Coding Schemes for an Erasure Relay Channel

Abstract

This paper considers a simple network consisting of a source, a destination, and a relay. In this model, the source-relay and relay-destination links are lossless, while the source-destination link is subject to erasures. Four coding schemes for reliably conveying k symbols from the source to the destination are described. Three of these techniques are adapted directly from well-known point-to-point coding schemes - viz., the use of maximum-distance separable (MDS) codes and Luby Transform (LT) codes. The fourth approach is a new technique using uncoded transmission from the source in conjunction with a relay that transmits a sequence with this property: When the destination subtracts the effects of the unerased symbols from the sequence, what remains is an "LT-like" code for the erased symbols - and this property holds regardless of which symbols were erased on the source-destination link. The four approaches are compared in terms of their complexity and performance.

Full text

Coding Schemes for an Erasure Relay Channel
Srinath Puducheri, J¨org Kliewer, and Thomas E. Fuja
Department of Electrical Engineering,
University of Notre Dame, Notre Dame, IN 46556, USA
Email: {spuduche, jkliewer, tfuja}@nd.edu
Abstract— This paper considers a simple network consisting
of a source, a destination, and a relay. In this model, the source-
relay and relay-destination links are lossless, while the source-
destination link is subject to erasures. Four coding schemes for
reliably conveying ksymbols from the source to the destination
are described. Three of these techniques are adapted directly
from well-known point-to-point coding schemes - viz., the use of
maximum-distance separable (MDS) codes and Luby Transform
(LT) codes. The fourth approach is a new technique using
uncoded transmission from the source in conjunction with a
relay that transmits a sequence with this property: When the
destination subtracts the effects of the unerased symbols from
the sequence, what remains is an “LT-like” code for the erased
symbols - and this property holds regardless of which symbols
were erased on the source-destination link. The four approaches
are compared in terms of their complexity and performance.
I. INTRODUCTION
Wireless relay networks modelled as erasure relay channels
have been well studied in the literature. For example, in [1]
and [2], the capacities of certain relay channels are derived,
and practical coding schemes based on maximum-distance
separable (MDS) codes are given. In [3] a max-flow min-cut
capacity result is obtained for a particular class of interference-
free wireless erasure networks assuming the decoder has
perfect knowledge about the erasure pattern.
This paper considers a three-node relay network comprised
of a source s, a destination dand a relay r. (See Figure 1.) The
source-relay and source-destination links together constitute
a (physically degraded) broadcast channel, while the relay-
destination link is a point-to-point channel. We assume time-
slotted packetized transmissions (of fixed duration/size) at
both the source and relay. Moreover, the source and relay
transmissions are assumed to occur in different time slots –
i.e., a time-division multiplexing (TDM) strategy is used to
control interference between the source and relay transmis-
sions. Finally, the source-relay and relay-destination links are
assumed to be lossless, while the source-destination channel
is modelled as an erasure channel in which each symbol
(or packet) is independently erased with probability ε. In
the presence of physical-layer error-correction, this is a good
higher-layer model for the case when the source-relay and
relay-destination distances are much smaller than the source-
destination distance.
The goal is to communicate kinformation symbols from
the source to the destination in no more than k·(1 + ∆) time
This work was supported in part by the Center for Applied Mathematics,
University of Notre Dame. It was also supported by NSF grant CCF 02-05310,
by German Research Foundation grant KL 1080/3-1, and by the University
of Notre Dame Faculty Research Program.
s
r
d
00
ε
C1C2
Fig. 1. A three-node erasure relay network
slots, where ∆is close to (but necessarily larger than) ǫ. The
feasibility of this task is established in Section II. Then, in
Section III, we discuss two possible approaches – one based
on MDS codes, and the other based on LT codes [4]. The
LT-based scheme requires encoding at the source, whereas
the MDS-based scheme does not – i.e., the MDS scheme
is systematic. However, the complexity of the MDS scheme
suffers in comparison to that of the LT-based approach.
Motivated by this observation, we propose two systematic
schemes based on LT codes. The first approach adapts sys-
tematic LT codes as developed by Shokrollahi [5]. The second
technique encodes the symbols at the relay in such a way that
the effect of the unerased symbols can be subtracted off at the
destination – and what remains resembles an LT sequence for
the erased symbols, regardless of which symbols were erased.
These new schemes are analyzed in Sections III-C and III-D,
respectively.
II. A CUT-SET BOUND
The network in Fig. 1 is a particular kind of degraded
erasure relay channel – one in which the destination receives
only the source symbols that are also received by the relay. The
capacity of the general degraded relay channel was derived in
[6]. This section develops an achievable cut-set bound ([7])
for the channel under consideration. Similar bounds for more
general erasure relay channels are in [1], [2].
Let θbe the fraction of time for which the source is
transmitting. Then, across the cut C1(broadcast cut) in
Figure 1, the maximum rate of information transfer is bounded
by θsymbols/time slot. This follows from the fact that every
symbol received by the destination is also received by the
relay, and the capacity of the ideal source-relay channel, when
active, is just one symbol/time slot. Similarly, across the cut
C2(multiple-access cut), the maximum rate of information

transfer is bounded by (1−θ)+θ(1−ε) = 1−θε. Consequently,
the maximum achievable end-to-end rate Ris bounded as
R≤max
θ{min(θ, 1−θε)}(1)
=1
1 + εsymbols / time-slot (2)
The maximizing value of θoccurs when the two terms
inside min(·,·)are equal and is hence given by θ∗= 1/(1+ε).
This bound is achievable, and a coding scheme demonstrating
this is discussed in the next section.
Note that, without the relay, the capacity is 1−ε < 1
1+ε.
Further, a capacity of 1
1+εimplies that at least k(1 + ε)time
slots are needed to convey ksymbols of information, and so
it is imperative that the value of ∆for any scheme satisfies
∆≥ε. Finally, ∆≥1allows routing of all the information
through the relay, thus trivially solving the problem. Hence,
the range of interest for ∆is ε≤∆<1.
III. FOUR CODING SCHEMES
Throughout this section, we assume that if nsymbols are
transmitted by the source, then exactly n(1 −ε)of those
symbols are successfully received at the destination. Of course,
this is not accurate; rather, the law of large numbers can be
invoked to guarantee that the fraction of symbols successfully
received is arbitrarily close to (1−ε). However, assuming that
the fraction is exactly (1 −ε)does not substantively change
the coding techniques described, and it simplifies notation.
Since the source-relay and relay-destination links are loss-
less, the optimal TDM strategy is for the source to first
complete its entire transmission and thereby convey all of its
information to the relay, before the relay begins transmitting
to the sink.
Finally, we refer to a scheme in which the source transmits
uncoded information symbols as a systematic scheme.
A. MDS-based scheme
We first consider a simple coding scheme that demonstrates
the achievability of the cut-set bound. Assume that the symbols
to be transmitted are drawn from a finite field. Then the
scheme is as follows:
1) The source first transmits kuncoded information sym-
bols. The relay successfully receives all the symbols,
whereas the destination receives k(1 −ε)of them.
2) The relay now encodes the kinformation symbols into
kε code symbols such that any set of kε information
symbols erased on the source-destination link can be
recovered from these code symbols. In other words, the
k×kε generator matrix Gused by the relay must
have the property that every set of kε rows are linearly
independent. This implies that GTmust be the generator
matrix of a (k, k ε)maximum-distance separable (MDS)
code - e.g., a Reed-Solomon code.
The total number of transmissions in this scheme is k(1+ε),
exactly what is indicated by the cut-set bound. Since this
scheme does not involve any encoding at the source, it is a
systematic scheme. Consequently, all this scheme’s computa-
tional complexity is localized to the relay and the destination.
To estimate complexity, we assume standard Reed-Solomon
encoder/decoder implementations. This means that encoding
requires O(k2ε)symbol operations; this is also the total com-
plexity at the relay. For decoding at the destination, the contri-
bution of the k(1 −ε)known symbols is first subtracted from
the kε symbols sent by the relay – this requires O(k2ε(1 −ε))
operations; then, we have kε linear equations in kε unknowns
and these can be solved with quadratic complexity O(k2ε2)
(as these equations involve a Vandermonde matrix)1. Thus, the
total complexity at the destination is O(k2ε).
Finally, note that the overall code describing symbols sent
from both the source and the relay – the code with generator
[Ik×k|G]– is not necessarily MDS, i.e., it need not have
the property that every k×ksub-matrix is invertible. So this
scheme explicitly exploits the assumption that all erasures will
occur on the source-to-destination link.
B. LT-based scheme
We now consider a scheme that is non-systematic, i.e., in
which the source transmits encoded symbols.
The codes we consider here are Luby Transform (LT) codes
[4]. LT codes have the property that slightly more than k
code symbols are needed to recover kinformation symbols –
unlike MDS codes which require exactly kcode symbols. The
symbols are binary strings and symbol operations consist of
bit-wise XORs. The ith code symbol is generated by XORing
dirandomly chosen information symbols. The integer di,
called the degree of the code symbol, is itself chosen randomly
according to a degree distribution – the choice of which is
crucial to the performance of the code. The degree distribution
µ(i),1≤i≤k, used in [4] is the robust soliton distribution
(RSD), defined as follows:
µ(i) = ρ(i) + τ(i)
β,for 1≤i≤k, (3)
with:
β=
k
X
i=1 ρ(i) + τ(i),(4)
ρ(i) = 1/k, for i= 1,
1/(i(i−1)),for 2≤i≤k, (5)
τ(i) = 


S/ik, for 1≤i≤k
S−1,
Slog(S/δ)/k, for i=k
S,
0,otherwise, (6)
S=c√klog k
δ,(7)
for constants c > 0and δ∈[0,1]. It is shown in [4] that
k+O(√k·log2(k/δ)) LT code symbols suffice to recover k
information symbols with probability at least 1−δ. Further-
more, the average encoding complexity per code symbol is
O(log(k/δ)), and the average complexity for decoding all the
information is O(k·log(k/δ)).
1There are asymptotically faster algorithms for this class of MDS codes;
however, in [4] and [5] it is observed that these are often in practice more
complicated to implement than the quadratic complexity schemes.

We now describe a coding scheme for our relay network
based on LT codes:
1) The source transmits ns=k+c√klog2(k/δ)LT code
symbols constructed from the kinformation symbols; of
these the destination receives ns(1 −ε), while the relay
receives all nssymbols.
2) From the received symbols, the relay decodes the in-
formation and re-encodes it into nr=nsεLT code
symbols, which are transmitted.
3) The destination decodes the information from the nsLT
code symbols received from both the source and relay.
The relay fails to decode with probability at most δ;
conditioned on successful decoding at the relay, the destination
fails with probability at most δ. Thus, the overall failure
probability is at most 2δ−δ2. Further, the average complexity
at the source is O(nslog(k/δ)) = O(klog(k/δ)). The average
complexity at the relay (decoding + encoding) is given by
O((1 + ε)·klog(k/δ)). Similarly, the average complexity at
the destination is O(klog(k/δ)).
Another important aspect of this coding scheme is the
overhead – defined as the number of transmissions required
in excess of the theoretical minimum (k(1 + ε)) indicated
by the cut-set bound. It can be verified that the overhead
of this scheme is given by c(1 + ε)√klog2(k/δ). However,
since the overhead is of the form o(k), for large k, the
normalized overhead (= overhead/k) can be made smaller than
its maximum permitted value ∆−ε.
Despite their overhead and non-zero failure probability, LT
codes may be preferred over MDS codes owing to their ease of
implementation, especially since they operate over the binary
field while the field size needs to scale as the blocklength
for Reed-Solomon codes. Also, for a given blocklength, the
complexity of encoding and decoding LT codes is significantly
lower than the standard quadratic complexity of MDS codes.
To illustrate the performance of the LT-based scheme,
Figure 2 shows the total number of source/relay transmissions
required for successful delivery of kinformation symbols.
Specifically, these results indicate that, to convey k= 2000
information symbols with a source-to-destination erasure prob-
ability of ǫ= 0.25, an average (mean) of about 2750 trans-
missions are required – which results in an average overhead
of about 250 symbols.
A naive systematic scheme based on LT codes could be
implemented by transmitting uncoded symbols from the source
and then constructing the relay’s sequence as before - i.e., by
taking randomly chosen linear combinations of the data, with
the degree of each relay-transmitted symbol selected according
to the RSD. However,this approach performs quite poorly. For
the parameters considered above (k= 2000 and ǫ= 0.25),
about 3280 transmitted symbols are required (on average) to
recover all the source symbols, much worse than the 2750
required with non-systematic LT encoding. This motivates the
use of more sophisticated systematic schemes based on LT
codes, provided in the next two sections.
2500 2700 2900 3100 3300 3500
0
0.02
0.04
0.06
0.08
Total number of transmissions required
Fraction of 50,000 simulations
Fig. 2. Histogram of the number of source and relay LT symbols required
to decode. Parameters: k= 2000,ε= 0.25.
C. Systematic scheme based on LT codes and pre-processing
at the relay
We now describe a scheme based on a less naive approach
to systematic LT coding – one proposed by Shokrollahi in [5].
The scheme is as follows:
1) The source transmits kuncoded symbols. Denote these
symbols by the 1×kvector u.
2) The relay performs a decode-and-reencode operation on
the received vector uas follows:
•An invertible k×kbinary matrix Gis precomputed
at the relay: First, k+o(k)vectors (each of length
k) are generated by sampling the RSD k+o(k)
times and then randomly selecting from the k-
tuples of appropriate weight. With high probability,
these vectors contain a linearly independent set of
size k, which may be identified using the belief-
propagation algorithm. The identified vectors are the
columns of G.
•Given the ksymbols from the source, the relay
generates a 1×kvector vsuch that u=v·G.
This can be done without inverting Gby merely
using the BP algorithm to solve for vfrom u, as
the columns of Gcan be permuted to obtain an
upper-triangular matrix [5].
•The result is that the vector u“looks like” an LT
code formed from v. The relay now generates and
transmits an additional kε +o(k)LT code symbols
derived from v.
3) The sink receives a total of k+o(k)code symbols from
which it attempts to decode v. From v, the sink can
recover the kε erased symbols of u, using the relation
u=v·G.
In practice, the performance of this code can be seen in
Figure 3, a histogram of the number of relay symbols required
for successful decoding when k= 2000 and ǫ= 0.25.
Figure 3 indicates than an average of 650 relay symbols
are required under these circumstances. This translates to an
average overhead of about 150 symbols.
In terms of complexity, this scheme does not involve any
computations at the source. At the relay, the complexity lies

500 700 900 1100 1300 1500
0
0.02
0.04
0.06
0.08
0.1
0.12
Number of relay symbols needed to decode
Fraction of 50,000 simulations
Fig. 3. Histogram of the number of relay symbols required for the scheme
based on systematic LT encoding [5]; parameters: k= 2000,ε= 0.25.
in generating v(given G) and kε LT symbols – and is
O(k(1+ε) log(k)). Likewise, the complexity at the destination
is O(k(1 + ε) log(k)). (The dependence on δis omitted.) The
computation of Gis a one-time operation and is hence not
considered here.
D. A new systematic scheme
This section describes a novel approach to the problem – a
systematic scheme in which the relay transmits an “LT-like”
sequence that the destination uses to recover the erased sym-
bols. This approach uses a different (non-RSD) distribution to
generate the code symbol degrees at the relay, but it does so
in such a way that the “target” distribution – the distribution
of the code symbol degrees over the erased symbols – has
properties similar to the RSD.
Assume that the source transmits kuncoded symbols and
the relay transmits m=k∆code symbols. Ideally, the m
code symbols would constitute an LT sequence for exactly the
kε symbols that were erased on the source-destination link,
so the destination could recover the lost symbols with high
probability for monly slightly larger than kε. However, this
is impossible, as the relay does not know what symbols were
erased on the source-destination channel.
An alternative is to generate mcoded symbols at the relay in
such a way that after subtracting the contribution of the known
symbols at the destination, the mcode symbols constitute an
LT code over the kε unknown symbols. Assume the relay
follows the LT encoding procedure but uses a different degree
distribution p(i),1≤i≤k. Then, ideally, we would want the
corresponding degree distribution viewed over any set of kε
information symbols to be the RSD.
Without loss of generality, assume that the first ℓ=kε
symbols transmitted on the source-destination channel are
erased. Now consider a code symbol of degree igenerated
by the relay, with ichosen according to p(i). Let the reduced
degree jbe the number of symbols in the first ℓpositions
that contribute to i. Letting Iand Jdenote the corresponding
random variables, we have:
P r(J=j|I=i) = (ℓ
j)(k−ℓ
i−j)
(k
i),(8)
for 0≤j≤ℓ, j ≤i≤(k−ℓ+j). Then, the reduced degree
distribution is given by:
pr(j) = P r(J=j)(9)
=
k
X
i=1
P r(J=j|I=i)·p(i).(10)
So pr(·)is the distribution on the number of information
symbols (among the first ℓsuch symbols) that contribute to a
particular code symbol. Ideally, we would like to choose p(i)
such that pr(j)is the RSD µ(j)for 1≤j≤l. However, this
is clearly impossible since pr(0) 6= 0. Instead we consider the
class of distributions p(i)such that, when pr(·)is computed
using (10), it satisfies
pr(j)≥1
α·µ(j),1≤j≤ℓ, (11)
for some α > 1. If (11) is satisfied, then, by following
Chernoff bound-type arguments, with high probability the
number of symbols required to decode the information is at
most αtimes the number required if the RSD had been used
to select the degrees. The main challenge is to obtain a value
of αthat is as close to one as possible to minimize overhead.
For this purpose, we focus on a choice of p(i)with support
over only those degrees of the form i=j/ε for 1≤j≤ℓ.
Using Stirling’s approximation n!≈nne−n√2πn in (8), then
P rJ=j|I=j
ε≥1
√2πj .(12)
Hence:
pr(j)> P rJ=j|I=j
ε·pj
ε(13)
>1
√2πj ·pj
ε.(14)
Note from (3)-(6) that the RSD µ(j)is of the form:
µ(j) = 


1
βℓ1
j(j−1) +1
sℓj,2≤j < sℓ,
1
βℓ·1
j(j−1) , sℓ< j ≤ℓ,
(15)
where βℓand sℓare constants that depend on ℓ. Specifically
the degree sℓis given by
sℓ=√ℓ
clog(ℓ/δ),(16)
and βℓ>1.
The degrees 1and sℓhave been omitted in (15) and also
in the following development; they are dealt with separately.
Consequently, if we choose p(i)to be of the form
p(i) = 






1
αℓβℓ1
√j(j−1) +1
sℓ√j, i =j
ε,2≤j < sℓ,
1
αℓβℓ·1
√j(j−1) , i =j
ε, sℓ< j ≤ℓ,
0,otherwise.(17)

500 700 900 1100 1300 1500
0
0.02
0.04
0.06
0.08
Number of relay symbols needed to decode
Fraction of 50,000 simulations
Fig. 4. Histogram of the number of relay symbols required for the new
systematic scheme; parameters: k= 2000,ε= 0.25.
where αℓis a normalization constant, then from (14), it is
easily seen that (11) is satisfied for all j(except 1, sℓ) with
α≤αℓ√2π. (18)
We now obtain an upper bound for αℓ. Note that
αℓ<1
βℓ
∞
X
j=2
1
√j(j−1) +1
βℓsℓ
sℓ
X
j=2
1
√j(19)
<
∞
X
j=2
1
√j(j−1) +1
sℓ
sℓ
X
j=2
1
√j(20)
since 1/βℓ<1. Further, clearly 1/(√j(j−1)) ∼j−3/2whose
infinite sum converges. Also, note that
1
sℓ
sℓ
X
j=2
1
√j<1
√2.(21)
Summarizing, αℓ<3, which gives α < 3√2π.
Recall that degrees 1and sℓwere not included in (15) –
correspondingly, the degrees 1/ε and sℓ/ε were not included
in (17). However, µ(1) and µ(sℓ)tend to zero for large ℓ, so
incorporating symbols of these degrees in our code requires
negligible overhead. More precisely: It can be shown that
enough symbols of reduced degree one can be generated using
p(·)as given in (17). Similarly, it can be shown that symbols
of reduced degree sℓcan be accounted for by incorporating a
small additional contribution to p(i)at i=sℓ/ε.
The bound α < 3√2πsuggests that as many as 7ℓrelay
symbols may required so that the destination can recover from
ℓerasures. This is clearly impractical. However, the bounding
techniques are quite loose, and, in practice, far fewer symbols
are actually needed. An example is given in Figure 4.
Figure 4 suggests that, on average, the new scheme requires
about 860 relay symbols to recover from ℓ=ǫk = 500
erasures when k= 2000 and ǫ= 0.25; this is a larger average
overhead – 360 symbols – compared to the 250 symbols
required by the LT-based scheme (Section III-B) and the 150
symbols required by the scheme based on systematic LT codes
(Section III-C). On the positive side, there is a smaller “tail”
of large redundancy for the new scheme; this is illustrated
in Figure 5, which shows the frame erasure rate (FER) as a
0 100 200 300 400 500 600 700 800 900
10−4
10−3
10−2
10−1
100
Overhead
Frame Erasure Rate
Non−systematic LT scheme
Systematic LT code [5]
New systematic scheme
Fig. 5. Frame erasure rate as a function of overhead for k= 2000,ε= 0.25.
function of the overhead for all three schemes, when k= 2000
and ǫ= 0.25. This figure demonstrates that for frame erasure
rates less than (approximately) 10−2, the new scheme requires
less overhead.
Regarding asymptotic complexity: Using p(i)in (17) as the
degree distribution, the average degree per code symbol is
O(√ℓ/ε)(compared to an average degree of O(log(ℓ)) for
the RSD). Consequently, letting ℓ=kε, the average encoding
and decoding complexity scale as O(k√kε).
IV. REGARDING AN EXTENSION TO RAPTOR CODES
This paper focused on LT codes because they are a key
component in the design of modern erasure correcting codes.
Raptor codes ([5]) consist of an outer “pre-code” and an inner
LT code. The pre-code ensures that, even with a constant
average degree per LT code symbol, good performance is
obtained. Specifically, the degree distribution may be capped
at a maximum degree independent of k. Thus in the context
of the relay network, by using a pre-code at the source, the
degree distribution in (17) could also be capped, ensuring that
the encoding complexity at the relay remains constant.
REFERENCES
[1] R. Khalili and K. Salamatian, “On the achievability of cut-set bound
for a class of erasure relay channels,” in Proc. IEEE Int. Workshop on
Wireless Ad-Hoc Networks, Oulu, Finland, May-June 2004, pp. 275–280.
[2] R. Khalili and K. Salamatian, “On the capacity of erasure relay channel:
multi-relay case,” in Proc. IEEE Inform. Theory Workshop, Rotorua, New
Zealand, Aug.-Sept. 2005, pp. 103–107.
[3] A. F. Dana, R. Gowaikar, R. Palanki, B. Hassibi, and M. Effros, “Capacity
of wireless erasure networks,” IEEE Trans. Inform. Theory, vol. 52, no.
3, pp. 789–804, Mar. 2006.
[4] M. Luby, “LT codes,” in Proc. of the 43rd Annual IEEE Symp. on
Foundations of Comp. Sc., Vancouver, Canada, Nov. 2002, pp. 271–280.
[5] A. Shokrollahi, “Raptor codes,” IEEE Trans. Inf. Theory, vol. 52, no. 6,
pp. 2551–2567, June 2006.
[6] T. Cover and A. A. El Gamal, “Capacity theorems for the relay channel,”
IEEE Trans. Inf. Theory, vol. 25, no. 5, pp. 572–584, Sept. 1979.
[7] T. M. Cover and J. A. Thomas, Elements of Information Theory, John
Wiley & Sons, New York, 1991.