Lattice Network Coding via Signal Codes

Chen Feng1 Danilo Silva2 Frank R. Kschischang1

1Department of Electrical and Computer EngineeringUniversity of Toronto, Canada

2Department of Electrical EngineeringFederal University of Santa Catarina (UFSC), Brazil

IEEE International Symposium on Information TheorySaint Petersburg, Russia, August 5, 2011

2011 IEEE International Symposium on Information Theory Lattice Network Coding via Signal Codes 1 / 27

Compute-and-ForwardNazer & Gastpar (2006)

w1

w2

2011 IEEE International Symposium on Information Theory Lattice Network Coding via Signal Codes 2 / 27

Compute-and-ForwardNazer & Gastpar (2006)

w1

w2

x1

x2

2011 IEEE International Symposium on Information Theory Lattice Network Coding via Signal Codes 2 / 27

Compute-and-ForwardNazer & Gastpar (2006)

y3 = h13x1 + h23x2 + z3

y4 = h14x1 + h24x2 + z4

w1

w2

x1

x2

2011 IEEE International Symposium on Information Theory Lattice Network Coding via Signal Codes 2 / 27

Compute-and-ForwardNazer & Gastpar (2006)

y3 = h13x1 + h23x2 + z3

y4 = h14x1 + h24x2 + z4

w4 = a14w1 + a24w2

w3 = a13w1 + a23w2w1

w2

2011 IEEE International Symposium on Information Theory Lattice Network Coding via Signal Codes 2 / 27

Compute-and-ForwardNazer & Gastpar (2006)

w4 = a14w1 + a24w2

w3 = a13w1 + a23w2w1

w2

x3

x4

w3

w4

2011 IEEE International Symposium on Information Theory Lattice Network Coding via Signal Codes 2 / 27

Compute-and-ForwardNazer & Gastpar (2006)

y5 = h35x3 + h45x4 + z5

w4 = a14w1 + a24w2

w3 = a13w1 + a23w2w1

w2

x3

x4

w3

w4

2011 IEEE International Symposium on Information Theory Lattice Network Coding via Signal Codes 2 / 27

Compute-and-ForwardNazer & Gastpar (2006)

y5 = h35x3 + h45x4 + z5

w5 = a35w3 + a45w4

w4 = a14w1 + a24w2

w3 = a13w1 + a23w2w1

w2

w3

w4

2011 IEEE International Symposium on Information Theory Lattice Network Coding via Signal Codes 2 / 27

Introduction

Lattice network coding

• Lattice-based compute-and-forward (Nazer & Gastpar, 2006)

• Linear physical-layer network coding

• R. Zamir, “Lattices are everywhere”

Related approaches• Narayanan-Wilson-Sprintson (2007)

• Nam-Chung-Lee (2008)

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Introduction

Nazer & Gastpar’s approach• Lattice partitions based on Erez-Zamir’s construction

• Main result: achievable rates for one-hop networks

• CSI only at the receivers but not at the transmitters

• However: asymptotically long block length and unboundedcomplexity

Our goal: practical codes for compute-and-forward• Finite block length and low complexity

• Example: wireless fading channel with short coherence time

Related workOrdentlich et al ISIT 2011, Hern & Narayanan ISIT 2011

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Our Previous Work

“An algebraic approach to physical-layer network coding” (ISIT 2010)• Lattice partition→ a module structure on the message space

• Fundamental theorem of finitely generated modules over a PID

• Generalized constructions over complex numbers

• Allows working with Eisenstein as well as Gaussian integers

“Design criteria for lattice network coding” (CISS 2011)

• Choice of receiver parameters (α, a)

• Shortest vector problem

• Upper bound on error probability for hypercube shaping

For more details:“An algebraic approach to physical-layer network coding” (submittedto IEEE Transactions on Information Theory, July 2011)

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This Work

Signal codes (Shalvi, Sommer & Feder 2003)

• Convolutional lattice codes

• Reasonably high coding gain

• Efficient encoding and decoding methods

• Relatively short packet length

• Issue: how to deal with the convolution tail

• Approach: send side information to reconstruct the tail

Contributions• Lattice partitions based on signal codes

• An efficient approach to transmit side information for multipleusers in a way that is compatible with lattice network coding

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Lattice Network Coding

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Lattice Network Coding

Key concepts

Fine lattice Λ, coarse lattice Λ′ ⊆ Λ, and lattice partition Λ/Λ′

GΛ =

[√3 1

0 2

]

Λ = {rGΛ : r ∈ Z2}

Λ′ = 3Λ

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Lattice Network Coding

Key concepts

Message space W (with |W | = |Λ/Λ′|)Labeling ϕ : Λ→W (consistent with Λ/Λ′)

Embedding map ϕ : W → Λ such that ϕ(ϕ(w)) = w

(1,1)

(1,0)

(0,1)

(0,0)

(1,2)

(0,2)

(2,2)

(2,0)

(2,1)W = Z3 × Z3

ϕ(wGΛ) = w mod 3

ϕ(w) = wGΛ

2011 IEEE International Symposium on Information Theory Lattice Network Coding via Signal Codes 9 / 27

Lattice Network Coding

Natural projection

For any discrete subring R ⊆ C such that Λ/Λ′ is an R-module, wecan make W into an R-module such that ϕ : Λ→W is a surjectiveR-module homomorphism with kernel Λ′.

(1,1)

(1,0)

(0,1)

(0,0)

(1,2)

(0,2)

(2,2)

(2,0)

(2,1)

R = Z

W = Z3 × Z3

ϕ(ϕ(2, 1) + ϕ(1, 0)

)

= (0, 1)

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Encoding and Decoding

GaussianMAC

w1 ! W

w2 ! W

wL ! WxL ! Cn

x2 ! Cn

x1 ! Cn

yu

u =L!

!=1

a!w!

......

y =L!

!=1

h!x! + z

Scaling D!

!y

! =L!

!=1

a!x!

Mapping!

(h1, . . . , hL)

(a1, . . . , aL)

Transmitter ` sends x` = ϕ(w`) + λ′`

Receiver computes u = ϕ(DΛ(αy))

Error Probability

Pr[error] = Pr[DΛ(neff) /∈ Λ′]

where neff ,∑

`(αh` − a`)x` + αz is the effective noise.

Remark: hypercube shaping =⇒ UBE based on lattice parameters

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Signal Codes

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Signal Codes: Convolutional Lattice CodesShalvi, Sommer & Feder (2003)

Generator matrix

Gk×(k+m)Λ =

1 g1 · · · gm 0 · · · 0 0

0 1 · · · gm−1 gm · · · 0 0...

... · · · ...... · · · ...

...0 0 · · · 0 0 · · · gm−1 gm

where gi ∈ C, for i = 1, . . . ,m.

Encoding and decoding operations• Encoding: convolution + shaping + termination

x = wGΛ + λ′ + d, where GΛ′ = πGΛ

• Decoding: sequential decoding

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Encoding

Convolution: wGΛ, or equivalently, (w1, . . . , wk) ∗ (1, g1, . . . , gm)

-1

0

1

2

3

0 1 2 3 4

Message Vector

-1

0

1

2

3

0 1 2 3 4

g(D)

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Encoding

Convolution: wGΛ, or equivalently, (w1, . . . , wk) ∗ (1, g1, . . . , gm)

-4

-2

0

2

4

6

8

0 1 2 3 4 5 6

Convolution

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Encoding

Shaping: wGΛ + λ′, where λ′ = w′GΛ′ = w′πGΛ and

w′i = −⌊(wi +

∑mj=1 gj(wi−j + πw′i−j))/π

⌉

-4

-2

0

2

4

6

8

0 1 2 3 4 5 6

Add −3Dg(D)

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Encoding

Shaping: wGΛ + λ′, where λ′ = w′GΛ′ = w′πGΛ and

w′i = −⌊(wi +

∑mj=1 gj(wi−j + πw′i−j))/π

⌉

-4

-2

0

2

4

6

8

0 1 2 3 4 5 6

Add −3D2g(D)

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Encoding

Shaping: wGΛ + λ′, where λ′ = w′GΛ′ = w′πGΛ and

w′i = −⌊(wi +

∑mj=1 gj(wi−j + πw′i−j))/π

⌉

-4

-2

0

2

4

6

8

0 1 2 3 4 5 6

Add 3D3g(D)

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Encoding

Shaping: wGΛ + λ′, where λ′ = w′GΛ′ = w′πGΛ and

w′i = −⌊(wi +

∑mj=1 gj(wi−j + πw′i−j))/π

⌉

-4

-2

0

2

4

6

8

0 1 2 3 4 5 6

Add −3D4g(D)

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Encoding

Shaping: wGΛ + λ′, where λ′ = w′GΛ′ = w′πGΛ and

w′i = −⌊(wi +

∑mj=1 gj(wi−j + πw′i−j))/π

⌉

-4

-2

0

2

4

6

8

0 1 2 3 4 5 6

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Encoding

Termination: wGΛ + λ′ + d, where d is a function of wGΛ + λ′

-4

-2

0

2

4

6

8

0 1 2 3 4 5 6

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Encoding

Termination: wGΛ + λ′ + d, where d is a function of wGΛ + λ′

-4

-2

0

2

4

6

8

0 1 2 3 4 5 6

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Lattice Network Coding viaSignal Codes

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Encoding and Decoding

Encoder

Transmitter ` sends x` = w`GΛ + λ′` + d`, where λ′` = w′`πGΛ

and d` is a function of w`GΛ + λ′`

Decoder

Receiver computes DΛ(αy −∑` a`d`)

Error Probability

Pr[error] = Pr[DΛ(neff) /∈ Λ′], where the effective noise neff is

neff ,∑

`(αh` − a`)x` + αz

Achieving coding gains

From (CISS 2011), coding gains of signal codes are achievable aslong as side information

∑` a`d` is available at the receiver

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Sending Side Information

Question

How shall transmitters send their side information d1, . . . ,dL?

Possible solutions• use side channels

• use a central coordinator

Our solutionuse the idea behind compute-and-forward

transmitter ` sends f(d`) concurrently

receiver only computes∑

` a`d`

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Methods for Sending Side Information

Recall that in our solution...

Transmitter ` sends f(d`), where d` ∈ Z[i]m

Receiver computes∑

` a`d`

Recall that in lattice network coding...

Transmitter ` sends x` = E(w`), where W = (Z[i]/(q))m

Receiver computes∑

` a`w`

Key observation

If q is large enough, then W is as good as Z[i]m for d`and the problem reduces to a lattice design problem

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Methods for Sending Side Information

A mathematical formulation for our design problem

Let Λq/Λ′q be a lattice partition with message space W

minimize dimension of Λq

subject to d2(Λq/Λ′q) ≥ γ

|q| ≥ 2θ

dimension of Λq corresponds to the extra channel uses;γ captures the reliability of decoding

∑` a`d`

θ captures the dynamic range of∑

` a`d`/π

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Overhead of Sending Side Information

Recall the design problem

minimize dimension of Λq

subject to d2(Λq/Λ′q) ≥ γ

|q| ≥ 2θ

We give a constructive method showing that

dimension of Λq . m√

1.57γ(|p1|+ |p2|+ . . .+ |ps|),where pi are Gaussian primes, and |p1p2 · · · ps| ≥ 2θ

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Overhead of Sending Side Information

A concrete example

For signal codes Λ and Λ′

• dimension: 2000

• parameters: {g1 = 1.96eiπ/8, g2 = 0.982eiπ/4}• Λ′ = 3Λ

• coding gain: 6.4 dB (over uncoded 9-QAM)

For the design problem

• γ = 1: corresponds to 9.5 dB protection

• θ = 20000: covers the dynamic range for two users

For our constructive method

• analysis: 90 extra channel uses

• simulation: 92 extra channel uses

Thus, overhead ≈ 5%

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Simulation Results

Setup• two-transmitter, single-receiver multiple-access

• Rayleigh faded channel gains

• receiver decodes a1w1 + a2w2 (a1, a2 6= 0)

• 9-QAM LNC: Λ/Λ′ = Z[i]2000/3Z[i]2000

• 9-QAM PNC: Zhang, Liew & Lam; Popovski & Yomo

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Simulation Results

25 30 35 40 45 5010−3

10−2

10−1

100

SNR [dB]

Fra

me−

Err

or R

ate

Nazer−GastparSignal−Code9−QAM LNC9−QAM PNC

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Conclusions

1. Potential Performance Gaincoding gains of signal codes can be achieved

but side information is required

2. Methods for Sending Side Informationa generic schemea lattice design problem

3. Overhead of Sending Side Informationan upper bounda concrete example

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Thank You!

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Backup Slides

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Decoding

Recall that...

Transmitter sends x = wGΛ + λ′ + d, where λ′ = w′πGΛ

and d is a function of wGΛ + λ′

Decoding

Receiver computes DΛ(y − d) = wGΛ + w′πGΛ +DΛ(z)

If DΛ(z) = 0, then the receiver obtains w + πw′

Note that [w + πw′] mod π = w mod π

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Advantages of Signal Codes

Coding gains

Recall that a signal code is “generated by” [1, g1, . . . , gm]

when m = 2, coding gain = 6.4dB

when m = 3, coding gain = 8.3dB

when m = 4, coding gain = 9.6dB

Achieving coding gains

coding gains of signal codes are achievable as long as sideinformation is available at the receiver

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