Asymptotic Analysis on Spatial Coupling Coding
for Two-Way Relay Channels
Satoshi Takabe∗ , Yuta Ishimatsu∗ , Tadashi Wadayama∗ , and Masahito Hayashi†‡
∗

arXiv:1801.06328v1 [] 19 Jan 2018

Department of Computer Science, Faculty of Engineering, Nagoya Institute of Technology
† Graduate School of Mathematics, Nagoya University
‡ Centre for Quantum Technologies, National University of Singapore
Email: s_takabe@nitech.ac.jp, 26115016@stn.nitech.ac.jp, wadayama@nitech.ac.jp, masahito@math.nagoya-u.ac.jp

Abstract—Compute-and-forward relaying is effective to increase bandwidth efficiency of wireless two-way relay channels.
In a compute-and-forward scheme, a relay tries to decode a
linear combination composed of transmitted messages from other
terminals or relays. Design for error correcting codes and its
decoding algorithms suitable for compute-and-forward relaying
schemes are still important issue to be studied. In this paper, we
will present an asymptotic performance analysis on LDPC codes
over two-way relay channels based on density evolution (DE).
Because of the asymmetric nature of the channel, we employ
the population dynamics DE combined with DE formulas for
asymmetric channels to obtain BP thresholds. In addition, we also
evaluate the asymptotic performance of spatially coupled LDPC
codes for two-way relay channels. The results indicate that the
spatial coupling codes yield improvements in the BP threshold
compared with corresponding uncoupled codes for two-way relay
channels.

I. I NTRODUCTION
A relaying with an appropriate signal processing and decoding are ubiquitous in wireless communications such as
satellite communications, mobile wireless communications,
and wireless local area networks. Increasing demand for band
width efficiency in wireless communications promotes spread
of research activities on relaying and forwarding techniques.
For example, theoretical limits of efficiencies of relaying
techniques such as decode-and-forward [1] and amplify-andforward [2] have been deeply studied. Recently, Nazar and
Gastpar presented a novel concept of compute-and-forward
relaying [3]. In a compute-and-forward scheme, a relay tries
to decode a linear combination composed of transmitted
messages from other terminals (or relays) and then the relay
forwards a decoded linear combination to another relay or a
terminal. That is, the repeater has no intention to decode each
message separately. The concept is also termed as wireless
network coding or physical layer network coding that has
invoked huge research interests [4], [5]. Recently, Sula [6]
et al. presented a practical decoding scheme for LDPC codes
in compute-forward multiple access (CFMA) systems. Ullah
et al. [7] derived the random coding error exponent for the
uplink phase of a two-way relay channel.
The simplest scenario for a compute-and-forward scheme
may be wireless two-way relay channels [8]. Two terminals
A, B and a relay R are involved in this channel. The terminal
A has own message and wishes to send it to the terminal B.

Similarly, the terminal B wishes to send own message to A.
There is no direct wireless connection between A and B, but
a relay R has bi-directional wireless connections to both of
A and B. When the relay R can decode a linear combination
successfully, it is broadcasted to A and B in the next time slot.
The terminals A and B can recover an intended message by
subtracting own message from the received message.
In order to obtain a highly reliable estimate of linear
combination at the relay, appropriate error correcting codes
should be exploited because the received signal is distorted by
additive noises. In such a case, the relay R intends to decode
a sum of two codewords sent from A and B. One possible
candidate of error correcting codes for such a situation is
low-density parity-check (LDPC) codes [9]. A combination of
LDPC codes and belief propagation (BP) decoding has been
proved to be very powerful and effective for additive noise
channels [10]. Sula et al. [6] discussed an appropriate modified
BP decoding for the two-way relay channels. They presented
a performance analysis on LDPC codes over two-way relay
channel based on computer simulations.
The goal of this work is twofold. The first goal is to provide
an asymptotic performance analysis for LDPC codes over twoway relay channels based on density evolution (DE). DE [11]
is a common theoretical tool to study the asymptotic typical
behavior of a BP decoder and it provides BP thresholds of
the target channel. Although the BP threshold is below the
Shannon limit, a BP threshold indicates a practical achievable rate with low complexity encoding and decoding. One
technical challenge for evaluating the BP threshold of twoway relay channels comes from an asymmetric nature of
the channel. That is, we cannot rely on the zero codeword
assumption commonly used in DE analysis for binary-input
memoryless output-symmetric channels [11]. In order to overcome this difficulty, we will employ population dynamics DE
[12] combined with the DE formula derived by Wang et al.
for asymmetric channels [13].
The second goal of this paper is to provide DE analysis for
spatially coupled LDPC (SC-LDPC) codes over two-way relay
channels. It is known that appropriately designed spatially
coupled codes yield improvements in BP thresholds compared
with those of uncoupled regular LDPC codes with comparable
parameters [14] [15]. In many cases, we can observe threshold

saturation [16], i.e., a phenomenon that the BP threshold
converges to the MAP threshold. The same is true for the
spatially coupling coding for two-way erasure multiple access
channels for a joint compute-and-forward scheme [17]. As
far as the authors know, typical behavior of BP decoding
of spatially coupled LDPC codes over the two-way relay
channels except for erasure ones is unknown. We consider
that it is worth studying not only from practical interests but
also from theoretical interests to provide an example of the
DE analysis for general asymmetric channels. In this work,
we will extend the population dynamics DE to protograph
codes [18] and perform numerical evaluations. Recent work
by Hayashi et al. [19] shows that efficient codes for two-way
relay channels are useful to establish secure communication
with untrusted relay. The spatially coupled codes presented
here can be regarded as promising candidates for such uses.
II. P RELIMINARIES
A. Problem setting

where w represents additive white Gaussian noise vector. A
decoder, possibly a BP decoder, tries to recover xA ⊕ xB
from the received word y. In this paper, we focus on decoding
methods for recovering xA ⊕ xB .
C. IID assumption-based belief propagation
Assume that two stochastic processes
(1)

(2)

(t)

(t)

Y (t) = µ(XA ) + µ(XB ) + W (t) ,

(1)

where W (t) is a zero mean Gaussian random variable with
variance σ 2 . The channel model (1) is justified under the
assumption such that perfect symbol/phase synchronization
and perfect power control are achieved at R. The relay R tries
(t)
(t)
to infer XA ⊕ XB from Y (t) as correct as possible, where
the operator ⊕ represents the addition over F2 . The channel
model is called the two-way relay channel in this paper.
If no error correcting code is used, then symbol by symbol
(t)
estimation can be applied. In the next phase, the estimate x̂A ⊕
(t)
(t)
x̂B is then broadcasted to A and B. If the estimate x̂A ⊕
(t)
(t)
(t)
x̂B equals to the true value xA ⊕ xB , then the terminal A
(t)
(t)
(t)
(t)
(resp. B) can retrieve xB (resp. xA ) from x̂A ⊕ x̂B . This
protocol, i.e., wireless network coding [4] [8], can be seen as
the simplest case of the compute-and-forward technique [3],
and it increases bandwidth efficiency of the two-way relay
channel.
B. LDPC coding
Let C ⊂ Fn2 be an LDPC code used in terminals A and B.
The terminals A and B independently select own codewords
xA = (xA,1 , . . . , xA,N ) ∈ C and xB = (xB,1 , . . . , xB,N ) ∈
C according to their own message. From the channel model
(1), the received word is given by
y = (µ(xA,1 ), . . . , µ(xA,N )) + (µ(xB,1 ), . . . , µ(xB,N )) + w,
(2)

(1)

(2)

(t)

(t)

(t)

are IID and that XA and XB are independent. For simplicity,
(t)
(t)
we here assume that Pr[XA = 1] = Pr[XB = 1] = 1/2
holds for any t. From these assumptions, we have the probability of events:
(t)

(t)

Pr[µ(XA ) + µ(XB ) = 0] =
(t)
Pr[µ(XA )
(t)
Pr[µ(XA )
(t)

The wireless channel model assumed here is described as
(t)
(t)
follows. Let XA (resp. XB ) be a binary random variable
where t = 1, 2, . . . represents time index. The binary-bipolar
conversion function µ : {0, 1} → {+1, −1}, µ(x) = 1 − 2x
(t)
(t)
is applied to XA and XB before their transmission. This
means that we assume binary phase shift keying (BPSK) as
modulation format. The terminals A and B then transmits the
(t)
(t)
modulated signals µ(XA ) and µ(XB ) to the air. The relay
R observes a received symbol

(t)

{XA , XA , . . . , XA , . . .}, {XB , XB , . . . , XB , . . .}

+
+

(t)
µ(XB )
(t)
µ(XB )

= −2]

= +2]

1/2,

(3)

1/4,

(4)

= 1/4.

(5)

=

(t)

Let Z (t) = XA ⊕XB . From the IID assumption, Z (t) is also
a memoryless stochastic process. We now consider a virtual
channel whose input and output symbols are Z (t) and Y (t) ,
respectively. It is evident that the prior probability of Z (t)
is given by Pr(Z (t) = 0) = Pr(Z (t) = 1) = 1/2. Under the
IID assumptions, the conditional PDF representing the channel
statistics of the virtual channel is given by
Pr[Y (t) = y|Z (t) = 1] = F (y; 0, σ 2 ),
(6)
1
1
Pr[Y (t) = y|Z (t) = 0] = F (y; −2, σ 2 )+ F (y; +2, σ 2 ),
2
2
where F (y; m, σ 2 ) is the Gaussian distribution with mean m
and variance σ 2 defined by


1
−(y − m)2
2
F (y; m, σ ) = √
exp
.
2σ 2
2πσ 2
From this conditional PDF, symbol log likelihood ratio
(LLR) can be easily derived:


Pr[Y (t) = y|Z (t) = 0]
2y
2
λ(t) (y) = ln
=
ln
cosh
− 2.
2
(t)
(t)
σ
σ
Pr[Y = y|Z = 1]
(7)
If the IID assumption is valid, we can make the best estimation
on Z (t) only from λ(t) . Note that this LLR expression is a
special case of the LLR expression derived by Sula et al. [6].
Let us go back to the argument on the case where terminals
A and B employ a binary linear code C. Due to linearity of
the code C, it is clear that (Z (1) , . . . , Z (n) ) also belongs to
C. From this fact, IID-assumption based maximum likelihood
(ML) decoding can be defined as
(ẑ1 , . . . , ẑn ) = arg

max

(z1 ,...,zn )∈C

n
Y
t=1

L(yt |zt ),

(8)

where the likelihood functions are defined by
L[y|1] = F (y; 0, σ 2 ),
1
1
L[y|0] = F (y; −2, σ 2 ) + F (y; +2, σ 2 ).
2
2

(9)

This ML rule is sub-optimal because the likelihood is based
on the IID assumption. Regardless of its sub-optimality, the
IID assumption makes the structure of a decoder simple, and it
also makes easier to exploit known channel coding techniques
developed for memoryless channels.
Belief propagation (BP) decoding for LDPC codes can be
regarded as an approximation of ML decoding as a message
passing form. It would be natural to develop a BP decoding
algorithm for the binary compute-and-forward channel based
on the IID assumption-based ML rule (8). It is not hard
to see that the IID assumption-based BP coincides with the
conventional log-domain BP algorithm [11] with symbol LLR
expression (7). This type of BP decoder has already discussed
in [6] [7]. A significant advantage of the IID assumption-based
BP is that it can be easily implemented based on a practical
BP decoder for the additive white Gaussian noise (AWGN)
channel just by replacing an LLR computation unit.
III. D ENSITY EVOLUTION FOR BINARY TWO - WAY RELAY
CHANNELS WITH IID ASSUMPTION
We employ DE to study BP thresholds of binary two-way
relay channels with the IID assumption. In this section, we
first introduce the population dynamics DE and estimate the
BP threshold for uncoupled regular LDPC codes. The BP
threshold for SC-LDPC is then evaluated.
A. Density evolution for asymmetric channels
For simplicity, we here focus on (dl , dr )-regular LDPC
codes, where dl and dr represent the variable and check
node degrees, respectively. Extension to irregular codes is
straightforward. It is noteworthy that we need to handle signal
dependent noises (6) for two-way relay channels with the IID
assumption. This means that we cannot rely on the zero code
assumption in a DE analysis. In the following, we follow the
Wang’s DE formulation [13] to overcome this difficulty.
The conditional PDF P (l) (m|z) (resp. Q(l) (m̂|z)) denote
the PDF of a message m from a variable node to a check
node (resp. m̂ from a check node to a variable node) with
transmitted word z at the l-th step. The distribution of LLR
of the virtual channel is denoted by P (0) (z). Note that those
PDFs depend on a transmitted word because of the asymmetric
nature of the channel. For symmetric channels, in contrast, the
zero code assumption omits the dependence. Let Γ(PA ) ,
PA ◦ γ −1 be a density transformation for a random variable
A with distribution PA [13] where γ : R → {0, 1} × [0, ∞),
γ(m) = (1m≤0 , ln coth |m/2|) with an indicator function 1{ · } .
The DE equations for binary asymmetric channels [13] are
given by

⊗(dl −1)
P (l) (m|z) = P (0) (z) ⊗ Q(l−1) (m̂|z
,
(10)
⊗(dr −1)
  (l)
P (m|0)+P (l) (m|1)
Q(l) (m̂|z) = Γ−1
Γ
2
  (l)
⊗(dr −1) !
P (m|0)−P (l) (m|1)
z
+(−1) Γ
,
2
(11)

where ⊗ denotes the convolution operator on PDFs. Although
these convolutions of PDFs can be efficiently evaluated with
fast Fourier transformation, numerical evaluation requires huge
computational costs. We use an alternative approach, population dynamics [12], to reduce computational complexity
because the DE analysis for SC-LDPC codes deals with a
number of DE equations simultaneously.
Equations (10) and (11) have equivalent forms called the
replica-symmetric cavity equations [12], which read
Z
Z dY
l −1
(l)
P (m|z) = dyL[y|z]
dm̂(s) Q(l−1) (m̂(s) |z)
s=1
dX
l −1

× δ m − λ(y) −
(l)

Q (m̂|z) =

1
2dr −2

Z

X
{z (s) }∈S

−1

×δ m̂−2 tanh

!
m̂

s=1
dY
r −1
s=1

(s)

,

(12)

dm(s) P (l) (m(s) |z (s) )

"d −1
r
Y
s=1


tanh

m(s)
2

#!
, (13)

wheren λ(y) denotes the LLR defined as the r.h.s. oof (7) and
Ldr (s)
S , {z (s) } ∈ {0, 1}dr ; s=1
z = 0, z (dr ) = z .
In Algorithm 1, we describe a procedure of the population
dynamics DE. In population dynamics, the PDFs P (·|z) and
Q(·|z) (z ∈ {0, 1}) are approximated to histograms (populations) of N samples denoted by, e.g., {νi0 } (i ∈ [N ] ,
{1, . . . , N }). The parameter N is called the population size
and the DE equations are exactly solved in the large-N limit.
Each sample is recursively updated by an update rule written
in a delta function δ(·) in (12) or (13). After each iteration
finishes, we can estimate bit error rate (BER) at the step.
Although the recursion should continue until every population
converges, it stops at the maximum iteration step T in practice.
We evaluate a BP threshold defined as a threshold of σ in
the virtual channel (6) below which LDPC codes are typically
decodable by a BP decoder. As a MAP threshold, we use the
symmetric information rate σsym (R) defined as a solution of
Csym (σsym (R)) = R for code rate R, where
Z ∞
Csym (σ) = −
P (y) log2 P (y)dy
−∞
Z
1
1 ∞
L[y|0] log2 L[y|0]dy + log2 (2πσ 2 e), (14)
+
2 −∞
4
denotes the symmetric information rate of the two-way relay
channel under the IID assumption and P (y) = (1/2)L[y|0]+
(1/2)L[y|1] is the PDF of a received symbol under the
assumption.
The BP thresholds of various regular LDPC ensembles
versus the code rate are shown in Fig. 1. We search BP
thresholds by evaluating BER using the population dynamics
DE with N = 105 and T = 2000. It is confirmed that the
estimation is accurate up to the third decimal place. The
BP thresholds have a gap to symmetric information rate as
predicted in [6].

(b)

(a)

Algorithm 1 Population Dynamics DE
Input: Population size N , Maximum iteration T
Output: Populations {νi0 }, {νi1 }, {ν̂i0 }, and {ν̂i1 } (i ∈ [N ])
1: Initialization: νi0 = νi1 = 0
2: for l = 1 to T do
3:
for z = 0 to 1 do
. Update of {ν̂iz } (Q(·|z))
4:
for i = 1 to N do
5:
Draw
 r − 1) uniformly in {0, 1}
Lz(1), . . . , z(d
dr −1
z(s)
= 0.
to satisfy z ⊕
s=1
6:
Draw i(1), . . . , i(d
in i
[N ].
hQr − 1) uniformly

z(s)
dr −1
7:
ν̂iz ← 2 tanh−1
tanh
ν
/2
.
s=1
i(s)
8:
end for
9:
end for
10:
for z = 0 to 1 do
. Update of {νiz } (P (·|z))
11:
for i = 0 to N do
12:
Draw y from L[y|z].
13:
Draw i(1), . . .P
, i(dl − 1) uniformly in [N ].
dl −1 z
14:
νiz ← λ(y) + s=1
ν̂i(s) .
15:
end for
16:
end for
17: end for

0.9
Symmetric information rate
BP threshold

0.8
(2, 8)

0.7

Fig. 2. A protograph of (a) (3, 6)-LDPC codes and (b) (3, 6, 5)-LDPC codes.

“distance” dˆl , e.g., (b) of Fig. 2 where (dl , dr , L) = (3, 6, 5).
As a result, we have L bundles of k variable nodes labeled
by i ∈ [L], and L + 2dˆl check nodes labeled by a ∈
{−dˆl + 1, . . . , L + dˆl }, where check nodes labeled from 1
to L are derived from original protographs. The design rate is
given by 1−(L+2dˆl )/(kL), which recovers that of uncoupled
codes as L → ∞.
Let us evaluate BP thresholds for SC-LDPC codes with
finite L. In a protograph, each variable and check nodes
respectively have a PDF P (l) (m|z) and Q(l) (m̂|z) of messages
as in the last subsection. Those PDFs are propagated as
messages on a protograph. From a symmetric structure in
(l)
each bundle, Pi→a (m|z) denotes the PDF of message m as
a message from a variable node in the i-th bundle to a check
node a at the l-th step. Similarly, let us denote the PDF of
message m̂ as a message from a check node a to a variable
(l)
node in the i-th bundle by Qa→i (m̂|z). DE equations of binary
two-way relay channels and (dl , dr , L)-LDPC codes then read
Z
Z
Y
(l)
(l−1)
Pi→a (m|z) = dyL[y|z]
dm̂b Qb→i (m̂b |z)
b∈N (i)\a

(3, 12)

Rate


(2, 6)

0.6

(3, 9)



×δ m − λ(y) −

(3, 7)
0.5
(2, 4)

(3, 6)

0.4
(3, 5)
0.3

0.4

0.5

0.6

0.7
BP threshold

1

(l)

Qa→i (m̂|z) =
0.8

0.9

1.0

Fig. 1. BP thresholds for regular LDPC code ensembles over two-way relay
channels versus code rate. The solid line represents the symmetric information
rate of the channel.

×

2dr −2

Y

k
Y

j∈N (a)\i

s=1



X

X

Z k−1
Y

(s)

{zj }∈S 0

We now turn to SC-LDPC codes. In this paper, we examine
the simplest (dl , dr , L)-LDPC codes with chain length L
where k = dr /dl and dˆl = (dl − 1)/2 are integers. The
protograph is then uniquely defined [20], which makes the
structure of the population dynamics DE relatively simple. The
DE analysis for general protograph codes is left as open here.
A protograph of (dl , kdl )-LDPC codes is represented by
k variable nodes and one check node, e.g., (a) of Fig. 2.
To construct a protograph of SC-LDPC codes, we prepare L
copies of the protograph of an uncoupled code and attach dˆl
check nodes to each side of copies. Edges of the protograph
are then assigned from a variable node to check nodes within

(15)

s=1

(s)

(l)

(s)

(s)

dmi Pi→a (mi |zi )
!

(s) (l)
(s) (s)
dmj Pj→a (mj |zj )



× δ m̂ − 2 tanh−1 

(s)

Y

(j,s)6=(i,k)

B. Spatial coupling coding for two-way relay channels

m̂b  ,

b∈N (i)\a

tanh

mj
2

!
 ,

(16)

where Nn
(·) is a set of neighboring nodes in a protograph
o
L
(s) s∈[k]
(s)
(k)
0
and S , {zj }j∈N (a) ∈ {0, 1}dr ; j,s zj = 0, zi = z . A
protograph of uncoupled LDPC codes recovers (12) and (13).
Population dynamics is implemented as an extension of
Algorithm 1. In this case, we prepare 4Ldl populations with
(l)
(l)
size N to approximate PDFs Pi→a (·|z) and Qa→i (·|z). Fig. 3
shows dynamics of BER of each variable node in (3, 6, 25)LDPC codes when N = 104 and σ = 0.78. It is apparent
that they decrease from each side of the chain, as observed in
the symmetric channel case [20]. BERs vanish after the 169th
step indicating that the code is decodable.
Fig. 4 shows the BP threshold of (3, 6, L)-LDPC codes
and symmetric information rate corresponding to the de-

100

Combining the population dynamics DE with DE formulas for
asymmetric channels, BP thresholds of regular LDPC codes
are evaluated. In addition, we provide the DE equations of
(dl , dr , L)-LDPC codes and performed the population dynamics DE. The results show that the spatial coupling coding
successfully improves the BP thresholds of two-way relay
channels.

10−5

BER

10−10
step: 9
10−15

step: 29

10−20

step: 109

step: 49

ACKNOWLEDGEMENT

step: 149
step: 159

−25

10

This work is supported by JSPS Grant-in-Aid for Scientific
Research (A) Grant Number 17H01280.

step: 162

10−30

5

10
15
Index of variable nodes l

20

25

Fig. 3. BER of each variable node in (3, 6, 25)-LDPC codes with several
DE steps evaluated by DE equations for a two-way-relay channel.
1.00
Spatially coupled LDPC
Symmetric information rate

0.95

Symmetric information rate (L → ∞)

uncoupled LDPC

BP threshold

0.90

0.85

0.80

0.75

0.70

5

10

15
Chain length L

20

25

Fig. 4. BP threshold for a two-way-relay channel and (3, 6, L)-LDPC codes
as a function of chain length L. The long-dashed line represents the BP
threshold of uncoupled (3, 6)-LDPC codes. The solid line and short-dashed
line respectively represent the symmetric information rate of the SC-LDPC
codes and its limiting value as L → ∞.

sign rate. In population dynamics, we use N = 105 and
T = 2000. The results indicate that the BP threshold is
monotonously decreasing as L increases. The limiting value
is estimated as 0.785 by extrapolation, which lies between
the BP threshold 0.742 of the uncoupled (3, 6)-LDPC codes
and the correspondent symmetric information rate 0.805. The
same is true for (3, 9, L)-LDPC codes: the spatially coupling
coding achieves 0.647 (L → ∞) while the BP threshold
and the symmetric information rate of uncoupled codes are
respectively given by 0.624 and 0.666. It is noteworthy that
our evaluation underestimates BP thresholds because T = 2000
is not sufficient in general. It is known that a BP decoder
for spatially coupled codes needs a large number of iterations
before convergence [20] especially around the threshold. These
facts suggest that the spatial coupling coding successfully
improves BP thresholds although whether it achieves the MAP
threshold or not is still left to open.
IV. S UMMARY
In this paper, asymptotic behavior of LDPC codes and
SC-LDPC codes for two-way relay channels are studied.

R EFERENCES
[1] T. M. Cover and A. El Gamal, “Capacity theorems for the relay channel,”
IEEE Trans. Inf. Theory, vol. 25, no. 5, pp. 572-584, Sep. 1979.
[2] J. N. Laneman, D. N. C.Tse, and G. W. Wornell, “Cooperative diversity
in wireless networks: Efficient protocols and outage behavior,” IEEE
Trans. Inf. Theory, vol. 50, no. 12, pp. 3062-3080, Dec. 2004.
[3] B. Nazer and M. Gastpar, “Compute-and-forward: harnessing interference through structured codes,” IEEE Trans. Inf. Theory, vol. 57, no.
10, pp. 6463-6486, Oct. 2011.
[4] S. Katti, H. Rahul, W. Hu, D. Katabi, M. Medard, and J. Crowcroft,
“XORs in the air: practical wireless network coding, ” IEEE/ACM Trans.
Networking, vol. 16, no. 3, pp. 497-510, Jun. 2008.
[5] S. Zhang, and S.-C. Liew, “Channel coding and decoding in a relay
system operated with physical-layer network coding, ” IEEE J. Select.
Areas in Commun., vol. 27, no. 5, pp. 788-796, Jun. 2009.
[6] E. Sula, J. Zhu, A. Pastore, S. H.Lim, and M. Gastpar, “Computeforward multiple access (CFMA) with nested LDPC codes, ” Proc. IEEE
Int. Symp. Inf. Theory, Aachen, Jun. 2017, pp. 2935-2939.
[7] S. S. Ullah, G. Liva, and S. C. Liew, “Physical-layer network coding:
a random coding error exponent perspective, ” Proc. IEEE Inf. Theory
Workshop, Kaohsiung, Nov. 2017.
[8] K. Narayanan, M. P. Wilson, and A. Sprintson, “Joint physical layer
coding and network coding for bi-directional relaying, ” Proc. 45th Ann.
Allerton Conf. Commun., Contr. Comput., Monticello, IL, Sep. 2007, pp.
5641-5654.
[9] R. G. Gallager, Low-Density Parity-Check Codes, MIT Press, 1963.
[10] D. J. C. MacKay, “Good error correcting codes based on very sparse
matrices, ” IEEE Trans. Inf. Theory, vol. 45, no. 2, pp. 399-431, Mar.
1999.
[11] T. Richardson and R. Urbanke, Modern Coding Theory, Cambridge
University Press, 2008.
[12] M. Mézard and A. Montanari, Information, Physics, and Computation,
Oxford University press, 2009.
[13] C.-C. Wang, S.R. Kulkarni, and H.V. Poor, “Density evolution for
asymmetric memoryless channels, ” IEEE Trans. Inf. Theory, vol. 51,
no. 12, pp. 4216-4236, Dec. 2005.
[14] A. J. Felstrom and K. S. Zigangirov, “Time-varying periodic convolutional codes with low-density parity-check matrix, ” IEEE Trans. Inf.
Theory, vol. 45, no. 5, pp. 2181-2190, Mar. 1999.
[15] M. Lentmaier, A. Sridharan, K. S. Zigangirov, and D. J. Costello, Jr.,
“Iterative decoding threshold analysis for LDPC convolutional codes, ”
IEEE Trans. Inf. Theory, vol. 56, no. 10, pp. 5274-5289, Oct. 2010.
[16] S. Kudekar, T. Richardson, and R. L. Urbanke, “Spatially coupled
ensembles universally achieve capacity under belief propagation, ” IEEE
Trans. Inf. Theory, vol. 59, no. 12, pp. 7761-7813, Dec. 2013.
[17] B. Hern and K. Narayanan, “Joint compute and forward for the two way
relay channel with spatially coupled LDPC codes,” 2012 IEEE Global
Commun. Conf., Anaheim, CA, 2012, pp. 2340-2345.
[18] J. Thorpe, K. Andrews, and S. Dolinar, “Methodologies for designing
LDPC codes using protographs and circulants, ” Proc. IEEE Int. Symp.
Inf. Theory, Chicago, Jun. 2004, p. 238.
[19] M. Hayashi, T. Wadayama, and A. Vazquez-Castro, “Secure communication via untrusted relay with linear codes,” Submitted to ISIT 2018.
[20] S. Kudekar, T. J. Richardson, and R. L. Urbanke, “Threshold saturation
via spatial coupling: Why convolutional LDPC ensembles perform so
well over the BEC,” IEEE Trans. Inf. Theory, vol. 57, no. 2, pp. 803-834,
Feb. 2011.

