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Versions: (draft-luby-rmt-bb-fec-raptor-object) 00 01 02 03 04 05 06 07 08 09 RFC 5053

Reliable Multicast Transport                                     M. Luby
Internet-Draft                                          Digital Fountain
Expires: March 6, 2006                                    A. Shokrollahi
                                                                    EPFL
                                                               M. Watson
                                                        Digital Fountain
                                                          T. Stockhammer
                                                              Siemens AG
                                                       September 2, 2005


                 Raptor Forward Error Correction Scheme
                 draft-ietf-rmt-bb-fec-raptor-object-02

Status of this Memo

   By submitting this Internet-Draft, each author represents that any
   applicable patent or other IPR claims of which he or she is aware
   have been or will be disclosed, and any of which he or she becomes
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   This Internet-Draft will expire on March 6, 2006.

Copyright Notice

   Copyright (C) The Internet Society (2005).

Abstract

   This document describes a Fully-Specified FEC scheme, corresponding
   to FEC Encoding ID 1, for the Raptor forward error correction code
   and its application to reliable delivery of data objects.



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   Raptor is a fountain code, i.e., as many encoding symbols as needed
   can be generated by the encoder on-the-fly from the source symbols of
   a source block of data.  The decoder is able to recover the source
   block from any set of encoding symbols only slightly more in number
   than the number of source symbols.

   The Raptor code described here is a systematic code, meaning that all
   the source symbols are among the encoding symbols that can be
   generated.










































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Table of Contents

   1.  Introduction . . . . . . . . . . . . . . . . . . . . . . . . .  4
   2.  Requirements notation  . . . . . . . . . . . . . . . . . . . .  5
   3.  Formats and Codes  . . . . . . . . . . . . . . . . . . . . . .  6
     3.1.  FEC Payload IDs  . . . . . . . . . . . . . . . . . . . . .  6
     3.2.  FEC Object Transmission Information  . . . . . . . . . . .  6
       3.2.1.  Mandatory  . . . . . . . . . . . . . . . . . . . . . .  6
       3.2.2.  Common . . . . . . . . . . . . . . . . . . . . . . . .  6
       3.2.3.  Scheme-Specific  . . . . . . . . . . . . . . . . . . .  7
   4.  Procedures . . . . . . . . . . . . . . . . . . . . . . . . . .  8
     4.1.  Content Delivery Protocol Requirements . . . . . . . . . .  8
     4.2.  Example parameter derivation algorithm . . . . . . . . . .  8
   5.  Raptor FEC code specification  . . . . . . . . . . . . . . . . 11
     5.1.  Introduction . . . . . . . . . . . . . . . . . . . . . . . 11
     5.2.  Definitions, Symbols and abbreviations . . . . . . . . . . 11
       5.2.1.  Definitions  . . . . . . . . . . . . . . . . . . . . . 11
       5.2.2.  Symbols  . . . . . . . . . . . . . . . . . . . . . . . 12
       5.2.3.  Abbreviations  . . . . . . . . . . . . . . . . . . . . 14
     5.3.  Overview . . . . . . . . . . . . . . . . . . . . . . . . . 14
     5.4.  Object delivery  . . . . . . . . . . . . . . . . . . . . . 15
       5.4.1.  Source block construction  . . . . . . . . . . . . . . 15
       5.4.2.  Encoding packet construction . . . . . . . . . . . . . 17
     5.5.  Systematic Raptor encoder  . . . . . . . . . . . . . . . . 18
       5.5.1.  Encoding overview  . . . . . . . . . . . . . . . . . . 18
       5.5.2.  First encoding step: Intermediate Symbol Generation  . 19
       5.5.3.  Second encoding step: LT encoding  . . . . . . . . . . 23
       5.5.4.  Generators . . . . . . . . . . . . . . . . . . . . . . 24
     5.6.  Example FEC decoder  . . . . . . . . . . . . . . . . . . . 27
       5.6.1.  General  . . . . . . . . . . . . . . . . . . . . . . . 27
       5.6.2.  Decoding a source block  . . . . . . . . . . . . . . . 27
     5.7.  Random Numbers . . . . . . . . . . . . . . . . . . . . . . 32
       5.7.1.  The table V0 . . . . . . . . . . . . . . . . . . . . . 32
       5.7.2.  The table V1 . . . . . . . . . . . . . . . . . . . . . 33
     5.8.  Systematic Indicies J(K) . . . . . . . . . . . . . . . . . 34
   6.  Security Considerations  . . . . . . . . . . . . . . . . . . . 47
   7.  IANA Considerations  . . . . . . . . . . . . . . . . . . . . . 48
   8.  Intellectual Property  . . . . . . . . . . . . . . . . . . . . 49
   9.  Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . 50
   10. References . . . . . . . . . . . . . . . . . . . . . . . . . . 51
     10.1. Normative references . . . . . . . . . . . . . . . . . . . 51
     10.2. Informative references . . . . . . . . . . . . . . . . . . 51
   Authors' Addresses . . . . . . . . . . . . . . . . . . . . . . . . 52
   Intellectual Property and Copyright Statements . . . . . . . . . . 53







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1.  Introduction

   This document specifies an FEC Scheme for the Raptor forward error
   correction code for object delivery applications.  The concept of an
   FEC Scheme is defined in [2].

   The Raptor FEC Scheme is a Fully-Specified FEC Scheme corresponding
   to FEC Encoding ID 1.

   Raptor is a fountain code, i.e., as many encoding symbols as needed
   can be generated by the encoder on-the-fly from the source symbols of
   a block.  The decoder is able to recover the source block from any
   set of encoding symbols only slightly more in number than the number
   of source symbols.

   The code described in this document is a systematic code, that is,
   the original source symbols can be sent unmodified from sender to
   receiver, as well as a number of repair symbols.

   The code described here is identical to that described in [3]































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2.  Requirements notation

   The key words "MUST", "MUST NOT", "REQUIRED", "MUST", "MUST NOT",
   "SHOULD", "SHOULD NOT", "RECOMMENDED", "MAY", and "OPTIONAL" in this
   document are to be interpreted as described in [1].














































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3.  Formats and Codes

3.1.  FEC Payload IDs

   The FEC Payload ID MUST be a 4 octet field defined as follows:

   +---------------------------+---------------------------+
   | Source Block Number (SBN) | Encoding Symbol ID (ESI)  |
   +---------------------------+---------------------------+

   Figure 1: FEC Payload ID format

      Source Block Number (SBN), (16 bits): An integer identifier for
      the source block that the encoding symbols within the packet
      relate to.

      Encoding Symbol ID (ESI), (16 bits): An integer identifier for the
      encoding symbols within the packet.

   The interpretation of the Source Block Number and Encoding Symbol
   Identifier is defined in Section 5.

3.2.  FEC Object Transmission Information

3.2.1.  Mandatory

   The value of the FEC Encoding ID MUST be 1 (one).

3.2.2.  Common

   The Common FEC Object Transmission Information elements used by this
   FEC Scheme are:

      - Transfer Length (F)

      - Encoding Symbol Length (T)

   The Transfer Length is a non-negative integer less than 2^^48.  The
   Encoding Symbol Length is a non-negative integer less than 2^^16.

   Where a Content Delivery Protocol requires an encoding for these
   elements to be specified by the FEC Scheme then the Transfer Length
   MUST be encoded as a 6 octet field in network byte order (high order
   byte first) and the Encoding Symbol Length MUST be encoded as a 2
   octet field in network byte order (high order byte first).






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3.2.3.  Scheme-Specific

   The following parameters are carried in the Scheme-Specific FEC
   Object Transmission Information element for this FEC Scheme:

      - The number of source blocks (Z)

      - The number of sub-blocks (N)

      - A symbol alignment parameter (Al)

   These parameters are all non-negative integers and MUST be encoded as
   a 4-octet field as defined in Figure 2.

    3         2         1
   10987654321098765432109876543210
   +--------------+-------+-------+
   |      Z       |   N   |   Al  |
   +--------------+-------+-------+

   Figure 2: Encoding of Scheme-specific FEC Object Transmission
   Information





























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4.  Procedures

4.1.  Content Delivery Protocol Requirements

   This section describes the information exchange between the Raptor
   FEC Scheme and any Content Delivery Protocol (CDP) making use the
   Raptor FEC Scheme for object delivery.

   The Raptor encoder and decoder for object delivery require the
   following information from the CDP:

      - The transfer length of the object, F, in bytes

      - The symbol alignment parameter, Al

      - The symbol size, T, in bytes, which MUST be a multiple of Al

      - The number of source blocks, Z

      - The number of sub-blocks in each source block, N

   The Raptor encoder for object delivery additionally requires:

      - the object to be encoded, F bytes

   The Raptor encoder supplies the CDP with encoding packet information
   consisting, for each packet, of:

      - Source Block Number (SBN)

      - Encoding Symbol ID (ESI)

      - Encoding symbol(s)

   The CDP MUST communicate this information transparently to the Raptor
   decoder.

4.2.  Example parameter derivation algorithm

   This section provides recommendations for the derivation of the four
   transport parameters, Al, T, Z and N. This recommendation is based on
   the following input parameters:

      - F the object size, in bytes

      - W a target on the sub-block size, in bytes





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      - P the maximum packet payload size, in bytes, which is assumed to
      be a multiple of Al

      - Al the symbol alignment factor, in bytes

      - Kmax the maximum number of source symbols per source block.

         Note: Section 5.2.2 defines Kmax to be 8192.

      - Kmin a minimum target on the number of symbols per source block

      - Gmax a maximum target number of symbols per packet

   Based on the above inputs, the transport parameters T, Z and N are
   calculated as follows:

   Let,

      G = min{ceil(P*Kmin/F), P/Al, Gmax}

      T = floor(P/(Al*G))*Al

      Kt = ceil(F/T)

      Z = ceil(Kt/Kmax)

      N = min{ceil(ceil(Kt/Z)*T/W), T/Al}

   The values of G and N derived above should be considered as lower
   bounds.  It may be advantageous to increase these values, for example
   to the nearest power of two.  In particular, the above algorithm does
   not guarantee that the symbol size, T, divides the maximum packet
   size, P, and so it may not be possible to use the packets of size
   exactly P. If, instead, G is chosen to be a value which divides P/Al,
   then the symbol size, T, will be a divisor of P and packets of size P
   can be used.

   The algorithm above and that defined in Section 5.4.1.2 ensure that
   the sub-symbol sizes are a multiple of the symbol alignment
   parameter, Al.  This is useful because the XOR operations used for
   encoding and decoding are generally performed several bytes at a
   time, for example at least 4 bytes at a time on a 32 bit processor.
   Thus the encoding and decoding can be performed faster if the sub-
   symbol sizes are a multiple of this number of bytes.

   Recommended settings for the input parameters, Al, Kmin and Gmax are
   as follows: Al = 4, Kmin = 1024, Gmax = 10.




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   The parameter W can be used to generate encoded data which can be
   decoded efficiently with limited working memory at the decoder.  Note
   that the actual maximum decoder memory requirement for a given value
   of W depends on the implementation, but that it is possible to
   implement decoding using working memory only slightly larger than W.














































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5.  Raptor FEC code specification

5.1.  Introduction

   This Section specifies the Raptor forward error correction code and
   its application to object delivery.  Raptor is a fountain code, i.e.,
   as many encoding symbols as needed can be generated by the encoder
   on-the-fly from the source symbols of a block.  The decoder is able
   to recover the source block from any set of encoding symbols only
   slightly more in number than the number of source symbols.

   The code described in this document is a systematic code, that is,
   the original source symbols may be sent unmodified from sender to
   receiver, as well as a number of repair symbols.

5.2.  Definitions, Symbols and abbreviations

5.2.1.  Definitions

   For the purposes of this specification, the following terms and
   definitions apply.

      Source block: a block of K source symbols which are considered
      together for Raptor encoding purposes.

      Source symbol: the smallest unit of data used during the encoding
      process.  All source symbols within a source block have the same
      size.

      Encoding symbol: a symbol that is included in a data packet.  The
      encoding symbols consist of the source symbols and the repair
      symbols.  Repair symbols generated from a source block have the
      same size as the source symbols of that source block.

      Systematic code: a code in which the source symbols are included
      as part of the encoding symbols sent for a source block.

      Repair symbol: the encoding symbols sent for a source block that
      are not the source symbols.  The repair symbols are generated
      based on the source symbols.

      Intermediate symbols: symbols generated from the source symbols
      using an inverse encoding process .  The repair symbols are then
      generated directly from the intermediate symbols.  The encoding
      symbols do not include the intermediate symbols, i.e.,
      intermediate symbols are not included in data packets.





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      Symbol: a unit of data.  The size, in bytes, of a symbol is known
      as the symbol size.

      Encoding symbol group: a group of encoding symbols that are sent
      together, i.e., within the same packet whose relationship to the
      source symbols can be derived from a single Encoding Symbol ID.

      Encoding Symbol ID: information that defines the relationship
      between the symbols of an encoding symbol group and the source
      symbols.

      Encoding packet: data packets that contain encoding symbols

      Sub-block: a source block is sometimes broken into sub-blocks,
      each of which is sufficiently small to be decoded in working
      memory.  For a source block consisting of K source symbols, each
      sub-block consists of K sub-symbols, each symbol of the source
      block being composed of one sub-symbol from each sub-block.

      Sub-symbol: part of a symbol.  Each source symbol is composed of
      as many sub-symbols as there are sub-blocks in the source block.

      Source packet: data packets that contain source symbols.

      Repair packet: data packets that contain repair symbols.

5.2.2.  Symbols

   i, j, x, h, a, b, d, v, m represent positive integers

   ceil(x) denotes the smallest positive integer which is greater than
        or equal to x

   choose(i,j) denotes the number of ways j objects can be chosen from
        among i objects without repetition

   floor(x): denotes the largest positive integer which is less than or
        equal to x

   i % j denotes i modulo j

   X ^ Y denotes, for equal-length bit strings X and Y, the bitwise
        exclusive-or of X and Y

   Al   denotes a symbol alignment parameter.  Symbol and sub-symbol
        sizes are restricted to be multiples of Al.





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   A    denotes a matrix over GF(2).

   Transpose[A] denotes the transposed matrix of matrix A

   A^^-1 denotes the inverse matrix of matrix A

   K    denotes the number of symbols in a single source block

   Kmax denotes the maximum number of source symbols that can be in a
        single source block.  Set to 8192.

   L    denotes the number of pre-coding symbols for a single source
        block

   S    denotes the number of LDPC symbols for a single source block

   H    denotes the number of Half symbols for a single source block

   C    denotes an array of intermediate symbols, C[0], C[1], C[2],...,
        C[L-1]

   C'   denotes an array of source symbols, C'[0], C'[1], C'[2],...,
        C'[K-1]

   X    a non-negative integer value

   V0, V1 two arrays of 4-byte integers, V0[0], V0[1],..., V0[255] and
        V1[0], V1[1],..., V1[255]

   Rand[X, i, m] a pseudo-random number generator

   Deg[v] a degree generator

   LTEnc[K, C ,(d, a, b)] a LT encoding symbol generator

   Trip[K, X] a triple generator function

   G    the number of symbols within an encoding symbol group

   GF(n) The Galois field with n elements.

   N    the number of sub-blocks within a source block

   T    the symbol size in bytes.  If the source block is partitioned
        into sub-blocks, then T = T'*N.






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   T'   the sub-symbol size, in bytes.  If the source block is not
        partitioned into sub-blocks then T' is not relevant.

   F    the object size, for object delivery, in bytes

   I    the sub-block size in bytes

   P    for object delivery, the payload size of each packet, in bytes,
        that is used in the recommended derivation of the object
        delivery transport parameters.

   Q    Q = 65521, i.e., Q is the largest prime smaller than 2^^16

   Z    the number of source blocks, for object delivery

   J(K) the systematic index associated with K

   G    denotes any generator matrix

   I_S  denotes the SxS identity matrix

   0_SxH denotes the SxH zero matrix

   a ^^ b a raised to the power b

5.2.3.  Abbreviations

   For the purposes of the present document, the following abbreviations
   apply:

   ESI       Encoding Symbol ID

   LDPC      Low Density Parity Check

   LT        Luby Transform

   SBN       Source Block Number

   SBL       Source Block Length (in units of symbols)

5.3.  Overview

   The principle component of the systematic Raptor code is the basic
   encoder described in Section 5.5.  First, it is described how to
   derive values for a set of intermediate symbols from the original
   source symbols such that knowledge of the intermediate symbols is
   sufficient to reconstruct the source symbols.  Secondly, the encoder
   produces repair symbols which are each the exclusive OR of a number



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   of the intermediate symbols.  The encoding symbols are the
   combination of the source and repair symbols.  The repair symbols are
   produced in such a way that the intermediate symbols and therefore
   also the source symbols can be recovered from any sufficiently large
   set of encoding symbols.

   This document defines the systematic Raptor code encoder.  A number
   of possible decoding algorithms are possible.  An efficient decoding
   algorithm is provided in Section 5.6.

   The construction of the intermediate and repair symbols is based in
   part on a pseudo-random number generator described in
   Section 5.5.4.1.  This generator is based on a fixed set of 512
   random numbers which MUST be available to both sender and receiver.
   These are provided in Section 5.7.

   Finally, the construction of the intermediate symbols from the source
   symbols is governed by a 'systematic index', values of which are
   provided in Section 5.8 for source block sizes from 4 source symbols
   to Kmax = 8192 source symbols.

5.4.  Object delivery

5.4.1.  Source block construction

5.4.1.1.  General

   In order to apply the Raptor encoder to a source object, the object
   may be broken into Z >= 1 blocks, known as source blocks.  The Raptor
   encoder is applied independently to each source block.  Each source
   block is identified by a unique integer Source Block Number (SBN),
   where the first source block has SBN zero, the second has SBN one,
   etc.  Each source block is divided into a number, K, of source
   symbols of size T bytes each.  Each source symbol is identified by a
   unique integer Encoding Symbol Identifier (ESI), where the first
   source symbol of a source block has ESI zero, the second has ESI one,
   etc.

   Each source block with K source symbols is divided into N >= 1 sub-
   blocks, which are small enough to be decoded in the working memory.
   Each sub-block is divided into K sub-symbols of size T'.

   Note that the value of K is not necessarily the same for each source
   block of a object and the value of T' may not necessarily be the same
   for each sub-block of a source block.  However, the symbol size T is
   the same for all source blocks of an object and the number of
   symbols, K is the same for every sub-block of a source block.  Exact
   partitioning of the object into source blocks and sub-blocks is



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   described in Section 5.4.1.2 below.

5.4.1.2.  Source block and sub-block partitioning

   The construction of source blocks and sub-blocks is determined based
   on five input parameters, F, Al, T, Z and N and a function
   Partition[].  The five input parameters are defined as follows:

      - F the transfer length of the object, in bytes

      - Al a symbol alignment parameter, in bytes

      - T the symbol size, in bytes, which MUST be a multiple of Al

      - Z the number of source blocks

      - N the number of sub-blocks in each source block

   These parameters MUST be set so that ceil(ceil(F/T)/Z) <= Kmax.
   Recommendations for derivation of these parameters are provided in
   Section 4.2.

   The function Partition[] takes a pair of integers (I, J) as input and
   derives four integers (IL, IS, JL, JS) as output.  Specifically, the
   value of Partition[I, J] is a sequence of four integers (IL, IS, JL,
   JS), where IL = ceil(I/J), IS = floor(I/J), JL = I - IS * J and JS =
   J - JL.  Partition[] derives parameters for partitioning a block of
   size I into J approximately equal sized blocks.  Specifically, JL
   blocks of length IL and JS blocks of length IS.

   The source object MUST be partitioned into source blocks and sub-
   blocks as follows:

   Let,

      Kt = ceil(F/T)

      (KL, KS, ZL, ZS) = Partition[Kt, Z]

      (TL, TS, NL, NS) = Partition[T/Al, N]

   Then, the object MUST be partitioned into Z = ZL + ZS contiguous
   source blocks, the first ZL source blocks each having length KL*T
   bytes and the remaining ZS source blocks each having KS*T bytes.

   If Kt*T > F then for encoding purposes, the last symbol MUST be
   padded at the end with Kt*T - F zero bytes.




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   Next, each source block MUST be divided into N = NL + NS contiguous
   sub-blocks, the first NL sub-blocks each consisting of K contiguous
   sub-symbols of size of TL*Al and the remaining NS sub-blocks each
   consisting of K contiguous sub-symbols of size of TS*Al.  The symbol
   alignment parameter Al ensures that sub-symbols are always a multiple
   of A bytes.

   Finally, the m-th symbol of a source block consists of the
   concatenation of the m-th sub-symbol from each of the N sub-blocks.
   Note that this implies that when N > 1 then a symbol is NOT a
   contiguous portion of the object.

5.4.2.  Encoding packet construction

5.4.2.1.  General

   Each encoding packet contains the following information:

      - Source Block Number (SBN)

      - Encoding Symbol ID (ESI)

      - encoding symbol(s)

   Each source block is encoded independently of the others.  Source
   blocks are numbered consecutively from zero.

   Encoding Symbol ID values from 0 to K-1 identify the source symbols
   of a source block in sequential order, where K is the number of
   symbols in the source block.  Encoding Symbol IDs from K onwards
   identify repair symbols.

5.4.2.2.  Encoding packet construction

   Each encoding packet either consists entirely of source symbols
   (source packet) or entirely of repair symbols (repair packet).  A
   packet may contain any number of symbols from the same source block.
   In the case that the last source symbol in a source packet includes
   padding bytes added for FEC encoding purposes then these bytes need
   not be included in the packet.  Otherwise, only whole symbols MUST be
   included.

   The Encoding Symbol ID, X, carried in each source packet is the
   Encoding Symbol ID of the first source symbol carried in that packet.
   The subsequent source symbols in the packet have Encoding Symbol IDs,
   X+1 to X+G-1, in sequential order, where G is the number of symbols
   in the packet.




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   Similarly, the Encoding Symbol ID, X, placed into a repair packet is
   the Encoding Symbol ID of the first repair symbol in the repair
   packet and the subsequent repair symbols in the packet have Encoding
   Symbol IDs X+1 to X+G-1 in sequential order, where G is the number of
   symbols in the packet.

   Note that it is not necessary for the receiver to know the total
   number of repair packets.

   Associated with each symbol is a triple of integers (d, a, b).

   The G repair symbol triples (d[0], a[0], b[0]),..., (d[G-1], a[G-1],
   b[G-1]) for the repair symbols placed into a repair packet with ESI X
   are computed using the Triple generator defined in Section 5.5.4.4 as
   follows:

      For each i = 0, ..., G-1, (d[i], a[i], b[i]) = Trip[K,X+i]

   The G repair symbols to be placed in repair packet with ESI X are
   calculated based on the repair symbol triples as described in
   Section 5.5 using the intermediate symbols C and the LT encoder
   LTenc[K, C, (d[i], a[i], b[i])].

5.5.  Systematic Raptor encoder

5.5.1.  Encoding overview

   The systematic Raptor encoder is used to generate repair symbols from
   a source block that consists of K source symbols.

   Symbols are the fundamental data units of the encoding and decoding
   process.  For each source block (sub-block) all symbols (sub-symbols)
   are the same size.  The atomic operation performed on symbols (sub-
   symbols) for both encoding and decoding is the exclusive-or
   operation.

   Let C'[0],..., C'[K-1] denote the K source symbols.

   Let C[0],..., C[L-1] denote L intermediate symbols.

   The first step of encoding is to generate a number, L > K, of
   intermediate symbols from the K source symbols.  In this step, K
   source triples (d[0], a[0], b[0]), ..., (d[K-1], a[K-1], b[K-1]) are
   generated using the Trip[] generator as described in Section 5.5.2.2.
   The K source triples are associated with the K source symbols and are
   then used to determine the L intermediate symbols C[0],..., C[L-1]
   from the source symbols using an inverse encoding process.  This
   process can be can be realized by a Raptor decoding process.



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   Certain "pre-coding relationships" MUST hold within the L
   intermediate symbols.  Section 5.5.2.3 describes these relationships
   and how the intermediate symbols are generated from the source
   symbols.

   Once the intermediate symbols have been generated, repair symbols are
   produced and one or more repair symbols are placed as a group into a
   single data packet.  Each repair symbol group is associated with an
   Encoding Symbol ID (ESI) and a number, G, of encoding symbols.  The
   ESI is used to generate a triple of three integers, (d, a, b) for
   each repair symbol, again using the Trip[] generator as described in
   Section 5.4.2.  Then, each (d,a,b)-triple is used to generate the
   corresponding repair symbol from the intermediate symbols using the
   LTEnc[K, C[0],..., C[L-1], (d,a,b)] generator described in
   Section 5.5.4.3.

5.5.2.  First encoding step: Intermediate Symbol Generation

5.5.2.1.  General

   The first encoding step is a pre-coding step to generate the L
   intermediate symbols C[0], ..., C[L-1] from the source symbols C'[0],
   ..., C'[K-1].  The intermediate symbols are uniquely defined by two
   sets of constraints:

      1.  The intermediate symbols are related to the source symbols by
      a set of source symbol triples.  The generation of the source
      symbol triples is defined in Section 5.5.2.2 using the the Trip[]
      generator described in Section 5.5.4.4.

      2.  A set of pre-coding relationships hold within the intermediate
      symbols themselves.  These are defined in Section 5.5.2.3

   The generation of the L intermediate symbols is then defined in
   Section 5.5.2.4

5.5.2.2.  Source symbol triples

   Each of the K source symbols is associated with a triple (d[i], a[i],
   b[i]) for 0 <= i < K. The source symbol triples are determined using
   the Triple generator defined in Section 5.5.4.4 as:

      For each i, 0 <= i < K

         (d[i], a[i], b[i]) = Trip[K, i]






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5.5.2.3.  Pre-coding relationships

   The pre-coding relationships amongst the L intermediate symbols are
   defined by expressing the last L-K intermediate symbols in terms of
   the first K intermediate symbols.

   The last L-K intermediate symbols C[K],...,C[L-1] consist of S LDPC
   symbols and H Half symbols The values of S and H are determined from
   K as described below.  Then L = K+S+H.

   Let

      X be the smallest positive integer such that X*(X-1) >= 2*K.

      S be the smallest prime integer such that S >= ceil(0.01*K) + X

      H be the smallest integer such that choose(H,ceil(H/2)) >= K + S

      H' = ceil(H/2)

      L = K+S+H

      C[0],...,C[K-1] denote the first K intermediate symbols

      C[K],...,C[K+S-1] denote the S LDPC symbols, initialised to zero

      C[K+S],...,C[L-1] denote the H Half symbols, initialised to zero

   The S LDPC symbols are defined to be the values of C[K],...,C[K+S-1]
   at the end of the following process:

      For i = 0,...,K-1 do

         a = 1 + (floor(i/S) % (S-1))

         b = i % S

         C[K + b] = C[K + b] ^ C[i]

         b = (b + a) % S

         C[K + b] = C[K + b] ^ C[i]

         b = (b + a) % S

         C[K + b] = C[K + b] ^ C[i]

   The H Half symbols are defined as follows:



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   Let

      g[i] = i ^ (floor(i/2)) for all positive integers i

         Note: g[i] is the Gray sequence, in which each element differs
         from the previous one in a single bit position

      g[j,k] denote the jth element, j=0, 1, 2, ..., of the subsequence
      of g[i] whose elements have exactly k non-zero bits in their
      binary representation

   Then, the Half symbols are defined as the values of C[K+S],...,C[L-1]
   after the following process:

      For h = 0,...,H-1 do

         For j = 0,...,K+S-1 do

            If bit h of g[j,H'] is equal to 1 then C[h+K+S] = C[h+K+S] ^
            C[j].

5.5.2.4.  Intermediate symbols

5.5.2.4.1.  Definition

   Given the K source symbols C'[0], C'[1],..., C'[K-1] the L
   intermediate symbols C[0], C[1],..., C[L-1] are the uniquely defined
   symbol values that satisfy the following conditions:

      1.  The K source symbols C'[0], C'[1],..., C'[K-1] satisfy the K
      constraints C'[i] = LTEnc[K, (C[0],..., C[L-1]), (d[i], a[i],
      b[i])], for all i, 0 <= i < K.

      2.  The L intermediate symbols C[0], C[1],..., C[L-1] satisfy the
      pre-coding relationships defined in Section 5.5.2.3

5.5.2.4.2.  Example method for calculation of intermediate symbols

   This subsection describes a possible method for calculation of the L
   intermediate symbols C[0], C[1],..., C[L-1] satisfying the
   constraints in Section 5.5.2.4.1

   The generator matrix G for a code which generates N output symbols
   from K input symbols is an NxK matrix over GF(2), where each row
   corresponds to one of the output symbols and each column to one of
   the input symbols and where the ith output symbol is equal to the sum
   of those input symbols whose column contains a non-zero entry in row
   i.



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   Then, the L intermediate symbols can be calculated as follows:

   Let

      C denote the column vector of the L intermediate symbols, C[0],
      C[1],..., C[L-1].

      D denote the column vector consisting of S+H zero symbols followed
      by the K source symbols C'[0], C'[1], ..., C'[K-1]

   Then the above constraints define an LxL matrix over GF(2), A, such
   that:

      A*C = D

   The matrix A can be constructed as follows:

   Let:

      G_LDPC be the S x K generator matrix of the LDPC symbols.  So,

         G_LDPC * Transpose[(C[0],...., C[K-1])] = Transpose[(C[K], ...,
         C[K+S-1])]

      G_Half be the H x (K+S) generator matrix of the Half symbols, So,

         G_Half * Transpose[(C[0], ..., C[S+K-1])] = Transpose[(C[K+S],
         ..., C[K+S+H-1])]

      I_S be the S x S identity matrix

      I_H be the H x H identity matrix

      0_SxH be the S x H zero matrix

      G_LT be the KxL generator matrix of the encoding symbols generated
      by the LT Encoder.  So,

         G_LT * Transpose[(C[0], ..., C[L-1])] =
         Transpose[(C'[0],C'[1],...,C'[K-1])]

         i.e.  G_LT(i,j) = 1 if and only if C[j] is included in the
         symbols which are XORed to produce LTEnc[K, (C[0], ...,
         C[L-1]), (d[i], a[i], b[i])].

   Then:





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      The first S rows of A are equal to G_LDPC | I_S | 0_SxH.

      The next H rows of A are equal to G_Half | I_H.

      The remaining K rows of A are equal to G_LT.

   The matrix A is depicted in Figure 3 below:

                 K               S       H
     +-----------------------+-------+-------+
     |                       |       |       |
   S |        G_LDPC         |  I_S  | 0_SxH |
     |                       |       |       |
     +-----------------------+-------+-------+
     |                               |       |
   H |        G_Half                 |  I_H  |
     |                               |       |
     +-------------------------------+-------+
     |                                       |
     |                                       |
   K |                 G_LT                  |
     |                                       |
     |                                       |
     +---------------------------------------+

   Figure 3: The matrix A

   The intermediate symbols can then be calculated as:

      C = (A^^-1)*D

   The source triples are generated such that for any K matrix A has
   full rank and is therefore invertible.  This calculation can be
   realized by applying a Raptor decoding process to the K source
   symbols C'[0], C'[1],..., C'[K-1] to produce the L intermediate
   symbols C[0], C[1],..., C[L-1].

   To efficiently generate the intermediate symbols from the source
   symbols, it is recommended that an efficient decoder implementation
   such as that described in Section 5.6 be used.  The source symbol
   triples are designed to facilitate efficient decoding of the source
   symbols using that algorithm.

5.5.3.  Second encoding step: LT encoding

   In the second encoding step, the repair symbol with ESI X is
   generated by applying the generator LTEnc[K, (C[0], C[1],...,
   C[L-1]), (d, a, b)] defined in Section 5.5.4.3 to the L intermediate



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   symbols C[0], C[1],..., C[L-1] using the triple (d, a, b)=Trip[K,X]
   generated according to Section 5.4.2

5.5.4.  Generators

5.5.4.1.  Random Generator

   The random number generator Rand[X, i, m] is defined as follows,
   where X is a non-negative integer, i is a non-negative integer and m
   is a positive integer and the value produced is an integer between 0
   and m-1.  Let V0 and V1 be arrays of 256 entries each, where each
   entry is a 4-byte unsigned integer.  These arrays are provided in
   Section 5.7.

   Then,

      Rand[X, i, m] = (V0[(X + i) % 256] ^ V1[(floor(X/256)+ i) % 256])
      % m

5.5.4.2.  Degree Generator

   The degree generator Deg[v] is defined as follows, where v is an
   integer that is at least 0 and less than 2^^20 = 1048576.

      In Table 1, find the index j such that f[j-1] <= v < f[j]

      Then, Deg[v] = d[j]
























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                       +---------+---------+------+
                       | Index j | f[j]    | d[j] |
                       +---------+---------+------+
                       | 0       | 0       | --   |
                       |         |         |      |
                       | 1       | 10241   | 1    |
                       |         |         |      |
                       | 2       | 491582  | 2    |
                       |         |         |      |
                       | 3       | 712794  | 3    |
                       |         |         |      |
                       | 4       | 831695  | 4    |
                       |         |         |      |
                       | 5       | 948446  | 10   |
                       |         |         |      |
                       | 6       | 1032189 | 11   |
                       |         |         |      |
                       | 7       | 1048576 | 40   |
                       +---------+---------+------+

       Table 1: Defines the degree distribution for encoding symbols

5.5.4.3.  LT Encoding Symbol Generator

   The encoding symbol generator LTEnc[K, (C[0], C[1],..., C[L-1]), (d,
   a, b)] takes the following inputs:

      K is the number of source symbols (or sub-symbols) for the source
      block (sub-block).  Let L be derived from K as described in
      Section 5.5.2.3, and let L' be the smallest prime integer greater
      than or equal to L.

      (C[0], C[1],..., C[L-1]) is the array of L intermediate symbols
      (sub-symbols) generated as described in Section 5.5.2.4

      (d, a, b) is a source triple determined using the Triple generator
      defined in Section 5.5.4.4, whereby

         d is an integer denoting an encoding symbol degree

         a is an integer between 1 and L'-1 inclusive

         b is an integer between 0 and L'-1 inclusive

   The encoding symbol generator produces a single encoding symbol as
   output, according to the following algorithm:





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      While (b >= L) do b = (b + a) % L'

      LTEnc[K, (C[0], C[1],..., C[L-1]), (d, a, b)] = C[b].

      For j = 1,...,min(d-1,L-1) do

         b = (b + a) % L'

         While (b >= L) do b = (b + a) % L'

         LTEnc[K, (C[0], C[1],..., C[L-1]), (d, a, b)] = LTEnc[K, (C[0],
         C[1],..., C[L-1]), (d, a, b)] ^ C[b]

5.5.4.4.  Triple generator

   The triple generator Trip[K,X] takes the following inputs:

      K - The number of source symbols

      X - An encoding symbol ID

   Let

      L be determined from K as described in Section 5.5.2.3

      L' be the smallest prime that is greater than or equal to L

      Q = 65521, the largest prime smaller than 2^^16.

      J(K) be the systematic index associated with K, as defined in
      Section 5.8

   The output of the triple generator is a triple, (d, a, b) determined
   as follows:

      A = (53591 + J(K)*997) % Q

      B = 10267*(J(K)+1) % Q

      Y = (B + X*A) % Q

      v = Rand[Y, 0, 2^^20]

      d = Deg[v]

      a = 1 + Rand[Y, 1, L'-1]





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      b = Rand[Y, 2, L']

5.6.  Example FEC decoder

5.6.1.  General

   This section describes an efficient decoding algorithm for the Raptor
   codes described in this specification.  Note that each received
   encoding symbol can be considered as the value of an equation amongst
   the intermediate symbols.  From these simultaneous equations, and the
   known pre-coding relationships amongst the intermediate symbols, any
   algorithm for solving simultaneous equations can successfully decode
   the intermediate symbols and hence the source symbols.  However, the
   algorithm chosen has a major effect on the computational efficiency
   of the decoding.

5.6.2.  Decoding a source block

5.6.2.1.  General

   It is assumed that the decoder knows the structure of the source
   block it is to decode, including the symbol size, T, and the number K
   of symbols in the source block.

   From the algorithms described in Section 5.5, the Raptor decoder can
   calculate the total number L = K+S+H of pre-coding symbols and
   determine how they were generated from the source block to be
   decoded.  In this description it is assumed that the received
   encoding symbols for the source block to be decoded are passed to the
   decoder.  Furthermore, for each such encoding symbol it is assumed
   that the number and set of intermediate symbols whose exclusive-or is
   equal to the encoding symbol is passed to the decoder.  In the case
   of source symbols, the source symbol triples described in
   Section 5.5.2.2 indicate the number and set of intermediate symbols
   which sum to give each source symbol.

   Let N >= K be the number of received encoding symbols for a source
   block and let M = S+H+N. The following M by L bit matrix A can be
   derived from the information passed to the decoder for the source
   block to be decoded.  Let C be the column vector of the L
   intermediate symbols, and let D be the column vector of M symbols
   with values known to the receiver, where the first S+H of the M
   symbols are zero-valued symbols that correspond to LDPC and Half
   symbols (these are check symbols for the LDPC and Half symbols, and
   not the LDPC and Half symbols themselves), and the remaining N of the
   M symbols are the received encoding symbols for the source block.
   Then, A is the bit matrix that satisfies A*C = D, where here *
   denotes matrix multiplication over GF[2].  In particular, A[i,j] = 1



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   if the intermediate symbol corresponding to index j is exclusive-ORed
   into the LDPC, Half or encoding symbol corresponding to index i in
   the encoding, or if index i corresponds to a LDPC or Half symbol and
   index j corresponds to the same LDPC or Half symbol.  For all other i
   and j, A[i,j] = 0.

   Decoding a source block is equivalent to decoding C from known A and
   D. It is clear that C can be decoded if and only if the rank of A
   over GF[2] is L. Once C has been decoded, missing source symbols can
   be obtained by using the source symbol triples to determine the
   number and set of intermediate symbols which MUST be exclusive-ORed
   to obtain each missing source symbol.

   The first step in decoding C is to form a decoding schedule.  In this
   step A is converted, using Gaussian elimination (using row operations
   and row and column reorderings) and after discarding M - L rows, into
   the L by L identity matrix.  The decoding schedule consists of the
   sequence of row operations and row and column re-orderings during the
   Gaussian elimination process, and only depends on A and not on D. The
   decoding of C from D can take place concurrently with the forming of
   the decoding schedule, or the decoding can take place afterwards
   based on the decoding schedule.

   The correspondence between the decoding schedule and the decoding of
   C is as follows.  Let c[0] = 0, c[1] = 1,...,c[L-1] = L-1 and d[0] =
   0, d[1] = 1,...,d[M-1] = M-1 initially.

      - Each time row i of A is exclusive-ORed into row i' in the
      decoding schedule then in the decoding process symbol D[d[i]] is
      exclusive-ORed into symbol D[d[i']].

      - Each time row i is exchanged with row i' in the decoding
      schedule then in the decoding process the value of d[i] is
      exchanged with the value of d[i'].

      - Each time column j is exchanged with column j' in the decoding
      schedule then in the decoding process the value of c[j] is
      exchanged with the value of c[j'].

   From this correspondence it is clear that the total number of
   exclusive-ORs of symbols in the decoding of the source block is the
   number of row operations (not exchanges) in the Gaussian elimination.
   Since A is the L by L identity matrix after the Gaussian elimination
   and after discarding the last M - L rows, it is clear at the end of
   successful decoding that the L symbols D[d[0]], D[d[1]],...,
   D[d[L-1]] are the values of the L symbols C[c[0]], C[c[1]],...,
   C[c[L-1]].




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   The order in which Gaussian elimination is performed to form the
   decoding schedule has no bearing on whether or not the decoding is
   successful.  However, the speed of the decoding depends heavily on
   the order in which Gaussian elimination is performed.  (Furthermore,
   maintaining a sparse representation of A is crucial, although this is
   not described here).  The remainder of this section describes an
   order in which Gaussian elimination could be performed that is
   relatively efficient.

5.6.2.2.  First Phase

   The first phase of the Gaussian elimination the matrix A is
   conceptually partitioned into submatrices.  The submatrix sizes are
   parameterized by non-negative integers i and u which are initialized
   to 0.  The submatrices of A are:

      (1) The submatrix I defined by the intersection of the first i
      rows and first i columns.  This is the identity matrix at the end
      of each step in the phase.

      (2) The submatrix defined by the intersection of the first i rows
      and all but the first i columns and last u columns.  All entries
      of this submatrix are zero.

      (3) The submatrix defined by the intersection of the first i
      columns and all but the first i rows.  All entries of this
      submatrix are zero.

      (4) The submatrix U defined by the intersection of all the rows
      and the last u columns.

      (5) The submatrix V formed by the intersection of all but the
      first i columns and the last u columns and all but the first i
      rows.

   Figure 4 illustrates the submatrices of A. At the beginning of the
   first phase V = A. In each step, a row of A is chosen.














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   +-----------+-----------------+---------+
   |           |                 |         |
   |     I     |    All Zeros    |         |
   |           |                 |         |
   +-----------+-----------------+    U    |
   |           |                 |         |
   |           |                 |         |
   | All Zeros |       V         |         |
   |           |                 |         |
   |           |                 |         |
   +-----------+-----------------+---------+

   Figure 4: Submatrices of A in the first phase

   The following graph defined by the structure of V is used in
   determining which row of A is chosen.  The columns that intersect V
   are the nodes in the graph, and the rows that have exactly 2 ones in
   V are the edges of the graph that connect the two columns (nodes) in
   the positions of the two ones.  A component in this graph is a
   maximal set of nodes (columns) and edges (rows) such that there is a
   path between each pair of nodes/edges in the graph.  The size of a
   component is the number of nodes (columns) in the component.

   There are at most L steps in the first phase.  The phase ends
   successfully when i + u = L, i.e., when V and the all zeroes
   submatrix above V have disappeared and A consists of I, the all
   zeroes submatrix below I, and U. The phase ends unsuccessfully in
   decoding failure if at some step before V disappears there is no non-
   zero row in V to choose in that step.  In each step, a row of A is
   chosen as follows:

   o  If all entries of V are zero then no row is chosen and decoding
      fails.

   o  Let r be the minimum integer such that at least one row of A has
      exactly r ones in V.

      *  If r != 2 then choose a row with exactly r ones in V with
         minimum original degree among all such rows.

      *  If r = 2 then choose any row with exactly 2 ones in V that is
         part of a maximum size component in the graph defined by X.

   After the row is chosen in this step the first row of A that
   intersects V is exchanged with the chosen row so that the chosen row
   is the first row that intersects V. The columns of A among those that
   intersect V are reordered so that one of the r ones in the chosen row
   appears in the first column of V and so that the remaining r-1 ones



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   appear in the last columns of V. Then, the chosen row is exclusive-
   ORed into all the other rows of A below the chosen row that have a
   one in the first column of V. Finally, i is incremented by 1 and u is
   incremented by r-1, which completes the step.

5.6.2.3.  Second Phase

   The submatrix U is further partitioned into the first i rows,
   U_upper, and the remaining M - i rows, U_lower.  Gaussian elimination
   is performed in the second phase on U_lower to either determine that
   its rank is less than u (decoding failure) or to convert it into a
   matrix where the first u rows is the identity matrix (success of the
   second phase).  Call this u by u identity matrix I_u.  The M - L rows
   of A that intersect U_lower - I_u are discarded.  After this phase A
   has L rows and L columns.

5.6.2.4.  Third Phase

   After the second phase the only portion of A which needs to be zeroed
   out to finish converting A into the L by L identity matrix is
   U_upper.  The number of rows i of the submatrix U_upper is generally
   much larger than the number of columns u of U_upper.  To zero out
   U_upper efficiently, the following precomputation matrix U' is
   computed based on I_u in the third phase and then U' is used in the
   fourth phase to zero out U_upper.  The u rows of Iu are partitioned
   into ceil(u/8) groups of 8 rows each.  Then, for each group of 8 rows
   all non-zero combinations of the 8 rows are computed, resulting in
   2^^8 - 1 = 255 rows (this can be done with 2^^8-8-1 = 247 exclusive-
   ors of rows per group, since the combinations of Hamming weight one
   that appear in I_u do not need to be recomputed).  Thus, the
   resulting precomputation matrix U' has ceil(u/8)*255 rows and u
   columns.  Note that U' is not formally a part of matrix A, but will
   be used in the fourth phase to zero out U_upper.

5.6.2.5.  Fourth Phase

   For each of the first i rows of A, for each group of 8 columns in the
   U_upper submatrix of this row, if the set of 8 column entries in
   U_upper are not all zero then the row of the precomputation matrix U'
   that matches the pattern in the 8 columns is exclusive-ORed into the
   row, thus zeroing out those 8 columns in the row at the cost of
   exclusive-oring one row of U' into the row.

   After this phase A is the L by L identity matrix and a complete
   decoding schedule has been successfully formed.  Then, as explained
   in Section 5.6.2.1, the corresponding decoding consisting of
   exclusive-ORing known encoding symbols can be executed to recover the
   intermediate symbols based on the decoding schedule.  The triples



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   associated with all source symbols are computed according to
   Section 5.5.2.2.  The triples for received source symbols are used in
   the decoding.  The triples for missing source symbols are used to
   determine which intermediate symbols need to be exclusive-ORed to
   recover the missing source symbols.

5.7.  Random Numbers

   The two tables V0 and V1 described in Section 5.5.4.1 are given
   below.  Each entry is a 32-bit integer in decimal representation.

5.7.1.  The table V0

   251291136, 3952231631, 3370958628, 4070167936, 123631495, 3351110283,
   3218676425, 2011642291, 774603218, 2402805061, 1004366930,
   1843948209, 428891132, 3746331984, 1591258008, 3067016507,
   1433388735, 504005498, 2032657933, 3419319784, 2805686246,
   3102436986, 3808671154, 2501582075, 3978944421, 246043949,
   4016898363, 649743608, 1974987508, 2651273766, 2357956801, 689605112,
   715807172, 2722736134, 191939188, 3535520147, 3277019569, 1470435941,
   3763101702, 3232409631, 122701163, 3920852693, 782246947, 372121310,
   2995604341, 2045698575, 2332962102, 4005368743, 218596347,
   3415381967, 4207612806, 861117671, 3676575285, 2581671944,
   3312220480, 681232419, 307306866, 4112503940, 1158111502, 709227802,
   2724140433, 4201101115, 4215970289, 4048876515, 3031661061,
   1909085522, 510985033, 1361682810, 129243379, 3142379587, 2569842483,
   3033268270, 1658118006, 932109358, 1982290045, 2983082771,
   3007670818, 3448104768, 683749698, 778296777, 1399125101, 1939403708,
   1692176003, 3868299200, 1422476658, 593093658, 1878973865,
   2526292949, 1591602827, 3986158854, 3964389521, 2695031039,
   1942050155, 424618399, 1347204291, 2669179716, 2434425874,
   2540801947, 1384069776, 4123580443, 1523670218, 2708475297,
   1046771089, 2229796016, 1255426612, 4213663089, 1521339547,
   3041843489, 420130494, 10677091, 515623176, 3457502702, 2115821274,
   2720124766, 3242576090, 854310108, 425973987, 325832382, 1796851292,
   2462744411, 1976681690, 1408671665, 1228817808, 3917210003,
   263976645, 2593736473, 2471651269, 4291353919, 650792940, 1191583883,
   3046561335, 2466530435, 2545983082, 969168436, 2019348792,
   2268075521, 1169345068, 3250240009, 3963499681, 2560755113,
   911182396, 760842409, 3569308693, 2687243553, 381854665, 2613828404,
   2761078866, 1456668111, 883760091, 3294951678, 1604598575,
   1985308198, 1014570543, 2724959607, 3062518035, 3115293053,
   138853680, 4160398285, 3322241130, 2068983570, 2247491078,
   3669524410, 1575146607, 828029864, 3732001371, 3422026452,
   3370954177, 4006626915, 543812220, 1243116171, 3928372514,
   2791443445, 4081325272, 2280435605, 885616073, 616452097, 3188863436,
   2780382310, 2340014831, 1208439576, 258356309, 3837963200,
   2075009450, 3214181212, 3303882142, 880813252, 1355575717, 207231484,



Luby, et al.              Expires March 6, 2006                [Page 32]

Internet-Draft              Raptor FEC Scheme             September 2005


   2420803184, 358923368, 1617557768, 3272161958, 1771154147,
   2842106362, 1751209208, 1421030790, 658316681, 194065839, 3241510581,
   38625260, 301875395, 4176141739, 297312930, 2137802113, 1502984205,
   3669376622, 3728477036, 234652930, 2213589897, 2734638932,
   1129721478, 3187422815, 2859178611, 3284308411, 3819792700,
   3557526733, 451874476, 1740576081, 3592838701, 1709429513,
   3702918379, 3533351328, 1641660745, 179350258, 2380520112,
   3936163904, 3685256204, 3156252216, 1854258901, 2861641019,
   3176611298, 834787554, 331353807, 517858103, 3010168884, 4012642001,
   2217188075, 3756943137, 3077882590, 2054995199, 3081443129,
   3895398812, 1141097543, 2376261053, 2626898255, 2554703076,
   401233789, 1460049922, 678083952, 1064990737, 940909784, 1673396780,
   528881783, 1712547446, 3629685652, 1358307511

5.7.2.  The table V1

   807385413, 2043073223, 3336749796, 1302105833, 2278607931, 541015020,
   1684564270, 372709334, 3508252125, 1768346005, 1270451292,
   2603029534, 2049387273, 3891424859, 2152948345, 4114760273,
   915180310, 3754787998, 700503826, 2131559305, 1308908630, 224437350,
   4065424007, 3638665944, 1679385496, 3431345226, 1779595665,
   3068494238, 1424062773, 1033448464, 4050396853, 3302235057,
   420600373, 2868446243, 311689386, 259047959, 4057180909, 1575367248,
   4151214153, 110249784, 3006865921, 4293710613, 3501256572, 998007483,
   499288295, 1205710710, 2997199489, 640417429, 3044194711, 486690751,
   2686640734, 2394526209, 2521660077, 49993987, 3843885867, 4201106668,
   415906198, 19296841, 2402488407, 2137119134, 1744097284, 579965637,
   2037662632, 852173610, 2681403713, 1047144830, 2982173936, 910285038,
   4187576520, 2589870048, 989448887, 3292758024, 506322719, 176010738,
   1865471968, 2619324712, 564829442, 1996870325, 339697593, 4071072948,
   3618966336, 2111320126, 1093955153, 957978696, 892010560, 1854601078,
   1873407527, 2498544695, 2694156259, 1927339682, 1650555729,
   183933047, 3061444337, 2067387204, 228962564, 3904109414, 1595995433,
   1780701372, 2463145963, 307281463, 3237929991, 3852995239,
   2398693510, 3754138664, 522074127, 146352474, 4104915256, 3029415884,
   3545667983, 332038910, 976628269, 3123492423, 3041418372, 2258059298,
   2139377204, 3243642973, 3226247917, 3674004636, 2698992189,
   3453843574, 1963216666, 3509855005, 2358481858, 747331248,
   1957348676, 1097574450, 2435697214, 3870972145, 1888833893,
   2914085525, 4161315584, 1273113343, 3269644828, 3681293816,
   412536684, 1156034077, 3823026442, 1066971017, 3598330293,
   1979273937, 2079029895, 1195045909, 1071986421, 2712821515,
   3377754595, 2184151095, 750918864, 2585729879, 4249895712,
   1832579367, 1192240192, 946734366, 31230688, 3174399083, 3549375728,
   1642430184, 1904857554, 861877404, 3277825584, 4267074718,
   3122860549, 666423581, 644189126, 226475395, 307789415, 1196105631,
   3191691839, 782852669, 1608507813, 1847685900, 4069766876,
   3931548641, 2526471011, 766865139, 2115084288, 4259411376,



Luby, et al.              Expires March 6, 2006                [Page 33]

Internet-Draft              Raptor FEC Scheme             September 2005


   3323683436, 568512177, 3736601419, 1800276898, 4012458395, 1823982,
   27980198, 2023839966, 869505096, 431161506, 1024804023, 1853869307,
   3393537983, 1500703614, 3019471560, 1351086955, 3096933631,
   3034634988, 2544598006, 1230942551, 3362230798, 159984793, 491590373,
   3993872886, 3681855622, 903593547, 3535062472, 1799803217, 772984149,
   895863112, 1899036275, 4187322100, 101856048, 234650315, 3183125617,
   3190039692, 525584357, 1286834489, 455810374, 1869181575, 922673938,
   3877430102, 3422391938, 1414347295, 1971054608, 3061798054,
   830555096, 2822905141, 167033190, 1079139428, 4210126723, 3593797804,
   429192890, 372093950, 1779187770, 3312189287, 204349348, 452421568,
   2800540462, 3733109044, 1235082423, 1765319556, 3174729780,
   3762994475, 3171962488, 442160826, 198349622, 45942637, 1324086311,
   2901868599, 678860040, 3812229107, 19936821, 1119590141, 3640121682,
   3545931032, 2102949142, 2828208598, 3603378023, 4135048896

5.8.  Systematic Indicies J(K)

   For each value of K the systematic index J(K) is designed to have the
   property that the set of source symbol triples (d[0], a[0], b[0]),
   ..., (d[L-1], a[L-1], b[L-1]) are such that the L intermediate
   symbols are uniquely defined, i.e. the matrix A in Section 5.5.2.4.2
   has full rank and is therefore invertible.

   The following is the list of the systematic indices for values of K
   between 4 and 8192 inclusive,

   18, 14, 61, 46, 14, 22, 20, 40, 48, 1, 29, 40, 43, 46, 18, 8, 20, 2,
   61, 26, 13, 29, 36, 19, 58, 5, 58, 0, 54, 56, 24, 14, 5, 67, 39, 31,
   25, 29, 24, 19, 14, 56, 49, 49, 63, 30, 4, 39, 2, 1, 20, 19, 61, 4,
   54, 70, 25, 52, 9, 26, 55, 69, 27, 68, 75, 19, 64, 57, 45, 3, 37, 31,
   100, 41, 25, 41, 53, 23, 9, 31, 26, 30, 30, 46, 90, 50, 13, 90, 77,
   61, 31, 54, 54, 3, 21, 66, 21, 11, 23, 11, 29, 21, 7, 1, 27, 4, 34,
   17, 85, 69, 17, 75, 93, 57, 0, 53, 71, 88, 119, 88, 90, 22, 0, 58,
   41, 22, 96, 26, 79, 118, 19, 3, 81, 72, 50, 0, 32, 79, 28, 25, 12,
   25, 29, 3, 37, 30, 30, 41, 84, 32, 31, 61, 32, 61, 7, 56, 54, 39, 33,
   66, 29, 3, 14, 75, 75, 78, 84, 75, 84, 25, 54, 25, 25, 107, 78, 27,
   73, 0, 49, 96, 53, 50, 21, 10, 73, 58, 65, 27, 3, 27, 18, 54, 45, 69,
   29, 3, 65, 31, 71, 76, 56, 54, 76, 54, 13, 5, 18, 142, 17, 3, 37,
   114, 41, 25, 56, 0, 23, 3, 41, 22, 22, 31, 18, 48, 31, 58, 37, 75,
   88, 3, 56, 1, 95, 19, 73, 52, 52, 4, 75, 26, 1, 25, 10, 1, 70, 31,
   31, 12, 10, 54, 46, 11, 74, 84, 74, 8, 58, 23, 74, 8, 36, 11, 16, 94,
   76, 14, 57, 65, 8, 22, 10, 36, 36, 96, 62, 103, 6, 75, 103, 58, 10,
   15, 41, 75, 125, 58, 15, 10, 34, 29, 34, 4, 16, 29, 18, 18, 28, 71,
   28, 43, 77, 18, 41, 41, 41, 62, 29, 96, 15, 106, 43, 15, 3, 43, 61,
   3, 18, 103, 77, 29, 103, 19, 58, 84, 58, 1, 146, 32, 3, 70, 52, 54,
   29, 70, 69, 124, 62, 1, 26, 38, 26, 3, 16, 26, 5, 51, 120, 41, 16, 1,
   43, 34, 34, 29, 37, 56, 29, 96, 86, 54, 25, 84, 50, 34, 34, 93, 84,
   96, 29, 29, 50, 50, 6, 1, 105, 78, 15, 37, 19, 50, 71, 36, 6, 54, 8,



Luby, et al.              Expires March 6, 2006                [Page 34]

Internet-Draft              Raptor FEC Scheme             September 2005


   28, 54, 75, 75, 16, 75, 131, 5, 25, 16, 69, 17, 69, 6, 96, 53, 96,
   41, 119, 6, 6, 88, 50, 88, 52, 37, 0, 124, 73, 73, 7, 14, 36, 69, 79,
   6, 114, 40, 79, 17, 77, 24, 44, 37, 69, 27, 37, 29, 33, 37, 50, 31,
   69, 29, 101, 7, 61, 45, 17, 73, 37, 34, 18, 94, 22, 22, 63, 3, 25,
   25, 17, 3, 90, 34, 34, 41, 34, 41, 54, 41, 54, 41, 41, 41, 163, 143,
   96, 18, 32, 39, 86, 104, 11, 17, 17, 11, 86, 104, 78, 70, 52, 78, 17,
   73, 91, 62, 7, 128, 50, 124, 18, 101, 46, 10, 75, 104, 73, 58, 132,
   34, 13, 4, 95, 88, 33, 76, 74, 54, 62, 113, 114, 103, 32, 103, 69,
   54, 53, 3, 11, 72, 31, 53, 102, 37, 53, 11, 81, 41, 10, 164, 10, 41,
   31, 36, 113, 82, 3, 125, 62, 16, 4, 41, 41, 4, 128, 49, 138, 128, 74,
   103, 0, 6, 101, 41, 142, 171, 39, 105, 121, 81, 62, 41, 81, 37, 3,
   81, 69, 62, 3, 69, 70, 21, 29, 4, 91, 87, 37, 79, 36, 21, 71, 37, 41,
   75, 128, 128, 15, 25, 3, 108, 73, 91, 62, 114, 62, 62, 36, 36, 15,
   58, 114, 61, 114, 58, 105, 114, 41, 61, 176, 145, 46, 37, 30, 220,
   77, 138, 15, 1, 128, 53, 50, 50, 58, 8, 91, 114, 105, 63, 91, 37, 37,
   13, 169, 51, 102, 6, 102, 23, 105, 23, 58, 6, 29, 29, 19, 82, 29, 13,
   36, 27, 29, 61, 12, 18, 127, 127, 12, 44, 102, 18, 4, 15, 206, 53,
   127, 53, 17, 69, 69, 69, 29, 29, 109, 25, 102, 25, 53, 62, 99, 62,
   62, 29, 62, 62, 45, 91, 125, 29, 29, 29, 4, 117, 72, 4, 30, 71, 71,
   95, 79, 179, 71, 30, 53, 32, 32, 49, 25, 91, 25, 26, 26, 103, 123,
   26, 41, 162, 78, 52, 103, 25, 6, 142, 94, 45, 45, 94, 127, 94, 94,
   94, 47, 209, 138, 39, 39, 19, 154, 73, 67, 91, 27, 91, 84, 4, 84, 91,
   12, 14, 165, 142, 54, 69, 192, 157, 185, 8, 95, 25, 62, 103, 103, 95,
   71, 97, 62, 128, 0, 29, 51, 16, 94, 16, 16, 51, 0, 29, 85, 10, 105,
   16, 29, 29, 13, 29, 4, 4, 132, 23, 95, 25, 54, 41, 29, 50, 70, 58,
   142, 72, 70, 15, 72, 54, 29, 22, 145, 29, 127, 29, 85, 58, 101, 34,
   165, 91, 46, 46, 25, 185, 25, 77, 128, 46, 128, 46, 188, 114, 46, 25,
   45, 45, 114, 145, 114, 15, 102, 142, 8, 73, 31, 139, 157, 13, 79, 13,
   114, 150, 8, 90, 91, 123, 69, 82, 132, 8, 18, 10, 102, 103, 114, 103,
   8, 103, 13, 115, 55, 62, 3, 8, 154, 114, 99, 19, 8, 31, 73, 19, 99,
   10, 6, 121, 32, 13, 32, 119, 32, 29, 145, 30, 13, 13, 114, 145, 32,
   1, 123, 39, 29, 31, 69, 31, 140, 72, 72, 25, 25, 123, 25, 123, 8, 4,
   85, 8, 25, 39, 25, 39, 85, 138, 25, 138, 25, 33, 102, 70, 25, 25, 31,
   25, 25, 192, 69, 69, 114, 145, 120, 120, 8, 33, 98, 15, 212, 155, 8,
   101, 8, 8, 98, 68, 155, 102, 132, 120, 30, 25, 123, 123, 101, 25,
   123, 32, 24, 94, 145, 32, 24, 94, 118, 145, 101, 53, 53, 25, 128,
   173, 142, 81, 81, 69, 33, 33, 125, 4, 1, 17, 27, 4, 17, 102, 27, 13,
   25, 128, 71, 13, 39, 53, 13, 53, 47, 39, 23, 128, 53, 39, 47, 39,
   135, 158, 136, 36, 36, 27, 157, 47, 76, 213, 47, 156, 25, 25, 53, 25,
   53, 25, 86, 27, 159, 25, 62, 79, 39, 79, 25, 145, 49, 25, 143, 13,
   114, 150, 130, 94, 102, 39, 4, 39, 61, 77, 228, 22, 25, 47, 119, 205,
   122, 119, 205, 119, 22, 119, 258, 143, 22, 81, 179, 22, 22, 143, 25,
   65, 53, 168, 36, 79, 175, 37, 79, 70, 79, 103, 70, 25, 175, 4, 96,
   96, 49, 128, 138, 96, 22, 62, 47, 95, 105, 95, 62, 95, 62, 142, 103,
   69, 103, 30, 103, 34, 173, 127, 70, 127, 132, 18, 85, 22, 71, 18,
   206, 206, 18, 128, 145, 70, 193, 188, 8, 125, 114, 70, 128, 114, 145,
   102, 25, 12, 108, 102, 94, 10, 102, 1, 102, 124, 22, 22, 118, 132,
   22, 116, 75, 41, 63, 41, 189, 208, 55, 85, 69, 8, 71, 53, 71, 69,



Luby, et al.              Expires March 6, 2006                [Page 35]

Internet-Draft              Raptor FEC Scheme             September 2005


   102, 165, 41, 99, 69, 33, 33, 29, 156, 102, 13, 251, 102, 25, 13,
   109, 102, 164, 102, 164, 102, 25, 29, 228, 29, 259, 179, 222, 95, 94,
   30, 30, 30, 142, 55, 142, 72, 55, 102, 128, 17, 69, 164, 165, 3, 164,
   36, 165, 27, 27, 45, 21, 21, 237, 113, 83, 231, 106, 13, 154, 13,
   154, 128, 154, 148, 258, 25, 154, 128, 3, 27, 10, 145, 145, 21, 146,
   25, 1, 185, 121, 0, 1, 95, 55, 95, 95, 30, 0, 27, 95, 0, 95, 8, 222,
   27, 121, 30, 95, 121, 0, 98, 94, 131, 55, 95, 95, 30, 98, 30, 0, 91,
   145, 66, 179, 66, 58, 175, 29, 0, 31, 173, 146, 160, 39, 53, 28, 123,
   199, 123, 175, 146, 156, 54, 54, 149, 25, 70, 178, 128, 25, 70, 70,
   94, 224, 54, 4, 54, 54, 25, 228, 160, 206, 165, 143, 206, 108, 220,
   234, 160, 13, 169, 103, 103, 103, 91, 213, 222, 91, 103, 91, 103, 31,
   30, 123, 13, 62, 103, 50, 106, 42, 13, 145, 114, 220, 65, 8, 8, 175,
   11, 104, 94, 118, 132, 27, 118, 193, 27, 128, 127, 127, 183, 33, 30,
   29, 103, 128, 61, 234, 165, 41, 29, 193, 33, 207, 41, 165, 165, 55,
   81, 157, 157, 8, 81, 11, 27, 8, 8, 98, 96, 142, 145, 41, 179, 112,
   62, 180, 206, 206, 165, 39, 241, 45, 151, 26, 197, 102, 192, 125,
   128, 67, 128, 69, 128, 197, 33, 125, 102, 13, 103, 25, 30, 12, 30,
   12, 30, 25, 77, 12, 25, 180, 27, 10, 69, 235, 228, 343, 118, 69, 41,
   8, 69, 175, 25, 69, 25, 125, 41, 25, 41, 8, 155, 146, 155, 146, 155,
   206, 168, 128, 157, 27, 273, 211, 211, 168, 11, 173, 154, 77, 173,
   77, 102, 102, 102, 8, 85, 95, 102, 157, 28, 122, 234, 122, 157, 235,
   222, 241, 10, 91, 179, 25, 13, 25, 41, 25, 206, 41, 6, 41, 158, 206,
   206, 33, 296, 296, 33, 228, 69, 8, 114, 148, 33, 29, 66, 27, 27, 30,
   233, 54, 173, 108, 106, 108, 108, 53, 103, 33, 33, 33, 176, 27, 27,
   205, 164, 105, 237, 41, 27, 72, 165, 29, 29, 259, 132, 132, 132, 364,
   71, 71, 27, 94, 160, 127, 51, 234, 55, 27, 95, 94, 165, 55, 55, 41,
   0, 41, 128, 4, 123, 173, 6, 164, 157, 121, 121, 154, 86, 164, 164,
   25, 93, 164, 25, 164, 210, 284, 62, 93, 30, 25, 25, 30, 30, 260, 130,
   25, 125, 57, 53, 166, 166, 166, 185, 166, 158, 94, 113, 215, 159, 62,
   99, 21, 172, 99, 184, 62, 259, 4, 21, 21, 77, 62, 173, 41, 146, 6,
   41, 128, 121, 41, 11, 121, 103, 159, 164, 175, 206, 91, 103, 164, 72,
   25, 129, 72, 206, 129, 33, 103, 102, 102, 29, 13, 11, 251, 234, 135,
   31, 8, 123, 65, 91, 121, 129, 65, 243, 10, 91, 8, 65, 70, 228, 220,
   243, 91, 10, 10, 30, 178, 91, 178, 33, 21, 25, 235, 165, 11, 161,
   158, 27, 27, 30, 128, 75, 36, 30, 36, 36, 173, 25, 33, 178, 112, 162,
   112, 112, 112, 162, 33, 33, 178, 123, 123, 39, 106, 91, 106, 106,
   158, 106, 106, 284, 39, 230, 21, 228, 11, 21, 228, 159, 241, 62, 10,
   62, 10, 68, 234, 39, 39, 138, 62, 22, 27, 183, 22, 215, 10, 175, 175,
   353, 228, 42, 193, 175, 175, 27, 98, 27, 193, 150, 27, 173, 17, 233,
   233, 25, 102, 123, 152, 242, 108, 4, 94, 176, 13, 41, 219, 17, 151,
   22, 103, 103, 53, 128, 233, 284, 25, 265, 128, 39, 39, 138, 42, 39,
   21, 86, 95, 127, 29, 91, 46, 103, 103, 215, 25, 123, 123, 230, 25,
   193, 180, 30, 60, 30, 242, 136, 180, 193, 30, 206, 180, 60, 165, 206,
   193, 165, 123, 164, 103, 68, 25, 70, 91, 25, 82, 53, 82, 186, 53, 82,
   53, 25, 30, 282, 91, 13, 234, 160, 160, 126, 149, 36, 36, 160, 149,
   178, 160, 39, 294, 149, 149, 160, 39, 95, 221, 186, 106, 178, 316,
   267, 53, 53, 164, 159, 164, 165, 94, 228, 53, 52, 178, 183, 53, 294,
   128, 55, 140, 294, 25, 95, 366, 15, 304, 13, 183, 77, 230, 6, 136,



Luby, et al.              Expires March 6, 2006                [Page 36]

Internet-Draft              Raptor FEC Scheme             September 2005


   235, 121, 311, 273, 36, 158, 235, 230, 98, 201, 165, 165, 165, 91,
   175, 248, 39, 185, 128, 39, 39, 128, 313, 91, 36, 219, 130, 25, 130,
   234, 234, 130, 234, 121, 205, 304, 94, 77, 64, 259, 60, 60, 60, 77,
   242, 60, 145, 95, 270, 18, 91, 199, 159, 91, 235, 58, 249, 26, 123,
   114, 29, 15, 191, 15, 30, 55, 55, 347, 4, 29, 15, 4, 341, 93, 7, 30,
   23, 7, 121, 266, 178, 261, 70, 169, 25, 25, 158, 169, 25, 169, 270,
   270, 13, 128, 327, 103, 55, 128, 103, 136, 159, 103, 327, 41, 32,
   111, 111, 114, 173, 215, 173, 25, 173, 180, 114, 173, 173, 98, 93,
   25, 160, 157, 159, 160, 159, 159, 160, 320, 35, 193, 221, 33, 36,
   136, 248, 91, 215, 125, 215, 156, 68, 125, 125, 1, 287, 123, 94, 30,
   184, 13, 30, 94, 123, 206, 12, 206, 289, 128, 122, 184, 128, 289,
   178, 29, 26, 206, 178, 65, 206, 128, 192, 102, 197, 36, 94, 94, 155,
   10, 36, 121, 280, 121, 368, 192, 121, 121, 179, 121, 36, 54, 192,
   121, 192, 197, 118, 123, 224, 118, 10, 192, 10, 91, 269, 91, 49, 206,
   184, 185, 62, 8, 49, 289, 30, 5, 55, 30, 42, 39, 220, 298, 42, 347,
   42, 234, 42, 70, 42, 55, 321, 129, 172, 173, 172, 13, 98, 129, 325,
   235, 284, 362, 129, 233, 345, 175, 261, 175, 60, 261, 58, 289, 99,
   99, 99, 206, 99, 36, 175, 29, 25, 432, 125, 264, 168, 173, 69, 158,
   273, 179, 164, 69, 158, 69, 8, 95, 192, 30, 164, 101, 44, 53, 273,
   335, 273, 53, 45, 128, 45, 234, 123, 105, 103, 103, 224, 36, 90, 211,
   282, 264, 91, 228, 91, 166, 264, 228, 398, 50, 101, 91, 264, 73, 36,
   25, 73, 50, 50, 242, 36, 36, 58, 165, 204, 353, 165, 125, 320, 128,
   298, 298, 180, 128, 60, 102, 30, 30, 53, 179, 234, 325, 234, 175, 21,
   250, 215, 103, 21, 21, 250, 91, 211, 91, 313, 301, 323, 215, 228,
   160, 29, 29, 81, 53, 180, 146, 248, 66, 159, 39, 98, 323, 98, 36, 95,
   218, 234, 39, 82, 82, 230, 62, 13, 62, 230, 13, 30, 98, 0, 8, 98, 8,
   98, 91, 267, 121, 197, 30, 78, 27, 78, 102, 27, 298, 160, 103, 264,
   264, 264, 175, 17, 273, 273, 165, 31, 160, 17, 99, 17, 99, 234, 31,
   17, 99, 36, 26, 128, 29, 214, 353, 264, 102, 36, 102, 264, 264, 273,
   273, 4, 16, 138, 138, 264, 128, 313, 25, 420, 60, 10, 280, 264, 60,
   60, 103, 178, 125, 178, 29, 327, 29, 36, 30, 36, 4, 52, 183, 183,
   173, 52, 31, 173, 31, 158, 31, 158, 31, 9, 31, 31, 353, 31, 353, 173,
   415, 9, 17, 222, 31, 103, 31, 165, 27, 31, 31, 165, 27, 27, 206, 31,
   31, 4, 4, 30, 4, 4, 264, 185, 159, 310, 273, 310, 173, 40, 4, 173, 4,
   173, 4, 250, 250, 62, 188, 119, 250, 233, 62, 121, 105, 105, 54, 103,
   111, 291, 236, 236, 103, 297, 36, 26, 316, 69, 183, 158, 206, 129,
   160, 129, 184, 55, 179, 279, 11, 179, 347, 160, 184, 129, 179, 351,
   179, 353, 179, 129, 129, 351, 11, 111, 93, 93, 235, 103, 173, 53, 93,
   50, 111, 86, 123, 94, 36, 183, 60, 55, 55, 178, 219, 253, 321, 178,
   235, 235, 183, 183, 204, 321, 219, 160, 193, 335, 121, 70, 69, 295,
   159, 297, 231, 121, 231, 136, 353, 136, 121, 279, 215, 366, 215, 353,
   159, 353, 353, 103, 31, 31, 298, 298, 30, 30, 165, 273, 25, 219, 35,
   165, 259, 54, 36, 54, 54, 165, 71, 250, 327, 13, 289, 165, 196, 165,
   165, 94, 233, 165, 94, 60, 165, 96, 220, 166, 271, 158, 397, 122, 53,
   53, 137, 280, 272, 62, 30, 30, 30, 105, 102, 67, 140, 8, 67, 21, 270,
   298, 69, 173, 298, 91, 179, 327, 86, 179, 88, 179, 179, 55, 123, 220,
   233, 94, 94, 175, 13, 53, 13, 154, 191, 74, 83, 83, 325, 207, 83, 74,
   83, 325, 74, 316, 388, 55, 55, 364, 55, 183, 434, 273, 273, 273, 164,



Luby, et al.              Expires March 6, 2006                [Page 37]

Internet-Draft              Raptor FEC Scheme             September 2005


   213, 11, 213, 327, 321, 21, 352, 185, 103, 13, 13, 55, 30, 323, 123,
   178, 435, 178, 30, 175, 175, 30, 481, 527, 175, 125, 232, 306, 232,
   206, 306, 364, 206, 270, 206, 232, 10, 30, 130, 160, 130, 347, 240,
   30, 136, 130, 347, 136, 279, 298, 206, 30, 103, 273, 241, 70, 206,
   306, 434, 206, 94, 94, 156, 161, 321, 321, 64, 161, 13, 183, 183, 83,
   161, 13, 169, 13, 159, 36, 173, 159, 36, 36, 230, 235, 235, 159, 159,
   335, 312, 42, 342, 264, 39, 39, 39, 34, 298, 36, 36, 252, 164, 29,
   493, 29, 387, 387, 435, 493, 132, 273, 105, 132, 74, 73, 206, 234,
   273, 206, 95, 15, 280, 280, 280, 280, 397, 273, 273, 242, 397, 280,
   397, 397, 397, 273, 397, 280, 230, 137, 353, 67, 81, 137, 137, 353,
   259, 312, 114, 164, 164, 25, 77, 21, 77, 165, 30, 30, 231, 234, 121,
   234, 312, 121, 364, 136, 123, 123, 136, 123, 136, 150, 264, 285, 30,
   166, 93, 30, 39, 224, 136, 39, 355, 355, 397, 67, 67, 25, 67, 25,
   298, 11, 67, 264, 374, 99, 150, 321, 67, 70, 67, 295, 150, 29, 321,
   150, 70, 29, 142, 355, 311, 173, 13, 253, 103, 114, 114, 70, 192, 22,
   128, 128, 183, 184, 70, 77, 215, 102, 292, 30, 123, 279, 292, 142,
   33, 215, 102, 468, 123, 468, 473, 30, 292, 215, 30, 213, 443, 473,
   215, 234, 279, 279, 279, 279, 265, 443, 206, 66, 313, 34, 30, 206,
   30, 51, 15, 206, 41, 434, 41, 398, 67, 30, 301, 67, 36, 3, 285, 437,
   136, 136, 22, 136, 145, 365, 323, 323, 145, 136, 22, 453, 99, 323,
   353, 9, 258, 323, 231, 128, 231, 382, 150, 420, 39, 94, 29, 29, 353,
   22, 22, 347, 353, 39, 29, 22, 183, 8, 284, 355, 388, 284, 60, 64, 99,
   60, 64, 150, 95, 150, 364, 150, 95, 150, 6, 236, 383, 544, 81, 206,
   388, 206, 58, 159, 99, 231, 228, 363, 363, 121, 99, 121, 121, 99,
   422, 544, 273, 173, 121, 427, 102, 121, 235, 284, 179, 25, 197, 25,
   179, 511, 70, 368, 70, 25, 388, 123, 368, 159, 213, 410, 159, 236,
   127, 159, 21, 373, 184, 424, 327, 250, 176, 176, 175, 284, 316, 176,
   284, 327, 111, 250, 284, 175, 175, 264, 111, 176, 219, 111, 427, 427,
   176, 284, 427, 353, 428, 55, 184, 493, 158, 136, 99, 287, 264, 334,
   264, 213, 213, 292, 481, 93, 264, 292, 295, 295, 6, 367, 279, 173,
   308, 285, 158, 308, 335, 299, 137, 137, 572, 41, 137, 137, 41, 94,
   335, 220, 36, 224, 420, 36, 265, 265, 91, 91, 71, 123, 264, 91, 91,
   123, 107, 30, 22, 292, 35, 241, 356, 298, 14, 298, 441, 35, 121, 71,
   63, 130, 63, 488, 363, 71, 63, 307, 194, 71, 71, 220, 121, 125, 71,
   220, 71, 71, 71, 71, 235, 265, 353, 128, 155, 128, 420, 400, 130,
   173, 183, 183, 184, 130, 173, 183, 13, 183, 130, 130, 183, 183, 353,
   353, 183, 242, 183, 183, 306, 324, 324, 321, 306, 321, 6, 6, 128,
   306, 242, 242, 306, 183, 183, 6, 183, 321, 486, 183, 164, 30, 78,
   138, 158, 138, 34, 206, 362, 55, 70, 67, 21, 375, 136, 298, 81, 298,
   298, 298, 230, 121, 30, 230, 311, 240, 311, 311, 158, 204, 136, 136,
   184, 136, 264, 311, 311, 312, 312, 72, 311, 175, 264, 91, 175, 264,
   121, 461, 312, 312, 238, 475, 350, 512, 350, 312, 313, 350, 312, 366,
   294, 30, 253, 253, 253, 388, 158, 388, 22, 388, 22, 388, 103, 321,
   321, 253, 7, 437, 103, 114, 242, 114, 114, 242, 114, 114, 242, 242,
   242, 306, 242, 114, 7, 353, 335, 27, 241, 299, 312, 364, 506, 409,
   94, 462, 230, 462, 243, 230, 175, 175, 462, 461, 230, 428, 426, 175,
   175, 165, 175, 175, 372, 183, 572, 102, 85, 102, 538, 206, 376, 85,
   85, 284, 85, 85, 284, 398, 83, 160, 265, 308, 398, 310, 583, 289,



Luby, et al.              Expires March 6, 2006                [Page 38]

Internet-Draft              Raptor FEC Scheme             September 2005


   279, 273, 285, 490, 490, 211, 292, 292, 158, 398, 30, 220, 169, 368,
   368, 368, 169, 159, 368, 93, 368, 368, 93, 169, 368, 368, 443, 368,
   298, 443, 368, 298, 538, 345, 345, 311, 178, 54, 311, 215, 178, 175,
   222, 264, 475, 264, 264, 475, 478, 289, 63, 236, 63, 299, 231, 296,
   397, 299, 158, 36, 164, 164, 21, 492, 21, 164, 21, 164, 403, 26, 26,
   588, 179, 234, 169, 465, 295, 67, 41, 353, 295, 538, 161, 185, 306,
   323, 68, 420, 323, 82, 241, 241, 36, 53, 493, 301, 292, 241, 250, 63,
   63, 103, 442, 353, 185, 353, 321, 353, 185, 353, 353, 185, 409, 353,
   589, 34, 271, 271, 34, 86, 34, 34, 353, 353, 39, 414, 4, 95, 95, 4,
   225, 95, 4, 121, 30, 552, 136, 159, 159, 514, 159, 159, 54, 514, 206,
   136, 206, 159, 74, 235, 235, 312, 54, 312, 42, 156, 422, 629, 54,
   465, 265, 165, 250, 35, 165, 175, 659, 175, 175, 8, 8, 8, 8, 206,
   206, 206, 50, 435, 206, 432, 230, 230, 234, 230, 94, 299, 299, 285,
   184, 41, 93, 299, 299, 285, 41, 285, 158, 285, 206, 299, 41, 36, 396,
   364, 364, 120, 396, 514, 91, 382, 538, 807, 717, 22, 93, 412, 54,
   215, 54, 298, 308, 148, 298, 148, 298, 308, 102, 656, 6, 148, 745,
   128, 298, 64, 407, 273, 41, 172, 64, 234, 250, 398, 181, 445, 95,
   236, 441, 477, 504, 102, 196, 137, 364, 60, 453, 137, 364, 367, 334,
   364, 299, 196, 397, 630, 589, 589, 196, 646, 337, 235, 128, 128, 343,
   289, 235, 324, 427, 324, 58, 215, 215, 461, 425, 461, 387, 440, 285,
   440, 440, 285, 387, 632, 325, 325, 440, 461, 425, 425, 387, 627, 191,
   285, 440, 308, 55, 219, 280, 308, 265, 538, 183, 121, 30, 236, 206,
   30, 455, 236, 30, 30, 705, 83, 228, 280, 468, 132, 8, 132, 132, 128,
   409, 173, 353, 132, 409, 35, 128, 450, 137, 398, 67, 432, 423, 235,
   235, 388, 306, 93, 93, 452, 300, 190, 13, 452, 388, 30, 452, 13, 30,
   13, 30, 306, 362, 234, 721, 635, 809, 784, 67, 498, 498, 67, 353,
   635, 67, 183, 159, 445, 285, 183, 53, 183, 445, 265, 432, 57, 420,
   432, 420, 477, 327, 55, 60, 105, 183, 218, 104, 104, 475, 239, 582,
   151, 239, 104, 732, 41, 26, 784, 86, 300, 215, 36, 64, 86, 86, 675,
   294, 64, 86, 528, 550, 493, 565, 298, 230, 312, 295, 538, 298, 295,
   230, 54, 374, 516, 441, 54, 54, 323, 401, 401, 382, 159, 837, 159,
   54, 401, 592, 159, 401, 417, 610, 264, 150, 323, 452, 185, 323, 323,
   185, 403, 185, 423, 165, 425, 219, 407, 270, 231, 99, 93, 231, 631,
   756, 71, 364, 434, 213, 86, 102, 434, 102, 86, 23, 71, 335, 164, 323,
   409, 381, 4, 124, 41, 424, 206, 41, 124, 41, 41, 703, 635, 124, 493,
   41, 41, 487, 492, 124, 175, 124, 261, 600, 488, 261, 488, 261, 206,
   677, 261, 308, 723, 908, 704, 691, 723, 488, 488, 441, 136, 476, 312,
   136, 550, 572, 728, 550, 22, 312, 312, 22, 55, 413, 183, 280, 593,
   191, 36, 36, 427, 36, 695, 592, 19, 544, 13, 468, 13, 544, 72, 437,
   321, 266, 461, 266, 441, 230, 409, 93, 521, 521, 345, 235, 22, 142,
   150, 102, 569, 235, 264, 91, 521, 264, 7, 102, 7, 498, 521, 235, 537,
   235, 6, 241, 420, 420, 631, 41, 527, 103, 67, 337, 62, 264, 527, 131,
   67, 174, 263, 264, 36, 36, 263, 581, 253, 465, 160, 286, 91, 160, 55,
   4, 4, 631, 631, 608, 365, 465, 294, 427, 427, 335, 669, 669, 129, 93,
   93, 93, 93, 74, 66, 758, 504, 347, 130, 505, 504, 143, 505, 550, 222,
   13, 352, 529, 291, 538, 50, 68, 269, 130, 295, 130, 511, 295, 295,
   130, 486, 132, 61, 206, 185, 368, 669, 22, 175, 492, 207, 373, 452,
   432, 327, 89, 550, 496, 611, 527, 89, 527, 496, 550, 516, 516, 91,



Luby, et al.              Expires March 6, 2006                [Page 39]

Internet-Draft              Raptor FEC Scheme             September 2005


   136, 538, 264, 264, 124, 264, 264, 264, 264, 264, 535, 264, 150, 285,
   398, 285, 582, 398, 475, 81, 694, 694, 64, 81, 694, 234, 607, 723,
   513, 234, 64, 581, 64, 124, 64, 607, 234, 723, 717, 367, 64, 513,
   607, 488, 183, 488, 450, 183, 550, 286, 183, 363, 286, 414, 67, 449,
   449, 366, 215, 235, 95, 295, 295, 41, 335, 21, 445, 225, 21, 295,
   372, 749, 461, 53, 481, 397, 427, 427, 427, 714, 481, 714, 427, 717,
   165, 245, 486, 415, 245, 415, 486, 274, 415, 441, 456, 300, 548, 300,
   422, 422, 757, 11, 74, 430, 430, 136, 409, 430, 749, 191, 819, 592,
   136, 364, 465, 231, 231, 918, 160, 589, 160, 160, 465, 465, 231, 157,
   538, 538, 259, 538, 326, 22, 22, 22, 179, 22, 22, 550, 179, 287, 287,
   417, 327, 498, 498, 287, 488, 327, 538, 488, 583, 488, 287, 335, 287,
   335, 287, 41, 287, 335, 287, 327, 441, 335, 287, 488, 538, 327, 498,
   8, 8, 374, 8, 64, 427, 8, 374, 417, 760, 409, 373, 160, 423, 206,
   160, 106, 499, 160, 271, 235, 160, 590, 353, 695, 478, 619, 590, 353,
   13, 63, 189, 420, 605, 427, 643, 121, 280, 415, 121, 415, 595, 417,
   121, 398, 55, 330, 463, 463, 123, 353, 330, 582, 309, 582, 582, 405,
   330, 550, 405, 582, 353, 309, 308, 60, 353, 7, 60, 71, 353, 189, 183,
   183, 183, 582, 755, 189, 437, 287, 189, 183, 668, 481, 384, 384, 481,
   481, 481, 477, 582, 582, 499, 650, 481, 121, 461, 231, 36, 235, 36,
   413, 235, 209, 36, 689, 114, 353, 353, 235, 592, 36, 353, 413, 209,
   70, 308, 70, 699, 308, 70, 213, 292, 86, 689, 465, 55, 508, 128, 452,
   29, 41, 681, 573, 352, 21, 21, 648, 648, 69, 509, 409, 21, 264, 21,
   509, 514, 514, 409, 21, 264, 443, 443, 427, 160, 433, 663, 433, 231,
   646, 185, 482, 646, 433, 13, 398, 172, 234, 42, 491, 172, 234, 234,
   832, 775, 172, 196, 335, 822, 461, 298, 461, 364, 1120, 537, 169,
   169, 364, 694, 219, 612, 231, 740, 42, 235, 321, 279, 960, 279, 353,
   492, 159, 572, 321, 159, 287, 353, 287, 287, 206, 206, 321, 287, 159,
   321, 492, 159, 55, 572, 600, 270, 492, 784, 173, 91, 91, 443, 443,
   582, 261, 497, 572, 91, 555, 352, 206, 261, 555, 285, 91, 555, 497,
   83, 91, 619, 353, 488, 112, 4, 592, 295, 295, 488, 235, 231, 769,
   568, 581, 671, 451, 451, 483, 299, 1011, 432, 422, 207, 106, 701,
   508, 555, 508, 555, 125, 870, 555, 589, 508, 125, 749, 482, 125, 125,
   130, 544, 643, 643, 544, 488, 22, 643, 130, 335, 544, 22, 130, 544,
   544, 488, 426, 426, 4, 180, 4, 695, 35, 54, 433, 500, 592, 433, 262,
   94, 401, 401, 106, 216, 216, 106, 521, 102, 462, 518, 271, 475, 365,
   193, 648, 206, 424, 206, 193, 206, 206, 424, 299, 590, 590, 364, 621,
   67, 538, 488, 567, 51, 51, 513, 194, 81, 488, 486, 289, 567, 563,
   749, 563, 338, 338, 502, 563, 822, 338, 563, 338, 502, 201, 230, 201,
   533, 445, 175, 201, 175, 13, 85, 960, 103, 85, 175, 30, 445, 445,
   175, 573, 196, 877, 287, 356, 678, 235, 489, 312, 572, 264, 717, 138,
   295, 6, 295, 523, 55, 165, 165, 295, 138, 663, 6, 295, 6, 353, 138,
   6, 138, 169, 129, 784, 12, 129, 194, 605, 784, 445, 234, 627, 563,
   689, 627, 647, 570, 627, 570, 647, 206, 234, 215, 234, 816, 627, 816,
   234, 627, 215, 234, 627, 264, 427, 427, 30, 424, 161, 161, 916, 740,
   180, 616, 481, 514, 383, 265, 481, 164, 650, 121, 582, 689, 420, 669,
   589, 420, 788, 549, 165, 734, 280, 224, 146, 681, 788, 184, 398, 784,
   4, 398, 417, 417, 398, 636, 784, 417, 81, 398, 417, 81, 185, 827,
   420, 241, 420, 41, 185, 185, 718, 241, 101, 185, 185, 241, 241, 241,



Luby, et al.              Expires March 6, 2006                [Page 40]

Internet-Draft              Raptor FEC Scheme             September 2005


   241, 241, 185, 324, 420, 420, 1011, 420, 827, 241, 184, 563, 241,
   183, 285, 529, 285, 808, 822, 891, 822, 488, 285, 486, 619, 55, 869,
   39, 567, 39, 289, 203, 158, 289, 710, 818, 158, 818, 355, 29, 409,
   203, 308, 648, 792, 308, 308, 91, 308, 6, 592, 792, 106, 106, 308,
   41, 178, 91, 751, 91, 259, 734, 166, 36, 327, 166, 230, 205, 205,
   172, 128, 230, 432, 623, 838, 623, 432, 278, 432, 42, 916, 432, 694,
   623, 352, 452, 93, 314, 93, 93, 641, 88, 970, 914, 230, 61, 159, 270,
   159, 493, 159, 755, 159, 409, 30, 30, 836, 128, 241, 99, 102, 984,
   538, 102, 102, 273, 639, 838, 102, 102, 136, 637, 508, 627, 285, 465,
   327, 327, 21, 749, 327, 749, 21, 845, 21, 21, 409, 749, 1367, 806,
   616, 714, 253, 616, 714, 714, 112, 375, 21, 112, 375, 375, 51, 51,
   51, 51, 393, 206, 870, 713, 193, 802, 21, 1061, 42, 382, 42, 543,
   876, 42, 876, 382, 696, 543, 635, 490, 353, 353, 417, 64, 1257, 271,
   64, 377, 127, 127, 537, 417, 905, 353, 538, 465, 605, 876, 427, 324,
   514, 852, 427, 53, 427, 557, 173, 173, 7, 1274, 563, 31, 31, 31, 745,
   392, 289, 230, 230, 230, 91, 218, 327, 420, 420, 128, 901, 552, 420,
   230, 608, 552, 476, 347, 476, 231, 159, 137, 716, 648, 716, 627, 740,
   718, 679, 679, 6, 718, 740, 6, 189, 679, 125, 159, 757, 1191, 409,
   175, 250, 409, 67, 324, 681, 605, 550, 398, 550, 931, 478, 174, 21,
   316, 91, 316, 654, 409, 425, 425, 699, 61, 699, 321, 698, 321, 698,
   61, 425, 699, 321, 409, 699, 299, 335, 321, 335, 61, 698, 699, 654,
   698, 299, 425, 231, 14, 121, 515, 121, 14, 165, 81, 409, 189, 81,
   373, 465, 463, 1055, 507, 81, 81, 189, 1246, 321, 409, 886, 104, 842,
   689, 300, 740, 380, 656, 656, 832, 656, 380, 300, 300, 206, 187, 175,
   142, 465, 206, 271, 468, 215, 560, 83, 215, 83, 215, 215, 83, 175,
   215, 83, 83, 111, 206, 756, 559, 756, 1367, 206, 559, 1015, 559, 559,
   946, 1015, 548, 559, 756, 1043, 756, 698, 159, 414, 308, 458, 997,
   663, 663, 347, 39, 755, 838, 323, 755, 323, 159, 159, 717, 159, 21,
   41, 128, 516, 159, 717, 71, 870, 755, 159, 740, 717, 374, 516, 740,
   51, 148, 335, 148, 335, 791, 120, 364, 335, 335, 51, 120, 251, 538,
   251, 971, 1395, 538, 78, 178, 538, 538, 918, 129, 918, 129, 538, 538,
   656, 129, 538, 538, 129, 538, 1051, 538, 128, 838, 931, 998, 823,
   1095, 334, 870, 334, 367, 550, 1061, 498, 745, 832, 498, 745, 716,
   498, 498, 128, 997, 832, 716, 832, 130, 642, 616, 497, 432, 432, 432,
   432, 642, 159, 432, 46, 230, 788, 160, 230, 478, 46, 693, 103, 920,
   230, 589, 643, 160, 616, 432, 165, 165, 583, 592, 838, 784, 583, 710,
   6, 583, 583, 6, 35, 230, 838, 592, 710, 6, 589, 230, 838, 30, 592,
   583, 6, 583, 6, 6, 583, 30, 30, 6, 375, 375, 99, 36, 1158, 425, 662,
   417, 681, 364, 375, 1025, 538, 822, 669, 893, 538, 538, 450, 409,
   632, 527, 632, 563, 632, 527, 550, 71, 698, 550, 39, 550, 514, 537,
   514, 537, 111, 41, 173, 592, 173, 648, 173, 173, 173, 1011, 514, 173,
   173, 514, 166, 648, 355, 161, 166, 648, 497, 327, 327, 550, 650, 21,
   425, 605, 555, 103, 425, 605, 842, 836, 1011, 636, 138, 756, 836,
   756, 756, 353, 1011, 636, 636, 1158, 741, 741, 842, 756, 741, 1011,
   677, 1011, 770, 366, 306, 488, 920, 920, 665, 775, 502, 500, 775,
   775, 648, 364, 833, 207, 13, 93, 500, 364, 500, 665, 500, 93, 295,
   183, 1293, 313, 272, 313, 279, 303, 93, 516, 93, 1013, 381, 6, 93,
   93, 303, 259, 643, 168, 673, 230, 1261, 230, 230, 673, 1060, 1079,



Luby, et al.              Expires March 6, 2006                [Page 41]

Internet-Draft              Raptor FEC Scheme             September 2005


   1079, 550, 741, 741, 590, 527, 741, 741, 442, 741, 442, 848, 741,
   590, 925, 219, 527, 925, 335, 442, 590, 239, 590, 590, 590, 239, 527,
   239, 1033, 230, 734, 241, 741, 230, 549, 548, 1015, 1015, 32, 36,
   433, 465, 724, 465, 73, 73, 73, 465, 808, 73, 592, 1430, 250, 154,
   154, 250, 538, 353, 353, 353, 353, 353, 175, 194, 206, 538, 632,
   1163, 960, 175, 175, 538, 452, 632, 1163, 175, 538, 960, 194, 175,
   194, 632, 960, 632, 94, 632, 461, 960, 1163, 1163, 461, 632, 960,
   755, 707, 105, 382, 625, 382, 382, 784, 707, 871, 559, 387, 387, 871,
   784, 559, 784, 88, 36, 570, 314, 1028, 975, 335, 335, 398, 573, 573,
   573, 21, 215, 562, 738, 612, 424, 21, 103, 788, 870, 912, 23, 186,
   757, 73, 818, 23, 73, 563, 952, 262, 563, 137, 262, 1022, 952, 137,
   1273, 442, 952, 604, 137, 308, 384, 913, 235, 325, 695, 398, 95, 668,
   776, 713, 309, 691, 22, 10, 364, 682, 682, 578, 481, 1252, 1072,
   1252, 825, 578, 825, 1072, 1149, 592, 273, 387, 273, 427, 155, 1204,
   50, 452, 50, 1142, 50, 367, 452, 1142, 611, 367, 50, 50, 367, 50,
   1675, 99, 367, 50, 1501, 1099, 830, 681, 689, 917, 1089, 453, 425,
   235, 918, 538, 550, 335, 161, 387, 859, 324, 21, 838, 859, 1123, 21,
   723, 21, 335, 335, 206, 21, 364, 1426, 21, 838, 838, 335, 364, 21,
   21, 859, 920, 838, 838, 397, 81, 639, 397, 397, 588, 933, 933, 784,
   222, 830, 36, 36, 222, 1251, 266, 36, 146, 266, 366, 581, 605, 366,
   22, 966, 681, 681, 433, 730, 1013, 550, 21, 21, 938, 488, 516, 21,
   21, 656, 420, 323, 323, 323, 327, 323, 918, 581, 581, 830, 361, 830,
   364, 259, 364, 496, 496, 364, 691, 705, 691, 475, 427, 1145, 600,
   179, 427, 527, 749, 869, 689, 335, 347, 220, 298, 689, 1426, 183,
   554, 55, 832, 550, 550, 165, 770, 957, 67, 1386, 219, 683, 683, 355,
   683, 355, 355, 738, 355, 842, 931, 266, 325, 349, 256, 1113, 256,
   423, 960, 554, 554, 325, 554, 508, 22, 142, 22, 508, 916, 767, 55,
   1529, 767, 55, 1286, 93, 972, 550, 931, 1286, 1286, 972, 93, 1286,
   1392, 890, 93, 1286, 93, 1286, 972, 374, 931, 890, 808, 779, 975,
   975, 175, 173, 4, 681, 383, 1367, 173, 383, 1367, 383, 173, 175, 69,
   238, 146, 238, 36, 148, 888, 238, 173, 238, 148, 238, 888, 185, 925,
   925, 797, 925, 815, 925, 469, 784, 289, 784, 925, 797, 925, 925,
   1093, 925, 925, 925, 1163, 797, 797, 815, 925, 1093, 784, 636, 663,
   925, 187, 922, 316, 1380, 709, 916, 916, 187, 355, 948, 916, 187,
   916, 916, 948, 948, 916, 355, 316, 316, 334, 300, 1461, 36, 583,
   1179, 699, 235, 858, 583, 699, 858, 699, 1189, 1256, 1189, 699, 797,
   699, 699, 699, 699, 427, 488, 427, 488, 175, 815, 656, 656, 150, 322,
   465, 322, 870, 465, 1099, 582, 665, 767, 749, 635, 749, 600, 1448,
   36, 502, 235, 502, 355, 502, 355, 355, 355, 172, 355, 355, 95, 866,
   425, 393, 1165, 42, 42, 42, 393, 939, 909, 909, 836, 552, 424, 1333,
   852, 897, 1426, 1333, 1446, 1426, 997, 1011, 852, 1198, 55, 32, 239,
   588, 681, 681, 239, 1401, 32, 588, 239, 462, 286, 1260, 984, 1160,
   960, 960, 486, 828, 462, 960, 1199, 581, 850, 663, 581, 751, 581,
   581, 1571, 252, 252, 1283, 264, 430, 264, 430, 430, 842, 252, 745,
   21, 307, 681, 1592, 488, 857, 857, 1161, 857, 857, 857, 138, 374,
   374, 1196, 374, 1903, 1782, 1626, 414, 112, 1477, 1040, 356, 775,
   414, 414, 112, 356, 775, 435, 338, 1066, 689, 689, 1501, 689, 1249,
   205, 689, 765, 220, 308, 917, 308, 308, 220, 327, 387, 838, 917, 917,



Luby, et al.              Expires March 6, 2006                [Page 42]

Internet-Draft              Raptor FEC Scheme             September 2005


   917, 220, 662, 308, 220, 387, 387, 220, 220, 308, 308, 308, 387,
   1009, 1745, 822, 279, 554, 1129, 543, 383, 870, 1425, 241, 870, 241,
   383, 716, 592, 21, 21, 592, 425, 550, 550, 550, 427, 230, 57, 483,
   784, 860, 57, 308, 57, 486, 870, 447, 486, 433, 433, 870, 433, 997,
   486, 443, 433, 433, 997, 486, 1292, 47, 708, 81, 895, 394, 81, 935,
   81, 81, 81, 374, 986, 916, 1103, 1095, 465, 495, 916, 667, 1745, 518,
   220, 1338, 220, 734, 1294, 741, 166, 828, 741, 741, 1165, 1371, 1371,
   471, 1371, 647, 1142, 1878, 1878, 1371, 1371, 822, 66, 327, 158, 427,
   427, 465, 465, 676, 676, 30, 30, 676, 676, 893, 1592, 93, 455, 308,
   582, 695, 582, 629, 582, 85, 1179, 85, 85, 1592, 1179, 280, 1027,
   681, 398, 1027, 398, 295, 784, 740, 509, 425, 968, 509, 46, 833, 842,
   401, 184, 401, 464, 6, 1501, 1501, 550, 538, 883, 538, 883, 883, 883,
   1129, 550, 550, 333, 689, 948, 21, 21, 241, 2557, 2094, 273, 308, 58,
   863, 893, 1086, 409, 136, 1086, 592, 592, 830, 830, 883, 830, 277,
   68, 689, 902, 277, 453, 507, 129, 689, 630, 664, 550, 128, 1626,
   1626, 128, 902, 312, 589, 755, 755, 589, 755, 407, 1782, 589, 784,
   1516, 1118, 407, 407, 1447, 589, 235, 755, 1191, 235, 235, 407, 128,
   589, 1118, 21, 383, 1331, 691, 481, 383, 1129, 1129, 1261, 1104,
   1378, 1129, 784, 1129, 1261, 1129, 947, 1129, 784, 784, 1129, 1129,
   35, 1104, 35, 866, 1129, 1129, 64, 481, 730, 1260, 481, 970, 481,
   481, 481, 481, 863, 481, 681, 699, 863, 486, 681, 481, 481, 55, 55,
   235, 1364, 944, 632, 822, 401, 822, 952, 822, 822, 99, 550, 2240,
   550, 70, 891, 860, 860, 550, 550, 916, 1176, 1530, 425, 1530, 916,
   628, 1583, 916, 628, 916, 916, 628, 628, 425, 916, 1062, 1265, 916,
   916, 916, 280, 461, 916, 916, 1583, 628, 1062, 916, 916, 677, 1297,
   924, 1260, 83, 1260, 482, 433, 234, 462, 323, 1656, 997, 323, 323,
   931, 838, 931, 1933, 1391, 367, 323, 931, 1391, 1391, 103, 1116,
   1116, 1116, 769, 1195, 1218, 312, 791, 312, 741, 791, 997, 312, 334,
   334, 312, 287, 287, 633, 1397, 1426, 605, 1431, 327, 592, 705, 1194,
   592, 1097, 1118, 1503, 1267, 1267, 1267, 618, 1229, 734, 1089, 785,
   1089, 1129, 1148, 1148, 1089, 915, 1148, 1129, 1148, 1011, 1011,
   1229, 871, 1560, 1560, 1560, 563, 1537, 1009, 1560, 632, 985, 592,
   1308, 592, 882, 145, 145, 397, 837, 383, 592, 592, 832, 36, 2714,
   2107, 1588, 1347, 36, 36, 1443, 1453, 334, 2230, 1588, 1169, 650,
   1169, 2107, 425, 425, 891, 891, 425, 2532, 679, 274, 274, 274, 325,
   274, 1297, 194, 1297, 627, 314, 917, 314, 314, 1501, 414, 1490, 1036,
   592, 1036, 1025, 901, 1218, 1025, 901, 280, 592, 592, 901, 1461, 159,
   159, 159, 2076, 1066, 1176, 1176, 516, 327, 516, 1179, 1176, 899,
   1176, 1176, 323, 1187, 1229, 663, 1229, 504, 1229, 916, 1229, 916,
   1661, 41, 36, 278, 1027, 648, 648, 648, 1626, 648, 646, 1179, 1580,
   1061, 1514, 1008, 1741, 2076, 1514, 1008, 952, 1089, 427, 952, 427,
   1083, 425, 427, 1089, 1083, 425, 427, 425, 230, 920, 1678, 920, 1678,
   189, 189, 953, 189, 133, 189, 1075, 189, 189, 133, 1264, 725, 189,
   1629, 189, 808, 230, 230, 2179, 770, 230, 770, 230, 21, 21, 784,
   1118, 230, 230, 230, 770, 1118, 986, 808, 916, 30, 327, 918, 679,
   414, 916, 1165, 1355, 916, 755, 733, 433, 1490, 433, 433, 433, 605,
   433, 433, 433, 1446, 679, 206, 433, 21, 2452, 206, 206, 433, 1894,
   206, 822, 206, 2073, 206, 206, 21, 822, 21, 206, 206, 21, 383, 1513,



Luby, et al.              Expires March 6, 2006                [Page 43]

Internet-Draft              Raptor FEC Scheme             September 2005


   375, 1347, 432, 1589, 172, 954, 242, 1256, 1256, 1248, 1256, 1256,
   1248, 1248, 1256, 842, 13, 592, 13, 842, 1291, 592, 21, 175, 13, 592,
   13, 13, 1426, 13, 1541, 445, 808, 808, 863, 647, 219, 1592, 1029,
   1225, 917, 1963, 1129, 555, 1313, 550, 660, 550, 220, 660, 552, 663,
   220, 533, 220, 383, 550, 1278, 1495, 636, 842, 1036, 425, 842, 425,
   1537, 1278, 842, 554, 1508, 636, 554, 301, 842, 792, 1392, 1021, 284,
   1172, 997, 1021, 103, 1316, 308, 1210, 848, 848, 1089, 1089, 848,
   848, 67, 1029, 827, 1029, 2078, 827, 1312, 1029, 827, 590, 872, 1312,
   427, 67, 67, 67, 67, 872, 827, 872, 2126, 1436, 26, 2126, 67, 1072,
   2126, 1610, 872, 1620, 883, 883, 1397, 1189, 555, 555, 563, 1189,
   555, 640, 555, 640, 1089, 1089, 610, 610, 1585, 610, 1355, 610, 1015,
   616, 925, 1015, 482, 230, 707, 231, 888, 1355, 589, 1379, 151, 931,
   1486, 1486, 393, 235, 960, 590, 235, 960, 422, 142, 285, 285, 327,
   327, 442, 2009, 822, 445, 822, 567, 888, 2611, 1537, 323, 55, 1537,
   323, 888, 2611, 323, 1537, 323, 58, 445, 593, 2045, 593, 58, 47, 770,
   842, 47, 47, 842, 842, 648, 2557, 173, 689, 2291, 1446, 2085, 2557,
   2557, 2291, 1780, 1535, 2291, 2391, 808, 691, 1295, 1165, 983, 948,
   2000, 948, 983, 983, 2225, 2000, 983, 983, 705, 948, 2000, 1795,
   1592, 478, 592, 1795, 1795, 663, 478, 1790, 478, 592, 1592, 173, 901,
   312, 4, 1606, 173, 838, 754, 754, 128, 550, 1166, 551, 1480, 550,
   550, 1875, 1957, 1166, 902, 1875, 550, 550, 551, 2632, 551, 1875,
   1875, 551, 2891, 2159, 2632, 3231, 551, 815, 150, 1654, 1059, 1059,
   734, 770, 555, 1592, 555, 2059, 770, 770, 1803, 627, 627, 627, 2059,
   931, 1272, 427, 1606, 1272, 1606, 1187, 1204, 397, 822, 21, 1645,
   263, 263, 822, 263, 1645, 280, 263, 605, 1645, 2014, 21, 21, 1029,
   263, 1916, 2291, 397, 397, 496, 270, 270, 1319, 264, 1638, 264, 986,
   1278, 1397, 1278, 1191, 409, 1191, 740, 1191, 754, 754, 387, 63, 948,
   666, 666, 1198, 548, 63, 1248, 285, 1248, 169, 1248, 1248, 285, 918,
   224, 285, 1426, 1671, 514, 514, 717, 514, 51, 1521, 1745, 51, 605,
   1191, 51, 128, 1191, 51, 51, 1521, 267, 513, 952, 966, 1671, 897, 51,
   71, 592, 986, 986, 1121, 592, 280, 2000, 2000, 1165, 1165, 1165,
   1818, 222, 1818, 1165, 1252, 506, 327, 443, 432, 1291, 1291, 2755,
   1413, 520, 1318, 227, 1047, 828, 520, 347, 1364, 136, 136, 452, 457,
   457, 132, 457, 488, 1087, 1013, 2225, 32, 1571, 2009, 483, 67, 483,
   740, 740, 1013, 2854, 866, 32, 2861, 866, 887, 32, 2444, 740, 32, 32,
   866, 2225, 866, 32, 1571, 2627, 32, 850, 1675, 569, 1158, 32, 1158,
   1797, 2641, 1565, 1158, 569, 1797, 1158, 1797, 55, 1703, 42, 55,
   2562, 675, 1703, 42, 55, 749, 488, 488, 347, 1206, 1286, 1286, 488,
   488, 1206, 1286, 1206, 1286, 550, 550, 1790, 860, 550, 2452, 550,
   550, 2765, 1089, 1633, 797, 2244, 1313, 194, 2129, 194, 194, 194,
   818, 32, 194, 450, 1313, 2387, 194, 1227, 2387, 308, 2232, 526, 476,
   278, 830, 830, 194, 830, 194, 278, 194, 714, 476, 830, 714, 830, 278,
   830, 2532, 1218, 1759, 1446, 960, 1747, 187, 1446, 1759, 960, 105,
   1446, 1446, 1271, 1446, 960, 960, 1218, 1446, 1446, 105, 1446, 960,
   488, 1446, 427, 534, 842, 1969, 2460, 1969, 842, 842, 1969, 427, 941,
   2160, 427, 230, 938, 2075, 1675, 1675, 895, 1675, 34, 129, 1811, 239,
   749, 1957, 2271, 749, 1908, 129, 239, 239, 129, 129, 2271, 2426,
   1355, 1756, 194, 1583, 194, 194, 1583, 194, 1355, 194, 1628, 2221,



Luby, et al.              Expires March 6, 2006                [Page 44]

Internet-Draft              Raptor FEC Scheme             September 2005


   1269, 2425, 1756, 1355, 1355, 1583, 1033, 427, 582, 30, 582, 582,
   935, 1444, 1962, 915, 733, 915, 938, 1962, 767, 353, 1630, 1962,
   1962, 563, 733, 563, 733, 353, 822, 1630, 740, 2076, 2076, 2076, 589,
   589, 2636, 866, 589, 947, 1528, 125, 273, 1058, 1058, 1161, 1635,
   1355, 1161, 1161, 1355, 1355, 650, 1206, 1206, 784, 784, 784, 784,
   784, 412, 461, 412, 2240, 412, 679, 891, 461, 679, 679, 189, 189,
   1933, 1651, 2515, 189, 1386, 538, 1386, 1386, 1187, 1386, 2423, 2601,
   2285, 175, 175, 2331, 194, 3079, 384, 538, 2365, 2294, 538, 2166,
   1841, 3326, 1256, 3923, 976, 85, 550, 550, 1295, 863, 863, 550, 1249,
   550, 1759, 146, 1069, 920, 2633, 885, 885, 1514, 1489, 166, 1514,
   2041, 885, 2456, 885, 2041, 1081, 1948, 362, 550, 94, 324, 2308, 94,
   2386, 94, 550, 874, 1329, 1759, 2280, 1487, 493, 493, 2099, 2599,
   1431, 1086, 1514, 1086, 2099, 1858, 368, 1330, 2599, 1858, 2846,
   2846, 2907, 2846, 713, 713, 1854, 1123, 713, 713, 3010, 1123, 3010,
   538, 713, 1123, 447, 822, 555, 2011, 493, 508, 2292, 555, 1736, 2135,
   2704, 555, 2814, 555, 2000, 555, 555, 822, 914, 327, 679, 327, 648,
   537, 2263, 931, 1496, 537, 1296, 1745, 1592, 1658, 1795, 650, 1592,
   1745, 1745, 1658, 1592, 1745, 1592, 1745, 1658, 1338, 2124, 1592,
   1745, 1745, 1745, 837, 1726, 2897, 1118, 1118, 230, 1118, 1118, 1118,
   1388, 1748, 514, 128, 1165, 931, 514, 2974, 2041, 2387, 2041, 979,
   185, 36, 1269, 550, 173, 812, 36, 1165, 2676, 2562, 1473, 2885, 1982,
   1578, 1578, 383, 383, 2360, 383, 1578, 2360, 1584, 1982, 1578, 1578,
   1578, 2019, 1036, 355, 724, 2023, 205, 303, 355, 1036, 1966, 355,
   1036, 401, 401, 401, 830, 401, 849, 578, 401, 849, 849, 578, 1776,
   1123, 552, 2632, 808, 1446, 1120, 373, 1529, 1483, 1057, 893, 1284,
   1430, 1529, 1529, 2632, 1352, 2063, 1606, 1352, 1606, 2291, 3079,
   2291, 1529, 506, 838, 1606, 1606, 1352, 1529, 1529, 1483, 1529, 1606,
   1529, 259, 902, 259, 902, 612, 612, 284, 398, 2991, 1534, 1118, 1118,
   1118, 1118, 1118, 734, 284, 2224, 398, 734, 284, 734, 398, 3031, 398,
   734, 1707, 2643, 1344, 1477, 475, 1818, 194, 1894, 691, 1528, 1184,
   1207, 1501, 6, 2069, 871, 2069, 3548, 1443, 2069, 2685, 3265, 1350,
   3265, 2069, 2069, 128, 1313, 128, 663, 414, 1313, 414, 2000, 128,
   2000, 663, 1313, 699, 1797, 550, 327, 550, 1526, 699, 327, 1797,
   1526, 550, 550, 327, 550, 1426, 1426, 1426, 2285, 1123, 890, 728,
   1707, 728, 728, 327, 253, 1187, 1281, 1364, 1571, 2170, 755, 3232,
   925, 1496, 2170, 2170, 1125, 443, 902, 902, 925, 755, 2078, 2457,
   902, 2059, 2170, 1643, 1129, 902, 902, 1643, 1129, 606, 36, 103, 338,
   338, 1089, 338, 338, 338, 1089, 338, 36, 340, 1206, 1176, 2041, 833,
   1854, 1916, 1916, 1501, 2132, 1736, 3065, 367, 1934, 833, 833, 833,
   2041, 3017, 2147, 818, 1397, 828, 2147, 398, 828, 818, 1158, 818,
   689, 327, 36, 1745, 2132, 582, 1475, 189, 582, 2132, 1191, 582, 2132,
   1176, 1176, 516, 2610, 2230, 2230, 64, 1501, 537, 1501, 173, 2230,
   2988, 1501, 2694, 2694, 537, 537, 173, 173, 1501, 537, 64, 173, 173,
   64, 2230, 537, 2230, 537, 2230, 2230, 2069, 3142, 1645, 689, 1165,
   1165, 1963, 514, 488, 1963, 1145, 235, 1145, 1078, 1145, 231, 2405,
   552, 21, 57, 57, 57, 1297, 1455, 1988, 2310, 1885, 2854, 2014, 734,
   1705, 734, 2854, 734, 677, 1988, 1660, 734, 677, 734, 677, 677, 734,
   2854, 1355, 677, 1397, 2947, 2386, 1698, 128, 1698, 3028, 2386, 2437,



Luby, et al.              Expires March 6, 2006                [Page 45]

Internet-Draft              Raptor FEC Scheme             September 2005


   2947, 2386, 2643, 2386, 2804, 1188, 335, 746, 1187, 1187, 861, 2519,
   1917, 2842, 1917, 675, 1308, 234, 1917, 314, 314, 2339, 2339, 2592,
   2576, 902, 916, 2339, 916, 2339, 916, 2339, 916, 1089, 1089, 2644,
   1221, 1221, 2446, 308, 308, 2225, 2225, 3192, 2225, 555, 1592, 1592,
   555, 893, 555, 550, 770, 3622, 2291, 2291, 3419, 465, 250, 2842,
   2291, 2291, 2291, 935, 160, 1271, 308, 325, 935, 1799, 1799, 1891,
   2227, 1799, 1598, 112, 1415, 1840, 2014, 1822, 2014, 677, 1822, 1415,
   1415, 1822, 2014, 2386, 2159, 1822, 1415, 1822, 179, 1976, 1033, 179,
   1840, 2014, 1415, 1970, 1970, 1501, 563, 563, 563, 462, 563, 1970,
   1158, 563, 563, 1541, 1238, 383, 235, 1158, 383, 1278, 383, 1898,
   2938, 21, 2938, 1313, 2201, 2059, 423, 2059, 1313, 872, 1313, 2044,
   89, 173, 3327, 1660, 2044, 1623, 173, 1114, 1114, 1592, 1868, 1651,
   1811, 383, 3469, 1811, 1651, 869, 383, 383, 1651, 1651, 3223, 2166,
   3469, 767, 383, 1811, 767, 2323, 3355, 1457, 3341, 2640, 2976, 2323,
   3341, 2323, 2640, 103, 103, 1161, 1080, 2429, 370, 2018, 2854, 2429,
   2166, 2429, 2094, 2207, 871, 1963, 1963, 2023, 2023, 2336, 663, 2893,
   1580, 691, 663, 705, 2046, 2599, 409, 2295, 1118, 2494, 1118, 1950,
   549, 2494, 2453, 2046, 2494, 2453, 2046, 2453, 2046, 409, 1118, 4952,
   2291, 2225, 1894, 1423, 2498, 567, 4129, 1475, 1501, 795, 463, 2084,
   828, 828, 232, 828, 232, 232, 1818, 1818, 666, 463, 232, 220, 220,
   2162, 2162, 833, 4336, 913, 35, 913, 21, 2927, 886, 3037, 383, 886,
   876, 1747, 383, 916, 916, 916, 2927, 916, 1747, 837, 1894, 717, 423,
   481, 1894, 1059, 2262, 3206, 4700, 1059, 3304, 2262, 871, 1831, 871,
   3304, 1059, 1158, 1934, 1158, 756, 1511, 41, 978, 1934, 2603, 720,
   41, 756, 41, 325, 2611, 1158, 173, 1123, 1934, 1934, 1511, 2045,
   2045, 2045, 1423, 3206, 3691, 2512, 3206, 2512, 2000, 1811, 2504,
   2504, 2611, 2437, 2437, 2437, 1455, 893, 150, 2665, 1966, 605, 398,
   2331, 1177, 516, 1962, 4241, 94, 1252, 760, 1292, 1962, 1373, 2000,
   1990, 3684, 42, 1868, 3779, 1811, 1811, 2041, 3010, 5436, 1780, 2041,
   1868, 1811, 1780, 1811, 1868, 1811, 2041, 1868, 1811, 5627, 4274,
   1811, 1868, 4602, 1811, 1811, 1474, 2665, 235, 1474, 2665




















Luby, et al.              Expires March 6, 2006                [Page 46]

Internet-Draft              Raptor FEC Scheme             September 2005


6.  Security Considerations

   The security considerations for this document are the same as they
   are for [2].















































Luby, et al.              Expires March 6, 2006                [Page 47]

Internet-Draft              Raptor FEC Scheme             September 2005


7.  IANA Considerations

   Values of FEC Encoding IDs and FEC Instance IDs are subject to IANA
   registration.  For general guidelines on IANA considerations as they
   apply to this document, see [2].  This document assigns the Fully-
   Specified FEC Encoding ID 1 under the ietf:rmt:fec:encoding name-
   space to "Raptor Code".












































Luby, et al.              Expires March 6, 2006                [Page 48]

Internet-Draft              Raptor FEC Scheme             September 2005


8.  Intellectual Property

   Digital Fountain does have intellectual property rights associated
   with the technology described in this document, and has provided a
   full IPR statement specific to this document to the IETF in
   accordinace with the requirements of the IETF.













































Luby, et al.              Expires March 6, 2006                [Page 49]

Internet-Draft              Raptor FEC Scheme             September 2005


9.  Acknowledgements

   Numerous editorial improvements and clarifications were made to this
   specification duing the review process within 3GPP.  Thanks are due
   to the members of 3GPP Technical Specification Group SA, Working
   Group 4, for these.













































Luby, et al.              Expires March 6, 2006                [Page 50]

Internet-Draft              Raptor FEC Scheme             September 2005


10.  References

10.1.  Normative references

   [1]  Bradner, S., "Key words for use in RFCs to Indicate Requirement
        Levels", BCP 14, RFC 2119, March 1997.

   [2]  Watson, M., "Forward Error Correction (FEC) Building Block",
        draft-ietf-rmt-fec-bb-revised-00 (work in progress), May 2005.

10.2.  Informative references

   [3]  3GPP, "Multimedia Broadcast/Multicast Service (MBMS); Protocols
        and codecs", 3GPP TS 26.346 6.1.0, June 2005.





































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Authors' Addresses

   Michael Luby
   Digital Fountain
   39141 Civic Center Drive
   Suite 300
   Fremont, CA  94538
   U.S.A.

   Email: luby@digitalfountain.com


   Amin Shokrollahi
   EPFL
   Laboratory of Algorithmic Mathematics
   IC-IIF-ALGO
   PSE-A
   Lausanne  1015
   Switzerland

   Email: amin.shokrollahi@epfl.ch


   Mark Watson
   Digital Fountain
   39141 Civic Center Drive
   Suite 300
   Fremont, CA  94538
   U.S.A.

   Email: mark@digitalfountain.com


   Thomas Stockhammer
   Siemens AG
   Mobile Devices
   Munich  81667
   Germany

   Email: stockhammer@nomor.de











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