Physical-layer Network Coding via Low Density Lattice Codes

Yi Wang and Alister Burr

Abstract—We present a new PLNC scheme based on therecently developed compute-and-forward (C&F) paradigm andlow density lattice codes (LDLC). LDLC possesses high codinggain and good algebraic structure which is inherently suitablefor C&F. We also show that the ring-based constellation can beused to improve the average rate per dimension.

I. INTRODUCTION

Physical-layer network coding (PLNC) has been shown to

be very effective in improving the throughput of a two-way

relay channel (TWRC) by some research groups in 2006 (e.g.

Zhang et al. [1] and Popovski et al. [2]). The core idea is

that the intermediate relay attempts to infer and forward linear

combinations of the simultaneously received signals, instead of

decoding the transmitted signals individually. The destination

node is able to recover the original signals successfully pro-

vided that the number of such linear combinations is sufficient.

The remarkable potential of PLNC was further exploited by

Nazer and Gastpar [3] who proposed a new approach to PLNC,

namely compute-and-forward (C&F) which extends TWRC

to a more general network topology. In this novel scheme,

the transmitted signals at each source node are lattice points

in a multi-dimensional lattice over integers, and the relay

decodes an integer combination of these lattice points based

on the noisy observations, which is again a lattice point. Their

approach relies on Loeliger’s type A construction of random

lattice ensembles which requires both a large field size pand infinite sequences of lattices, making C&F essentially an

information-theoretic approach, and hence less straightforward

for practical implementation. In a recent work, Feng et al. [4]

formulated a general algebraic framework to generate a large

class of efficient and finite-dimensional lattice partitions. They

showed that the lattice partitions for C&F closely related to

the algebraic theory of finite modules over a principal ideal

domain (PID), and hence the resulting lattice structure allows

the coset leaders to be employed as C&F codewords inherently

and efficiently.

One possible solution for the C&F design problem is to

use existing linear channel codes to construct lattices based

on finite dimensional R-lattice partitions, where R is a PID

of C. For example, by employing complex construction A

over the ring of Eisenstein integers Z[ω], Narayanan et al. [5]

implement the C&F protocol using low density parity check

(LDPC) codes over Fp and Sun et al. [6] implement it using

convolutional, BCH and Reed-Solomon codes over GF(4).

Another solution for C&F design is to employ practical

high coding gain lattice codes. Constructions of lattices on

the basis of the classical low-dimensional lattices or channel

codes attracted research attention in the past, but there was

less progress toward designing lattice codes directly in the

Euclidean space. Recent works from Sommer et al. filled this

gap by developing practically decodable lattice codes (e.g.

The work described in this paper was supported by the European Commis-sion Framework 7 Programme under grant agreement 318177 (DIWINE).

signal codes [7]). These lattice codes have a special algebraic

structure and this inherent merit makes them easily recovered

from a linear system over the finite field in a multiple access

channel (MAC), and promotes their use as a suitable and

practical coding approach for PLNC under the C&F paradigm.

In [8] [9], the authors have shown that signal codes applied

in PLNC achieve a good network throughput as expected. In

comparison to the lattices constructed from channel codes,

LDLC incorporates channel coding and modulation in a single

entity, and also its codeword is inherently a low-density lattice

point, which greatly simplifies the design in the PLNC system.

An interesting alternative to signal codes is the low density

lattice codes (LDLC) [10], in which the lattice is generated by

the inverse of a sparse matrix. LDLC was shown to be capable

of achieving error-free decoding within 0.6dB of the Shannon

bound. Inspired by its high coding gain and the algebraic

property, we use a modified Gaussian mixture model for low-

complexity LDLC detection, and present a demonstrator for

the LDLC-based C&F scheme with performance comparison

for both non-fading and fading cases, which constitute the

main contributions of this paper. To the best of our knowledge,

LDLC-based PLNC has not been proposed before, partly

because efficient shaping and decoding operations were not

well understood [8]. Our work here shows the remarkable

potential of LDLC-based PLNC, and may motivate further

research toward the remaining unsolved problems.

A. Notation Definitions

Throughout this paper, we use R and Z to denote the fields

of real numbers and integer, respectively and Fq , q > 1, q ∈ Z

to denote the algebraic ring with size q (which is a finite field

when q = p) where p is a prime number. We also use boldface

lowercase and boldface uppercase to denote column vector and

matrices, respectively (e.g. h = [h1, · · · , hn]T ). We denote

h\j = [h1, · · · , hj−1, hj+1, hn] and⊙

for element-wise

multiplication. Fnq denotes an n-dimensional ring where the

ring size of the ith dimension i ∈ {1, 2, · · · , n} is determined

by qj , and similarly for Fnp. Functions g(·) : Fn

p → Zn and

gr(·) : Fnq → Zn are used to map between the finite field

and the corresponding subset of integers, {0, 1, · · · , p− 1}n,

and between the ring and the corresponding subset of integers,

{0, 1, · · · , q − 1}n, respectively.

II. LOW DENSITY LATTICE CODES

A. Lattice Definitions

Definition 1 (Lattice): An n-dimensional lattice Λ is defined

as a discrete subgroup of Euclidean space Rn such that if λ1,

λ2 ∈ Λ, we have a1λ1 + a2λ2 ∈ Λ for any a1, a2 ∈ Z. A

lattice can be written in the form of:

Λ = {λ = Gb : b ∈ Zn} (1)

where G ∈ Rn×n including basis of n linearly independent

vectors in Rn.

Definition 2 (Quantizer): A lattice quantizer, QΛ: Rn → Λ,

maps a point s in Rn to the nearest point in Λ in Euclidean

distance:

QΛ(s) = argminλ∈Λ

‖s− λ‖ (2)

Definition 3 (Voronoi Region): The Voronoi region of a

lattice point λ, denoted by V(λ), is the set of points in Rn

closest to this point, i.e.,

V(λ) = {s : s ∈ Rn,QΛ(s) = λ} (3)

The fundamental voronoi region is defined as V(0), or simply

represented as V . Let Vol(V) denote the volume of V and

Vol(V) = Vol(Λ).Definition 4 (Goodness of Lattices): Let z be an n-

dimensional i.i.d. Gaussian vector, z � N (0, σ2In×n), a

sequence of lattices Λ is Poltyrev-good if

Pr(z /∈ V) ≤ e−nEp(γ) (4)

where Ep(·) is the Poltyrev exponent and γ is the volumn-to-

noise ratio (VNR),

γ =Vol(Λ)

2n

2πeσ2. (5)

B. LDLC Encoding and Shaping

Following Definition 1, we define an n-dimensional LDLC

in terms of its non-singular lattice generator matrix G ∈ Rn×n

such that its parity check matrix H = G−1 is sparse. Let

b denote a sequence of integer message vectors with each

element bi ∈ {0, 1, · · · ,m−1}. The integer linear combination

of the independent columns of G forms the unshaped LDLC

codeword over Rn which can be seen as a subset of an

infinite lattice over Rn. An (n, d) LDLC can be constructed

by designing a sparse parity check matrix H having constant

row and column weight d. Let

h = [1, τ, τ, · · · , τ︸ ︷︷ ︸(d−1) τ in total

, 0, · · · , 0] (6)

be a row/column vector with d−1 τs (τ > 0) and τ is designed

to ensure full convergence [10]. H should be designed such

that each row and column is a permutation of h followed by a

random sign change, and of course H must be cycle-free.The

encoding operation x = Gb has a computational complexity

of O(n2) based on the designed H, but the iterative Jacobi

method can be used to obtain the codewords by solving a

sparse linear system H · x = b.

Compared with finite-alphabet codes, the LDLC codeword

x = Gb may have very large energy and hence an efficient

shaping scheme for LDLC is necessary to limit the average

power of x. Tomlinson-Harashima precoding scheme allows us

to restrict the LDLC codewords into a hypercube. Recall that

hypercube shaping for each dimension of the signal codes is

based on the monic causal filter coefficients and their generator

matrix has a Toeplitz structure, which is similar to lower-

triangular structure [7]. Thus, signal codes are naturally suit-

able for shaping and the constellation size of each dimension

remains the same. However, to perform hypercube shaping

[11] for LDLC, we must reconstruct the H matrix based

−20 −15 −10 −5 0 5 10 15 200

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

x

GM

R

Fig. 1. A Gaussian mixture (black dash-dot) comprised of N = 10 Gaussiancomponents (blue dashed lines) is approximated by another Gaussian mixture(red thick line) with Nmax = 3.

on h to have a lower triangular structure. Obviously, this

means that the H matrix no longer has constant degree-dnow. Hence the row degree should be 1 at the top row and

gradually increases until it reaches d (similarly for column

degree). Since codewords corresponding to the low-degree row

in H are less protected, the messages whose check equations

have weak protection capability should belong to a small size

constellation mi, i ∈ {1, 2, · · · , n}. The shaping operation

maps the message integer bi of the ith dimension into ai by

performing

ai = bi −miki (7)

where ki is an integer which should be computed such that

the shaped codewords |xs,i| ≤ mi

2 . since Hxs = a, we have

xs,i = bi −miki −i−1∑l=1

Hi,lxs,l (8)

Let ki = ��i�, and hence |�i| ≤ 12 + ki, then we have

|(�i − ki)mi| ≤ mi

2, mi > 0 (9)

Following the shaping condition, (�i − ki)mi = xs,i, and

combining (8) and (9), ki is obtained from,

ki = ��i� =⌊bi −

∑i−1l=1 Hi,lxs,l

mi

⌉(10)

Another method which works for hypercube and nested

lattice shaping is based on the QR decomposition [11] of an

arbitrary structure H, where H = JQ can be decomposed

such that J is a lower triangular matrix and Q is orthonormal.

In this case, J has a different form from the lower triangular

H matrix described above, and ki should be calculated as

ki = �ϑi� =⌊(

bi −∑i−1

l=1 Ji,lxs,l

)/mi

⌉, and xs,i is

xs,i =bi −miki −

∑i−1l=1 Ji,lxs,l

Ji,i(11)

The derivation of the approach is similar to (7)-(10), except

that |(ϑi − ki)mi/Ji,i| ≤ mi

2·Ji,i, and hence, we need to define

φ1

a1L1

x1

S1

b1

E1

φ2

a2L2

x2S2

E2

+

z

R

αy

β

g(·)

b2g(·)

L−1v

φ−1

QΛ(y)

g−1(·)

u =

⊕2

�=1

(q′

�� b�

)

E−1

D1

D2

Fig. 2. Encoder and Decoder for LDLC-based PLNC at MAC.

a nominal constellation size m such that mi = �Ji,im� for

each dimension of b, to guarantee that the codewords are

constrained in a hypercube. In the rest of the paper, we will

consider QR decomposition-based hypercube shaping.

C. Decoding

We assume the noisy LDLC codeword y = x + z where

z is i.i.d. Gaussian noise z ∼ N (0, σ2In×1) . A sequential

maximum likelihood (ML) detector b = argminb ‖y−Gb‖2

obviously has large computational costs. Hence an iterative

detection method employing the belief propagation algorithm

over the bipartite graph of H was proposed [10]. Unlike

LDPC, the metrics exchanged between the variable nodes

(VN) and check nodes (CN) are real continuous functions over

(∞,−∞) rather than scalar values. At the check node: the ith

CN calculates function ηj(ω) for the jth VN connected to

it, based on the functions μ\j(ω) transmitted from all VNs

(except the jth one) that are connected to this CN. ηj(ω)should be computed in terms of the check equations, where

all functions in μ\j(ω) needs to be expanded, convolved and

stretched, followed by a periodic extension with period 1

hi,j. At

the variable node: the ith VN calculates the joint probability

density function (PDF) for the jth CN connected to it, with

pj(ω) ∝1

√2πσ2

e−(yj−xj)

2

2σ2

(⊙η\j(ω)

)(12)

The final decision occurs at the variable node where the ith

VN estimates the intrinsic messages pfinal(x) using all CN

functions connected to it, and xi should correspond to the

peak value of xi = argmaxxipfinal(x).

This decoder has been implemented in [11] where a quanti-

zation method, using Fourier transforms to combine messages,

was used to tackle the problem of the continuous function.

However, the discrete Fourier transform (e.g. 1024 point quan-

tization) taken at CN needs large computational complexity

and storage. To consider a practically feasible decoder, the

authors in [12] [13] use Gaussian mixture distributions as

the messages exchanged between CN and VN. Thus the

convolution and multiplication operations at CN and VN can

be performed in the form of Gaussian mixtures and hence

the output is another Gaussian mixture. The core problem is

to limit the exponential growth in the number of Gaussians as

the iteration proceeds. The Gaussian mixture reduction (GMR)

algorithm approximates a mixture of N Gaussians by another

mixture of Nmax Gaussians, where Nmax < N . In [13], the

GMR algorithm compares the distance metric of all possible

pairs of single Gaussians in the input list, and replaces the

pair having the minimum metric by a single Gaussian using

the second moment matching method.

In this paper we use a modified GMR algorithm based

on [13]. The LDLC decoder used for C&F has greater com-

putational cost than a traditional point-to-point system since

the periodic extension occurs over possible integers Z which

has a larger cardinality |Z|. Our modified GMR algorithm

further reduces the computational complexities of the metric

comparisons, and makes our LDLC decoder more suitable for

the demanding work at the intermediate relay using PLNC.

Fig.1 shows an example of the result of the GMR algorithm.

III. LDLC IN PLNC VIA C&F

We consider here a TWRC network as shown in Fig.2 where

two source nodes S1 and S2 transmit information messages to

the destination nodes D1 and D2 via a single relay node R. We

are mainly concerned with the MAC phase since it dominates

the overall system performance compared to the broadcast

phase. The n-dimensional message vector at the node S�,

� ∈ {1, 2} is denoted as b� = {b�,1, b�,2, · · · , b�,n} b� ∈ Fnp.

Without loss of generality, we consider that the two source

nodes have the same message length n. Each message vector

b� will be encoded to an n-dimensional LDLC codeword by

the transmitter encoder, E� : Fnp → Rn,

E�(b�) = L

(φ(g(b�)

))(13)

where E� is equipped with the mapping function g(·) : Fnp →

Zn, a shaping operator defined in (7) - (11) and a LDLC

encoder L� : Zn → Rn. The transmitted codeword is subject

to the average power constraint,

E[‖ E�(b�) ‖

2]� nP (14)

where P = Ω(Λ) is the normalized second moment over the

voronoi region V of the generated low density lattices Λ. Relay

R observes the noisy superimposed signals

y = x1 + βx2 + z (15)

where β is the relative channel fading coefficient β =|β| exp(jθ) between the two MACs. θ is the channel phase

difference, and z ∼ N (0, σ2In).

The relay attempts to decode y to a linear combination u

over Fnp,

u = (q′1 � b1)⊕ (q′2 � b2) (16)

where � and ⊕ are referred to as element-wise multiplication

and element-wise addition over Fnp, and q′� ∈ Fn

p. The

relay first attempts to decode a lattice equation v which is

an integer combination of lattice codewords based on some

integer coefficients c�, v =∑

2

�=1c�x�. In order to estimate

the lattice equation v, the LDLC decoder quantizes αy to the

closest lattice points in Λ, where α is the optimal scaling factor

minimizing the effective noise variance. Following Definition

2, the LDLC decoder is a lattice quantizer with respect to the

low density lattices Λ,

v = QΛ(αy) (17)

Then, the relay maps v to u which is the estimate of the

ideal linear combination u. We derive that:

u = E−1(v) = g−1(φ−1

(G−1v

))(18)

= g−1

(φ−1

(G−1

2∑�=1

c�x�

))(19)

(a)= g−1

(φ−1

(2∑

�=1

c� [g(b�)−m�k�]

))(20)

(b)= g−1

([2∑

�=1

c�g(b�)

]mod m�

)(21)

(c)= g−1

[g

(2⊕

�=1

(q′� � b�

))]mod m� (22)

=

2⊕�=1

(q′� � b�

)(23)

where (a) follows from the LDLC encoding and shaping

operations defined in section II-B, (b) follows from the inverse

shaping, and (c) follows from the algebraic property of the

bijective function g(·), and q′� = g−1([c�] mod p).⊕

denotes

element-wise summation over Fnp.

Non-Fading Case: Thus, β = 1, and y = x1 + x2 + z. In

this case, the original message b�,i at the ith dimension is not

necessarily a value of the prime-sized finite field Fp. It is easy

to verify that the algebra ring (thus b� ∈ Fnq) will suffice to

validate the derivations of (18) - (23). Now, the estimate u of

the linear combination is over ring Fnq, thus,

u = b1 � b1 (24)

where � denotes addition over Fnq.

This in fact relaxes the constraints of the information mes-

sages, and hence increases the flexibility of the constellation

selection. As mentioned in section II-B, the constellation size

at each dimension should be carefully selected in terms of

the shaping parity check matrix to guarantee the optimised

trade-off between the average transmission rate per dimension

and the reliability of the lattice decoding. In other words, a

message vector from an n-dimensional ring provides a more

flexible way in the design of the constellation size to reduce

the rate loss. Note that the hypercube-shaped signal codes are

not sensitive to the constellation selection since the shaping is

directly implemented in its generator matrix which naturally

has a toeplitz form. We also notice that the authors in [8] use

signal codes for C&F at TWRC based on the non-fading case.

Fading case: Due to the effects of the fading coefficients,

the linear combination of the transmitted lattice codewords is

distorted. A good solution [3] is to scale the received signal

by an MMSE factor α and hence to maximize the possibility

of correctly quantizing an integer linear combination of the

LDLC codewords. Following (15), the scaled y is

y = αy

=

2∑�=1

c�x� +

2∑�=1

(αβ� − c�)x� + αz (25)

The optimal scaling factor α is selected based on the MMSE

criterion such that the mean squared norm ε of the effective

noise zeff �∑2

�=1(αβ� − c�)x� + αz is minimized. By

simply setting the first derivative to zero with respect to α,

the optimal scaling factor is given by:

αopt =βT c�P

‖ β ‖2 P + σ2(26)

where β = [1, β] and c� = [1, c�,2] in terms of (15). The

integer coefficient vector c� is determined by maximizing the

computational rate

R(β, c�) = maxc�∈Z

2log

2

(P

‖ αoptβ − c� ‖2 P + |α|2σ2

)

(27)

which is equivalent to the shortest vector problem: c� =argminc�∈Z

2,c� �=0

{cT� Dc�

}, where D is a positive definite

matrix derived from (27).In the fading case, we find that the message vector from a

ring b� ∈ Fnq cannot lead to equation (22), and hence a linear

combination in the form of (23) cannot be obtained. The infor-

mation message must be the finite field value in order to get the

linear combination of (23). Due to the special characteristic of

LDLC as mentioned above, it would be of interest to examine

the possibility of relaxing the constraint from the finite field to

ring for the original messages. Apparently the relay needs to

recover another form of linear combination. Following LDLC

quantizer output, the estimate of the linear combination can

be described as:

ur = E−1

r(v)

= G−1v mod |c�,2|m�, |c�,2| > 1 (28)

=

[2∑

�=1

c�φ(gr(b�)

)]

mod |c�,2|m� (29)

where gr(·) : Fnq → Zn. Thus, the estimate of the fine lattice Λ

is mapped to a linear combination over a ring with increased

cardinality |c�,2|m�. This provides protection for those code-

words that have been servely faded. At the destination nodes

D1 and D2, they attempt to recover the original messages

(assuming error-free transmission over the broadcast phase).

Node D2 has the knowledge of the information message b2

and recovers b1 using:

b1 =[[ur − c�,2gr(b2)] mod |c�,2|m�

]mod m� (30)

17 18 19 20 21 22 23 24 2510−5

10−4

10−3

10−2

10−1

SNR [dB]

SE

R

Capacity of Hypercube Shapingβ=1, (1000,7) LDLC, 50 Iterationsβ=1.9, (1000,7) LDLC, 50 Iterationsβ=1, (100,5) LDLC, 10 Iterations

2.9 bits/dimension

Fig. 3. SER performance of the LDLC-based PLNC scheme.

Node D1 has the knowledge of b1 and attempts to recover b2

using:

b2 =

[ur −

([φ(gr(b1))] mod c�,2m�

)]mod |c�,2|m�

|c�,2|(31)

Note that node D1 needs to reshape gr(b1) for the successful

recovery of b2.

IV. SIMULATION RESULTS AND DISCUSSIONS

In this section we show and analyse the performance of

our LDLC-based PLNC scheme. Simulations were carried

out for the ring-type constellation b� ∈ Fnq where the con-

stellation size qi at the ith dimension is determined by the

shaping parity check matrix aforementioned in section II-B.

We set the nominal constellation size m = 8 and hence

qi = mi = �|Ji,im|� which gives an average transmission rate

of R = 2.90 bits/dimension. The symbol-to-noise ratio (SNR)

is defined as SNR = E[‖xi‖2]

σ2 . Fig.3 shows the symbol error

rate (SER) performance for non-fading and fading cases. In the

non-fading case, we observe that there is a loss around 1.7 dB

from the hypercube shaping capacity (HSC) at SER = 10−5

for (1000, 7) LDLC with 50 iterations, and around 4.1 dB

at SER = 10−5 for (100, 5) LDLC with 10 iterations. Note

that the number of iterations increases for (1000, 7) LDLC

to ensure full convergence at the BP decoder, and hence its

complexity is higher than (100, 5) LDLC. Since the SER

performance of LDLC improves as the length n increases, we

expect that the loss could be further reduced by 0.6 ∼ 0.7 dB

for (104, 7) LDLC. In the fading case, we set a fixed relative

fading coefficient β = 1.9 for the simulation. Fig.3 shows that

the gap to capacity at SER = 10−5 is 2.9 dB for (1000, 7)LDLC with 50 iterations. Thus a further 1.2 dB is required in

fading.

From Definition 4, we state that the necessary condition for

reliable lattice decoding at relay is:

γ =Vol(λ∑

2�=1 c�x�

)2n

2πeσ2zeff

> 1 (32)

where Vol(λ∑2�=1 c�x�

)2n is the volume of the Voronoi region

of the LDLC lattice point v =∑

2

�=1c�x�, and σ2

zeffis

the variance of the effective noise. This ensures the error

probability

Pr(u = u) = Pr(zeff /∈ V(λ∑2

�=1 c�x�

)(33)

goes to zero exponentially as n → ∞. When the LDLC

codewords are faded by non-integers, the probability that the

effective noise leaves the Voronoi region of V

(λ∑2

�=1 c�x�

)

increases. This requires a smaller receiver noise variance σ2,

and hence a larger SNR, to fulfil the condition of (33).

V. CONCLUSIONS

We have demonstrated that LDLC is a class of good lattice

codes suitable for PLNC via C&F. We proved that finite field

messages at each dimension works well, but also showed the

feasibility using ring-based constellation for the linear system

recovery. This effectively reduces the rate loss of hypercube-

shaped LDLC. Simulations were carried out for the ring-type

constellation, and showed good SER performance for both

non-fading and fading case, using a modified GMR model.

Our work in the future includes the extension of the proposed

PLNC scheme to a more general network topology.

VI. ACKNOWLEDGEMENTS

The authors would like to thank Chen Feng from University

of Toronto for useful discussions on lattice network coding.

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