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Versions: 00 01 02 03 04 05 06 RFC 6330

Reliable Multicast Transport                                     M. Luby
Internet-Draft                                            Qualcomm, Inc.
Intended status: Standards Track                          A. Shokrollahi
Expires: August 14, 2010                                            EPFL
                                                               M. Watson
                                                          Qualcomm, Inc.
                                                          T. Stockhammer
                                                          Nomor Research
                                                       February 10, 2010


      RaptorQ Forward Error Correction Scheme for Object Delivery
                    draft-ietf-rmt-bb-fec-raptorq-01

Abstract

   This document describes a Fully-Specified FEC scheme, corresponding
   to FEC Encoding ID 6 (to be confirmed (tbc)), for the RaptorQ forward
   error correction code and its application to reliable delivery of
   data objects.

   RaptorQ codes are a new family of codes that provide superior
   flexibility, support for larger source block sizes and better coding
   efficiency than Raptor codes in RFC5053.  RaptorQ is also 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 for most cases equal to the number of source
   symbols and in rare cases with slightly more than the number of
   source symbols.

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

Status of this Memo

   This Internet-Draft is submitted to IETF in full conformance with the
   provisions of BCP 78 and BCP 79.

   Internet-Drafts are working documents of the Internet Engineering
   Task Force (IETF), its areas, and its working groups.  Note that
   other groups may also distribute working documents as Internet-
   Drafts.

   Internet-Drafts are draft documents valid for a maximum of six months
   and may be updated, replaced, or obsoleted by other documents at any
   time.  It is inappropriate to use Internet-Drafts as reference



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   Drafts.

   Internet-Drafts are draft documents valid for a maximum of six months
   and may be updated, replaced, or obsoleted by other documents at any
   time.  It is inappropriate to use Internet-Drafts as reference
   material or to cite them other than as "work in progress."

   The list of current Internet-Drafts can be accessed at
   http://www.ietf.org/ietf/1id-abstracts.txt.

   The list of Internet-Draft Shadow Directories can be accessed at
   http://www.ietf.org/shadow.html.

   This Internet-Draft will expire on August 14, 2010.

Copyright Notice

   Copyright (c) 2010 IETF Trust and the persons identified as the
   document authors.  All rights reserved.

   This document is subject to BCP 78 and the IETF Trust's Legal
   Provisions Relating to IETF Documents
   (http://trustee.ietf.org/license-info) in effect on the date of
   publication of this document.  Please review these documents
   carefully, as they describe your rights and restrictions with respect
   to this document.  Code Components extracted from this document must
   include Simplified BSD License text as described in Section 4.e of
   the Trust Legal Provisions and are provided without warranty as
   described in the BSD License.



























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Table of Contents

   1.  Introduction  . . . . . . . . . . . . . . . . . . . . . . . . . 2
   2.  Requirements notation . . . . . . . . . . . . . . . . . . . . . 3
   3.  Formats and Codes . . . . . . . . . . . . . . . . . . . . . . . 4
     3.1.  FEC Payload IDs . . . . . . . . . . . . . . . . . . . . . . 4
     3.2.  FEC Object Transmission Information . . . . . . . . . . . . 4
       3.2.1.  Mandatory . . . . . . . . . . . . . . . . . . . . . . . 4
       3.2.2.  Common  . . . . . . . . . . . . . . . . . . . . . . . . 4
       3.2.3.  Scheme-Specific . . . . . . . . . . . . . . . . . . . . 5
   4.  Procedures  . . . . . . . . . . . . . . . . . . . . . . . . . . 7
     4.1.  Content Delivery Protocol Requirements  . . . . . . . . . . 7
     4.2.  Example Parameter Derivation Algorithm  . . . . . . . . . . 7
     4.3.  Object Delivery . . . . . . . . . . . . . . . . . . . . . . 9
       4.3.1.  Source block construction . . . . . . . . . . . . . . . 9
       4.3.2.  Encoding packet construction  . . . . . . . . . . . .  11
   5.  RaptorQ FEC Code Specification  . . . . . . . . . . . . . . .  12
     5.1.  Definitions, Symbols and Abbreviations  . . . . . . . . .  12
       5.1.1.  Definitions . . . . . . . . . . . . . . . . . . . . .  12
       5.1.2.  Symbols . . . . . . . . . . . . . . . . . . . . . . .  13
       5.1.3.  Abbreviations . . . . . . . . . . . . . . . . . . . .  15
     5.2.  Overview  . . . . . . . . . . . . . . . . . . . . . . . .  15
     5.3.  Systematic RaptorQ encoder  . . . . . . . . . . . . . . .  17
       5.3.1.  Introduction  . . . . . . . . . . . . . . . . . . . .  17
       5.3.2.  Encoding overview . . . . . . . . . . . . . . . . . .  18
       5.3.3.  First encoding step: Intermediate Symbol Generation .  19
       5.3.4.  Second encoding step: Encoding  . . . . . . . . . . .  27
       5.3.5.  Generators  . . . . . . . . . . . . . . . . . . . . .  27
     5.4.  Example FEC decoder . . . . . . . . . . . . . . . . . . .  30
       5.4.1.  General . . . . . . . . . . . . . . . . . . . . . . .  30
       5.4.2.  Decoding an extended source block . . . . . . . . . .  32
     5.5.  Random Numbers  . . . . . . . . . . . . . . . . . . . . .  36
       5.5.1.  The table V0  . . . . . . . . . . . . . . . . . . . .  36
       5.5.2.  The table V1  . . . . . . . . . . . . . . . . . . . .  37
       5.5.3.  The table V2  . . . . . . . . . . . . . . . . . . . .  38
       5.5.4.  The table V3  . . . . . . . . . . . . . . . . . . . .  39
     5.6.  Systematic indices and other parameters . . . . . . . . .  40
     5.7.  Arithmetic in GF(256) . . . . . . . . . . . . . . . . . .  44
       5.7.1.  Introduction  . . . . . . . . . . . . . . . . . . . .  44
       5.7.2.  The table GF256_EXP . . . . . . . . . . . . . . . . .  44
       5.7.3.  The table GF256_LOG . . . . . . . . . . . . . . . . .  45
   6.  Security Considerations . . . . . . . . . . . . . . . . . . .  48
   7.  IANA Considerations . . . . . . . . . . . . . . . . . . . . .  49
   8.  Acknowledgements  . . . . . . . . . . . . . . . . . . . . . .  50
   9.  References  . . . . . . . . . . . . . . . . . . . . . . . . .  51
     9.1.  Normative references  . . . . . . . . . . . . . . . . . .  51
     9.2.  Informative references  . . . . . . . . . . . . . . . . .  51
   Authors' Addresses  . . . . . . . . . . . . . . . . . . . . . . .  52



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1.  Introduction

   This document specifies an FEC Scheme for the RaptorQ forward error
   correction code for object delivery applications.  The concept of an
   FEC Scheme is defined in RFC5052 [RFC5052] and this document follows
   the format prescribed there and uses the terminology of that
   document.  The RaptorQ code described herein is a next generation of
   the Raptor code described in RFC5053 [RFC5053].  The RaptorQ code
   provides superior reliability, better coding efficiency, and support
   for larger source block sizes than the Raptor code of RFC5053
   [RFC5053].  These improvements simplify the usage of the RaptorQ code
   in an object delivery Content Delivery Protocol compared to RFC5053
   [RFC5053].

   The RaptorQ FEC Scheme is a Fully-Specified FEC Scheme corresponding
   to FEC Encoding ID 6 (tbc).

      Editor's Note: The finalized FEC encoding ID is still to be
      defined, but '6 (tbc)' is used as temporary value in this Internet
      Draft expecting sequential use of FEC encoding IDs in the IANA
      registration process.

   RaptorQ 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 backgound
   on the use of Forward Error Correction codes in reliable multicast,
   see [RFC3453].


















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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].














































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3.  Formats and Codes

3.1.  FEC Payload IDs

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

        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
       +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
       |     SBN       |               Encoding Symbol ID              |
       +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+

                      Figure 1: FEC Payload ID format

      Source Block Number (SBN), (8 bits): An integer identifier for the
      source block that the encoding symbols within the packet relate
      to.

      Encoding Symbol ID (ESI), (24 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 4.

3.2.  FEC Object Transmission Information

3.2.1.  Mandatory

   The value of the FEC Encoding ID MUST be 6, as assigned by IANA (see
   Section 7).

3.2.2.  Common

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

   o  Transfer Length (F)

   o  Symbol Size (T)

   The Transfer Length is a non-negative integer that is at most
   946287651840, which can be represented by 40 bits.  The Symbol Size
   is a non-negative integer less than 2^^16.

   The Transfer Length is a field of 40 bits in its definition, and the
   Symbol Size field is 16 bits, and both length units are bytes.

   The encoded Common FEC Object Transmission Information format is



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   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 (F)                      |
      +               +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
      |               |    Reserved   |           Symbol Size (T)     |
      +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+


          Figure 2: Encoded Common FEC OTI for RaptorQ FEC Scheme

      NOTE 1: The limit of 946287651840 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 56404 and
      the limitation on the number of source blocks to 2^^8.

3.2.3.  Scheme-Specific

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

   o  The number of source blocks (Z)

   o  The number of sub-blocks (N)

   o  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 (12 bits), N (12 bits) and Al (8 bits)
   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 12-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



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   described in Section 4.3.1.2


















































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4.  Procedures

4.1.  Content Delivery Protocol Requirements

   This section describes the information exchange between the RaptorQ
   FEC Scheme and any Content Delivery Protocol (CDP) that makes use of
   the RaptorQ FEC Scheme for object delivery.

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

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

   o  A symbol alignment parameter, Al

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

   o  The number of source blocks, Z

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

   The RaptorQ encoder scheme for object delivery additionally requires:

      - the object to be encoded, F bytes

   The RaptorQ encoder scheme supplies the CDP with the following
   information for each packet to be sent:

   o  Source Block Number (SBN)

   o  Encoding Symbol ID (ESI)

   o  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:

   o  F the transfer length of the object, in bytes

   o  WS the maximum size block that is decodable in working memory, in
      bytes





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

   o  Al the symbol alignment parameter, in bytes

   o  SS a parameter where the desired lower bound on the sub-symbol
      size is SS*Al

   o  K'_max the maximum number of source symbols per source block.

         Note: Section 5.1.2 defines K'_max to be 56404

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

   Let,

   o  T = P'

   o  Kt = ceil(F/T)

   o  N_max = floor(T/(SS*Al))

   o  for all n=1, ..., N_max

      *  KL(n) is the maximum K' value in Table 2 in Section 5.6 such
         that

            K' <= floor (WS/(Al*(ceil(T/(Al*n)))))

   o  Z = ceil(Kt/KL(N_max))

   o  N is the minimum n=1, ..., N_max such that ceil (Kt/Z) <= KL(n)

   It is RECOMMENDED that each packet contains exactly one symbol.
   However, receivers SHALL support the reception of packets that
   contain multiple symbols.

   The value Kt is the total number of symbols required to represent the
   source data of the object.

   The algorithm above and that defined in Section 4.3.1.2 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.



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   The recommended settings for the input parameter Al is 4.

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

4.3.  Object Delivery

4.3.1.  Source block construction

4.3.1.1.  General

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

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

   Note that the value of K is not necessarily the same for each source
   block of a object and the value of T' may not necessarily be the same
   for each sub-block of a source block.  However, the symbol size T is
   the same for all source blocks of an object and the number of
   symbols, K is the same for every sub-block of a source block.  Exact
   partitioning of the object into source blocks and sub-blocks is
   described in Section 4.3.1.2 below.

4.3.1.2.  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:

   o  F the transfer length of the object, in bytes

   o  Al a symbol alignment parameter, in bytes





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

   o  Z the number of source blocks

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

   These parameters MUST be set so that ceil(ceil(F/T)/Z) <= K'_max.
   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,

   o  Kt = ceil(F/T)

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

   o  (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 KL*T bytes,
   i.e.  KL source symbols of T bytes each, and the remaining ZS source
   blocks each having KS*T bytes, i.e.  KS source symbols of T bytes
   each.

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

   Next, each source block with K source symbols 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 bytes and the
   remaining NS sub-blocks each consisting of K contiguous sub-symbols
   of size of TS*Al bytes.  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



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   contiguous portion of the object.

4.3.2.  Encoding packet construction

   Each encoding packet contains the following information:

   o  Source Block Number (SBN)

   o  Encoding Symbol ID (ESI)

   o  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
   source symbols in the source block.  Encoding Symbol IDs K onwards
   identify repair symbols generated from the source symbols using the
   RaptorQ encoder.

   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.








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5.  RaptorQ FEC Code Specification

5.1.  Definitions, Symbols and Abbreviations

   For the purpose of the RaptorQ FEC code specification in this
   section, the following definitions, symbols and abbreviations apply.

5.1.1.  Definitions

   o  Source block: a block of K source symbols which are considered
      together for RaptorQ encoding and decoding purposes.

   o  Extended Source Block: a block of K' source symbols, where K' >= K
      constructed from a source block and zero or more padding symbols.

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

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

   o  Padding symbol: a symbol with all zero bits that is added to the
      source block to form the extended source block.

   o  Encoding symbol: a symbol that can be sent as part of the encoding
      of a source block.  The encoding symbols of a source block consist
      of the source symbols of the source block and the repair symbols
      generated from the source block.  Repair symbols generated from a
      source block have the same size as the source symbols of that
      source block.

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

   o  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 sent as part of the encoding of a
      source block.  The intermediate symbols are partitioned into LT
      symbols and PI symbols.

   o  LT symbols: The subset of the intermediate symbols that can be LT
      neighbors of an encoding symbol.





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   o  PI symbols: The subset of the intermediate symbols that can be PI
      neighbors of an encoding symbol.

   o  Systematic code: a code in which all source symbols are included
      as part of the encoding symbols of a source block.  The RaptorQ
      code as described herein is a systematic code.

   o  Encoding Symbol ID: information that uniquely identifies each
      encoding symbol associated with a source block for sending and
      receiving purposes.

   o  Internal Symbol ID: information that uniquely identifies each
      symbol associated with an extended source block for encoding and
      decoding purposes.

5.1.2.  Symbols

   i, j, u, v, h, d, a, b, d1, a1, b1, v, m, x, y   represent values or
        variables of one type or another, depending on the context.

   X    denotes a non-negative integer value that is either an ISI value
        or an ESI value, depending on the context.

   ceil(x)  denotes the smallest integer which is greater than or equal
        to x, where x is a real value.

   floor(x)  denotes the largest positive integer which is less than or
        equal to x, where x is a real value.

   min(x,y)  denotes the minimum value of the values x and y, and in
        general the minimum value of all the argument values.

   max(x,y)  denotes the maximum vaue of the values x and y, and in
        general the maximum value of all the argument values.

   i % j  denotes i modulo j.

   u ^ v  denotes, for equal-length bit strings u and v, the bitwise
        exclusive-or of u and v.

   A    denotes a matrix A.

   Transpose[A]  denotes the transposed matrix of matrix A.

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






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   K    denotes the number of symbols in a single source block.

   K'   denotes the number of source plus padding symbols in an extended
        source block.  For the majority of this specification, the
        padding symbols are considered to be additional source symbols.

   K'_max  denotes the maximum number of source symbols that can be in a
        single source block.  Set to 56404.

   L    denotes the number of intermediate symbols for a single extended
        source block.

   S    denotes the number of LDPC symbols for a single extended source
        block.  These are LT symbols.  For each value of K' shown in
        Table 2 in Section 5.6, the corresponding value of S is a prime
        number.

   H    denotes the number of HDPC symbols for a single extended source
        block.  These are PI symbols.

   B    denotes the number of intermediate symbols that are LT symbols
        excluding the LDPC symbols.

   W    denotes the number of intermediate symbols that are LT symbols.
        For each value of K' in Table 2 shown in Section 5.6, the
        corresponding value of W is a prime number.

   P    denotes the number of intermediate symbols that are PI symbols.
        These contain all HDPC symbols.

   P1   denotes the smallest prime number greater than or equal to P.

   U    denotes the number of non-HDPC intermediate symbols that are PI
        symbols.

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

   C'   denotes an array of the symbols of the extended source block,
        where C'[0], C'[1], C'[2],..., C'[K-1] are the source symbols of
        the source block and C'[K], C'[K+1],..., C'[K'-1] are padding
        symbols.

   V0, V1, V2, V3  denote four arrays of 4-byte integers, V0[0],
        V0[1],..., V0[255] ; V1[0], V1[1],..., V1[255]; V2[0],
        V2[1],..., V2[255]; and V3[0], V3[1],..., V3[255] as shown in
        Section 5.5.




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

   Deg[v]  a degree generator

   Enc[K', C ,(d, a, b, d1, a1, b1)]  an encoding symbol generator

   Tuple[K', X]  a tuple generator function

   GF(n)  denotes the Galois field with n elements.

   T    denotes the symbol size in bytes.

   J(K')  denotes the systematic index associated with K'.

   G    denotes any generator matrix.

   I_S  denotes the SxS identity matrix.

   a ^^ b  denotes the operation a raised to the power b.

5.1.3.  Abbreviations

   ESI       Encoding Symbol ID

   GF        Galois Field

   HDPC      High Density Parity Check

   ISI       Internal Symbol ID

   LDPC      Low Density Parity Check

   LT        Luby Transform

   PI        Permanently Interactive

   SBN       Source Block Number

   SBL       Source Block Length (in units of symbols)

5.2.  Overview

   This section defines the systematic RaptorQ FEC code.

   Symbols are the fundamental data units of the encoding and decoding
   process.  For each source block all symbols are the same size,
   referred to as the symbol size T. The atomic operations performed on
   symbols for both encoding and decoding are the exclusive-or operation



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   between symbols and an operation of the elements of the finite field
   GF(256) upon symbols.

   The basic encoder is described in Section 5.3.  The encoder first
   derives a block of intermediate symbols from the source symbols of a
   source block.  This intermediate block has the property that both
   source and repair symbols can be generated from it using the same
   process.  The encoder produces repair symbols from the intermediate
   block using an efficient process, where each such repair symbol is
   the exclusive OR of a small number of intermediate symbols from the
   block.  Source symbols can also be reproduced from the intermediate
   block using the same process.  The encoding symbols are the
   combination of the source and repair symbols.

   An example of a decoder is described in Section 5.4.  The process for
   producing source and repair symbols from the intermediate block is
   designed so that the intermediate block can be recovered from any
   sufficiently large set of encoding symbols, independent of the mix of
   source and repair symbols in the set.  Once the intermediate block is
   recovered, missing source symbols of the source block can be
   recovered using the encoding process.

   If a RaptorQ compliant decoding algorithm receives a mathematically
   sufficient set of encoding symbols generated according to the encoder
   specification in Section 5.3 for reconstruction of a source block
   then such a decoder SHALL recover the entire source block.  A number
   of decoding algorithms are possible to achieve this optimal behavior.
   An efficient decoding algorithm to achieve this is provided in
   Section 5.4.

   The construction of the intermediate and repair symbols is based in
   part on a pseudo-random number generator described in Section 5.3.
   This generator is based on a fixed set of 1024 random numbers which
   must be available to both sender and receiver.  These numbers are
   provided in Section 5.5.  Encoding and decoding operations for
   RaptorQ use operations in the field GF(256).  Section 5.7 provides a
   recommended way to perform these operations.

   Finally, the construction of the intermediate symbols from the source
   symbols is governed by "systematic indices", values of which are
   provided in Section 5.6 for specific extended source block sizes
   between 6 and K'_max = 56404 source symbols.  Thus, the RaptorQ code
   supports source blocks with between 1 and 56404 source symbols.

5.3.  Systematic RaptorQ encoder






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5.3.1.  Introduction

   For a given source block of K source symbols, for encoding and
   decoding purposes the source block is augmented with K'-K additional
   padding symbols, where K' is the smallest value that is at least K in
   the systematic index Table 2 of Section 5.6.  The reason for padding
   out a source block to a multiple of K' is to enable faster encoding
   and decoding, and to minimize the amount of table information that
   needs to be stored in the encoder and decoder.

   For purposes of transmitting and receiving data, the value of K is
   used to determine the number of source symbols in a source block, and
   thus K needs to be known at the sender and the receiver.  In this
   case the sender and receiver can compute K' from K and the K'-K
   padding symbols can be automatically added to the source block
   without any additional communication.  The encoding symbol ID (ESI)
   is used by a sender and receiver to identify the encoding symbols of
   a source block, where the encoding symbols of a source block consist
   of the source symbols and the repair symbols associated with the
   source block.  For a source block with K source symbols, the ESIs for
   the source symbols are 0,1,2,...,K-1 and the ESIs for the repair
   symbols are K, K+1, K+2,... .  Using the ESI for identifying encoding
   symbols in transport ensures that the ESI values continue
   consecutively between the source and repair symbols.

   For purposes of encoding and decoding data, the value of K' derived
   from K is used as the number of source symbols of the extended source
   block upon which encoding and decoding operations are performed,
   where the K' source symbols consist of the original K source symbols
   and an additional K'-K padding symbols.  The internal symbol ID (ISI)
   is used by the encoder and decoder to identify the symbols associated
   with the extended source block, i.e., for generating encoding symbols
   and for decoding.  For a source block with K original source symbols,
   the ISIs for the original source symbols are 0,1,2,...,K-1, the ISIs
   for the K'-K padding symbols are K, K+1, K+2,..., K'-1, and the ISIs
   for the repair symbols are K', K'+1, K'+2,... .  Using the ISI for
   encoding and decoding allows the padding symbols of the extended
   source block to be treated the same way as other source symbols of
   the extended source block, and that a given prefix of repair symbols
   are generated in a consistent way for a given number K' of source
   symbols in the extended source block independent of K.

   The relationship between the ESIs and the ISIs is simple: the ESIs
   and the ISIs for the original K source symbols are the same, the K'-K
   padding symbols have an ISI but do not have a corresponding ESI
   (since they are symbols that are neither sent nor received), and a
   repair symbol ISI is simply the repair symbol ESI plus K'-K.  The
   translation between ESIs used to identify encoding symbols sent and



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   received and the corresponding ISIs used for encoding and decoding,
   and the proper padding of the extended source block with padding
   symbols used for encoding and decoding, is the responsibility of the
   padding function in the RaptorQ encoder/decoder.

5.3.2.  Encoding overview

   The systematic RaptorQ encoder is used to generate any number of
   repair symbols from a source block that consists of K source symbols
   placed into an extended source block C'.  Figure 4 shows the encoding
   overview.

   The first step of encoding is to construct an extended source block
   by adding zero or more padding symbols such that the total number of
   symbols, K', is one of the values listed in Section 5.6.  Each
   padding symbol consists of T bytes where the value of each byte is
   zero.  K' MUST be selected as the smallest value of K' from the table
   of Section 5.6 which is greater than or equal to K.



         -----------------------------------------------------------+
         |                                                          |
         |    +-----------+    +--------------+    +-------------+  |
      C' |    |           | C' | Intermediate | C  |             |  |
     ----+--->|  Padding  |--->|    Symbol    |--->|   Encoding  |--+-->
      K  |    |           | K' |  Generation  | L  |             |  |
         |    +-----------+    +--------------+    +-------------+  |
         |           |                             (d,a,b, ^        |
         |           |                            d1,a1,b1)|        |
         |           |                              +------------+  |
         |           |              K'              |   Tuple    |  |
         |           +----------------------------->|            |  |
         |                                          | Generation |  |
         |                                          +------------+  |
         |                                                 ^        |
         +-------------------------------------------------+--------+
                                                           |
                                                         ISI X


                        Figure 4: Encoding Overview

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

   Let C'[K], ..., C'[K'-1] denote the K'-K padding symbols, which are
   all set to zero bits.  Then, C'[0],..., C'[K'-1] are the symbols of
   the extended source block upon which encoding and decoding are



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   performed.

   In the remainder of this description these padding symbols will be
   considered as additional source symbols and referred to as such.
   However, these padding symbols are not part of the encoding symbols,
   i.e., they are not sent as part of the encoding.  At a receiver, the
   value of K' can be computed based on K, then the receiver can insert
   K'-K padding symbols at the end of a source block of K' source
   symbols and recover the remaining K source symbols of the source
   block from received encoding symbols.

   The second step of encoding is to generate a number, L > K', of
   intermediate symbols from the K' source symbols.  In this step, K'
   source tuples (d[0], a[0], b[0], d1[0], a1[0], b1[0]), ..., (d[K'-1],
   a[K'-1], b[K'-1], d1[K'-1], a1[K'-1], b1[K'-1]) are generated using
   the Tuple[] generator as described in Section 5.3.5.4.  The K' source
   tuples and the ISIs associated with the K' source symbols are used to
   determine L intermediate symbols C[0],..., C[L-1] from the source
   symbols using an inverse encoding process.  This process can be
   realized by a RaptorQ decoding process.

   Certain "pre-coding relationships" must hold within the L
   intermediate symbols.  Section 5.3.3.3 describes these relationships.
   Section 5.3.3.4 describes how the intermediate symbols are generated
   from the source symbols.

   Once the intermediate symbols have been generated, repair symbols can
   be produced.  For a repair symbol with ISI X>K', the tuple of
   integers, (d, a, b, d1, a1, b1) can be generated, using the Tuple[]
   generator as described in Section 5.3.5.4.  Then, the (d, a, b, d1,
   a1, b1)-tuple and the ISI X is used to generate the corresponding
   repair symbol from the intermediate symbols using the Enc[] generator
   described in Section 5.3.5.3.  The corresponding ESI for this repair
   symbol is then X-(K'-K).  Note that source symbols of the extended
   source block can also be generated using the same process, i.e., for
   any X < K', the symbol generated using this process has the same
   value as C'(X).

5.3.3.  First encoding step: Intermediate Symbol Generation

5.3.3.1.  General

   This 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], , where L > K is defined in Section 5.3.3.3.  The
   intermediate symbols are uniquely defined by two sets of constraints:





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   1.  The intermediate symbols are related to the source symbols by a
       set of source symbol tuples and by the ISIs of the source
       symbols.  The generation of the source symbol tuples is defined
       in Section 5.3.3.2 using the the Tuple[] generator as described
       in Section 5.3.5.4.

   2.  A set of pre-coding relationships hold within the intermediate
       symbols themselves.  These are defined in Section 5.3.3.3

   The generation of the L intermediate symbols is then defined in
   Section 5.3.3.4

5.3.3.2.  Source symbol tuples

   Each of the K' source symbols is associated with a source symbol
   tuple (d[X], a[X], b[X], d1[X], a1[X], b1[X]) for 0 <= X < K'.  The
   source symbol tuples are determined using the Tuple generator defined
   in Section 5.3.5.4 as:

      For each X, 0 <= X < K'

         (d[X], a[X], b[X], d1[X], a1[X], b1[X]) = Tuple[K, X]

5.3.3.3.  Pre-coding relationships

   The pre-coding relationships amongst the L intermediate symbols are
   defined by requiring that a set of S+H linear combinations of the
   intermediate symbols evaluate to zero.  There are S LDPC and H HDPC
   symbols, and thus L = K'+S+H. Another partition of the L intermediate
   symbols is into two sets, one set of W LT symbols and another set of
   P PI symbols, and thus it is also the case that L = W+P. The P PI
   symbols are treated differently than the W LT symbols in the encoding
   process.  The P PI symbols consist of the H HDPC symbols together
   with a set of U = P-H of the other K' intermediate symbols.  The W LT
   symbols consist of the S LDPC symbols together with W-S of the other
   K' intermediate symbols.  The values of these parameters are
   determined from K' as described below where H(K'), S(K'), and W(K')
   are derived from Table 2 in Section 5.6.

   Let

   o  S = S(K')

   o  H = H(K')

   o  W = W(K')





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   o  L = K' + S + H

   o  P = L - W

   o  P1 denote the smallest prime number greater than or equal to P

   o  U =P - H

   o  B = W - S

   o  C[0], ..., C[B-1] denote the intermediate symbols that are LT
      symbols but not LDPC symbols.

   o  C[B], ..., C[B+S-1] denote the S LDPC symbols that are also LT
      symbols.

   o  C[W], ..., C[W+U-1] denote the intermediate symbols that are PI
      symbols but not HDPC symbols.

   o  C[L-H], ..., C[L-1] denote the H HDPC symbols that are also PI
      symbols.

   The first set of pre-coding relations, called LDPC relations, is
   described below and requires that at the end of this process the set
   of symbols D[0] , ..., D[S-1] are all zero:

   o  Initialize the symbols D[0] = C[B], ... , D[S-1] = C[B+S-1].

   o  For i = 0, ..., B-1 do

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

      *  b = i % S

      *  D[b] = D[b] ^ C[i]

      *  b = (b + a) % S

      *  D[b] = D[b] ^ C[i]

      *  b = (b + a) % S

      *  D[b] = D[b] ^ C[i]

   o  For i = 0, ..., S-1 do

      *  a = i % P




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      *  b = (i+1) % P

      *  D[i] = D[i] ^ C[W+a] ^ C[W+b]

   The second set of relations, called HDPC relations, is obtained by
   considering each intermediate symbol as a sequence of elements from
   the finite field GF(256).  We represent elements of GF(256) in the
   usual way as polynomials in one variable, x, with coefficients from
   the finite field GF(2) modulo an irreducible polynomial f(x).  A
   single byte of data from a symbol, b7,b6,b5,b4,b3,b2,b1,b0, where b7
   is the highest order bit and b0 is the lowest order bit, corresponds
   to the finite field element

      b7 x^^7 + b6 x^^6 + b5 x^^5 + b4 x^^4 + b3 x^^3 + b2 x^^2 + b1 x +
      b0 mod f(x)

   The irreducible polynomial f(x) is defined to be:

      f(x) = x^^8 + x^^4 + x^^3 + x^^2 + 1.

   We then define the operation of elements of GF(256) on symbols as
   follows:

   Let

   o  beta denote an element of GF(256).

   o  C denote a symbol of length T bytes.

   o  c[0], ..., c[T-1] denote the bytes of C.

   o  gamma[0], ..., gamma[T-1] denote the elements of GF(256)
      corresponding to c[0], ..., c[T-1] respectively.

   Then we define

      delta[i] = beta*gamma[i] for i=0, ..., T-1

   where '*' represents the usual multiplication operation in GF(256).
   A multiplication table for GF(256) and a recommended way to perform
   calculations in GF(256) is provided in Section 5.7.  Then the
   operation of beta on C is defined as follows:

      beta*C = d[0], ..., d[T-1]

   where d[i] is the byte value corresponding to delta[i] for i=0,...,
   T-1.




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   The set of HDPC relations among the intermediate symbols C[0], ...,
   C[K'+S+H-1] is defined as follows:

   Let

   o  alpha denote a generator element of GF(256), specifically the
      element represented by the polynomial x mod f(x).

   o  T denote an H x (K' +S ) matrix with elements from GF(256), where
      for j=0,...,K'+S-2 the entry T[i,j] is the identity element if i=
      Rand[j,6,H] or i = (Rand[j,6,H] + Rand[j,7,H-1] + 1) % H and
      T[i,j] is the zero element for all other values of i, and for
      j=K'+S-1, T[i,j] = alpha^^i for i=0,...,H-1.

   o  GAMMA denote a (K'+S ) x (K'+S ) matrix with elements from
      GF(256), where

   o

         GAMMA[i,j] =

            alpha ^^ (i-j) for i <= j

            0 otherwise

   Then the relationship between the first K'+S intermediate symbols
   C[0], ..., C[K'+S-1] and the H HDPC symbols C[K'+S], ..., C[K'+S+H-1]
   is given by:

      Transpose[C[K'+S], ..., C[K'+S+H-1]] + T* GAMMA * Transpose[C[0],
      ..., C[K'+S-1]] = 0

   where '*' represents standard matrix multiplication utilizing the
   above defined operation to define the multiplication between a matrix
   over GF(256) and a matrix of symbols (in particular the column vector
   of symbols).

   The H HDPC relations may be conveniently described using the
   following algorithm, where u is a working register containing a
   single symbol.  These relations require that the values of the
   symbols D[S], ..., D[S+H-1] are zero at the end of the following
   process.

   o  Initialize the symbols D[S] = C[K'+S], ..., D[S+H-1] = C[K'+S+H-1]

   o  u = C[0]





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   o  For j = 1, ..., K'+S-1 do

      *  pos1 = Rand[j,6,H]

      *  pos2 = (pos1 + Rand[j,7,H-1] + 1) % H

      *  D[S+pos1] = D[S+pos1] ^ u

      *  D[S+pos2] = D[S+pos2] ^ u

      *  u = (alpha*u) ^ C[j]

   o  For i = 0, ..., H-1

      *  D[S+i] = D[S+i] ^ u

      *  u = (alpha*u)

5.3.3.4.  Intermediate symbols

5.3.3.4.1.  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'[X] = Enc[K', (C[0],..., C[L-1]), (d[X], a[X], b[X], d1[X],
          a1[X], b1[X])], for all X, 0 <= X < K',

       where (d[X], a[X], b[X], d1[X], a1[X], b1[X])) = Tuple[K',X],
       Tuple[] is defined in Section 5.3.5.4 and Enc[] is described in
       Section 5.3.5.3.

   2.  The L intermediate symbols C[0], C[1],..., C[L-1] satisfy the
       pre-coding relationships defined in Section 5.3.3.3

5.3.3.4.2.  Example method for calculation of intermediate symbols

   This section describes a possible method for calculation of the L
   intermediate symbols C[0], C[1],..., C[L-1] satisfying the
   constraints in Section 5.3.3.4.1

   The generator matrix G for a code which generates n output symbols
   from k input symbols is an n x k matrix over GF(256), where each row
   corresponds to one of the output symbols and each column to one of



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   the input symbols and where the i-th output symbol is equal to the
   sum of the product of each input symbol with the corresponding entry
   in row i.

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

   Let

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

   o  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 L x L matrix A over GF(256) such
   that:

      A*C = D

   The matrix A can be constructed as follows:

   Let:

   o  G_LDPC,1 and G_LDPC,2 be S x B and S x P matrices such that

         G_LDPC,1 * Transpose[(C[0],...., C[B-1])] + G_LDPC,2 *
         Transpose(C[W], ..., C[W+P-1]) + Transpose[(C[B], ...,
         C[B+S-1])] = 0

      and "+" is the component-wise XOR of the vectors involved.

   o  G_HDPC be the H x (K'+S) generator matrix of the HDPC symbols, So,

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

         i.e.  G_HDPC = H*GAMMA

   o  I_S be the S x S identity matrix

   o  I_H be the H x H identity matrix

   o  G_ENC be the K'x L generator matrix of the encoding symbols
      generated by the Encoder.  So,

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




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         i.e.  G_ENC[i,j] = 1 if and only if C[j] is included in the
         symbols which are XORed to produce Enc[K', (C[0], ..., C[L-1]),
         (d[i], a[i], b[i], d1[i], a1[i], b1[i])] and G_ENC[i,j] = 0
         otherwise.

   Then:

   o  The first S rows of A are equal to G_LDPC,1 | I_S | G_LDPC,2.

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

   o  The remaining K' rows of A are equal to G_ENC.

   The matrix A is depicted in Figure (Figure 5) below:

                 B               S         U         H
      +-----------------------+-------+------------------+
      |                       |       |                  |
    S |        G_LDPC,1       |  I_S  |      G_LDPC,2    |
      |                       |       |                  |
      +-----------------------+-------+----------+-------+
      |                                          |       |
    H |                G_HDPC                    |  I_H  |
      |                                          |       |
      +------------------------------------------+-------+
      |                                                  |
      |                                                  |
   K' |                      G_ENC                       |
      |                                                  |
      |                                                  |
      +--------------------------------------------------+

                          Figure 5: The matrix A

   The intermediate symbols can then be calculated as:

      C = (A^^-1)*D

   The source tuples are generated such that for any K' matrix A has
   full rank and is therefore invertible.  This calculation can be
   realized by applying a RaptorQ 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 5.4 be used.




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5.3.4.  Second encoding step: Encoding

   In the second encoding step, the repair symbol with ISI X (X >= K')
   is generated by applying the generator Enc[K', (C[0], C[1],...,
   C[L-1]), (d, a, b, d1, a1, b1)] defined in Section 5.3.5.3 to the L
   intermediate symbols C[0], C[1],..., C[L-1] using the tuple (d, a, b,
   d1, a1, b1)=Tuple[K',X].

5.3.5.  Generators

5.3.5.1.  Random Generator

   The random number generator Rand[y, i, m] is defined as follows,
   where y is a non-negative integer, i is a non-negative integer less
   than 256, and m is a positive integer and the value produced is an
   integer between 0 and m-1.  Let V0, V1, V2 and V3 be arrays of 256
   entries each, where each entry is a 4-byte unsigned integer.  These
   arrays are provided in Section 5.5.

   Let

   o  x0 = (y + i) mod 2^^8

   o  x1 = (floor(y / 2^^8) + i) mod 2^^8

   o  x2 = (floor(y / 2^^16) + i) mod 2^^8

   o  x3 = (floor(y / 2^^24) + i) mod 2^^8

   Then,

      Rand[y, i, m] = (V0[x0] ^ V1[x1] ^ V2[x2] ^ V3[x3]) % m

5.3.5.2.  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.  Given v,
   find index d in Table 1 such that f[d-1] <= v < f[d], and set Deg[v]
   = min(d, W-2).












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                 +---------+---------+---------+---------+
                 | Index d | f[d]    | Index d | f[d]    |
                 +---------+---------+---------+---------+
                 | 0       | 0       | 1       | 5243    |
                 +---------+---------+---------+---------+
                 | 2       | 529531  | 3       | 704294  |
                 +---------+---------+---------+---------+
                 | 4       | 791675  | 5       | 844104  |
                 +---------+---------+---------+---------+
                 | 6       | 879057  | 7       | 904023  |
                 +---------+---------+---------+---------+
                 | 8       | 922747  | 9       | 937311  |
                 +---------+---------+---------+---------+
                 | 10      | 948962  | 11      | 958494  |
                 +---------+---------+---------+---------+
                 | 12      | 966438  | 13      | 973160  |
                 +---------+---------+---------+---------+
                 | 14      | 978921  | 15      | 983914  |
                 +---------+---------+---------+---------+
                 | 16      | 988283  | 17      | 992138  |
                 +---------+---------+---------+---------+
                 | 18      | 995565  | 19      | 998631  |
                 +---------+---------+---------+---------+
                 | 20      | 1001391 | 21      | 1003887 |
                 +---------+---------+---------+---------+
                 | 22      | 1006157 | 23      | 1008229 |
                 +---------+---------+---------+---------+
                 | 24      | 1010129 | 25      | 1011876 |
                 +---------+---------+---------+---------+
                 | 26      | 1013490 | 27      | 1014983 |
                 +---------+---------+---------+---------+
                 | 28      | 1016370 | 29      | 1017662 |
                 +---------+---------+---------+---------+
                 | 30      | 1048576 |         |         |
                 +---------+---------+---------+---------+

       Table 1: Defines the degree distribution for encoding symbols

5.3.5.3.  Encoding Symbol Generator

   The encoding symbol generator Enc[K', (C[0], C[1],..., C[L-1]), (d,
   a, b, d1, a1, b1)] takes the following inputs:

   o  K' is the number of source symbols for the extended source block.
      Let L, W, B, S, P and P1 be derived from K' as described in
      Section 5.3.3.3.





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   o  (C[0], C[1],..., C[L-1]) is the array of L intermediate symbols
      (sub-symbols) generated as described in Section 5.3.3.4

   o  (d, a, b, d1, a1, b1) is a source tuple determined from ISI X
      using the Tuple generator defined in Section 5.3.5.4, whereby

      *  d is an integer denoting an encoding symbol LT degree

      *  a is an integer between 1 and W-1 inclusive

      *  b is an integer between 0 and W-1 inclusive

      *  d1 is an integer between 2 and 3 inclusive denoting an encoding
         symbol PI degree

      *  a1 is an integer between 1 and P1-1 inclusive

      *  b1 is an integer between 0 and P1-1 inclusive

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

   o  result = C[b].

   o  For j = 1, ..., d-1 do

      *  b = (b + a) % W

      *  result = result ^ C[b]

   o  while (b1 >= P) do b1 = (b1+a1) % P1

   o  result = result ^ C[W+b1]

   o  For j = 1, ..., d1-1 do

      *  b1 = (b1 + a1) % P1

      *  while (b1 >= P) do b1 = (b1+a1) % P1

      *  result = result ^ C[W+b1]

   o  Return result








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5.3.5.4.  Tuple generator

   The tuple generator Tuple[K',X] takes the following inputs:

   o  K' - The number of source symbols in the extended source block

   o  X - An Intermediate symbol ID (ISI)

   Let

   o  L be determined from K' as described in Section 5.3.3.3

   o  J=J(K') be the systematic index associated with K', as defined
      inTable 2 inSection 5.6

   The output of the source symbol tuple generator is a tuple, (d, a, b,
   d1, a1, b1) determined as follows:

   o  A = 53591 + J*997

   o  If ((A % 2) = 0) { A = A + 1 }

   o  B = 10267*(J+1)

   o  y = (B + X*A) % 2^^32

   o  v = Rand[y, 0, 2^^20]

   o  d = Deg[v]

   o  a = 1 + Rand[y, 1, W-1]

   o  b = Rand[y, 2, W]

   o  if (d<4) { d1 = 2 + Rand[y, 3, 2] } else { d1 = 2 }

   o  a1 = 1 + Rand[y, 4, P1-1]

   o  b1 = Rand[y, 5, P1]

5.4.  Example FEC decoder

5.4.1.  General

   This section describes an efficient decoding algorithm for the
   RaptorQ code introduced 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



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   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.

5.4.2.  Decoding an extended source block

5.4.2.1.  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 and the number K' of source symbols in
   the extended source block.

   From the algorithms described in Sections Section 5.3, the RaptorQ
   decoder can calculate the total number L = K'+S+H of intermediate
   symbols and determine how they were generated from the extended
   source block to be decoded.  In this description it is assumed that
   the received encoding symbols for the extended source block to be
   decoded are passed to the decoder.  Furthermore, for each such
   encoding symbol it is assumed that the number and set of intermediate
   symbols whose exclusive-or is equal to the encoding symbol is passed
   to the decoder.  In the case of source symbols, including padding
   symbols, the source symbol tuples described in Section 5.3.3.2
   indicate the number and set of intermediate symbols which sum to give
   each source symbol.

   Let N >= K' be the number of received encoding symbols to be used for
   decoding, including padding symbols for an extended source block and
   let M = S+H+N. Then with the notation of Section 5.3.3.4.2 we have
   A*C=D.

   Decoding an extended 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 is L. Once C has been decoded, missing source symbols can
   be obtained by using the source symbol tuples to determine the number
   and set of intermediate symbols which must be exclusive-ORed to
   obtain each missing source symbol.

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



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   based on the decoding schedule.

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

   o  Each time a multiple, beta, of row i of A is added to row i' in
      the decoding schedule then in the decoding process the symbol
      beta*D[d[i]] is added to symbol D[d[i']] .

   o  Each time a row i of A is multiplied by a field element beta, then
      in the decoding process the symbol D[d[i]] is also multiplied by
      beta.

   o  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'].

   o  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
   operations on symbols in the decoding of the extended 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.

5.4.2.2.  First Phase

   The first phase of the Gaussian elimination the matrix A is
   conceptually partitioned into submatrices and additionally, a matrix
   X is created.  This matrix has as many rows and columns as A, and it
   will be a lower triangular matrix throughout the first phase.  At the
   beginning of this phase, the matrix A is copied into the matrix X.
   The submatrix sizes are parameterized by non-negative integers i and



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   u which are initialized to 0 and P, the number of PI symbols,
   respectively.  The submatrices of A are:

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

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

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

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

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

   Figure 6 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 6: 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 non-zero
   entries in V and are not HDPC rows 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.




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

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

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

      *  If r != 2 then choose a row with exactly r ones in V with
         minimum original degree among all such rows, except that HDPC
         rows should not be chosen until all non-HDPC rows have been
         processed.

      *  If r = 2 then choose any row with exactly 2 ones in V that is
         part of a maximum size component in the graph described above
         which is 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. The same row and column operations
   are also performed on the matrix X. Then, an appropriate multiple of
   the chosen row is added to all the other rows of A below the chosen
   row that have a non-zero entry in the first column of V.
   Specifically, if a row below the chosen row has entry beta in the
   first column of V, and the chosen row has entry alpha in the first
   column of V, then beta/alpha multiplied by the chosen row is added to
   this row to leave a zero value in the first column of V. Finally, i
   is incremented by 1 and u is incremented by r-1, which completes the
   step.

   Note that efficiency can be improved if the row operations identified
   above are not actually performed until the affected row is itself
   chosen during the decoding process.  This avoids processing of row
   operations for rows which are not eventually used in the decoding
   process and in particular avoid those rows for which beta!=1 until
   they are actually required.  Furthermore, the row operations required
   for the HDPC rows may be performed for all such rows in one process,
   by using the algorithm described in Section 5.3.3.3.




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5.4.2.3.  Second Phase

   At this point, all the entries of X outside the first i rows and i
   columns are discarded, so that X has lower triangular form.  The last
   i rows and columns of X are discarded, so that X now has i rows i
   columns.  The submatrix U is further partitioned into the first i
   rows, U_upper, and the remaining M - i rows, Ulower.  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.

5.4.2.4.  Third Phase

   After the second phase the only portion of A which needs to be zeroed
   out to finish converting A into the L by L identity matrix is
   U_upper.  The number of rows i of the submatrix U_upper is generally
   much larger than the number of columns u of U_upper.  Moreover, at
   this time, the matrix U_upper is typically dense, i.e., the number of
   nonzero entries of this matrix is large.  To reduce this matrix to a
   sparse form, the sequence of operations performed to obtain the
   matrix U_lower needs to be inverted.  To this end, the matrix X is
   multiplied with the submatrix of A consisting of the first i rows of
   A. After this operation the submatrix of A consisting of the
   intersection of the first i rows and columns equals to X, whereas the
   matrix U_upper is transformed to a sparse form.

5.4.2.5.  Fourth Phase

   For each of the first i rows of U_upper do the following: if the row
   has a nonzero entry at position j, and if the value of that nonzero
   entry is b, then add to this row b times row j of I_u.  After this
   step, the submatrix of A consisting of the intersection of the first
   i rows and columns is equal to X, the submatrix U_upper consists of
   zeros, the submatrix consisting of the intersection of the last u
   rows and the first i columns consists of zeros, and the submatrix
   consisting of the last u rows and columns is is the matrix I_u.

5.4.2.6.  Fifth Phase

   For j from 1 to i perform the following operations:

   1.  if A[j,j] is not one, then divide row j of A by A[j,j].

   2.  For l from 1 to j-1, if A[j,l] is nonzero, then add A[j,l]
       multiplied with row l of A to row j of A.



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   After this phase A is the L by L identity matrix and a complete
   decoding schedule has been successfully formed.  Then, the
   corresponding decoding consisting of exclusive-ORing known encoding
   symbols can be executed to recover the intermediate symbols based on
   the decoding schedule.  The tuples associated with all source symbols
   are computed according to Section 5.3.3.2.  The tuples for received
   source symbols are used in the decoding.  The tuples for missing
   source symbols are used to determine which intermediate symbols need
   to be exclusive-ORed to recover the missing source symbols.

5.5.  Random Numbers

   The four tables V0, V1, V2 and V3 described in Section 5.3.5.1 are
   given below.  Each entry is a 32-bit integer in decimal
   representation.

5.5.1.  The table V0

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



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   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

5.5.2.  The table V1

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



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   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

5.5.3.  The table V2

   1629829892, 282540176, 2794583710, 496504798, 2990494426, 3070701851,
   2575963183, 4094823972, 2775723650, 4079480416, 176028725,
   2246241423, 3732217647, 2196843075, 1306949278, 4170992780,
   4039345809, 3209664269, 3387499533, 293063229, 3660290503,
   2648440860, 2531406539, 3537879412, 773374739, 4184691853,
   1804207821, 3347126643, 3479377103, 3970515774, 1891731298,
   2368003842, 3537588307, 2969158410, 4230745262, 831906319,
   2935838131, 264029468, 120852739, 3200326460, 355445271, 2296305141,
   1566296040, 1760127056, 20073893, 3427103620, 2866979760, 2359075957,
   2025314291, 1725696734, 3346087406, 2690756527, 99815156, 4248519977,
   2253762642, 3274144518, 598024568, 3299672435, 556579346, 4121041856,
   2896948975, 3620123492, 918453629, 3249461198, 2231414958,
   3803272287, 3657597946, 2588911389, 242262274, 1725007475,
   2026427718, 46776484, 2873281403, 2919275846, 3177933051, 1918859160,
   2517854537, 1857818511, 3234262050, 479353687, 200201308, 2801945841,
   1621715769, 483977159, 423502325, 3689396064, 1850168397, 3359959416,
   3459831930, 841488699, 3570506095, 930267420, 1564520841, 2505122797,
   593824107, 1116572080, 819179184, 3139123629, 1414339336, 1076360795,
   512403845, 177759256, 1701060666, 2239736419, 515179302, 2935012727,
   3821357612, 1376520851, 2700745271, 966853647, 1041862223, 715860553,
   171592961, 1607044257, 1227236688, 3647136358, 1417559141,
   4087067551, 2241705880, 4194136288, 1439041934, 20464430, 119668151,
   2021257232, 2551262694, 1381539058, 4082839035, 498179069, 311508499,
   3580908637, 2889149671, 142719814, 1232184754, 3356662582,
   2973775623, 1469897084, 1728205304, 1415793613, 50111003, 3133413359,
   4074115275, 2710540611, 2700083070, 2457757663, 2612845330,



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   3775943755, 2469309260, 2560142753, 3020996369, 1691667711,
   4219602776, 1687672168, 1017921622, 2307642321, 368711460,
   3282925988, 213208029, 4150757489, 3443211944, 2846101972,
   4106826684, 4272438675, 2199416468, 3710621281, 497564971, 285138276,
   765042313, 916220877, 3402623607, 2768784621, 1722849097, 3386397442,
   487920061, 3569027007, 3424544196, 217781973, 2356938519, 3252429414,
   145109750, 2692588106, 2454747135, 1299493354, 4120241887,
   2088917094, 932304329, 1442609203, 952586974, 3509186750, 753369054,
   854421006, 1954046388, 2708927882, 4047539230, 3048925996,
   1667505809, 805166441, 1182069088, 4265546268, 4215029527,
   3374748959, 373532666, 2454243090, 2371530493, 3651087521,
   2619878153, 1651809518, 1553646893, 1227452842, 703887512,
   3696674163, 2552507603, 2635912901, 895130484, 3287782244,
   3098973502, 990078774, 3780326506, 2290845203, 41729428, 1949580860,
   2283959805, 1036946170, 1694887523, 4880696, 466000198, 2765355283,
   3318686998, 1266458025, 3919578154, 3545413527, 2627009988,
   3744680394, 1696890173, 3250684705, 4142417708, 915739411,
   3308488877, 1289361460, 2942552331, 1169105979, 3342228712,
   698560958, 1356041230, 2401944293, 107705232, 3701895363, 903928723,
   3646581385, 844950914, 1944371367, 3863894844, 2946773319,
   1972431613, 1706989237, 29917467, 3497665928

5.5.4.  The table V3

   1191369816, 744902811, 2539772235, 3213192037, 3286061266,
   1200571165, 2463281260, 754888894, 714651270, 1968220972, 3628497775,
   1277626456, 1493398934, 364289757, 2055487592, 3913468088,
   2930259465, 902504567, 3967050355, 2056499403, 692132390, 186386657,
   832834706, 859795816, 1283120926, 2253183716, 3003475205, 1755803552,
   2239315142, 4271056352, 2184848469, 769228092, 1249230754,
   1193269205, 2660094102, 642979613, 1687087994, 2726106182, 446402913,
   4122186606, 3771347282, 37667136, 192775425, 3578702187, 1952659096,
   3989584400, 3069013882, 2900516158, 4045316336, 3057163251,
   1702104819, 4116613420, 3575472384, 2674023117, 1409126723,
   3215095429, 1430726429, 2544497368, 1029565676, 1855801827,
   4262184627, 1854326881, 2906728593, 3277836557, 2787697002,
   2787333385, 3105430738, 2477073192, 748038573, 1088396515,
   1611204853, 201964005, 3745818380, 3654683549, 3816120877,
   3915783622, 2563198722, 1181149055, 33158084, 3723047845, 3790270906,
   3832415204, 2959617497, 372900708, 1286738499, 1932439099,
   3677748309, 2454711182, 2757856469, 2134027055, 2780052465,
   3190347618, 3758510138, 3626329451, 1120743107, 1623585693,
   1389834102, 2719230375, 3038609003, 462617590, 260254189, 3706349764,
   2556762744, 2874272296, 2502399286, 4216263978, 2683431180,
   2168560535, 3561507175, 668095726, 680412330, 3726693946, 4180630637,
   3335170953, 942140968, 2711851085, 2059233412, 4265696278,
   3204373534, 232855056, 881788313, 2258252172, 2043595984, 3758795150,
   3615341325, 2138837681, 1351208537, 2923692473, 3402482785,



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   2105383425, 2346772751, 499245323, 3417846006, 2366116814,
   2543090583, 1828551634, 3148696244, 3853884867, 1364737681,
   2200687771, 2689775688, 232720625, 4071657318, 2671968983,
   3531415031, 1212852141, 867923311, 3740109711, 1923146533,
   3237071777, 3100729255, 3247856816, 906742566, 4047640575,
   4007211572, 3495700105, 1171285262, 2835682655, 1634301229,
   3115169925, 2289874706, 2252450179, 944880097, 371933491, 1649074501,
   2208617414, 2524305981, 2496569844, 2667037160, 1257550794,
   3399219045, 3194894295, 1643249887, 342911473, 891025733, 3146861835,
   3789181526, 938847812, 1854580183, 2112653794, 2960702988,
   1238603378, 2205280635, 1666784014, 2520274614, 3355493726,
   2310872278, 3153920489, 2745882591, 1200203158, 3033612415,
   2311650167, 1048129133, 4206710184, 4209176741, 2640950279,
   2096382177, 4116899089, 3631017851, 4104488173, 1857650503,
   3801102932, 445806934, 3055654640, 897898279, 3234007399, 1325494930,
   2982247189, 1619020475, 2720040856, 885096170, 3485255499,
   2983202469, 3891011124, 546522756, 1524439205, 2644317889,
   2170076800, 2969618716, 961183518, 1081831074, 1037015347,
   3289016286, 2331748669, 620887395, 303042654, 3990027945, 1562756376,
   3413341792, 2059647769, 2823844432, 674595301, 2457639984,
   4076754716, 2447737904, 1583323324, 625627134, 3076006391, 345777990,
   1684954145, 879227329, 3436182180, 1522273219, 3802543817,
   1456017040, 1897819847, 2970081129, 1382576028, 3820044861,
   1044428167, 612252599, 3340478395, 2150613904, 3397625662,
   3573635640, 3432275192

5.6.  Systematic indices and other parameters

   Table 2 below specifies the supported values of K'.  The table also
   specifies for each supported value of K' the systematic index J(K'),
   the number H(K') of HDPC symbols, the number S(K') of LDPC symbols,
   and the number W(K') of LT symbols.  For each value of K', the
   corresponding values of S(K') and W(K') are prime numbers.

   The systematic index J(K') is designed to have the property that the
   set of source symbol tuples (d[0], a[0], b[0], d1[0], a1[0], b1[0]),
   ..., (d[K'-1], a[K'-1], b[K'-1], d1[K'-1], a1[K'-1], b1[K'-1]) are
   such that the L intermediate symbols are uniquely defined, i.e., the
   matrix A in Figure 6 has full rank and is therefore invertible.

   +-------+-------+-------+-------+-------+
   | K'    | J(K') | S(K') | H(K') | W(K') |
   +-------+-------+-------+-------+-------+
   | 6     | 3     | 5     | 10    | 11    |
   +-------+-------+-------+-------+-------+
   | 12    | 57    | 7     | 10    | 19    |
   +-------+-------+-------+-------+-------+
   | 18    | 27    | 11    | 10    | 29    |



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   | 26    | 96    | 11    | 10    | 37    |
   +-------+-------+-------+-------+-------+
   | 32    | 959   | 11    | 10    | 43    |
   +-------+-------+-------+-------+-------+
   | 36    | 564   | 11    | 10    | 47    |
   +-------+-------+-------+-------+-------+
   | 42    | 39    | 11    | 10    | 53    |
   +-------+-------+-------+-------+-------+
   | 48    | 10    | 13    | 10    | 61    |
   +-------+-------+-------+-------+-------+
   | 55    | 531   | 13    | 10    | 67    |
   +-------+-------+-------+-------+-------+
   | 62    | 55    | 13    | 10    | 73    |
   +-------+-------+-------+-------+-------+
   | 69    | 235   | 13    | 10    | 79    |
   +-------+-------+-------+-------+-------+
   | 75    | 234   | 17    | 10    | 89    |
   +-------+-------+-------+-------+-------+
   | 88    | 113   | 17    | 10    | 101   |
   +-------+-------+-------+-------+-------+
   | 101   | 8     | 17    | 10    | 113   |
   +-------+-------+-------+-------+-------+
   | 114   | 8     | 19    | 10    | 127   |
   +-------+-------+-------+-------+-------+
   | 127   | 184   | 19    | 10    | 139   |
   +-------+-------+-------+-------+-------+
   | 140   | 7     | 19    | 10    | 151   |
   +-------+-------+-------+-------+-------+
   | 160   | 39    | 23    | 10    | 173   |
   +-------+-------+-------+-------+-------+
   | 185   | 751   | 23    | 10    | 197   |
   +-------+-------+-------+-------+-------+
   | 213   | 1     | 23    | 10    | 223   |
   +-------+-------+-------+-------+-------+
   | 242   | 10    | 29    | 10    | 257   |
   +-------+-------+-------+-------+-------+
   | 267   | 195   | 29    | 10    | 281   |
   +-------+-------+-------+-------+-------+
   | 295   | 572   | 29    | 10    | 307   |
   +-------+-------+-------+-------+-------+
   | 324   | 447   | 31    | 10    | 337   |
   +-------+-------+-------+-------+-------+
   | 362   | 751   | 31    | 10    | 373   |
   +-------+-------+-------+-------+-------+
   | 403   | 234   | 37    | 10    | 419   |
   +-------+-------+-------+-------+-------+
   | 443   | 974   | 37    | 10    | 457   |
   +-------+-------+-------+-------+-------+



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   +-------+-------+-------+-------+-------+
   | 497   | 115   | 37    | 10    | 509   |
   +-------+-------+-------+-------+-------+
   | 555   | 17    | 41    | 10    | 569   |
   +-------+-------+-------+-------+-------+
   | 619   | 75    | 41    | 10    | 631   |
   +-------+-------+-------+-------+-------+
   | 685   | 476   | 47    | 10    | 701   |
   +-------+-------+-------+-------+-------+
   | 759   | 112   | 47    | 10    | 773   |
   +-------+-------+-------+-------+-------+
   | 839   | 454   | 53    | 10    | 857   |
   +-------+-------+-------+-------+-------+
   | 932   | 424   | 53    | 10    | 947   |
   +-------+-------+-------+-------+-------+
   | 1032  | 34    | 59    | 10    | 1051  |
   +-------+-------+-------+-------+-------+
   | 1144  | 600   | 61    | 11    | 1163  |
   +-------+-------+-------+-------+-------+
   | 1281  | 75    | 67    | 11    | 1303  |
   +-------+-------+-------+-------+-------+
   | 1420  | 726   | 67    | 11    | 1439  |
   +-------+-------+-------+-------+-------+
   | 1575  | 39    | 73    | 11    | 1597  |
   +-------+-------+-------+-------+-------+
   | 1734  | 83    | 79    | 11    | 1759  |
   +-------+-------+-------+-------+-------+
   | 1906  | 394   | 83    | 11    | 1931  |
   +-------+-------+-------+-------+-------+
   | 2103  | 75    | 89    | 11    | 2131  |
   +-------+-------+-------+-------+-------+
   | 2315  | 772   | 97    | 11    | 2347  |
   +-------+-------+-------+-------+-------+
   | 2550  | 726   | 97    | 11    | 2579  |
   +-------+-------+-------+-------+-------+
   | 2812  | 683   | 103   | 11    | 2843  |
   +-------+-------+-------+-------+-------+
   | 3101  | 512   | 113   | 11    | 3137  |
   +-------+-------+-------+-------+-------+
   | 3411  | 650   | 127   | 11    | 3457  |
   +-------+-------+-------+-------+-------+
   | 3751  | 838   | 127   | 11    | 3793  |
   +-------+-------+-------+-------+-------+
   | 4086  | 547   | 131   | 11    | 4127  |
   +-------+-------+-------+-------+-------+
   | 4436  | 305   | 139   | 11    | 4481  |
   +-------+-------+-------+-------+-------+
   | 4780  | 3     | 149   | 11    | 4831  |



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   | 5134  | 518   | 157   | 11    | 5189  |
   +-------+-------+-------+-------+-------+
   | 5512  | 229   | 163   | 11    | 5569  |
   +-------+-------+-------+-------+-------+
   | 6070  | 980   | 173   | 11    | 6131  |
   +-------+-------+-------+-------+-------+
   | 6688  | 596   | 191   | 11    | 6761  |
   +-------+-------+-------+-------+-------+
   | 7360  | 960   | 197   | 11    | 7433  |
   +-------+-------+-------+-------+-------+
   | 8087  | 85    | 211   | 11    | 8167  |
   +-------+-------+-------+-------+-------+
   | 8886  | 479   | 223   | 11    | 8971  |
   +-------+-------+-------+-------+-------+
   | 9793  | 200   | 239   | 11    | 9887  |
   +-------+-------+-------+-------+-------+
   | 10779 | 290   | 257   | 11    | 10883 |
   +-------+-------+-------+-------+-------+
   | 11864 | 543   | 277   | 12    | 11981 |
   +-------+-------+-------+-------+-------+
   | 13046 | 893   | 293   | 12    | 13171 |
   +-------+-------+-------+-------+-------+
   | 14355 | 527   | 311   | 12    | 14489 |
   +-------+-------+-------+-------+-------+
   | 15786 | 601   | 337   | 12    | 15937 |
   +-------+-------+-------+-------+-------+
   | 17376 | 479   | 359   | 12    | 17539 |
   +-------+-------+-------+-------+-------+
   | 19126 | 518   | 389   | 12    | 19309 |
   +-------+-------+-------+-------+-------+
   | 21044 | 933   | 419   | 13    | 21247 |
   +-------+-------+-------+-------+-------+
   | 23177 | 85    | 449   | 13    | 23399 |
   +-------+-------+-------+-------+-------+
   | 25491 | 710   | 479   | 13    | 25733 |
   +-------+-------+-------+-------+-------+
   | 28035 | 11    | 521   | 13    | 28319 |
   +-------+-------+-------+-------+-------+
   | 30898 | 738   | 557   | 14    | 31219 |
   +-------+-------+-------+-------+-------+
   | 33988 | 602   | 599   | 14    | 34351 |
   +-------+-------+-------+-------+-------+
   | 37372 | 545   | 647   | 14    | 37783 |
   +-------+-------+-------+-------+-------+
   | 41127 | 11    | 701   | 15    | 41593 |
   +-------+-------+-------+-------+-------+
   | 45245 | 639   | 757   | 15    | 45767 |
   +-------+-------+-------+-------+-------+



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   +-------+-------+-------+-------+-------+
   | 49791 | 249   | 821   | 15    | 50377 |
   +-------+-------+-------+-------+-------+
   | 54768 | 300   | 877   | 16    | 55411 |
   +-------+-------+-------+-------+-------+
   | 56404 | 733   | 907   | 16    | 57077 |
   +-------+-------+-------+-------+-------+

             Table 2: Systematic indices and other parameters

5.7.  Arithmetic in GF(256)

5.7.1.  Introduction

   Elements of GF(256) are represented by bytes.  In this section, we
   opt to represent them by integers in the range 0 through 255.  For
   ease of exposition, operations in GF(256) are facilitated by two
   tables: GF256_EXP, and GF256_LOG.  GF256_EXP has 512 entries, whereas
   GF256_LOG has 256 entries.  For an integer i between 0 and 511,
   GF256_EXP[i] is the binary value of the polynomial x^^i modulo x^^8 +
   x^^4 + x^^3 + x^^2 + 1, whereas for i between 1 and 255 the value of
   GF256_LOG[i] is the integer j such that the binary value of x^^j
   modulo x^^8 + x^^4 + x^^3 + x^^2 + 1 is i.  In this representation we
   have

      i + j = i ^ j, and

      i * j =

         0, if either i or j are 0,

         GF256_EXP[GF256_LOG[i] + GF256_LOG[j]] otherwise

5.7.2.  The table GF256_EXP

   1, 2, 4, 8, 16, 32, 64, 128, 29, 58, 116, 232, 205, 135, 19, 38, 76,
   152, 45, 90, 180, 117, 234, 201, 143, 3, 6, 12, 24, 48, 96, 192, 157,
   39, 78, 156, 37, 74, 148, 53, 106, 212, 181, 119, 238, 193, 159, 35,
   70, 140, 5, 10, 20, 40, 80, 160, 93, 186, 105, 210, 185, 111, 222,
   161, 95, 190, 97, 194, 153, 47, 94, 188, 101, 202, 137, 15, 30, 60,
   120, 240, 253, 231, 211, 187, 107, 214, 177, 127, 254, 225, 223, 163,
   91, 182, 113, 226, 217, 175, 67, 134, 17, 34, 68, 136, 13, 26, 52,
   104, 208, 189, 103, 206, 129, 31, 62, 124, 248, 237, 199, 147, 59,
   118, 236, 197, 151, 51, 102, 204, 133, 23, 46, 92, 184, 109, 218,
   169, 79, 158, 33, 66, 132, 21, 42, 84, 168, 77, 154, 41, 82, 164, 85,
   170, 73, 146, 57, 114, 228, 213, 183, 115, 230, 209, 191, 99, 198,
   145, 63, 126, 252, 229, 215, 179, 123, 246, 241, 255, 227, 219, 171,
   75, 150, 49, 98, 196, 149, 55, 110, 220, 165, 87, 174, 65, 130, 25,



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   50, 100, 200, 141, 7, 14, 28, 56, 112, 224, 221, 167, 83, 166, 81,
   162, 89, 178, 121, 242, 249, 239, 195, 155, 43, 86, 172, 69, 138, 9,
   18, 36, 72, 144, 61, 122, 244, 245, 247, 243, 251, 235, 203, 139, 11,
   22, 44, 88, 176, 125, 250, 233, 207, 131, 27, 54, 108, 216, 173, 71,
   142, 1, 2, 4, 8, 16, 32, 64, 128, 29, 58, 116, 232, 205, 135, 19, 38,
   76, 152, 45, 90, 180, 117, 234, 201, 143, 3, 6, 12, 24, 48, 96, 192,
   157, 39, 78, 156, 37, 74, 148, 53, 106, 212, 181, 119, 238, 193, 159,
   35, 70, 140, 5, 10, 20, 40, 80, 160, 93, 186, 105, 210, 185, 111,
   222, 161, 95, 190, 97, 194, 153, 47, 94, 188, 101, 202, 137, 15, 30,
   60, 120, 240, 253, 231, 211, 187, 107, 214, 177, 127, 254, 225, 223,
   163, 91, 182, 113, 226, 217, 175, 67, 134, 17, 34, 68, 136, 13, 26,
   52, 104, 208, 189, 103, 206, 129, 31, 62, 124, 248, 237, 199, 147,
   59, 118, 236, 197, 151, 51, 102, 204, 133, 23, 46, 92, 184, 109, 218,
   169, 79, 158, 33, 66, 132, 21, 42, 84, 168, 77, 154, 41, 82, 164, 85,
   170, 73, 146, 57, 114, 228, 213, 183, 115, 230, 209, 191, 99, 198,
   145, 63, 126, 252, 229, 215, 179, 123, 246, 241, 255, 227, 219, 171,
   75, 150, 49, 98, 196, 149, 55, 110, 220, 165, 87, 174, 65, 130, 25,
   50, 100, 200, 141, 7, 14, 28, 56, 112, 224, 221, 167, 83, 166, 81,
   162, 89, 178, 121, 242, 249, 239, 195, 155, 43, 86, 172, 69, 138, 9,
   18, 36, 72, 144, 61, 122, 244, 245, 247, 243, 251, 235, 203, 139, 11,
   22, 44, 88, 176, 125, 250, 233, 207, 131, 27, 54, 108, 216, 173, 71,
   142, 1, 2

5.7.3.  The table GF256_LOG

   255, 0, 1, 25, 2, 50, 26, 198, 3, 223, 51, 238, 27, 104, 199, 75, 4,
   100, 224, 14, 52, 141, 239, 129, 28, 193, 105, 248, 200, 8, 76, 113,
   5, 138, 101, 47, 225, 36, 15, 33, 53, 147, 142, 218, 240, 18, 130,
   69, 29, 181, 194, 125, 106, 39, 249, 185, 201, 154, 9, 120, 77, 228,
   114, 166, 6, 191, 139, 98, 102, 221, 48, 253, 226, 152, 37, 179, 16,
   145, 34, 136, 54, 208, 148, 206, 143, 150, 219, 189, 241, 210, 19,
   92, 131, 56, 70, 64, 30, 66, 182, 163, 195, 72, 126, 110, 107, 58,
   40, 84, 250, 133, 186, 61, 202, 94, 155, 159, 10, 21, 121, 43, 78,
   212, 229, 172, 115, 243, 167, 87, 7, 112, 192, 247, 140, 128, 99, 13,
   103, 74, 222, 237, 49, 197, 254, 24, 227, 165, 153, 119, 38, 184,
   180, 124, 17, 68, 146, 217, 35, 32, 137, 46, 55, 63, 209, 91, 149,
   188, 207, 205, 144, 135, 151, 178, 220, 252, 190, 97, 242, 86, 211,
   171, 20, 42, 93, 158, 132, 60, 57, 83, 71, 109, 65, 162, 31, 45, 67,
   216, 183, 123, 164, 118, 196, 23, 73, 236, 127, 12, 111, 246, 108,
   161, 59, 82, 41, 157, 85, 170, 251, 96, 134, 177, 187, 204, 62, 90,
   203, 89, 95, 176, 156, 169, 160, 81, 11, 245, 22, 235, 122, 117, 44,
   215, 79, 174, 213, 233, 230, 231, 173, 232, 116, 214, 244, 234, 168,
   80, 88, 175








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6.  Security Considerations

   Data delivery can be subject to denial-of-service attacks by
   attackers which 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.















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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 6 (tbc) under the ietf:rmt:fec:
   encoding name-space to "RaptorQ Code".












































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8.  Acknowledgements

   Thanks are due to Lorenz Minder and Ranganathan (Ranga) Krishnan.
   Lorenz Minder did the original implementation of RaptorQ, supervised
   by Amin Shokrollahi.  Ranga Krishnan has been very supportive in
   finding and resolving implementation details and in finding the
   systematic indices.












































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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.

   [SHA1]     "Secure Hash Standard", Federal Information Processing
              Standards Publication          (FIPS PUB) 180-1,
              April 2005.

   [RFC5052]  Watson, M., Luby, M., and L. Vicisano, "Forward Error
              Correction (FEC) Building Block", RFC 5052, August 2007.

   [RFC5053]  Luby, M., Shokrollahi, A., Watson, M., and T. Stockhammer,
              "Raptor Forward Error Correction Scheme for Object
              Delivery", RFC 5053, October 2007.

9.2.  Informative references

   [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.























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Authors' Addresses

   Michael Luby
   Qualcomm, Inc.
   3165 Kifer Road
   Santa Clara, 95051  94538
   U.S.A.

   Email: luby@qualcomm.com


   Amin Shokrollahi
   EPFL
   Laboratoire d'algorithmique
   EPFL
   Station 14
   Batiment BC
   Lausanne  1015
   Switzerland

   Email: amin.shokrollahi@epfl.ch


   Mark Watson
   Qualcomm, Inc.
   3165 Kifer Road
   Santa Clara, CA  95051
   U.S.A.

   Email: watson@qualcomm.com


   Thomas Stockhammer
   Nomor Research
   Brecherspitzstrasse 8
   Munich  81541
   Germany

   Email: stockhammer@nomor.de












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