Andrew Liau, Shahram Yousefi, Senior Member, IEEE,

and Il-Min Kim Senior Member, IEEE

Binary Soliton-Like Rateless Coding for the Y-Network

IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 59, NO. 12, DECEMBER 2011

OutlineIntroductionSystem modelSoliton-like rateless codingSimulation results

IntroductionIn today’s telecommunication applications,

content can originate from multiple sources and may travel through many transport nodes to reach one or more receivers.

Currently, intermediate nodes in a communications network perform Buffer-and-forward (BF)

Not an optimal strategy in the sense of overall network throughput

Network coding (NC)Each transport node linearly combines packets receivedProvides the maximum throughput for all users

simultaneouslyThe complexity increases on the decoder side

LT code and Raptor codeProvide practical capacity-achieving solutions

by way of carefully-designed encoding degree distributions

The complexities for these rateless codes are very low (logarithmic to linear scale).

For multicast scenarios for the binary erasure channel (BEC)

When the encoder uses the Robust Soliton Distribution (RSD)Capacity over the BEC is achieved universallyThe erasure rate of the channel does not need to

be known a priori

The original LT codes provide optimality for single-source, single-hop, and single-sink networks.

MotivationWe want a scheme that has good information

diffusion Using a channel code providing Maximum Distance

Separable (MDS)-type (every coded bit is the same) properties Loss resilience

NC linearly combines packets at intermediate nodes

Fountain codes linearly combine packets at the sources and provides

low decoding complexity

Advantages of marrying NC and fountain codesThe low complexity decoderThe ability to increase the effective length of the

fountain code

Previous works[8] describes a system where encoding is

superimposed at each transport node resulting in multi-layer fountain coding.The performance of the code is equivalent to a single

hop as the RSD is preserved.Multi-layer fountain decoding might be impractical to

use due to its high complexity.LT Network Codes [4]

Generalizing the setting to any network with a single source and sink.

Using complex data structures, transport nodes selectively combine packets to form the RSD at each hop.

NP-hard problem at each transport nodeOther shortcomings

Not resilient to nodes churn ratesNot scalable (complexity and dependencies on the

network configurations)

[4] M. Champel, K. Huguenin, A. Kermarrec, and N. Le Scouarnec, “LTnetwork codes,” in Proc. ICDCS, 2010.

[8] R. Gummadi and R. S. Sreenivas, “Relaying a fountain code acrossmultiple nodes,” in Proc. ITW, 2008, pp. 49–153.

System model Soliton-like degree distribution

Allowing each source to use the RSD regardless of the number of total sources.

We consider a two-user, two-hop , single-sink network.(Y-network)

System model At each source (S1 and S2): The information is

encoded by an LT code.At the relay ( ): Either BF or NC is performed.𝑅At the sink ( ): After successful decoding, the 𝐷

sink transmits a single acknowledgment (ACK) bit indicating the termination of the session.

System model Each performs LT coding [5]

Over the sets To produce the packets

R : If NC is applied, re-encode

and to generateIf BF is applied , forwards packets from S1 in

even time slots and packets from S2 in odd time slots.

System model A key component of a fountain code is the packet

degree distribution, which characterizes the decoding efficiency and throughput optimality.

RSD

RSD : The literature scale poorly with network size Sensitive to node churn rates =>SLRC

With the RSD at each source , we need a intelligent NC at R to preserve important properties of the RSD.=> NC at R

Soliton-like rateless coding

𝑝(⋅) is an aggregate degree distribution seen from D.

The probability of degree-two packet is the maximum of the distribution =>=> (fountain code ,in single-

source, single-hop)=> (in more practical scenarios)

For BP decoding to start, degree-one packets are required

=>=> (too many of them cause

inefficient decoding)p(1)<<p(2) (Otherwise , distributions result in

significantly larger minimum overhead)

Some attributes of the best distribution

Soliton-like distribution

We protect degree-one and two packets by forwarding them with probability λ ,where λ will be optimized.

If the packets are not forwarded by R, then they are buffered for

future use. The memory of R is restricted to K for each

source.R is restricted to form a new packet by

combining a single packet from S1 with a single packet from S2.

Although a Soliton-like distribution is generated at R, redundancy must also be addressed.

Soliton-like rateless coding : At R

Soliton-like rateless coding

Definition 3 (Soliton-like rateless coding (SLRC)): The SLRC protocol requires LT coding at each

source Combining at R according to Algorithm 1 where

and are innovative. This means that Algorithm 1 reuses a packet

or more than once only if there are no unused packets in the corresponding buffers.

Soliton-like rateless coding

Theorem 1: The aggregate distribution produced by the SLRC with ≥ 0.67 is Soliton-𝜆like.

Proof : We can determine the degree distribution, ( ), 𝜇 𝑘

seen at from𝐷the set of packets forwarded from either source:

𝑞(𝑘) is the probability of a packet of degree k being forwarded:

Soliton-like rateless coding

The degree distribution, ,of innovative buffered S1 packets will be:

Where and are the probabilities that a packet of degree one and two are not forwarded, respectively:

When a packet is not forwarded, the relay distribution due to only linear combining is :

The aggregate distribution in is a mixture of forwarded and linearly combined packets :

The probability, , that a packet is from either 𝜃distribution is defined as :

4) : is satisfied when ≥ 0.67 ( By letting 𝜆 )

=> 5)

6) : Satisfied at each source encodes => also satisfies=> maintains 6)

Soliton-like distribution

Since the RSD is used at each source

Corollary 1: The aggregate distribution produced by the SLRC protocol in the presence of a single source in a session is the RSD.

Proof: Suppose that S2 has left the network. In this case, R can assume that only degree-zero packets have been received from S2.

Soliton-like distribution

which results in the aggregate distribution being equal to the RSD.

Simulation resultsThe DLT [9]code is based on the RSD :

With values of , , and a message length of 2𝑐 𝛿 KThe proposed SLRC :

With values of , , and a message length of 𝑐 𝛿 KThe SDLT [10]:

With values of , , and a message length of 𝑐 𝛿 K A coding distribution, Λ( ), at 𝑥 R

BF

With K =100, an optimum value of = 0.95 𝜆was found for SLRC.

[9] S. Puducheri, J. Kliewer, and T. E. Fuja, “The design and performance of distributed LT codes,” IEEE Trans. Inf. Theory, vol. 53, no. 10, pp.3740–3754, Oct. 2007.

[10] D. Sejdinovic, R. Piechocki, and A. Doufexi, “AND-OR tree analysisof distributed LT codes,” in Proc. ITW, 2009, pp. 261–265.

Simulation results

Simulation results

Simulation results

Conclusion We propose a scheme that exploits the benefits

of network coding and fountain coding SLRC

Not affected by node churn rates in that if a source node left, no changes to the protocol are needed.

By preserving key properties of the RSD as packets travel through the network, we show that the aggregate distribution is Soliton-like

Better at reliable success rates when compared to the DLT and SDLT codes.