1. CprE 545 project proposal Long

2.  Introduction  Random linear code  LT-code  Application  Future work

3.  A file is divided into equal length pieces and placed into packets  Receivers acknowledge each received packet and senders retransmit the packets lost  Low efficiency over networks with high latencies and high loss rates  Capacity is wasted by feedback messages according to the Shannon Theory  Wastefulness is terrible in the case of a multicast channel

4.  An ideal/paradigm for data transmission, without need for receivers to send any feedback message and for senders to resend any packet.  k packets of a file can potentially generate limitless encoded packets; once you receive any k(1+α) packets regardless of the order ( α is a small fraction < 1, especially when k is large ), you can quickly reconstruct the original file.

5.  It doesn’t matter what is received or lost  It only matters that enough is received

6.  One to One transport  Design the flow and congestion control mechanisms independently of reliability of transmission links.  One-to-many transport  Reliability without feedback  High/unknown/variable loss  Massive scalability  Many-to-one transport  Minimize delivery of redundant packets

7.  Digital fountains can be constructed by using fountain codes  Fountain Codes  Rateless  Delivery and recovery regardless of the network reliability  Small encoding and decoding complexities  Several types of fountain codes:  Random linear fountain  LT-code(Luby, 98)  Raptor-code(Shokrollahi, 01)

8.  A file is divided into K packets s 1, s 2, … s k, each packet is composed of a whole number of bits  At each clock cycle, labelled by n, the encoder generates K random bits {G kn }  The encoded packet E n is set to the bitwise sum (modulo 2) of the source packets for which G kn is 1, which is:

9.  Let K=3, n=1 ksksk G kn s k G kn 110111 2111000000 300011

10. The generator matrix of a random linear code and packets transmission [Mack 05]

11.  A receiver collects N packets.  Let us assume that he knows the generator matrix G kn by some means  If N<K, the receiver has not got enough information to recover the file  If N=K, the receiver has 0.289 probability to recover the file  If N>K, the receiver can recover the file if and only if a k-by-K invertible matrix exists in G, so that the receiver can compute G -1 and recover

12.  Let N=K+L and the probability that a receiver can recover the original file is 1-δ, then for any K, the probability of failure of recovery is bounded by  The number of packets required to have probability 1-δ of success is  Pros: get arbitrarily close to the Shannon limit  Cons:  Encoding complexity:  Decoding complexity: Is there any better solution with lower computational cost?

13.  Encoder  Randomly choose the degree d n of the packet from a degree distribution p(d); the appropriate choice of p depends on the source file size K  Choose uniformly at random d distinct input packets and set E n equal to the bitwise sum (modulo 2) of those d n packets  The encoding process can be demonstrated by a graph in which each encoded packet is connected to the corresponding original packets { s n }, and the graph is sparse if the mean degree E(d) is greatly smaller than K.

14.  Decoder  Recover s from E=sG, where G is the matrix associated with graph, supposing that the receiver somehow knows G  G is much simpler than the generator matrix in random linear fountain and is determined by degree distribution p(d) and uniform distribution such that a simple way can be used to decoding by message passing.

15. 1) Find an encoded packet E n that is connected to only one source packet s k (if there is no such encoded packet, this decoding algorithm halts at this point, and fails to recover all the source packets).  Set s k = E n  Substract s k from all encoded packets that are connected to s k so that E n ’ = E n – s k  Remove all the edges connected to the source packet s k 2) Repeat (1) until all s k are determined.

16. 0 00 1 1 1  Take an example in which there are K=3 original packets s 1,s 2,s 3 where each packet is just one bit for brevity. The receiver received four encoded packets E 1,E 2,E 3,E 4 = 1011 at the start of the algorithm. s1s1 s2s2 s3s3 1011 S 1 = 1 S 2 = 1 S 3 = 0

17.  A small portion of encoded packets must have high degree  Majority packets must have low degree  Thus, an appropriate degree distribution should be chosen:  The cost of encoding and decoding: Ideal Solition Distribution: for d = 2, 3, … k

18.  Reliable Multicast  Downloading in Parallel  Point-to-Point Data Transmission  One-to-Many TCP  Distribution on Overlay Networks  Video Streaming  Other applications out of network: Storage systems etc…

19.  Implementation of LT-code  Encoding and decoding  C/C++  Testify the computation complexity  Network application (If time permits)  One to one transmission  One to many transmission  Test the recovery success ratio with different number of packets loss during the transmission

20.  [1] D.J.C. MacKay, “Fountain Codes”, IEE Proceedings – Commun. Vol. 152, No 6, Dec 2005.  [2] Michael Luby, “LT Codes”, Proceedings of the 43 rd Annual IEEE symposium on Foundations of Computer Science (FOCS’02)  [3] Michael Mitzenmacher, “Digital Fountains: A Survey and Look Forward”.