Rateless Codes with Optimum Intermediate Performance Ali Talari and Nazanin Rahnavard Oklahoma State University, USA IEEE GLOBECOM 2009 & IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 60, NO. 5, MAY

Outlines Introduction Rateless codes design Evaluation results Conclusion 2

Introduction Intermediate recovery rate is important in video or voice transmission applications where partial recovery of the source packets from received encoded packets is beneficial. This motivates the design of forward error correction (FEC) codes with high intermediate performance. In rateless coding, the employed degree distribution significantly affects the packet recovery rate. 3

Intermediate Performance of Rateless Codes Sujay Sanghavi LIDS, MIT IEEE ITW(Information Theory Workshop)

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Introduction Sanghavi in [4] – z ∈ [0, ½] the optimum degree distribution has degree one packets only, – the optimum degree distribution has degree two packets only, 6 [4] S. Sanghavi, “Intermediate performance of rateless codes,” IEEE Information Theory Workshop (ITW), pp. 478–482, Sept

Introduction – For an integer m, where It is shown that the optimum degree distribution in this region is given by 7

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Introduction The number of source packets : k Number of received coded packets : n Received overhead by γ, where γ = n / k The ratio of number of recovered packets at the receiver to k by z Finding degree distributions with maximal packet recovery rates in intermediate range, 0 < γ < 1. We define packet recovery rates at 3 values of γ as our conflicting objective functions and employ NSGA-II multi- objective genetic algorithms optimization method to find several degree distributions with optimum packet recovery rates. 10

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Growth Codes Degree of a codeword “grows” with time At each timepoint codeword of a specific degree has the most utility for a decoder (on average) This “most useful” degree grows monotonically with time R : Number of decoded symbols sink has R1R1 R3R3 R2R2 R4R4 d=1 d=2d=3d=4 Time -> [6] A. Kamra, V. Misra, J. Feldman, and D. Rubenstein, “Growth codes: Maximizing sensor network data persistence,” SIGCOMM Computer Communication Rev., vol. 36, no. 4, pp. 255–266, 2006.

Ideas of Proposed Method Method: – Growth Codes: Been designed for sensor networks in catastrophic or emergency scenarios. To make new received encoded packet useful. – Can be decoded immediately. To avoid new received encoded packet useless. – Cannot be decoded.

Ideas of Proposed Method Growth Codes: – A received encoded packet is immediately useful: if d - 1 of the data used to form this encoded packet are already decoded/known. y4y4 x3x5x6x3x5x6 already decoded data:new received packets: x1x1 x2x2 x3x3 x5x5 x3x3 x5x5   y4y4 x6x6 d = 3 d – 1 data are already decoded.

Ideas of Proposed Method Growth Codes: – A received encoded packet is useless: if all d data used to form a encoded packet are already known. y1y1 x1x3x1x3 already decoded data:new received packets: x1x1 x2x2 x3x3 x5x5 d = 2 d data are already decoded. new received packet is useless.

Ideas of Proposed Method Consider the degree of an encoded packet: – Decoder has decoded r original data. – The probability that new received encoded packet is immediately decodable to the decoder: Number of decoded original data: r Importance of Immediately Decodable Packet : Low Degree : High Degree

Rateless codes design We propose a novel approach that finds degree distributions for high recovery rates throughout intermediate range We select one γ from each region, i.e. γ ∈ {0.5, 0.75, 1}, and define 3 objective functions to be the packet recovery rates at these γ ’s 2 approaches – 1)We consider the infinite asymptotic case similar to existing studies. We formulate packet recovery rates using a technique called And-Or tree analysis [1, 7-9] – 2)We consider finite-length rateless codes with k = 100 and k = 1000 and show how degree distributions vary with k 17 [8] N. Rahnavard and F. Fekri, “Generalization of rateless codes for unequal error protection and recovery time: Asymptotic analysis,” IEEE International Symposium on Information Theory, 2006, pp. 523–527, July [9] N. Rahnavard, B. Vellambi, and F. Fekri, “Rateless codes with unequal error protection property,” IEEE Transactions on Information Theory, vol. 53, pp. 1521–1532, April 2007.

Rateless codes design In And-Or tree analysis technique [1, 7–9] the error rate of iterative decoding of rateless codes is probabilistically formulated for k = ∞ Consider a rateless code with parameters Ω(x) and γ. Let y l be the probability that a packet is not recovered after l decoding iterations 18 [1] P. Maymounkov, “Online codes,” NYU Technical Report TR , [7] M. G. Luby, M. Mitzenmacher, and M. A. Shokrollahi, “Analysis of random processes via And-Or tree evaluation,” (Philadelphia, PA, USA), pp. 364–373, Society for Industrial and Applied Mathematics, 1998.

Rateless codes design Let F denote the corresponding fixed point This fixed point is the final packet error rate of a rateless code with parameters Ω(x) and γ We define 3 objective functions given as fixed points of (2) for γ = 0.5, γ = 0.75, and γ = 1 19

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Evaluation Results The upper bound on rateless codes recovery rates at γ = 0.5, γ = 0.75, and γ = 1 are , and 1 We define F(Ω(x)) as where are the weights assigned to each objective function 22

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24 Fig. 4. Comparison of the performance of the rateless codes employing designed degree distributions for asymptotic case with the upper bound on rateless codes intermediate performance.

25 The designed degree distributions show a high performance and perform close to upper bound. These degree distributions are optimum in intermediate performance. According to the selected weights the resulting codes have the highest recovery rate at the γ with the highest weight.

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27 When k = 100, decoder requires larger fraction of degree one packets and lower degree packets are preferred. Degree two packets constitute a high percentage of encoded packets compared to packets of other degrees.

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29 Fig. 5. Comparison of the performance of the rateless codes employing designed degree distributions for k = 100 with the upper bound on rateless codes intermediate performance. K=100

30 Fig. 5. Comparison of the performance of the rateless codes employing designed degree distributions for k = 1000 with the upper bound on rateless codes intermediate performance. K=1000

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Conclusions In this paper, we studied the intermediate performance of rateless codes and proposed to employ multi-objective genetic algorithms to find several optimum degree distributions in intermediate range. We used the state-of-the-art optimization algorithm NSGA-II to find the set of optimum degree distributions which are called pareto optimal. 32

References [1] P. Maymounkov, “Online codes,” NYU Technical Report TR , [4] S. Sanghavi, “Intermediate performance of rateless codes,” Information Theory Workshop, ITW ’07. IEEE, pp. 478–482, Sept [5] K. Deb, A. Pratap, S. Agarwal, and T. Meyarivan, “A fast and elitist multiobjective genetic algorithm: NSGA-II,” IEEE Transactions on Evolutionary Computation, vol. 6, pp. 182–197, Apr [6] A. Kamra, V. Misra, J. Feldman, and D. Rubenstein, “Growth codes: Maximizing sensor network data persistence,” SIGCOMM Computer Communication Rev., vol. 36, no. 4, pp. 255– 266, [7] M. G. Luby, M. Mitzenmacher, and M. A. Shokrollahi, “Analysis of random processes via And-Or tree evaluation,” (Philadelphia, PA, USA), pp. 364–373, Society for Industrial and Applied Mathematics, [8] N. Rahnavard and F. Fekri, “Generalization of rateless codes for unequal error protection and recovery time: Asymptotic analysis,” IEEE International Symposium on Information Theory, 2006, pp. 523–527, July [9] N. Rahnavard, B. Vellambi, and F. Fekri, “Rateless codes with unequal error protection property,” IEEE Transactions on Information Theory, vol. 53, pp. 1521–1532, April [10] S. Kim and S. Lee, “Improved intermediate performance of rateless codes,” ICACT 2009, vol. 3, pp. 1682–1686, Feb Valerio Bioglio, Marco Grangetto, Rossano Gaeta, Matteo Sereno: An optimal partial decoding algorithm for rateless codes. IEEE ISIT 2011,pp

An Optimal Partial Decoding Algorithm for Rateless Codes V. Bioglio, M. Grangetto, R. Gaeta, M. Sereno Dipartimento di Informatica Universit`a di Torino IEEE ISIT(International Symposium on Information Theory)

35 Fig. 2. Partial decoding performance of LT codes.

36 Fig. 3. Algorithm complexity vs. k.