GENWiki

Premier IT Outsourcing and Support Services within the UK

User Tools

Site Tools

Problem, Formatting or Query -  Send Feedback

Was this page helpful?-10+1


rfc:rfc5053

Network Working Group M. Luby Request for Comments: 5053 Digital Fountain Category: Standards Track A. Shokrollahi

                                                                  EPFL
                                                             M. Watson
                                                      Digital Fountain
                                                        T. Stockhammer
                                                        Nomor Research
                                                          October 2007
     Raptor Forward Error Correction Scheme for Object Delivery

Status of This Memo

 This document specifies an Internet standards track protocol for the
 Internet community, and requests discussion and suggestions for
 improvements.  Please refer to the current edition of the "Internet
 Official Protocol Standards" (STD 1) for the standardization state
 and status of this protocol.  Distribution of this memo is unlimited.

Abstract

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

Luby, et al. Standards Track [Page 1] RFC 5053 Raptor FEC Scheme October 2007

Table of Contents

 1.  Introduction . . . . . . . . . . . . . . . . . . . . . . . . .  3
 2.  Requirements Notation  . . . . . . . . . . . . . . . . . . . .  3
 3.  Formats and Codes  . . . . . . . . . . . . . . . . . . . . . .  3
   3.1.  FEC Payload IDs  . . . . . . . . . . . . . . . . . . . . .  3
   3.2.  FEC Object Transmission Information (OTI)  . . . . . . . .  4
     3.2.1.  Mandatory  . . . . . . . . . . . . . . . . . . . . . .  4
     3.2.2.  Common . . . . . . . . . . . . . . . . . . . . . . . .  4
     3.2.3.  Scheme-Specific  . . . . . . . . . . . . . . . . . . .  5
 4.  Procedures . . . . . . . . . . . . . . . . . . . . . . . . . .  5
   4.1.  Content Delivery Protocol Requirements . . . . . . . . . .  5
   4.2.  Example Parameter Derivation Algorithm . . . . . . . . . .  6
 5.  Raptor FEC Code Specification  . . . . . . . . . . . . . . . .  8
   5.1.  Definitions, Symbols, and Abbreviations  . . . . . . . . .  8
     5.1.1.  Definitions  . . . . . . . . . . . . . . . . . . . . .  8
     5.1.2.  Symbols  . . . . . . . . . . . . . . . . . . . . . . .  9
     5.1.3.  Abbreviations  . . . . . . . . . . . . . . . . . . . . 11
   5.2.  Overview . . . . . . . . . . . . . . . . . . . . . . . . . 11
   5.3.  Object Delivery  . . . . . . . . . . . . . . . . . . . . . 12
     5.3.1.  Source Block Construction  . . . . . . . . . . . . . . 12
     5.3.2.  Encoding Packet Construction . . . . . . . . . . . . . 14
   5.4.  Systematic Raptor Encoder  . . . . . . . . . . . . . . . . 15
     5.4.1.  Encoding Overview  . . . . . . . . . . . . . . . . . . 15
     5.4.2.  First Encoding Step: Intermediate Symbol Generation  . 16
     5.4.3.  Second Encoding Step: LT Encoding  . . . . . . . . . . 20
     5.4.4.  Generators . . . . . . . . . . . . . . . . . . . . . . 21
   5.5.  Example FEC Decoder  . . . . . . . . . . . . . . . . . . . 23
     5.5.1.  General  . . . . . . . . . . . . . . . . . . . . . . . 23
     5.5.2.  Decoding a Source Block  . . . . . . . . . . . . . . . 23
   5.6.  Random Numbers . . . . . . . . . . . . . . . . . . . . . . 28
     5.6.1.  The Table V0 . . . . . . . . . . . . . . . . . . . . . 28
     5.6.2.  The Table V1 . . . . . . . . . . . . . . . . . . . . . 29
   5.7.  Systematic Indices J(K)  . . . . . . . . . . . . . . . . . 30
 6.  Security Considerations  . . . . . . . . . . . . . . . . . . . 43
 7.  IANA Considerations  . . . . . . . . . . . . . . . . . . . . . 43
 8.  Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . 44
 9.  References . . . . . . . . . . . . . . . . . . . . . . . . . . 44
   9.1.  Normative References . . . . . . . . . . . . . . . . . . . 44
   9.2.  Informative References . . . . . . . . . . . . . . . . . . 44

Luby, et al. Standards Track [Page 2] RFC 5053 Raptor FEC Scheme October 2007

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 [RFC5052] and this document follows the
 format prescribed there and uses the terminology of that document.
 Raptor Codes were introduced in [Raptor].  For an overview, see, for
 example, [CCNC].
 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.  For more background
 on the use of Forward Error Correction codes in reliable multicast,
 see [RFC3453].
 The code described here is identical to that described in [MBMS].

2. Requirements Notation

 The key words "MUST", "MUST NOT", "REQUIRED", "SHALL", "SHALL NOT",
 "SHOULD", "SHOULD NOT", "RECOMMENDED", "MAY", and "OPTIONAL" in this
 document are to be interpreted as described in [RFC2119].

3. Formats and Codes

3.1. FEC Payload IDs

 The FEC Payload ID MUST be a 4 octet field defined as follows:
      0                   1                   2                   3
      0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1
     +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
     |     Source Block Number       |      Encoding Symbol ID       |
     +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
                    Figure 1: FEC Payload ID format

Luby, et al. Standards Track [Page 3] RFC 5053 Raptor FEC Scheme October 2007

    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 (OTI)

3.2.1. Mandatory

 The value of the FEC Encoding ID MUST be 1 (one), as assigned by IANA
 (see Section 7).

3.2.2. Common

 The Common FEC Object Transmission Information elements used by this
 FEC Scheme are:
  1. Transfer Length (F)
  1. Encoding Symbol Length (T)
 The Transfer Length is a non-negative integer less than 2^^45.  The
 Encoding Symbol Length is a non-negative integer less than 2^^16.
 The encoded Common FEC Object Transmission Information format is
 shown in Figure 2.
     0                   1                   2                   3
     0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1
    +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
    |                      Transfer Length                          |
    +                               +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
    |                               |           Reserved            |
    +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
    |    Encoding Symbol Length     |
    +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
        Figure 2: Encoded Common FEC OTI for Raptor FEC Scheme
    NOTE 1: The limit of 2^^45 on the transfer length is a consequence
    of the limitation on the symbol size to 2^^16-1, the limitation on
    the number of symbols in a source block to 2^^13, and the

Luby, et al. Standards Track [Page 4] RFC 5053 Raptor FEC Scheme October 2007

    limitation on the number of source blocks to 2^^16.  However, the
    Transfer Length is encoded as a 48-bit field for simplicity.

3.2.3. Scheme-Specific

 The following parameters are carried in the Scheme-Specific FEC
 Object Transmission Information element for this FEC Scheme:
  1. The number of source blocks (Z)
  1. The number of sub-blocks (N)
  1. A symbol alignment parameter (Al)
 These parameters are all non-negative integers.  The encoded Scheme-
 specific Object Transmission Information is a 4-octet field
 consisting of the parameters Z (2 octets), N (1 octet), and Al (1
 octet) as shown in Figure 3.
      0                   1                   2                   3
      0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1
     +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
     |             Z                 |      N        |       Al      |
     +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
 Figure 3: Encoded Scheme-Specific FEC Object Transmission Information
 The encoded FEC Object Transmission Information is a 14-octet field
 consisting of the concatenation of the encoded Common FEC Object
 Transmission Information and the encoded Scheme-Specific FEC Object
 Transmission Information.
 These three parameters define the source block partitioning as
 described in Section 5.3.1.2.

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) that makes use of
 the Raptor FEC Scheme for object delivery.
 The Raptor encoder and decoder for object delivery require the
 following information from the CDP:
  1. The transfer length of the object, F, in bytes

Luby, et al. Standards Track [Page 5] RFC 5053 Raptor FEC Scheme October 2007

  1. A symbol alignment parameter, Al
  1. The symbol size, T, in bytes, which MUST be a multiple of Al
  1. The number of source blocks, Z
  1. The number of sub-blocks in each source block, N
 The Raptor encoder for object delivery additionally requires:
  1. the object to be encoded, F bytes
 The Raptor encoder supplies the CDP with the following information
 for each packet to be sent:
  1. Source Block Number (SBN)
  1. Encoding Symbol ID (ESI)
  1. Encoding symbol(s)
 The CDP MUST communicate this information to the receiver.

4.2. Example Parameter Derivation Algorithm

 This section provides recommendations for the derivation of the three
 transport parameters, T, Z, and N.  This recommendation is based on
 the following input parameters:
  1. F the transfer length of the object, in bytes
  1. W a target on the sub-block size, in bytes
  1. P the maximum packet payload size, in bytes, which is assumed to

be a multiple of Al

  1. Al the symbol alignment parameter, in bytes
  1. Kmax the maximum number of source symbols per source block.
           Note: Section 5.1.2 defines Kmax to be 8192.
  1. Kmin a minimum target on the number of symbols per source block
  1. Gmax a maximum target number of symbols per packet

Luby, et al. Standards Track [Page 6] RFC 5053 Raptor FEC Scheme October 2007

 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 value G represents the maximum number of symbols to be
 transported in a single packet.  The value Kt is the total number of
 symbols required to represent the source data of the object.  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 that 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 5.3.1.2 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.
 The parameter W can be used to generate encoded data that 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 it is possible to implement
 decoding using working memory only slightly larger than W.

Luby, et al. Standards Track [Page 7] RFC 5053 Raptor FEC Scheme October 2007

5. Raptor FEC Code Specification

5.1. Definitions, Symbols, and Abbreviations

5.1.1. Definitions

 For the purposes of this specification, the following terms and
 definitions apply.
    Source block: a block of K source symbols that 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 all the source symbols may be
    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.
    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

Luby, et al. Standards Track [Page 8] RFC 5053 Raptor FEC Scheme October 2007

    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.

5.1.2. Symbols

 i, j, x, h, a, b, d, v, m  represent positive integers.
 ceil(x)  denotes the smallest positive integer that 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 that 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.
 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.

Luby, et al. Standards Track [Page 9] RFC 5053 Raptor FEC Scheme October 2007

 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.
 T'   the sub-symbol size, in bytes.  If the source block is not
      partitioned into sub-blocks, then T' is not relevant.
 F    the transfer length of an object, 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

Luby, et al. Standards Track [Page 10] RFC 5053 Raptor FEC Scheme October 2007

 I_S  denotes the SxS identity matrix.
 0_SxH  denotes the SxH zero matrix.
 a ^^ b  a raised to the power b

5.1.3. 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)

5.2. Overview

 The principal component of the systematic Raptor code is the basic
 encoder described in Section 5.4.  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
 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 specifies the systematic Raptor code encoder.  A number
 of possible decoding algorithms are possible.  An efficient decoding
 algorithm is provided in Section 5.5.
 The construction of the intermediate and repair symbols is based in
 part on a pseudo-random number generator described in
 Section 5.4.4.1.  This generator is based on a fixed set of 512
 random numbers that MUST be available to both sender and receiver.
 These are provided in Section 5.6.

Luby, et al. Standards Track [Page 11] RFC 5053 Raptor FEC Scheme October 2007

 Finally, the construction of the intermediate symbols from the source
 symbols is governed by a 'systematic index', values of which are
 provided in Section 5.7 for source block sizes from 4 source symbols
 to Kmax = 8192 source symbols.

5.3. Object Delivery

5.3.1. Source Block Construction

5.3.1.1. 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 an 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
 described in Section 5.3.1.2 below.

5.3.1.2. 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:
  1. F the transfer length of the object, in bytes
  1. Al a symbol alignment parameter, in bytes
  1. T the symbol size, in bytes, which MUST be a multiple of Al
  1. Z the number of source blocks

Luby, et al. Standards Track [Page 12] RFC 5053 Raptor FEC Scheme October 2007

  1. 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.
 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 Al 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.

Luby, et al. Standards Track [Page 13] RFC 5053 Raptor FEC Scheme October 2007

5.3.2. Encoding Packet Construction

 Each encoding packet contains the following information:
  1. Source Block Number (SBN)
  1. Encoding Symbol ID (ESI)
  1. 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.
 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.
 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 5.4.4.4 as
 follows:

Luby, et al. Standards Track [Page 14] RFC 5053 Raptor FEC Scheme October 2007

    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 5.4, using the intermediate symbols C and the LT encoder
 LTEnc[K, C, (d[i], a[i], b[i])].

5.4. Systematic Raptor Encoder

5.4.1. 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 symbol 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 5.4.2.2.  The K source symbol 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 realized by a Raptor decoding
 process.
 Certain "pre-coding relationships" MUST hold within the L
 intermediate symbols.  Section 5.4.2.3 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 repair 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 5.4.4.4.  Then, each (d,a,b)-triple is used to generate the

Luby, et al. Standards Track [Page 15] RFC 5053 Raptor FEC Scheme October 2007

 corresponding repair symbol from the intermediate symbols using the
 LTEnc[K, C[0],..., C[L-1], (d,a,b)] generator described in
 Section 5.4.4.3.

5.4.2. First Encoding Step: Intermediate Symbol Generation

5.4.2.1. 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 5.4.2.2 using the Trip[]
    generator described in Section 5.4.4.4.
    2.  A set of pre-coding relationships hold within the intermediate
    symbols themselves.  These are defined in Section 5.4.2.3.
 The generation of the L intermediate symbols is then defined in
 Section 5.4.2.4

5.4.2.2. 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 5.4.4.4 as:
    For each i, 0 <= i < K
       (d[i], a[i], b[i]) = Trip[K, i]

5.4.2.3. 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.

Luby, et al. Standards Track [Page 16] RFC 5053 Raptor FEC Scheme October 2007

 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:
 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
    m[k] denote the subsequence of g[.] whose elements have exactly k
    non-zero bits in their binary representation.

Luby, et al. Standards Track [Page 17] RFC 5053 Raptor FEC Scheme October 2007

    m[j,k] denote the jth element of the sequence m[k], where j=0, 1,
    2, ...
 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 m[j,H'] is equal to 1 then C[h+K+S] = C[h+K+S] ^
          C[j].

5.4.2.4. Intermediate Symbols

5.4.2.4.1. 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 5.4.2.3.

5.4.2.4.2. 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 5.4.2.4.1.
 The 'generator matrix' for a code that 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.

Luby, et al. Standards Track [Page 18] RFC 5053 Raptor FEC Scheme October 2007

 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 that are XORed to produce LTEnc[K, (C[0], ..., C[L-1]),
       (d[i], a[i], b[i])].
 Then:
    The first S rows of A are equal to G_LDPC | I_S | 0_SxH.

Luby, et al. Standards Track [Page 19] RFC 5053 Raptor FEC Scheme October 2007

    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 4 below:
               K               S       H
   +-----------------------+-------+-------+
   |                       |       |       |
 S |        G_LDPC         |  I_S  | 0_SxH |
   |                       |       |       |
   +-----------------------+-------+-------+
   |                               |       |
 H |        G_Half                 |  I_H  |
   |                               |       |
   +-------------------------------+-------+
   |                                       |
   |                                       |
 K |                 G_LT                  |
   |                                       |
   |                                       |
   +---------------------------------------+
                        Figure 4: The matrix A
 The intermediate symbols can then be calculated as:
    C = (A^^-1)*D
 The source symbol 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 5.5 be used.  The source symbol
 triples are designed to facilitate efficient decoding of the source
 symbols using that algorithm.

5.4.3. 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 5.4.4.3 to the L intermediate
 symbols C[0], C[1],..., C[L-1] using the triple (d, a, b)=Trip[K,X]
 generated according to Section 5.3.2

Luby, et al. Standards Track [Page 20] RFC 5053 Raptor FEC Scheme October 2007

5.4.4. Generators

5.4.4.1. 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 5.6.
 Then,
    Rand[X, i, m] = (V0[(X + i) % 256] ^ V1[(floor(X/256)+ i) % 256])
    % m

5.4.4.2. 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]
                     +---------+---------+------+
                     | 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

5.4.4.3. LT Encoding Symbol Generator

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

Luby, et al. Standards Track [Page 21] RFC 5053 Raptor FEC Scheme October 2007

    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 5.4.2.3, 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 5.4.2.4.
    (d, a, b) is a source triple determined using the Triple generator
    defined in Section 5.4.4.4, 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:
    While (b >= L) do b = (b + a) % L'
    Let result = C[b].
    For j = 1,...,min(d-1,L-1) do
       b = (b + a) % L'
       While (b >= L) do b = (b + a) % L'
       result = result ^ C[b]
    Return result

5.4.4.4. 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 5.4.2.3
    L' be the smallest prime that is greater than or equal to L

Luby, et al. Standards Track [Page 22] RFC 5053 Raptor FEC Scheme October 2007

    Q = 65521, the largest prime smaller than 2^^16.
    J(K) be the systematic index associated with K, as defined in
    Section 5.7.
 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]
    b = Rand[Y, 2, L']

5.5. Example FEC Decoder

5.5.1. 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.

5.5.2. Decoding a Source Block

5.5.2.1. 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 5.4, 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

Luby, et al. Standards Track [Page 23] RFC 5053 Raptor FEC Scheme October 2007

 encoding symbols for the source block to be decoded are passed to the
 decoder.  Note that, as described in Section 5.3.2, the last source
 symbol of a source packet may have included padding bytes added for
 FEC encoding purposes.  These padding bytes may not be actually
 included in the packet sent and so must be reinserted at the received
 before passing the symbol to the decoder.
 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 also passed to the decoder.  In the case of source symbols,
 the source symbol triples described in Section 5.4.2.2 indicate the
 number and set of intermediate symbols that 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
 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 that 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 reorderings 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.

Luby, et al. Standards Track [Page 24] RFC 5053 Raptor FEC Scheme October 2007

 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.
  1. 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']].
  1. 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'].
  1. 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]].
 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.

5.5.2.2. 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.

Luby, et al. Standards Track [Page 25] RFC 5053 Raptor FEC Scheme October 2007

    (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 5 illustrates the submatrices of A.  At the beginning of the
 first phase, V = A.  In each step, a row of A is chosen.
 +-----------+-----------------+---------+
 |           |                 |         |
 |     I     |    All Zeros    |         |
 |           |                 |         |
 +-----------+-----------------+    U    |
 |           |                 |         |
 |           |                 |         |
 | All Zeros |       V         |         |
 |           |                 |         |
 |           |                 |         |
 +-----------+-----------------+---------+
             Figure 5: 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.  Whenever there are non-
 zero rows in V, then the next step starts by choosing a row of A as
 follows:

Luby, et al. Standards Track [Page 26] RFC 5053 Raptor FEC Scheme October 2007

 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 V.

 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 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.

5.5.2.3. 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.

5.5.2.4. Third Phase

 After the second phase, the only portion of A that 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.

Luby, et al. Standards Track [Page 27] RFC 5053 Raptor FEC Scheme October 2007

5.5.2.5. 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 5.5.2.1, 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
 associated with all source symbols are computed according to
 Section 5.4.2.2.  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.

5.6. Random Numbers

 The two tables V0 and V1 described in Section 5.4.4.1 are given
 below.  Each entry is a 32-bit integer in decimal representation.

5.6.1. 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,

Luby, et al. Standards Track [Page 28] RFC 5053 Raptor FEC Scheme October 2007

 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,
 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

5.6.2. 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,

Luby, et al. Standards Track [Page 29] RFC 5053 Raptor FEC Scheme October 2007

 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,
 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

5.7. Systematic Indices 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 5.4.2.4.2
 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,

Luby, et al. Standards Track [Page 30] RFC 5053 Raptor FEC Scheme October 2007

 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,
 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,

Luby, et al. Standards Track [Page 31] RFC 5053 Raptor FEC Scheme October 2007

 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,
 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,

Luby, et al. Standards Track [Page 32] RFC 5053 Raptor FEC Scheme October 2007

 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,
 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,

Luby, et al. Standards Track [Page 33] RFC 5053 Raptor FEC Scheme October 2007

 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,
 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,

Luby, et al. Standards Track [Page 34] RFC 5053 Raptor FEC Scheme October 2007

 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,
 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,

Luby, et al. Standards Track [Page 35] RFC 5053 Raptor FEC Scheme October 2007

 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,
 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,

Luby, et al. Standards Track [Page 36] RFC 5053 Raptor FEC Scheme October 2007

 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,
 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,

Luby, et al. Standards Track [Page 37] RFC 5053 Raptor FEC Scheme October 2007

 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,
 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,

Luby, et al. Standards Track [Page 38] RFC 5053 Raptor FEC Scheme October 2007

 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,
 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,

Luby, et al. Standards Track [Page 39] RFC 5053 Raptor FEC Scheme October 2007

 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,
 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,

Luby, et al. Standards Track [Page 40] RFC 5053 Raptor FEC Scheme October 2007

 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,
 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,

Luby, et al. Standards Track [Page 41] RFC 5053 Raptor FEC Scheme October 2007

 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,
 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. Standards Track [Page 42] RFC 5053 Raptor FEC Scheme October 2007

6. Security Considerations

 Data delivery can be subject to denial-of-service attacks by
 attackers that send corrupted packets that are accepted as legitimate
 by receivers.  This is particularly a concern for multicast delivery
 because a corrupted packet may be injected into the session close to
 the root of the multicast tree, in which case, the corrupted packet
 will arrive at many receivers.  This is particularly a concern when
 the code described in this document is used because the use of even
 one corrupted packet containing encoding data may result in the
 decoding of an object that is completely corrupted and unusable.  It
 is thus RECOMMENDED that source authentication and integrity checking
 are applied to decoded objects before delivering objects to an
 application.  For example, a SHA-1 hash [SHA1] of an object may be
 appended before transmission, and the SHA-1 hash is computed and
 checked after the object is decoded but before it is delivered to an
 application.  Source authentication SHOULD be provided, for example,
 by including a digital signature verifiable by the receiver computed
 on top of the hash value.  It is also RECOMMENDED that a packet
 authentication protocol, such as TESLA [RFC4082], be used to detect
 and discard corrupted packets upon arrival.  This method may also be
 used to provide source authentication.  Furthermore, it is
 RECOMMENDED that Reverse Path Forwarding checks be enabled in all
 network routers and switches along the path from the sender to
 receivers to limit the possibility of a bad agent successfully
 injecting a corrupted packet into the multicast tree data path.
 Another security concern is that some FEC information may be obtained
 by receivers out-of-band in a session description, and if the session
 description is forged or corrupted, then the receivers will not use
 the correct protocol for decoding content from received packets.  To
 avoid these problems, it is RECOMMENDED that measures be taken to
 prevent receivers from accepting incorrect session descriptions,
 e.g., by using source authentication to ensure that receivers only
 accept legitimate session descriptions from authorized senders.

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 [RFC5052].  This document assigns the
 Fully-Specified FEC Encoding ID 1 under the ietf:rmt:fec:encoding
 name-space to "Raptor Code".

Luby, et al. Standards Track [Page 43] RFC 5053 Raptor FEC Scheme October 2007

8. Acknowledgements

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

9. References

9.1. Normative References

 [RFC2119]  Bradner, S., "Key words for use in RFCs to Indicate
            Requirement Levels", BCP 14, RFC 2119, March 1997.
 [RFC4082]  Perrig, A., Song, D., Canetti, R., Tygar, J., and B.
            Briscoe, "Timed Efficient Stream Loss-Tolerant
            Authentication (TESLA): Multicast Source Authentication
            Transform Introduction", RFC 4082, June 2005.
 [RFC5052]  Watson, M., Luby, M., and L. Vicisano, "Forward Error
            Correction (FEC) Building Block", RFC 5052, August 2007.

9.2. Informative References

 [CCNC]     Luby, M., Watson, M., Gasiba, T., Stockhammer, T., and W.
            Xu, "Raptor Codes for Reliable Download Delivery in
            Wireless Broadcast Systems", CCNC 2006, Las Vegas, NV ,
            Jan 2006.
 [MBMS]     3GPP, "Multimedia Broadcast/Multicast Service (MBMS);
            Protocols and codecs", 3GPP TS 26.346 6.1.0, June 2005.
 [RFC3453]  Luby, M., Vicisano, L., Gemmell, J., Rizzo, L., Handley,
            M., and J. Crowcroft, "The Use of Forward Error Correction
            (FEC) in Reliable Multicast", RFC 3453, December 2002.
 [Raptor]   Shokrollahi, A., "Raptor Codes", IEEE Transactions on
            Information Theory no. 6, June 2006.
 [SHA1]     "Secure Hash Standard", Federal Information Processing
            Standards Publication (FIPS PUB) 180-1, April 2005.

Luby, et al. Standards Track [Page 44] RFC 5053 Raptor FEC Scheme October 2007

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
 Nomor Research
 Brecherspitzstrasse 8
 Munich  81541
 Germany
 EMail: stockhammer@nomor.de

Luby, et al. Standards Track [Page 45] RFC 5053 Raptor FEC Scheme October 2007

Full Copyright Statement

 Copyright (C) The IETF Trust (2007).
 This document is subject to the rights, licenses and restrictions
 contained in BCP 78, and except as set forth therein, the authors
 retain all their rights.
 This document and the information contained herein are provided on an
 "AS IS" basis and THE CONTRIBUTOR, THE ORGANIZATION HE/SHE REPRESENTS
 OR IS SPONSORED BY (IF ANY), THE INTERNET SOCIETY, THE IETF TRUST AND
 THE INTERNET ENGINEERING TASK FORCE DISCLAIM ALL WARRANTIES, EXPRESS
 OR IMPLIED, INCLUDING BUT NOT LIMITED TO ANY WARRANTY THAT THE USE OF
 THE INFORMATION HEREIN WILL NOT INFRINGE ANY RIGHTS OR ANY IMPLIED
 WARRANTIES OF MERCHANTABILITY OR FITNESS FOR A PARTICULAR PURPOSE.

Intellectual Property

 The IETF takes no position regarding the validity or scope of any
 Intellectual Property Rights or other rights that might be claimed to
 pertain to the implementation or use of the technology described in
 this document or the extent to which any license under such rights
 might or might not be available; nor does it represent that it has
 made any independent effort to identify any such rights.  Information
 on the procedures with respect to rights in RFC documents can be
 found in BCP 78 and BCP 79.
 Copies of IPR disclosures made to the IETF Secretariat and any
 assurances of licenses to be made available, or the result of an
 attempt made to obtain a general license or permission for the use of
 such proprietary rights by implementers or users of this
 specification can be obtained from the IETF on-line IPR repository at
 http://www.ietf.org/ipr.
 The IETF invites any interested party to bring to its attention any
 copyrights, patents or patent applications, or other proprietary
 rights that may cover technology that may be required to implement
 this standard.  Please address the information to the IETF at
 ietf-ipr@ietf.org.

Luby, et al. Standards Track [Page 46]

/data/webs/external/dokuwiki/data/pages/rfc/rfc5053.txt · Last modified: 2007/10/23 18:29 (external edit)