1. 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

2. Outline Introduction System model Soliton-like rateless coding Simulation results

3. Introduction In 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 received Provides the maximum throughput for all users simultaneously The complexity increases on the decoder side

4. LT code and Raptor code Provide 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 universally The 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.

5. Motivation We 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 codes The low complexity decoder The ability to increase the effective length of the fountain code

6. 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 node Other shortcomings Not resilient to nodes churn rates Not scalable (complexity and dependencies on the network configurations) [4] M. Champel, K. Huguenin, A. Kermarrec, and N. Le Scouarnec, “LT network codes,” in Proc. ICDCS, 2010. [8] R. Gummadi and R. S. Sreenivas, “Relaying a fountain code across multiple nodes,” in Proc. ITW, 2008, pp. 49–153.

7. 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)

8. System model At each source (S 1 and S 2 ): The information is encoded by an LT code. At the relay (): Either BF or NC is performed. At the sink (): After successful decoding, the sink transmits a single acknowledgment (ACK) bit indicating the termination of the session.

9. System model Each performs LT coding [5] Over the sets To produce the packets R : If NC is applied, re-encode and to generate If BF is applied, forwards packets from S 1 in even time slots and packets from S 2 in odd time slots.

10. System model A key component of a fountain code is the packet degree distribution, which characterizes the decoding efficiency and throughput optimality. RSD

11. 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

12. ( ⋅ ) 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

13. Soliton-like distribution

14. 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 S 1 with a single packet from S 2. Although a Soliton-like distribution is generated at R, redundancy must also be addressed. Soliton-like rateless coding : At R

15. Soliton-like rateless coding

16. 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

17. Theorem 1: The aggregate distribution produced by the SLRC with ≥ 0.67 is Soliton-like. Proof : We can determine the degree distribution, (), seen at from the set of packets forwarded from either source: () is the probability of a packet of degree k being forwarded: Soliton-like rateless coding

18. The degree distribution,,of innovative buffered S 1 packets will be: Where and are the probabilities that a packet of degree one and two are not forwarded, respectively:

19. 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 distribution is defined as :

20. 4) : is satisfied when ≥ 0.67 ( By letting ) => 5) 6) : Satisfied at each source encodes => also satisfies => maintains 6) Soliton-like distribution Since the RSD is used at each source

21. 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 S 2 has left the network. In this case, R can assume that only degree-zero packets have been received from S 2. Soliton-like distribution which results in the aggregate distribution being equal to the RSD.

22. Simulation results The DLT [9]code is based on the RSD : With values of,, and a message length of 2K The proposed SLRC : With values of,, and a message length of K The SDLT [10]: With values of,, and a message length of K A coding distribution, Λ (), at R BF With K =100, an optimum value of = 0.95 was 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 analysis of distributed LT codes,” in Proc. ITW, 2009, pp. 261–265.

23. Simulation results

26. 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.