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Unformatted text preview: Bangalore, December 2005 Raptor Codes Amin Shokrollahi Bangalore, December 2005 Synopsis 1. Some data Transmission Problems and their (conventional) solutions 2. Fountain Codes 2.1. Deﬁnition 2.2. Some type of fountain codes 2.3. LTCodes 2.4. Raptor Codes 2.5. Systematic codes 3. Analysis of Raptor Codes 3.1. Asymptotic analysis 3.2. Finite length analysis 3.3. Finite length design 4. Advanced Raptor Codes 4.1. ML decoding 4.2. Induced graph 4.3. Data structures 4.4. Standardized codes 2 1 20 03 Bangalore, December 2005 PointtoPoint Communication Data Transmission Problems Software Company XYZ with main ofﬁce in Sillicon Valley wants to exchange gigabytes of data daily with its branch ofﬁce in Bangalore. XYZ leases a guaranteed 10mbps link, and thinks that this way K gigabytes of data are transmitted in K × 2 33 10 7 seconds using standard software, like ftp. In reality, however, the transfer time is about 10 times slower. Productivity falls. Facit: XYZ may be a great software company, but it has poor knowledge of networking! What happened? Bangalore, December 2005 PointtoPoint Communication Data Transmission Problems ftp, like many other applications, uses the ubiquitous transmission control protocol (TCP). In essence, TCP needs acknowledgement of each packet to guarantee reliable delivery. A lot of time is spent on waiting for acknowledgements. The actual throughput of TCP (in its steady state) is inversely proportional to the roundtriptime (RTT) between the sender and the receiver. The RTT puts an upper bound on the transmission speed of TCP, no matter the size of the leased line is! The RTT between California and Cupertino dictates a transmission speed of about 1mbps, even if a lot more is available. XYZ needs to ﬁnd a different solution to increase productivity! Bangalore, December 2005 PointtoMultipoint Communication Data Transmission Problems Challenge: Cars see satellite at completely random times and experience massive amounts of loss. Moreover, no feedback exists between satellite and cars. Bangalore, December 2005 PointtoMultipoint Communication Data Transmission Problems Trivial solution: send the original data several times in a carousel manner. Original ﬁle consists of k packets; cars tune in at random times, and each time they receive b packets. Assume that a complete transmission takes one day, that every car tunes in 2 times per day. How many rounds t of transmission are needed to ensure that 99.99% of the cars have received all the packets? (Minimum is k/ 2 b. ) Throw tk balls at random into k bins. For a given bin, what is the probability that it has received at least one ball? k packets Bangalore, December 2005 PointtoMultipoint Communication Data Transmission Problems In every round, every bin receives a ball with probability 2 b / k . (This is a good approximation when b is much smaller than k. ) Probability that it is empty after t rounds is Want this quantity to be less than 0.0001; so t is roughly 4.6 is roughly 4....
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 Spring '11
 Frankdesouza
 Coding theory, Fountain Codes

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