Lattice Network Coding via Signal Codes Chen Feng 1 Danilo Silva 2 Frank R. Kschischang 1 1 Department of Electrical and Computer Engineering University of Toronto, Canada 2 Department of Electrical Engineering Federal University of Santa Catarina (UFSC), Brazil IEEE International Symposium on Information Theory Saint Petersburg, Russia, August 5, 2011 2011 IEEE International Symposium on Information Theory Lattice Network Coding via Signal Codes 1 / 27
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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
Related approaches• Narayanan-Wilson-Sprintson (2007)
• Nam-Chung-Lee (2008)
2011 IEEE International Symposium on Information Theory Lattice Network Coding via Signal Codes 3 / 27
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
2011 IEEE International Symposium on Information Theory Lattice Network Coding via Signal Codes 4 / 27
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)
2011 IEEE International Symposium on Information Theory Lattice Network Coding via Signal Codes 5 / 27
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
2011 IEEE International Symposium on Information Theory Lattice Network Coding via Signal Codes 6 / 27
Lattice Network Coding
2011 IEEE International Symposium on Information Theory Lattice Network Coding via Signal Codes 7 / 27
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
2011 IEEE International Symposium on Information Theory Lattice Network Coding via Signal Codes 8 / 27
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)
2011 IEEE International Symposium on Information Theory Lattice Network Coding via Signal Codes 10 / 27
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
2011 IEEE International Symposium on Information Theory Lattice Network Coding via Signal Codes 11 / 27
Signal Codes
2011 IEEE International Symposium on Information Theory Lattice Network Coding via Signal Codes 12 / 27
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
2011 IEEE International Symposium on Information Theory Lattice Network Coding via Signal Codes 13 / 27