Raptor%20codes%20for%20reliable%20multicast%20object%20delivery

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Raptor codes for reliable multicast object
delivery

• Michael Luby
• Digital Fountain

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Raptor codec overview

• Can generate as many encoding symbols as desired
  from a source block, on the fly rateless
  (called a fountain code in the Information Theory
  community)
• Can decode after the length of received encoding
  symbols is a very small amount above the source
  block length with very high probability,
  independent of loss patterns
• Linear time encoding and decoding, independent of
  loss patterns

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Overview of draft

• Raptor codec description
• Fully-Specified FEC scheme for object delivery
  using Raptor codec
• Systematic Raptor codec description
• Fully-Specified FEC scheme for object delivery
  using systematic Raptor codec

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Raptor codec

• Inputs
• T is symbol size
• K is the number of symbols in source block
• Raptor encoder
• Generate encoding symbols from source block
• Raptor decoder
• Generate source block from encoding symbols

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Raptor encoder
Source block

• Generate precoding block from source block
• Generate encoding symbols from precoding block

Precoding block
ES
ES
ES
ES
ES
ES
ES
ES
ES
ES
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Precoding
Precoding block
K
S
H

• S sqrt2K 0.01K
• H 2logK
• L K S H number of precoding syms.
• 3K XORS of symbols to generate LDPC
• 3K XORS of symbols to generate Half
• 6K XORS of symbols to generate precoding block

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Generators

• RAND(X,i,M) random number generator
• The seed X is a two byte integer
• The integer i selects a random generator
• The integer M is the range of possible outputs
• The result is a random integer in the range
  0,,M-1

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Generators

• DEG(v) degree distribution generator
• v is a random number in the range 0,,220-1
• The result is d(j), where j is the smallest index
  such that f(j-1) v lt f(j) in the following
  table

j f(j) d(j)
0 0 --
1 10,241 1
2 491,582 2
3 712,794 3
4 831,695 4
5 948,446 10
6 1,032,189 11
7 220 40
Average degree 4.63
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Generators

• ENC(d,a,b) encoding symbol generator
• d is the number of src symbols XORed
• a,b is used to choose the src symbols to XOR
• Let L be the smallest prime L
• Let C0, C1, , CL be the precoding symbols
• Let E be the resulting encoding symbol
• While (b L) b b a mod L
• E Cb
• For i 1,,d-1
• b b a mod L while (b L) b b a mod L
• E E XOR Cb

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Generating Encoding Symbols

• Exact details FEC scheme specific
• Example
• X is a two byte ESI
• E is resulting encoding symbol value
• d DEG(RAND(X,0,220))
• a RAND(X,1,L-1) 1
• b RAND(X,2,L)
• E ENC(d,a,b)

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Raptor decoder

• Known symbol values
• LDPC symbol and XOR of src symbols used to
  generate it 0
• Half symbol and XOR of srcLDPC symbols used to
  generate it 0
• XOR of precoding symbols used to generate each
  encoding symbol enc symbol value
• Unknown symbol values
• Precoding symbol values
• Matrix relating unknowns to knowns is known
• Can solve for precoding symbol values iff matrix
  is full rank
• Efficiency of decoding depends on the order of
  Gaussian elimination
• A good order leads to very small decoding
  complexity

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Raptor summary

• Precoding complexity
• XORS of bytes 6source block length
• Encoding symbol complexity
• XORS of bytes 4.63encoding length on average
• Decoding complexity
• XORS of bytes 10 source block length
• Decoding reception overhead/failure probability
• With 1 reception overhead 10-3 failure
  probability
• With 2 reception overhead 10-6 failure
  probability

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Raptor object delivery FEC scheme

• OTI
• The FEC Encoding ID
• F is the object size in bytes
• B is the maximum source block size in bytes
• W is the maximum block size that fits into
  working memory in bytes
• P is the packet payload size in bytes
• FEC Payload ID
• SBN two byte source block number
• ESI two byte encoding symbol identifier

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Raptor object delivery sender

• Based on F, B, W, P, describes the following
• The source block structure (based on FLUTE),
  i.e., the number of source blocks and the size of
  each source block
• N the number of subblocks each source block is
  partitioned into
• T the subsymbol size
• NT is the source symbol size
• G the number of encoding symbols to place into
  each packet
• G, N, T must satisfy P GNT
• Describes how to generate FEC Payload IDs (SBN,
  ESI pairs) for packets
• Describes how to generate encoding symbols in
  packets based on FEC Payload IDs
• Exact description
• Uses RAND, DEG, ENC functions

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Raptor object delivery sender

• Small objects (F W)
• Treated as one source block and no partitioning
  into subblocks
• Larger objects (W lt F B)
• Treated as one source block partitioned into more
  than one subblock
• Each subblock can be decoded at receiver
  separately
• Protection is provided over entire source block
  (same logical subsymbol from each subblock is in
  each packet)
• Large objects (B lt F)
• Object is first partitioned into source blocks of
  size at most B based on FLUTE partitioning
  algorithm
• Each source block is treated independently (each
  packet contains symbols from only one source
  block)

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Systematic Raptor encoder
Intermediate Precoding block
Raptor decoder
Raptor encoder
Source block
Repair symbols
K symbols K systematic triples
R symbols R triples
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Systematic Raptor decoder
Intermediate Precoding block
Source block
Repair symbols

• Use Raptor decoder to recover Intermediate
  Precoding block
• From received source and repair symbols
• Use Raptor encoder to recover missing source
  symbols from
• Intermediate Precoding block

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Systematic Raptor summary

• Encoding and Decoding complexities somewhat
  higher
• Decoding reception overhead/failure probability
  the same as for Raptor

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Systematic Raptor object delivery FEC scheme

• Similar to the Raptor object delivery scheme
• Uses systematic Raptor instead of Raptor
• Many small details different
• Two different types of packets
• Source packets containing source symbols
• Repair packets containing repair symbols
• May want to demultiplex based on higher level
  signaling

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IPR

• Digital Fountain does have IPR rights to all that
  is described in this draft
• Licensing on RAND terms

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Proposal

• Accept this as a RMT working group item