` = Nλ, m = N(1− λ), N →∞

R =log2 |Code|

C (ε) = 1− ε .

Achieving capacity

Construction and decoding algorithm

Achieving capacity

Equivalent descripton of the channel

x (1) x (2) x (3) x (m)

Achieving capacity

Equivalent descripton of the channel

x (1) x (2) x (m) x (1) x (2) x (m)

+ + +z(1) z(2) z(m)

Z ≡ uniformly random subspace ⊆ F`2

z(1), . . . , z(m) ∈ Z

Achieving capacity

000000

m − `ω `ω

m − `ω ‘symbols’ → LDPC

`ω ‘symbols’ → learn the error space Z

Achieving capacity

LDPC: Check nodes

x (1) x (2) x (3)

h1 h2 h3

h1x(1) + h2x

(2) + h3x(3) = 0

‘Edge labels’ hi ∈ F`×`2 : `× ` matrices

Achieving capacity

LDPC: Check nodes

x (1) x (2) x (3)

h1 h2 h3

h1x(1) + h2x

(2) + h3x(3) = 0

Achieving capacity

LDPC: Check nodes

x (1) x (2) x (3)

h1 h2 h3

h1x(1) + h2x

(2) + h3x(3) = 0

Achieving capacity

LDPC: Variable nodes

x (i) ∈ z(i) + Z

Degree = 2 !

Achieving capacity

Message passing decoder

Messages → Affine subspaces Vi→a ⊆ F`2

Operations → Subspace intersections/sums

Achieving capacity

Message passing decoder: Check nodes

h1x(1) + h2x

(2) + h3x(3) = 0

V3 = h−13

(h1V1 + h2V2

Achieving capacity

h1x(1) + h2x

(2) + h3x(3) = 0

V3 = h−13

(h1V1 + h2V2

Achieving capacity

h1x(1) + h2x

(2) + h3x(3) = 0

V3 = h−13

(h1V1 + h2V2

Achieving capacity

Message passing decoder: Variable nodes

x (i) ∈ z(i) + Z

V2 = V1 ∩ z(i) + X

Achieving capacity

x (i) ∈ z(i) + Z

V2 = V1 ∩ z(i) + X

Achieving capacity

x (i) ∈ z(i) + Z

V2 = V1 ∩ z(i) + X

Achieving capacity

Capacity achieving ensemble

Variable degree:

Check degree:

P(k)i =

2k(k − 1)

i(i − 1)(i − 2).

Not the soliton!

Achieving capacity

Variable degree:

Check degree:

P(k)i =

2k(k − 1)

i(i − 1)(i − 2).

Not the soliton!

Achieving capacity

Variable degree:

Check degree:

P(k)i =

2k(k − 1)

i(i − 1)(i − 2).

Not the soliton!

Achieving capacity

Variable degree:

Check degree:

P(k)i =

2k(k − 1)

i(i − 1)(i − 2).

Not the soliton!

Achieving capacity

Theorem

If ω = 1/k, then the ensemble (2,P(k)) has rate equal to thecapacity of the rank metric channel and achieves vanishing errorprobability under message passing decoding.Further, it achieves vanishing error probability over the erasurechannel.

Achieving capacity

Theorem

If ω = 1/k, then the ensemble (2,P(k)) has rate equal to thecapacity of the rank metric channel and achieves vanishing errorprobability under message passing decoding.Further, it achieves vanishing error probability over the erasurechannel.

Achieving capacity

Encoding complexity O(m`2)

Decoding complexity O(m`2)

Error probability exp[−Ω(γiter)], exp(−Ω(m, `))

Realistic example (from Chou et al.):

` ≈ 50 , m ≈ 1700 (over F28)

Achieving capacity

Encoding complexity O(m`2)

Decoding complexity O(m`2)

Error probability exp[−Ω(γiter)], exp(−Ω(m, `))

Realistic example (from Chou et al.):

` ≈ 50 , m ≈ 1700 (over F28)

Achieving capacity

The magic

1. Fix number of iterations.

2. For large m, message Vi uniformly random conditional ondimension.

3. For large `, output dimension determined by input dimension.

4. For ω = 1/k, only dimension `ω or 0 is possible.

Achieving capacity

The magic

Achieving capacity

The magic

Achieving capacity

The magic

Achieving capacity

“For large `, output dimension determined by inputdimension”

d1∩2 ≈ max(d1 + d2 − damb, 0) .

Achieving capacity

“For large `, output dimension determined by inputdimension”

d1∩2 ≈ max(d1 + d2 − damb, 0) .

Achieving capacity

Simulations at small packet sizes

m = 42, ` = 136 (LP optimized degree sequence)

20 21 22 23 24 25 26 27

threshold

Achieving capacity

Conclusion 1

q-ary LDPC’s (Davey, MacKay/ Burshtein, Miller/ Rathi, Urbanke)Mixed outcomes

Exact density evolution for large q

Capacity achieving ensembles

Achieving capacity

Conclusion 1

q-ary LDPC’s (Davey, MacKay/ Burshtein, Miller/ Rathi, Urbanke)Mixed outcomes

Exact density evolution for large q

Capacity achieving ensembles

Achieving capacity

Conclusion 2

LDPC codes achieve capacity (under message passing decoding)over the erasure channel (Luby et al. 97) . . .

. . . and the rank metric channel

Achieving capacity

Conclusion 2

Achieving capacity

Conclusion 2

Iterative Coding for Network Coding · Montanari, Rathi, Urbanke Iterative Coding for Network...

Documents

Embracing Wireless Interference : Analog Network Coding · Analog Network Coding Analog Network Coding Supports wireless interference. Nodes are allowed to transmit packets simultaneously

Chapter 1 Network Coding Techniques for Wireless and ...ostovari/publications_files/Network Coding... · Chapter 1 Network Coding Techniques for Wireless and Sensor Networks 1Pouya

Part I: Network Part I: Network Coding Coding

Network Coding Testbed

Institute of Network Coding

Quantum Network Coding - Texas A&M Universitystudents.cs.tamu.edu/salah/academia/QNetCoding_presentation.pdf · Quantum Network Coding Summary Quantum Network Coding Salah A. Aly

Organization Introduction to Network Coding Practical Network Coding Secure Network Coding Structured File Sharing Conclusion

Network Coding Chapter 5

l Iterative Decoding of Superposition Codingliping/Research... · Iterative Decoding of Superposition Coding l Jun Tong and Li Ping ... Encoder of a superposition coding system. The

Iterative method for improvement of coding and decryption

Optimized Noisy Network Coding for Gaussian Relay Networksisl.stanford.edu/~abbas/papers/Optimized Noisy Network Coding for... · Optimized Noisy Network Coding for Gaussian Relay

Network Coding Ppt

About Network coding

Correlated Coding: Efﬁcient Network Coding under ...song/paper/ICNP14_Shuai.pdf · Correlated Coding: Efﬁcient Network Coding under Correlated Unreliable Wireless Links Shuai

Network Coding Introduction

Partial Network Coding

Network Coding Tomography for Network Failures

Iterative methods for network alignment

1 Network Coding Random Linear Network Coding Time Division Duplexing Introduction to Network Coding in Wireless Networks - The Random Linear Network Coding

Iterative Transformer Network for 3D Point Cloud