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<pre>Internet Engineering Task Force (IETF)                           M. Luby
Request for Comments: 6330                         Qualcomm Incorporated
Category: Standards Track                                 A. Shokrollahi
ISSN: 2070-1721                                                     EPFL
                                                               M. Watson
                                                            Netflix Inc.
                                                          T. Stockhammer
                                                          Nomor Research
                                                               L. Minder
                                                   Qualcomm Incorporated
                                                             August 2011


      <span class="h1">RaptorQ Forward Error Correction Scheme for Object Delivery</span>

Abstract

   This document describes a Fully-Specified Forward Error Correction
   (FEC) scheme, corresponding to FEC Encoding ID 6, for the RaptorQ FEC
   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 <a href="./rfc5053">RFC 5053</a>.  RaptorQ is also a fountain
   code, i.e., as many encoding symbols as needed can be generated on
   the fly by the encoder from the source symbols of a source block of
   data.  The decoder is able to recover the source block from almost
   any set of encoding symbols of sufficient cardinality -- in most
   cases, a set of cardinality equal to the number of source symbols is
   sufficient; in rare cases, a set of cardinality slightly more than
   the number of source symbols is required.

   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 is an Internet Standards Track document.

   This document is a product of the Internet Engineering Task Force
   (IETF).  It represents the consensus of the IETF community.  It has
   received public review and has been approved for publication by the
   Internet Engineering Steering Group (IESG).  Further information on
   Internet Standards is available in <a href="./rfc5741#section-2">Section&nbsp;2 of RFC 5741</a>.






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<span class="grey"><a href="./rfc6330">RFC 6330</a>                   RaptorQ FEC Scheme                August 2011</span>


   Information about the current status of this document, any errata,
   and how to provide feedback on it may be obtained at
   <a href="http://www.rfc-editor.org/info/rfc6330">http://www.rfc-editor.org/info/rfc6330</a>.

Copyright Notice

   Copyright (c) 2011 IETF Trust and the persons identified as the
   document authors.  All rights reserved.

   This document is subject to <a href="https://www.rfc-editor.org/bcp/bcp78">BCP 78</a> and the IETF Trust's Legal
   Provisions Relating to IETF Documents
   (<a href="http://trustee.ietf.org/license-info">http://trustee.ietf.org/license-info</a>) 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 Simplified BSD License.

































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<span class="grey"><a href="./rfc6330">RFC 6330</a>                   RaptorQ FEC Scheme                August 2011</span>


Table of Contents

   <a href="#section-1">1</a>.  Introduction . . . . . . . . . . . . . . . . . . . . . . . . .  <a href="#page-4">4</a>
   <a href="#section-2">2</a>.  Requirements Notation  . . . . . . . . . . . . . . . . . . . .  <a href="#page-4">4</a>
   <a href="#section-3">3</a>.  Formats and Codes  . . . . . . . . . . . . . . . . . . . . . .  <a href="#page-5">5</a>
     <a href="#section-3.1">3.1</a>.  Introduction . . . . . . . . . . . . . . . . . . . . . . .  <a href="#page-5">5</a>
     <a href="#section-3.2">3.2</a>.  FEC Payload IDs  . . . . . . . . . . . . . . . . . . . . .  <a href="#page-5">5</a>
     <a href="#section-3.3">3.3</a>.  FEC Object Transmission Information  . . . . . . . . . . .  <a href="#page-5">5</a>
       <a href="#section-3.3.1">3.3.1</a>.  Mandatory  . . . . . . . . . . . . . . . . . . . . . .  <a href="#page-5">5</a>
       <a href="#section-3.3.2">3.3.2</a>.  Common . . . . . . . . . . . . . . . . . . . . . . . .  <a href="#page-5">5</a>
       <a href="#section-3.3.3">3.3.3</a>.  Scheme-Specific  . . . . . . . . . . . . . . . . . . .  <a href="#page-6">6</a>
   <a href="#section-4">4</a>.  Procedures . . . . . . . . . . . . . . . . . . . . . . . . . .  <a href="#page-7">7</a>
     <a href="#section-4.1">4.1</a>.  Introduction . . . . . . . . . . . . . . . . . . . . . . .  <a href="#page-7">7</a>
     <a href="#section-4.2">4.2</a>.  Content Delivery Protocol Requirements . . . . . . . . . .  <a href="#page-7">7</a>
     <a href="#section-4.3">4.3</a>.  Example Parameter Derivation Algorithm . . . . . . . . . .  <a href="#page-7">7</a>
     <a href="#section-4.4">4.4</a>.  Object Delivery  . . . . . . . . . . . . . . . . . . . . .  <a href="#page-9">9</a>
       <a href="#section-4.4.1">4.4.1</a>.  Source Block Construction  . . . . . . . . . . . . . .  <a href="#page-9">9</a>
       <a href="#section-4.4.2">4.4.2</a>.  Encoding Packet Construction . . . . . . . . . . . . . <a href="#page-11">11</a>
       <a href="#section-4.4.3">4.4.3</a>.  Example Receiver Recovery Strategies . . . . . . . . . <a href="#page-12">12</a>
   <a href="#section-5">5</a>.  RaptorQ FEC Code Specification . . . . . . . . . . . . . . . . <a href="#page-12">12</a>
     <a href="#section-5.1">5.1</a>.  Background . . . . . . . . . . . . . . . . . . . . . . . . <a href="#page-12">12</a>
       <a href="#section-5.1.1">5.1.1</a>.  Definitions  . . . . . . . . . . . . . . . . . . . . . <a href="#page-13">13</a>
       <a href="#section-5.1.2">5.1.2</a>.  Symbols  . . . . . . . . . . . . . . . . . . . . . . . <a href="#page-14">14</a>
     <a href="#section-5.2">5.2</a>.  Overview . . . . . . . . . . . . . . . . . . . . . . . . . <a href="#page-17">17</a>
     <a href="#section-5.3">5.3</a>.  Systematic RaptorQ Encoder . . . . . . . . . . . . . . . . <a href="#page-18">18</a>
       <a href="#section-5.3.1">5.3.1</a>.  Introduction . . . . . . . . . . . . . . . . . . . . . <a href="#page-18">18</a>
       <a href="#section-5.3.2">5.3.2</a>.  Encoding Overview  . . . . . . . . . . . . . . . . . . <a href="#page-19">19</a>
       5.3.3.  First Encoding Step: Intermediate Symbol Generation  . 21
       <a href="#section-5.3.4">5.3.4</a>.  Second Encoding Step: Encoding . . . . . . . . . . . . <a href="#page-27">27</a>
       <a href="#section-5.3.5">5.3.5</a>.  Generators . . . . . . . . . . . . . . . . . . . . . . <a href="#page-27">27</a>
     <a href="#section-5.4">5.4</a>.  Example FEC Decoder  . . . . . . . . . . . . . . . . . . . <a href="#page-30">30</a>
       <a href="#section-5.4.1">5.4.1</a>.  General  . . . . . . . . . . . . . . . . . . . . . . . <a href="#page-30">30</a>
       <a href="#section-5.4.2">5.4.2</a>.  Decoding an Extended Source Block  . . . . . . . . . . <a href="#page-31">31</a>
     <a href="#section-5.5">5.5</a>.  Random Numbers . . . . . . . . . . . . . . . . . . . . . . <a href="#page-36">36</a>
       <a href="#section-5.5.1">5.5.1</a>.  The Table V0 . . . . . . . . . . . . . . . . . . . . . <a href="#page-36">36</a>
       <a href="#section-5.5.2">5.5.2</a>.  The Table V1 . . . . . . . . . . . . . . . . . . . . . <a href="#page-37">37</a>
       <a href="#section-5.5.3">5.5.3</a>.  The Table V2 . . . . . . . . . . . . . . . . . . . . . <a href="#page-38">38</a>
       <a href="#section-5.5.4">5.5.4</a>.  The Table V3 . . . . . . . . . . . . . . . . . . . . . <a href="#page-40">40</a>
     <a href="#section-5.6">5.6</a>.  Systematic Indices and Other Parameters  . . . . . . . . . <a href="#page-41">41</a>
     <a href="#section-5.7">5.7</a>.  Operating with Octets, Symbols, and Matrices . . . . . . . <a href="#page-62">62</a>
       <a href="#section-5.7.1">5.7.1</a>.  General  . . . . . . . . . . . . . . . . . . . . . . . <a href="#page-62">62</a>
       <a href="#section-5.7.2">5.7.2</a>.  Arithmetic Operations on Octets  . . . . . . . . . . . <a href="#page-62">62</a>
       <a href="#section-5.7.3">5.7.3</a>.  The Table OCT_EXP  . . . . . . . . . . . . . . . . . . <a href="#page-63">63</a>
       <a href="#section-5.7.4">5.7.4</a>.  The Table OCT_LOG  . . . . . . . . . . . . . . . . . . <a href="#page-64">64</a>
       <a href="#section-5.7.5">5.7.5</a>.  Operations on Symbols  . . . . . . . . . . . . . . . . <a href="#page-65">65</a>
       <a href="#section-5.7.6">5.7.6</a>.  Operations on Matrices . . . . . . . . . . . . . . . . <a href="#page-65">65</a>
     <a href="#section-5.8">5.8</a>.  Requirements for a Compliant Decoder . . . . . . . . . . . <a href="#page-65">65</a>
   <a href="#section-6">6</a>.  Security Considerations  . . . . . . . . . . . . . . . . . . . <a href="#page-66">66</a>



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   <a href="#section-7">7</a>.  IANA Considerations  . . . . . . . . . . . . . . . . . . . . . <a href="#page-67">67</a>
   <a href="#section-8">8</a>.  Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . <a href="#page-67">67</a>
   <a href="#section-9">9</a>.  References . . . . . . . . . . . . . . . . . . . . . . . . . . <a href="#page-67">67</a>
     <a href="#section-9.1">9.1</a>.  Normative References . . . . . . . . . . . . . . . . . . . <a href="#page-67">67</a>
     <a href="#section-9.2">9.2</a>.  Informative References . . . . . . . . . . . . . . . . . . <a href="#page-68">68</a>

<span class="h2"><a class="selflink" id="section-1" href="#section-1">1</a>.  Introduction</span>

   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 <a href="./rfc5052">RFC 5052</a> [<a href="./rfc5052" title="&quot;Forward Error Correction (FEC) Building Block&quot;">RFC5052</a>], 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 <a href="./rfc5053">RFC 5053</a> [<a href="./rfc5053" title="&quot;Raptor Forward Error Correction Scheme for Object Delivery&quot;">RFC5053</a>].  The RaptorQ code
   provides superior reliability, better coding efficiency, and support
   for larger source block sizes than the Raptor code of <a href="./rfc5053">RFC 5053</a>
   [<a href="./rfc5053" title="&quot;Raptor Forward Error Correction Scheme for Object Delivery&quot;">RFC5053</a>].  These improvements simplify the usage of the RaptorQ code
   in an object delivery Content Delivery Protocol compared to <a href="./rfc5053">RFC 5053</a>
   <a href="./rfc5053">RFC 5053</a> [<a href="./rfc5053" title="&quot;Raptor Forward Error Correction Scheme for Object Delivery&quot;">RFC5053</a>].  A detailed mathematical design and analysis of
   the RaptorQ code together with extensive simulation results are
   provided in [<a href="#ref-RaptorCodes">RaptorCodes</a>].

   The RaptorQ FEC scheme is a Fully-Specified FEC scheme corresponding
   to FEC Encoding ID 6.

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

   The code described in this document is a systematic code; that is,
   the original unmodified source symbols, as well as a number of repair
   symbols, can be sent from sender to receiver.  For more background on
   the use of Forward Error Correction codes in reliable multicast, see
   [<a href="./rfc3453" title="&quot;The Use of Forward Error Correction (FEC) in Reliable Multicast&quot;">RFC3453</a>].

<span class="h2"><a class="selflink" id="section-2" href="#section-2">2</a>.  Requirements Notation</span>

   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 [<a href="./rfc2119" title="&quot;Key words for use in RFCs to Indicate Requirement Levels&quot;">RFC2119</a>].









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<span class="h2"><a class="selflink" id="section-3" href="#section-3">3</a>.  Formats and Codes</span>

<span class="h3"><a class="selflink" id="section-3.1" href="#section-3.1">3.1</a>.  Introduction</span>

   The octet order of all fields is network byte order, i.e., big-
   endian.

<span class="h3"><a class="selflink" id="section-3.2" href="#section-3.2">3.2</a>.  FEC Payload IDs</span>

   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

   o  Source Block Number (SBN): 8-bit unsigned integer.  A non-negative
      integer identifier for the source block that the encoding symbols
      within the packet relate to.

   o  Encoding Symbol ID (ESI): 24-bit unsigned integer.  A non-negative
      integer identifier for the encoding symbols within the packet.

   The interpretation of the Source Block Number and Encoding Symbol
   Identifier is defined in <a href="#section-4">Section 4</a>.

<span class="h3"><a class="selflink" id="section-3.3" href="#section-3.3">3.3</a>.  FEC Object Transmission Information</span>

<span class="h4"><a class="selflink" id="section-3.3.1" href="#section-3.3.1">3.3.1</a>.  Mandatory</span>

   The value of the FEC Encoding ID MUST be 6, as assigned by IANA (see
   <a href="#section-7">Section 7</a>).

<span class="h4"><a class="selflink" id="section-3.3.2" href="#section-3.3.2">3.3.2</a>.  Common</span>

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

   o  Transfer Length (F): 40-bit unsigned integer.  A non-negative
      integer that is at most 946270874880.  This is the transfer length
      of the object in units of octets.

   o  Symbol Size (T): 16-bit unsigned integer.  A positive integer that
      is less than 2^^16.  This is the size of a symbol in units of
      octets.



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   The encoded Common FEC Object Transmission Information (OTI) format
   is shown in Figure 2.

       0                   1                   2                   3
       0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1
      +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
      |                      Transfer Length (F)                      |
      +               +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
      |               |    Reserved   |           Symbol Size (T)     |
      +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+

          Figure 2: Encoded Common FEC OTI for RaptorQ FEC Scheme

      NOTE: The limit of 946270874880 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 56403,
      and the limitation on the number of source blocks to 2^^8.

<span class="h4"><a class="selflink" id="section-3.3.3" href="#section-3.3.3">3.3.3</a>.  Scheme-Specific</span>

   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): 8-bit unsigned integer.

   o  The number of sub-blocks (N): 16-bit unsigned integer.

   o  A symbol alignment parameter (Al): 8-bit unsigned integer.

   These parameters are all positive integers.  The encoded Scheme-
   specific Object Transmission Information is a 4-octet field
   consisting of the parameters Z, N, and Al 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
   described in <a href="#section-4.4.1.2">Section 4.4.1.2</a>.



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<span class="h2"><a class="selflink" id="section-4" href="#section-4">4</a>.  Procedures</span>

<span class="h3"><a class="selflink" id="section-4.1" href="#section-4.1">4.1</a>.  Introduction</span>

   For any undefined symbols or functions used in this section, in
   particular the functions "ceil" and "floor", refer to <a href="#section-5.1">Section 5.1</a>.

<span class="h3"><a class="selflink" id="section-4.2" href="#section-4.2">4.2</a>.  Content Delivery Protocol Requirements</span>

   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  F: the transfer length of the object, in octets

   o  Al: the symbol alignment parameter

   o  T: the symbol size in octets, 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

   The RaptorQ encoder scheme for object delivery additionally requires:

   -  the object to be encoded, which is F octets long

   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.

<span class="h3"><a class="selflink" id="section-4.3" href="#section-4.3">4.3</a>.  Example Parameter Derivation Algorithm</span>

   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 octets



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

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

   o  Al: the symbol alignment parameter, in octets

   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: <a href="#section-5.1.2">Section 5.1.2</a> defines K'_max to be 56403.

   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 <a href="#section-5.6">Section 5.6</a> such
         that

            K' &lt;= 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) &lt;= 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 <a href="#section-4.4.1.2">Section 4.4.1.2</a> ensure that
   the sub-symbol sizes are a multiple of the symbol alignment
   parameter, Al.  This is useful because the sum operations used for
   encoding and decoding are generally performed several octets at a



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   time, for example, at least 4 octets 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 octets.

   The recommended setting for the input parameter Al is 4.

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

<span class="h3"><a class="selflink" id="section-4.4" href="#section-4.4">4.4</a>.  Object Delivery</span>

<span class="h4"><a class="selflink" id="section-4.4.1" href="#section-4.4.1">4.4.1</a>.  Source Block Construction</span>

<span class="h5"><a class="selflink" id="section-4.4.1.1" href="#section-4.4.1.1">4.4.1.1</a>.  General</span>

   In order to apply the RaptorQ encoder to a source object, the object
   may be broken into Z &gt;= 1 blocks, known as source blocks.  The
   RaptorQ encoder is applied independently to each source block.  Each
   source block is identified by a unique 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 octets each.  Each source symbol is identified by a
   unique 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 &gt;= 1 sub-
   blocks, which are small enough to be decoded in the working memory.
   Each sub-block is divided into K sub-symbols of size T'.

   Note that the value of K is not necessarily the same for each source
   block of an object, and the value of T' may not necessarily be the
   same for each sub-block of a source block.  However, the symbol size
   T is the same for all source blocks of an object, and the number of
   symbols K is the same for every sub-block of a source block.  Exact
   partitioning of the object into source blocks and sub-blocks is
   described in <a href="#section-4.4.1.2">Section 4.4.1.2</a> below.

<span class="h5"><a class="selflink" id="section-4.4.1.2" href="#section-4.4.1.2">4.4.1.2</a>.  Source Block and Sub-Block Partitioning</span>

   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 octets




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   o  Al: a symbol alignment parameter, in octets

   o  T: the symbol size, in octets, 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) &lt;= K'_max.
   Recommendations for derivation of these parameters are provided in
   <a href="#section-4.3">Section 4.3</a>.

   The function Partition[I,J] derives parameters for partitioning a
   block of size I into J approximately equal-sized blocks.  More
   specifically, it partitions I into JL blocks of length IL and JS
   blocks of length IS.  The output of Partition[I, J] is the sequence
   (IL, IS, JL, JS), where IL = ceil(I/J), IS = floor(I/J), JL = I - IS
   * J, and JS = J - JL.

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

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

   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 octets and
   the remaining NS sub-blocks each consisting of K contiguous sub-
   symbols of size of TS*Al octets.  The symbol alignment parameter Al
   ensures that sub-symbols are always a multiple of Al octets.






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   Finally, the mth symbol of a source block consists of the
   concatenation of the mth sub-symbol from each of the N sub-blocks.
   Note that this implies that when N &gt; 1, a symbol is NOT a contiguous
   portion of the object.

<span class="h4"><a class="selflink" id="section-4.4.2" href="#section-4.4.2">4.4.2</a>.  Encoding Packet Construction</span>

   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.  Each
   encoding packet contains encoding symbols generated from the one
   source block identified by the SBN carried in the encoding packet.
   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 contains only source symbols (source
   packet) or contains only 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
   octets added for FEC encoding purposes, then these octets need not be
   included in the packet.  Otherwise, each packet MUST contain only
   whole symbols.

   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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<span class="grey"><a href="./rfc6330">RFC 6330</a>                   RaptorQ FEC Scheme                August 2011</span>


<span class="h4"><a class="selflink" id="section-4.4.3" href="#section-4.4.3">4.4.3</a>.  Example Receiver Recovery Strategies</span>

   A receiver can use the received encoding symbols for each source
   block of an object to recover the source symbols for that source
   block independently of all other source blocks.

   If there is one sub-block per source block, i.e., N = 1, then the
   portion of the data in the original object in its original order
   associated with a source block consists of the concatenation of the
   source symbols of a source block in consecutive ESI order.

   If there are multiple sub-blocks per source block, i.e., if N &gt; 1,
   then the portion of the data in the original object in its original
   order associated with a source block consists of the concatenation of
   the sub-blocks associated with the source block, where sub-symbols
   within each sub-block are in consecutive ESI order.  In this case,
   there are different receiver source block recovery strategies worth
   considering depending on the available amount of Random Access Memory
   (RAM) at the receiver, as outlined below.

   One strategy is to recover the source symbols of a source block using
   the decoding procedures applied to the received symbols for the
   source block to recover the source symbols as described in <a href="#section-5">Section 5</a>,
   and then to reorder the sub-symbols of the source symbols so that all
   consecutive sub-symbols of the first sub-block are first, followed by
   all consecutive sub-symbols of the second sub-block, etc., followed
   by all consecutive sub-symbols of the Nth sub-block.  This strategy
   is especially applicable if the receiver has enough RAM to decode an
   entire source block.

   Another strategy is to separately recover the sub-blocks of a source
   block.  For example, a receiver may demultiplex and store sub-symbols
   associated with each sub-block separately as packets containing
   encoding symbols arrive, and then use the stored sub-symbols received
   for a sub-block to recover that sub-block using the decoding
   procedures described in <a href="#section-5">Section 5</a>.  This strategy is especially
   applicable if the receiver has enough RAM to decode only one sub-
   block at a time.

<span class="h2"><a class="selflink" id="section-5" href="#section-5">5</a>.  RaptorQ FEC Code Specification</span>

<span class="h3"><a class="selflink" id="section-5.1" href="#section-5.1">5.1</a>.  Background</span>

   For the purpose of the RaptorQ FEC code specification in this
   section, the following definitions, symbols, and abbreviations apply.
   A basic understanding of linear algebra, matrix operations, and
   finite fields is assumed in this section.  In particular, matrix
   multiplication and matrix inversion operations over a mixture of the



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<span class="grey"><a href="./rfc6330">RFC 6330</a>                   RaptorQ FEC Scheme                August 2011</span>


   finite fields GF[2] and GF[256] are used.  A basic familiarity with
   sparse linear equations, and efficient implementations of algorithms
   that take advantage of sparse linear equations, is also quite
   beneficial to an implementer of this specification.

<span class="h4"><a class="selflink" id="section-5.1.1" href="#section-5.1.1">5.1.1</a>.  Definitions</span>

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

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

   o  Symbol: a unit of data.  The size, in octets, of a symbol is known
      as the symbol size.  The symbol size is always a positive 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 based on pre-coding
      relationships.  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 for the purposes of the encoding process.

   o  LT symbols: a process similar to that described in [<a href="#ref-LTCodes" title="&quot;LT codes&quot;">LTCodes</a>] is
      used to generate part of the contribution to each generated
      encoding symbol from the portion of the intermediate symbols
      designated as LT symbols.



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<span class="grey"><a href="./rfc6330">RFC 6330</a>                   RaptorQ FEC Scheme                August 2011</span>


   o  PI symbols: a process even simpler than that described in
      [<a href="#ref-LTCodes" title="&quot;LT codes&quot;">LTCodes</a>] is used to generate the other part of the contribution
      to each generated encoding symbol from the portion of the
      intermediate symbols designated as PI symbols.  In the decoding
      algorithm suggested in <a href="#section-5.4">Section 5.4</a>, the PI symbols are inactivated
      at the start, i.e., are placed into the matrix U at the beginning
      of the first phase of the decoding algorithm.  Because the symbols
      corresponding to the columns of U are sometimes called the
      "inactivated" symbols, and since the PI symbols are inactivated at
      the beginning, they are considered "permanently inactivated".

   o  HDPC symbols: there is a small subset of the intermediate symbols
      that are HDPC symbols.  Each HDPC symbol has a pre-coding
      relationship with a large fraction of the other intermediate
      symbols.  HDPC means "High Density Parity Check".

   o  LDPC symbols: there is a moderate-sized subset of the intermediate
      symbols that are LDPC symbols.  Each LDPC symbol has a pre-coding
      relationship with a small fraction of the other intermediate
      symbols.  LDPC means "Low Density Parity Check".

   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 (ESI): information that uniquely identifies
      each encoding symbol associated with a source block for sending
      and receiving purposes.

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

   o  Arithmetic operations on octets and symbols and matrices: the
      operations that are used to produce encoding symbols from source
      symbols and vice versa.  See <a href="#section-5.7">Section 5.7</a>.

<span class="h4"><a class="selflink" id="section-5.1.2" href="#section-5.1.2">5.1.2</a>.  Symbols</span>

   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 that is greater than or equal
        to x, where x is a real value.




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   floor(x)  denotes the largest integer that 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 value of the values x and y, and in
        general the maximum value of all the argument values.

   i % j  denotes i modulo j.

   i + j  denotes the sum of i and j.  If i and j are octets or symbols,
        this designates the arithmetic on octets or symbols,
        respectively, as defined in <a href="#section-5.7">Section 5.7</a>.  If i and j are
        integers, then it denotes the usual integer addition.

   i * j  denotes the product of i and j.  If i and j are octets, this
        designates the arithmetic on octets, as defined in <a href="#section-5.7">Section 5.7</a>.
        If i is an octet and j is a symbol, this denotes the
        multiplication of a symbol by an octet, as also defined in
        <a href="#section-5.7">Section 5.7</a>.  Finally, if i and j are integers, i * j denotes
        the usual product of integers.

   a ^^ b  denotes the operation a raised to the power b.  If a is an
        octet and b is a non-negative integer, this is understood to
        mean a*a*...*a (b terms), with '*' being the octet product as
        defined in <a href="#section-5.7">Section 5.7</a>.

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

   Transpose[A]  denotes the transposed matrix of matrix A.  In this
        specification, all matrices have entries that are octets.

   A^^-1  denotes the inverse matrix of matrix A.  In this
        specification, all the matrices have octets as entries, so it is
        understood that the operations of the matrix entries are to be
        done as stated in <a href="#section-5.7">Section 5.7</a> and A^^-1 is the matrix inverse of
        A with respect to octet arithmetic.

   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 56403.



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<span class="grey"><a href="./rfc6330">RFC 6330</a>                   RaptorQ FEC Scheme                August 2011</span>


   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 <a href="#section-5.6">Section 5.6</a>, 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 <a href="#section-5.6">Section 5.6</a>, 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 32-bit unsigned 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
        <a href="#section-5.5">Section 5.5</a>.

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

   Deg[v]  denotes a degree generator.

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

   Tuple[K', X]  denotes a tuple generator function.




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   T    denotes the symbol size in octets.

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

   G    denotes any generator matrix.

   I_S  denotes the S x S identity matrix.

<span class="h3"><a class="selflink" id="section-5.2" href="#section-5.2">5.2</a>.  Overview</span>

   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 arithmetic operations
   defined in <a href="#section-5.7">Section 5.7</a>.

   The basic encoder is described in <a href="#section-5.3">Section 5.3</a>.  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 <a href="#section-5.4">Section 5.4</a>.  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.

   Requirements for a RaptorQ-compliant decoder are provided in
   <a href="#section-5.8">Section 5.8</a>.  A number of decoding algorithms are possible to achieve
   these requirements.  An efficient decoding algorithm to achieve these
   requirements is provided in <a href="#section-5.4">Section 5.4</a>.

   The construction of the intermediate and repair symbols is based in
   part on a pseudo-random number generator described in <a href="#section-5.3">Section 5.3</a>.
   This generator is based on a fixed set of 1024 random numbers that
   must be available to both sender and receiver.  These numbers are





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<span class="grey"><a href="./rfc6330">RFC 6330</a>                   RaptorQ FEC Scheme                August 2011</span>


   provided in <a href="#section-5.5">Section 5.5</a>.  Encoding and decoding operations for
   RaptorQ use operations on octets.  <a href="#section-5.7">Section 5.7</a> describes how 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 <a href="#section-5.6">Section 5.6</a> for specific extended source block sizes
   between 6 and K'_max = 56403 source symbols.  Thus, the RaptorQ code
   supports source blocks with between 1 and 56403 source symbols.

<span class="h3"><a class="selflink" id="section-5.3" href="#section-5.3">5.3</a>.  Systematic RaptorQ Encoder</span>

<span class="h4"><a class="selflink" id="section-5.3.1" href="#section-5.3.1">5.3.1</a>.  Introduction</span>

   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 <a href="#section-5.6">Section 5.6</a>.  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



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<span class="grey"><a href="./rfc6330">RFC 6330</a>                   RaptorQ FEC Scheme                August 2011</span>


   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.  Also, it ensures 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
   received) and the corresponding ISIs (used for encoding and
   decoding), as well as determining the proper padding of the extended
   source block with padding symbols (used for encoding and decoding),
   is the internal responsibility of the RaptorQ encoder/decoder.

<span class="h4"><a class="selflink" id="section-5.3.2" href="#section-5.3.2">5.3.2</a>.  Encoding Overview</span>

   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 <a href="#section-5.6">Section 5.6</a>.  Each
   padding symbol consists of T octets where the value of each octet is
   zero.  K' MUST be selected as the smallest value of K' from the table
   of <a href="#section-5.6">Section 5.6</a> that is greater than or equal to K.





















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<span class="grey"><a href="./rfc6330">RFC 6330</a>                   RaptorQ FEC Scheme                August 2011</span>


         -----------------------------------------------------------+
         |                                                          |
         |    +-----------+    +--------------+    +-------------+  |
      C' |    |           | C' | Intermediate | C  |             |  |
     ----+---&gt;|  Padding  |---&gt;|    Symbol    |---&gt;|   Encoding  |--+--&gt;
      K  |    |           | K' |  Generation  | L  |             |  |
         |    +-----------+    +--------------+    +-------------+  |
         |           |                             (d,a,b, ^        |
         |           |                            d1,a1,b1)|        |
         |           |                              +------------+  |
         |           |              K'              |   Tuple    |  |
         |           +-----------------------------&gt;|            |  |
         |                                          | 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
   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 &gt; 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 <a href="#section-5.3.5.4">Section 5.3.5.4</a>.  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.





<span class="grey">Luby, et al.                 Standards Track                   [Page 20]</span></pre>
<hr class='noprint'/><!--NewPage--><pre class='newpage'><span id="page-21" ></span>
<span class="grey"><a href="./rfc6330">RFC 6330</a>                   RaptorQ FEC Scheme                August 2011</span>


   Certain "pre-coding relationships" must hold within the L
   intermediate symbols.  <a href="#section-5.3.3.3">Section 5.3.3.3</a> describes these relationships.
   <a href="#section-5.3.3.4">Section 5.3.3.4</a> 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 &gt; K', the tuple of non-
   negative integers (d, a, b, d1, a1, b1) can be generated, using the
   Tuple[] generator as described in <a href="#section-5.3.5.4">Section 5.3.5.4</a>.  Then, the (d, a,
   b, d1, a1, b1) tuple and the ISI X are used to generate the
   corresponding repair symbol from the intermediate symbols using the
   Enc[] generator described in <a href="#section-5.3.5.3">Section 5.3.5.3</a>.  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 &lt; K', the symbol generated using this
   process has the same value as C'[X].

<span class="h4"><a class="selflink" id="section-5.3.3" href="#section-5.3.3">5.3.3</a>.  First Encoding Step: Intermediate Symbol Generation</span>

<span class="h5"><a class="selflink" id="section-5.3.3.1" href="#section-5.3.3.1">5.3.3.1</a>.  General</span>

   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 &gt; K' is defined in <a href="#section-5.3.3.3">Section 5.3.3.3</a>.  The
   intermediate symbols are uniquely defined by two sets of constraints:

   1.  The intermediate symbols are related to the source symbols by a
       set of source symbol tuples and by the ISIs of the source
       symbols.  The generation of the source symbol tuples is defined
       in <a href="#section-5.3.3.2">Section 5.3.3.2</a> using the Tuple[] generator as described in
       <a href="#section-5.3.5.4">Section 5.3.5.4</a>.

   2.  A number of pre-coding relationships hold within the intermediate
       symbols themselves.  These are defined in <a href="#section-5.3.3.3">Section 5.3.3.3</a>.

   The generation of the L intermediate symbols is then defined in
   <a href="#section-5.3.3.4">Section 5.3.3.4</a>.

<span class="h5"><a class="selflink" id="section-5.3.3.2" href="#section-5.3.3.2">5.3.3.2</a>.  Source Symbol Tuples</span>

   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 &lt;= X &lt; K'.  The
   source symbol tuples are determined using the Tuple[] generator
   defined in <a href="#section-5.3.5.4">Section 5.3.5.4</a> as:

      For each X, 0 &lt;= X &lt; K'

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



<span class="grey">Luby, et al.                 Standards Track                   [Page 21]</span></pre>
<hr class='noprint'/><!--NewPage--><pre class='newpage'><span id="page-22" ></span>
<span class="grey"><a href="./rfc6330">RFC 6330</a>                   RaptorQ FEC Scheme                August 2011</span>


<span class="h5"><a class="selflink" id="section-5.3.3.3" href="#section-5.3.3.3">5.3.3.3</a>.  Pre-Coding Relationships</span>

   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 <a href="#section-5.6">Section 5.6</a>.

   Let

   o  S = S(K')

   o  H = H(K')

   o  W = W(K')

   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.






<span class="grey">Luby, et al.                 Standards Track                   [Page 22]</span></pre>
<hr class='noprint'/><!--NewPage--><pre class='newpage'><span id="page-23" ></span>
<span class="grey"><a href="./rfc6330">RFC 6330</a>                   RaptorQ FEC Scheme                August 2011</span>


   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)

      *  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

      *  b = (i+1) % P

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

   Recall that the addition of symbols is to be carried out as specified
   in <a href="#section-5.7">Section 5.7</a>.

   Note that the LDPC relations as defined in the algorithm above are
   linear, so there exists an S x B matrix G_LDPC,1 and an S x P matrix
   G_LDPC,2 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

   (The matrix G_LDPC,1 is defined by the first loop in the above
   algorithm, and G_LDPC,2 can be deduced from the second loop.)

   The second set of relations among the intermediate symbols C[0], ...,
   C[L-1] are the HDPC relations and they are defined as follows:





<span class="grey">Luby, et al.                 Standards Track                   [Page 23]</span></pre>
<hr class='noprint'/><!--NewPage--><pre class='newpage'><span id="page-24" ></span>
<span class="grey"><a href="./rfc6330">RFC 6330</a>                   RaptorQ FEC Scheme                August 2011</span>


   Let

   o  alpha denote the octet represented by integer 2 as defined in
      <a href="#section-5.7">Section 5.7</a>.

   o  MT denote an H x (K' + S) matrix of octets, where for j=0, ...,
      K'+S-2, the entry MT[i,j] is the octet represented by the integer
      1 if i= Rand[j+1,6,H] or i = (Rand[j+1,6,H] + Rand[j+1,7,H-1] + 1)
      % H, and MT[i,j] is the zero element for all other values of i,
      and for j=K'+S-1, MT[i,j] = alpha^^i for i=0, ..., H-1.

   o  GAMMA denote a (K'+S) x (K'+S) matrix of octets, where

         GAMMA[i,j] =

            alpha ^^ (i-j) for i &gt;= 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]] + MT * GAMMA *
      Transpose[C[0], ..., C[K'+S-1]] = 0,

   where '*' represents standard matrix multiplication utilizing the
   octet multiplication to define the multiplication between a matrix of
   octets and a matrix of symbols (in particular, the column vector of
   symbols), and '+' denotes addition over octet vectors.

<span class="h5"><a class="selflink" id="section-5.3.3.4" href="#section-5.3.3.4">5.3.3.4</a>.  Intermediate Symbols</span>

<span class="h6"><a class="selflink" id="section-5.3.3.4.1" href="#section-5.3.3.4.1">5.3.3.4.1</a>.  Definition</span>

   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 &lt;= X &lt; K',

       where (d[X], a[X], b[X], d1[X], a1[X], b1[X])) = Tuple[K',X],
       Tuple[] is defined in <a href="#section-5.3.5.4">Section 5.3.5.4</a>, and Enc[] is described in
       <a href="#section-5.3.5.3">Section 5.3.5.3</a>.



<span class="grey">Luby, et al.                 Standards Track                   [Page 24]</span></pre>
<hr class='noprint'/><!--NewPage--><pre class='newpage'><span id="page-25" ></span>
<span class="grey"><a href="./rfc6330">RFC 6330</a>                   RaptorQ FEC Scheme                August 2011</span>


   2.  The L intermediate symbols C[0], C[1], ..., C[L-1] satisfy the
       pre-coding relationships defined in <a href="#section-5.3.3.3">Section 5.3.3.3</a>.

<span class="h6"><a class="selflink" id="section-5.3.3.4.2" href="#section-5.3.3.4.2">5.3.3.4.2</a>.  Example Method for Calculation of Intermediate Symbols</span>

   This section describes a possible method for calculation of the L
   intermediate symbols C[0], C[1], ..., C[L-1] satisfying the
   constraints in <a href="#section-5.3.3.4.1">Section 5.3.3.4.1</a>.

   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 of octets 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 as defined in
      <a href="#section-5.3.3.3">Section 5.3.3.3</a>.

   o  G_HDPC be the H x (K'+S) matrix such that

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

         i.e., G_HDPC = MT*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 matrix such that

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





<span class="grey">Luby, et al.                 Standards Track                   [Page 25]</span></pre>
<hr class='noprint'/><!--NewPage--><pre class='newpage'><span id="page-26" ></span>
<span class="grey"><a href="./rfc6330">RFC 6330</a>                   RaptorQ FEC Scheme                August 2011</span>


         i.e., G_ENC[i,j] = 1 if and only if C[j] is included in the
         symbols that are summed 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 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 <a href="#section-5.4">Section 5.4</a> be used.




<span class="grey">Luby, et al.                 Standards Track                   [Page 26]</span></pre>
<hr class='noprint'/><!--NewPage--><pre class='newpage'><span id="page-27" ></span>
<span class="grey"><a href="./rfc6330">RFC 6330</a>                   RaptorQ FEC Scheme                August 2011</span>


<span class="h4"><a class="selflink" id="section-5.3.4" href="#section-5.3.4">5.3.4</a>.  Second Encoding Step: Encoding</span>

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

<span class="h4"><a class="selflink" id="section-5.3.5" href="#section-5.3.5">5.3.5</a>.  Generators</span>

<span class="h5"><a class="selflink" id="section-5.3.5.1" href="#section-5.3.5.1">5.3.5.1</a>.  Random Number Generator</span>

   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 the arrays
   provided in <a href="#section-5.5">Section 5.5</a>.

   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

<span class="h5"><a class="selflink" id="section-5.3.5.2" href="#section-5.3.5.2">5.3.5.2</a>.  Degree Generator</span>

   The degree generator Deg[v] is defined as follows, where v is a non-
   negative integer that is less than 2^^20 = 1048576.  Given v, find
   index d in Table 1 such that f[d-1] &lt;= v &lt; f[d], and set Deg[v] =
   min(d, W-2).  Recall that W is derived from K' as described in
   <a href="#section-5.3.3.3">Section 5.3.3.3</a>.












<span class="grey">Luby, et al.                 Standards Track                   [Page 27]</span></pre>
<hr class='noprint'/><!--NewPage--><pre class='newpage'><span id="page-28" ></span>
<span class="grey"><a href="./rfc6330">RFC 6330</a>                   RaptorQ FEC Scheme                August 2011</span>


                 +---------+---------+---------+---------+
                 | 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

<span class="h5"><a class="selflink" id="section-5.3.5.3" href="#section-5.3.5.3">5.3.5.3</a>.  Encoding Symbol Generator</span>

   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
      <a href="#section-5.3.3.3">Section 5.3.3.3</a>.





<span class="grey">Luby, et al.                 Standards Track                   [Page 28]</span></pre>
<hr class='noprint'/><!--NewPage--><pre class='newpage'><span id="page-29" ></span>
<span class="grey"><a href="./rfc6330">RFC 6330</a>                   RaptorQ FEC Scheme                August 2011</span>


   o  (C[0], C[1], ..., C[L-1]) is the array of L intermediate symbols
      (sub-symbols) generated as described in <a href="#section-5.3.3.4">Section 5.3.3.4</a>.

   o  (d, a, b, d1, a1, b1) is a source tuple determined from ISI X
      using the Tuple[] generator defined in <a href="#section-5.3.5.4">Section 5.3.5.4</a>, whereby

      *  d is a positive integer denoting an encoding symbol LT degree

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

      *  b is a non-negative integer between 0 and W-1 inclusive

      *  d1 is a positive integer that has value either 2 or 3 denoting
         an encoding symbol PI degree

      *  a1 is a positive integer between 1 and P1-1 inclusive

      *  b1 is a non-negative integer between 0 and P1-1 inclusive

   The encoding symbol generator produces a single encoding symbol as
   output (referred to as result), 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 &gt;= 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 &gt;= P) do b1 = (b1+a1) % P1

      *  result = result + C[W+b1]

   o  Return result








<span class="grey">Luby, et al.                 Standards Track                   [Page 29]</span></pre>
<hr class='noprint'/><!--NewPage--><pre class='newpage'><span id="page-30" ></span>
<span class="grey"><a href="./rfc6330">RFC 6330</a>                   RaptorQ FEC Scheme                August 2011</span>


<span class="h5"><a class="selflink" id="section-5.3.5.4" href="#section-5.3.5.4">5.3.5.4</a>.  Tuple Generator</span>

   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 ISI

   Let

   o  L be determined from K' as described in <a href="#section-5.3.3.3">Section 5.3.3.3</a>

   o  J = J(K') be the systematic index associated with K', as defined
      in Table 2 in <a href="#section-5.6">Section 5.6</a>

   The output of the 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 &lt; 4) { d1 = 2 + Rand[X, 3, 2] } else { d1 = 2 }

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

   o  b1 = Rand[X, 5, P1]

<span class="h3"><a class="selflink" id="section-5.4" href="#section-5.4">5.4</a>.  Example FEC Decoder</span>

<span class="h4"><a class="selflink" id="section-5.4.1" href="#section-5.4.1">5.4.1</a>.  General</span>

   This section describes an efficient decoding algorithm for the
   RaptorQ code introduced in this specification.  Note that each
   received encoding symbol is a known linear combination of the
   intermediate symbols.  So, each received encoding symbol provides a



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<hr class='noprint'/><!--NewPage--><pre class='newpage'><span id="page-31" ></span>
<span class="grey"><a href="./rfc6330">RFC 6330</a>                   RaptorQ FEC Scheme                August 2011</span>


   linear equation among the intermediate symbols, which, together with
   the known linear pre-coding relationships amongst the intermediate
   symbols, gives a system of linear equations.  Thus, any algorithm for
   solving systems of linear 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.

<span class="h4"><a class="selflink" id="section-5.4.2" href="#section-5.4.2">5.4.2</a>.  Decoding an Extended Source Block</span>

<span class="h5"><a class="selflink" id="section-5.4.2.1" href="#section-5.4.2.1">5.4.2.1</a>.  General</span>

   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 <a href="#section-5.3">Section 5.3</a>, 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
   sum is equal to the encoding symbol are passed to the decoder.  In
   the case of source symbols, including padding symbols, the source
   symbol tuples described in <a href="#section-5.3.3.2">Section 5.3.3.2</a> indicate the number and
   set of intermediate symbols that sum to give each source symbol.

   Let N &gt;= 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 <a href="#section-5.3.3.4.2">Section 5.3.3.4.2</a>, 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 that must be summed 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 x L identity matrix.  The decoding schedule consists of the
   sequence of row operations and row and column reorderings during the
   Gaussian elimination process, and it only depends on A and not on D.




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<hr class='noprint'/><!--NewPage--><pre class='newpage'><span id="page-32" ></span>
<span class="grey"><a href="./rfc6330">RFC 6330</a>                   RaptorQ FEC Scheme                August 2011</span>


   The decoding of C from D can take place concurrently with the forming
   of the decoding schedule, or the decoding can take place afterwards
   based on the decoding schedule.

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

   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 an octet 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 x 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.

<span class="h5"><a class="selflink" id="section-5.4.2.2" href="#section-5.4.2.2">5.4.2.2</a>.  First Phase</span>

   In 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.



<span class="grey">Luby, et al.                 Standards Track                   [Page 32]</span></pre>
<hr class='noprint'/><!--NewPage--><pre class='newpage'><span id="page-33" ></span>
<span class="grey"><a href="./rfc6330">RFC 6330</a>                   RaptorQ FEC Scheme                August 2011</span>


   The submatrix sizes are parameterized by non-negative integers i and
   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 consists of the first L-P columns of A, and U consists
   of the last P columns corresponding to the PI symbols.  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 nonzero
   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




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<hr class='noprint'/><!--NewPage--><pre class='newpage'><span id="page-34" ></span>
<span class="grey"><a href="./rfc6330">RFC 6330</a>                   RaptorQ FEC Scheme                August 2011</span>


   (rows) such that there is a path between each pair of nodes/edges in
   the graph.  The size of a component is the number of nodes (columns)
   in the component.

   There are at most L steps in the first phase.  The phase ends
   successfully when i + u = L, i.e., when V and the all zeros submatrix
   above V have disappeared, and A consists of I, the all zeros
   submatrix below I, and U.  The phase ends unsuccessfully in decoding
   failure if at some step before V disappears there is no nonzero 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 nonzeros in V.

      *  If r != 2, then choose a row with exactly r nonzeros 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 and there is a row with exactly 2 ones in V, then
         choose any row with exactly 2 ones in V that is part of a
         maximum size component in the graph described above that is
         defined by V.

      *  If r = 2 and there is no row with exactly 2 ones in V, then
         choose any row with exactly 2 nonzeros in 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 nonzeros in the
   chosen row appears in the first column of V and so that the remaining
   r-1 nonzeros 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 nonzero 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.





<span class="grey">Luby, et al.                 Standards Track                   [Page 34]</span></pre>
<hr class='noprint'/><!--NewPage--><pre class='newpage'><span id="page-35" ></span>
<span class="grey"><a href="./rfc6330">RFC 6330</a>                   RaptorQ FEC Scheme                August 2011</span>


   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 that are not eventually used in the decoding
   process, and in particular this avoids 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 <a href="#section-5.3.3.3">Section 5.3.3.3</a>.

<span class="h5"><a class="selflink" id="section-5.4.2.3" href="#section-5.4.2.3">5.4.2.3</a>.  Second Phase</span>

   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 and i
   columns.  The submatrix U is further partitioned into the first i
   rows, U_upper, and the remaining M - i rows, U_lower.  Gaussian
   elimination is performed in the second phase on U_lower either to
   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 x 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.

<span class="h5"><a class="selflink" id="section-5.4.2.4" href="#section-5.4.2.4">5.4.2.4</a>.  Third Phase</span>

   After the second phase, the only portion of A that needs to be zeroed
   out to finish converting A into the L x 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.

<span class="h5"><a class="selflink" id="section-5.4.2.5" href="#section-5.4.2.5">5.4.2.5</a>.  Fourth Phase</span>

   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 the matrix I_u.



<span class="grey">Luby, et al.                 Standards Track                   [Page 35]</span></pre>
<hr class='noprint'/><!--NewPage--><pre class='newpage'><span id="page-36" ></span>
<span class="grey"><a href="./rfc6330">RFC 6330</a>                   RaptorQ FEC Scheme                August 2011</span>


<span class="h5"><a class="selflink" id="section-5.4.2.6" href="#section-5.4.2.6">5.4.2.6</a>.  Fifth Phase</span>

   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.

   After this phase, A is the L x L identity matrix and a complete
   decoding schedule has been successfully formed.  Then, the
   corresponding decoding consisting of summing 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 <a href="#section-5.3.3.2">Section 5.3.3.2</a>.  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 summed to recover the missing source symbols.

<span class="h3"><a class="selflink" id="section-5.5" href="#section-5.5">5.5</a>.  Random Numbers</span>

   The four arrays V0, V1, V2, and V3 used in <a href="#section-5.3.5.1">Section 5.3.5.1</a> are
   provided below.  There are 256 entries in each of the four arrays.
   The indexing into each array starts at 0, and the entries are 32-bit
   unsigned integers.

<span class="h4"><a class="selflink" id="section-5.5.1" href="#section-5.5.1">5.5.1</a>.  The Table V0</span>

      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,



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<hr class='noprint'/><!--NewPage--><pre class='newpage'><span id="page-37" ></span>
<span class="grey"><a href="./rfc6330">RFC 6330</a>                   RaptorQ FEC Scheme                August 2011</span>


      1523670218, 2708475297, 1046771089, 2229796016, 1255426612,
      4213663089, 1521339547, 3041843489, 420130494, 10677091,
      515623176, 3457502702, 2115821274, 2720124766, 3242576090,
      854310108, 425973987, 325832382, 1796851292, 2462744411,
      1976681690, 1408671665, 1228817808, 3917210003, 263976645,
      2593736473, 2471651269, 4291353919, 650792940, 1191583883,
      3046561335, 2466530435, 2545983082, 969168436, 2019348792,
      2268075521, 1169345068, 3250240009, 3963499681, 2560755113,
      911182396, 760842409, 3569308693, 2687243553, 381854665,
      2613828404, 2761078866, 1456668111, 883760091, 3294951678,
      1604598575, 1985308198, 1014570543, 2724959607, 3062518035,
      3115293053, 138853680, 4160398285, 3322241130, 2068983570,
      2247491078, 3669524410, 1575146607, 828029864, 3732001371,
      3422026452, 3370954177, 4006626915, 543812220, 1243116171,
      3928372514, 2791443445, 4081325272, 2280435605, 885616073,
      616452097, 3188863436, 2780382310, 2340014831, 1208439576,
      258356309, 3837963200, 2075009450, 3214181212, 3303882142,
      880813252, 1355575717, 207231484, 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

<span class="h4"><a class="selflink" id="section-5.5.2" href="#section-5.5.2">5.5.2</a>.  The Table V1</span>

      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,



<span class="grey">Luby, et al.                 Standards Track                   [Page 37]</span></pre>
<hr class='noprint'/><!--NewPage--><pre class='newpage'><span id="page-38" ></span>
<span class="grey"><a href="./rfc6330">RFC 6330</a>                   RaptorQ FEC Scheme                August 2011</span>


      1047144830, 2982173936, 910285038, 4187576520, 2589870048,
      989448887, 3292758024, 506322719, 176010738, 1865471968,
      2619324712, 564829442, 1996870325, 339697593, 4071072948,
      3618966336, 2111320126, 1093955153, 957978696, 892010560,
      1854601078, 1873407527, 2498544695, 2694156259, 1927339682,
      1650555729, 183933047, 3061444337, 2067387204, 228962564,
      3904109414, 1595995433, 1780701372, 2463145963, 307281463,
      3237929991, 3852995239, 2398693510, 3754138664, 522074127,
      146352474, 4104915256, 3029415884, 3545667983, 332038910,
      976628269, 3123492423, 3041418372, 2258059298, 2139377204,
      3243642973, 3226247917, 3674004636, 2698992189, 3453843574,
      1963216666, 3509855005, 2358481858, 747331248, 1957348676,
      1097574450, 2435697214, 3870972145, 1888833893, 2914085525,
      4161315584, 1273113343, 3269644828, 3681293816, 412536684,
      1156034077, 3823026442, 1066971017, 3598330293, 1979273937,
      2079029895, 1195045909, 1071986421, 2712821515, 3377754595,
      2184151095, 750918864, 2585729879, 4249895712, 1832579367,
      1192240192, 946734366, 31230688, 3174399083, 3549375728,
      1642430184, 1904857554, 861877404, 3277825584, 4267074718,
      3122860549, 666423581, 644189126, 226475395, 307789415,
      1196105631, 3191691839, 782852669, 1608507813, 1847685900,
      4069766876, 3931548641, 2526471011, 766865139, 2115084288,
      4259411376, 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

<span class="h4"><a class="selflink" id="section-5.5.3" href="#section-5.5.3">5.5.3</a>.  The Table V2</span>

      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,



<span class="grey">Luby, et al.                 Standards Track                   [Page 38]</span></pre>
<hr class='noprint'/><!--NewPage--><pre class='newpage'><span id="page-39" ></span>
<span class="grey"><a href="./rfc6330">RFC 6330</a>                   RaptorQ FEC Scheme                August 2011</span>


      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,
      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





<span class="grey">Luby, et al.                 Standards Track                   [Page 39]</span></pre>
<hr class='noprint'/><!--NewPage--><pre class='newpage'><span id="page-40" ></span>
<span class="grey"><a href="./rfc6330">RFC 6330</a>                   RaptorQ FEC Scheme                August 2011</span>


<span class="h4"><a class="selflink" id="section-5.5.4" href="#section-5.5.4">5.5.4</a>.  The Table V3</span>

      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, 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,



<span class="grey">Luby, et al.                 Standards Track                   [Page 40]</span></pre>
<hr class='noprint'/><!--NewPage--><pre class='newpage'><span id="page-41" ></span>
<span class="grey"><a href="./rfc6330">RFC 6330</a>                   RaptorQ FEC Scheme                August 2011</span>


      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

<span class="h3"><a class="selflink" id="section-5.6" href="#section-5.6">5.6</a>.  Systematic Indices and Other Parameters</span>

   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') |
                 +-------+-------+-------+-------+-------+
                 | 10    | 254   | 7     | 10    | 17    |
                 +-------+-------+-------+-------+-------+
                 | 12    | 630   | 7     | 10    | 19    |
                 +-------+-------+-------+-------+-------+
                 | 18    | 682   | 11    | 10    | 29    |
                 +-------+-------+-------+-------+-------+
                 | 20    | 293   | 11    | 10    | 31    |
                 +-------+-------+-------+-------+-------+
                 | 26    | 80    | 11    | 10    | 37    |
                 +-------+-------+-------+-------+-------+
                 | 30    | 566   | 11    | 10    | 41    |
                 +-------+-------+-------+-------+-------+
                 | 32    | 860   | 11    | 10    | 43    |
                 +-------+-------+-------+-------+-------+
                 | 36    | 267   | 11    | 10    | 47    |
                 +-------+-------+-------+-------+-------+
                 | 42    | 822   | 11    | 10    | 53    |
                 +-------+-------+-------+-------+-------+
                 | 46    | 506   | 13    | 10    | 59    |
                 +-------+-------+-------+-------+-------+
                 | 48    | 589   | 13    | 10    | 61    |
                 +-------+-------+-------+-------+-------+
                 | 49    | 87    | 13    | 10    | 61    |
                 +-------+-------+-------+-------+-------+



<span class="grey">Luby, et al.                 Standards Track                   [Page 41]</span></pre>
<hr class='noprint'/><!--NewPage--><pre class='newpage'><span id="page-42" ></span>
<span class="grey"><a href="./rfc6330">RFC 6330</a>                   RaptorQ FEC Scheme                August 2011</span>


                 +-------+-------+-------+-------+-------+
                 | 55    | 520   | 13    | 10    | 67    |
                 +-------+-------+-------+-------+-------+
                 | 60    | 159   | 13    | 10    | 71    |
                 +-------+-------+-------+-------+-------+
                 | 62    | 235   | 13    | 10    | 73    |
                 +-------+-------+-------+-------+-------+
                 | 69    | 157   | 13    | 10    | 79    |
                 +-------+-------+-------+-------+-------+
                 | 75    | 502   | 17    | 10    | 89    |
                 +-------+-------+-------+-------+-------+
                 | 84    | 334   | 17    | 10    | 97    |
                 +-------+-------+-------+-------+-------+
                 | 88    | 583   | 17    | 10    | 101   |
                 +-------+-------+-------+-------+-------+
                 | 91    | 66    | 17    | 10    | 103   |
                 +-------+-------+-------+-------+-------+
                 | 95    | 352   | 17    | 10    | 107   |
                 +-------+-------+-------+-------+-------+
                 | 97    | 365   | 17    | 10    | 109   |
                 +-------+-------+-------+-------+-------+
                 | 101   | 562   | 17    | 10    | 113   |
                 +-------+-------+-------+-------+-------+
                 | 114   | 5     | 19    | 10    | 127   |
                 +-------+-------+-------+-------+-------+
                 | 119   | 603   | 19    | 10    | 131   |
                 +-------+-------+-------+-------+-------+
                 | 125   | 721   | 19    | 10    | 137   |
                 +-------+-------+-------+-------+-------+
                 | 127   | 28    | 19    | 10    | 139   |
                 +-------+-------+-------+-------+-------+
                 | 138   | 660   | 19    | 10    | 149   |
                 +-------+-------+-------+-------+-------+
                 | 140   | 829   | 19    | 10    | 151   |
                 +-------+-------+-------+-------+-------+
                 | 149   | 900   | 23    | 10    | 163   |
                 +-------+-------+-------+-------+-------+
                 | 153   | 930   | 23    | 10    | 167   |
                 +-------+-------+-------+-------+-------+
                 | 160   | 814   | 23    | 10    | 173   |
                 +-------+-------+-------+-------+-------+
                 | 166   | 661   | 23    | 10    | 179   |
                 +-------+-------+-------+-------+-------+
                 | 168   | 693   | 23    | 10    | 181   |
                 +-------+-------+-------+-------+-------+
                 | 179   | 780   | 23    | 10    | 191   |
                 +-------+-------+-------+-------+-------+




<span class="grey">Luby, et al.                 Standards Track                   [Page 42]</span></pre>
<hr class='noprint'/><!--NewPage--><pre class='newpage'><span id="page-43" ></span>
<span class="grey"><a href="./rfc6330">RFC 6330</a>                   RaptorQ FEC Scheme                August 2011</span>


                 +-------+-------+-------+-------+-------+
                 | 181   | 605   | 23    | 10    | 193   |
                 +-------+-------+-------+-------+-------+
                 | 185   | 551   | 23    | 10    | 197   |
                 +-------+-------+-------+-------+-------+
                 | 187   | 777   | 23    | 10    | 199   |
                 +-------+-------+-------+-------+-------+
                 | 200   | 491   | 23    | 10    | 211   |
                 +-------+-------+-------+-------+-------+
                 | 213   | 396   | 23    | 10    | 223   |
                 +-------+-------+-------+-------+-------+
                 | 217   | 764   | 29    | 10    | 233   |
                 +-------+-------+-------+-------+-------+
                 | 225   | 843   | 29    | 10    | 241   |
                 +-------+-------+-------+-------+-------+
                 | 236   | 646   | 29    | 10    | 251   |
                 +-------+-------+-------+-------+-------+
                 | 242   | 557   | 29    | 10    | 257   |
                 +-------+-------+-------+-------+-------+
                 | 248   | 608   | 29    | 10    | 263   |
                 +-------+-------+-------+-------+-------+
                 | 257   | 265   | 29    | 10    | 271   |
                 +-------+-------+-------+-------+-------+
                 | 263   | 505   | 29    | 10    | 277   |
                 +-------+-------+-------+-------+-------+
                 | 269   | 722   | 29    | 10    | 283   |
                 +-------+-------+-------+-------+-------+
                 | 280   | 263   | 29    | 10    | 293   |
                 +-------+-------+-------+-------+-------+
                 | 295   | 999   | 29    | 10    | 307   |
                 +-------+-------+-------+-------+-------+
                 | 301   | 874   | 29    | 10    | 313   |
                 +-------+-------+-------+-------+-------+
                 | 305   | 160   | 29    | 10    | 317   |
                 +-------+-------+-------+-------+-------+
                 | 324   | 575   | 31    | 10    | 337   |
                 +-------+-------+-------+-------+-------+
                 | 337   | 210   | 31    | 10    | 349   |
                 +-------+-------+-------+-------+-------+
                 | 341   | 513   | 31    | 10    | 353   |
                 +-------+-------+-------+-------+-------+
                 | 347   | 503   | 31    | 10    | 359   |
                 +-------+-------+-------+-------+-------+
                 | 355   | 558   | 31    | 10    | 367   |
                 +-------+-------+-------+-------+-------+
                 | 362   | 932   | 31    | 10    | 373   |
                 +-------+-------+-------+-------+-------+




<span class="grey">Luby, et al.                 Standards Track                   [Page 43]</span></pre>
<hr class='noprint'/><!--NewPage--><pre class='newpage'><span id="page-44" ></span>
<span class="grey"><a href="./rfc6330">RFC 6330</a>                   RaptorQ FEC Scheme                August 2011</span>


                 +-------+-------+-------+-------+-------+
                 | 368   | 404   | 31    | 10    | 379   |
                 +-------+-------+-------+-------+-------+
                 | 372   | 520   | 37    | 10    | 389   |
                 +-------+-------+-------+-------+-------+
                 | 380   | 846   | 37    | 10    | 397   |
                 +-------+-------+-------+-------+-------+
                 | 385   | 485   | 37    | 10    | 401   |
                 +-------+-------+-------+-------+-------+
                 | 393   | 728   | 37    | 10    | 409   |
                 +-------+-------+-------+-------+-------+
                 | 405   | 554   | 37    | 10    | 421   |
                 +-------+-------+-------+-------+-------+
                 | 418   | 471   | 37    | 10    | 433   |
                 +-------+-------+-------+-------+-------+
                 | 428   | 641   | 37    | 10    | 443   |
                 +-------+-------+-------+-------+-------+
                 | 434   | 732   | 37    | 10    | 449   |
                 +-------+-------+-------+-------+-------+
                 | 447   | 193   | 37    | 10    | 461   |
                 +-------+-------+-------+-------+-------+
                 | 453   | 934   | 37    | 10    | 467   |
                 +-------+-------+-------+-------+-------+
                 | 466   | 864   | 37    | 10    | 479   |
                 +-------+-------+-------+-------+-------+
                 | 478   | 790   | 37    | 10    | 491   |
                 +-------+-------+-------+-------+-------+
                 | 486   | 912   | 37    | 10    | 499   |
                 +-------+-------+-------+-------+-------+
                 | 491   | 617   | 37    | 10    | 503   |
                 +-------+-------+-------+-------+-------+
                 | 497   | 587   | 37    | 10    | 509   |
                 +-------+-------+-------+-------+-------+
                 | 511   | 800   | 37    | 10    | 523   |
                 +-------+-------+-------+-------+-------+
                 | 526   | 923   | 41    | 10    | 541   |
                 +-------+-------+-------+-------+-------+
                 | 532   | 998   | 41    | 10    | 547   |
                 +-------+-------+-------+-------+-------+
                 | 542   | 92    | 41    | 10    | 557   |
                 +-------+-------+-------+-------+-------+
                 | 549   | 497   | 41    | 10    | 563   |
                 +-------+-------+-------+-------+-------+
                 | 557   | 559   | 41    | 10    | 571   |
                 +-------+-------+-------+-------+-------+
                 | 563   | 667   | 41    | 10    | 577   |
                 +-------+-------+-------+-------+-------+




<span class="grey">Luby, et al.                 Standards Track                   [Page 44]</span></pre>
<hr class='noprint'/><!--NewPage--><pre class='newpage'><span id="page-45" ></span>
<span class="grey"><a href="./rfc6330">RFC 6330</a>                   RaptorQ FEC Scheme                August 2011</span>


                 +-------+-------+-------+-------+-------+
                 | 573   | 912   | 41    | 10    | 587   |
                 +-------+-------+-------+-------+-------+
                 | 580   | 262   | 41    | 10    | 593   |
                 +-------+-------+-------+-------+-------+
                 | 588   | 152   | 41    | 10    | 601   |
                 +-------+-------+-------+-------+-------+
                 | 594   | 526   | 41    | 10    | 607   |
                 +-------+-------+-------+-------+-------+
                 | 600   | 268   | 41    | 10    | 613   |
                 +-------+-------+-------+-------+-------+
                 | 606   | 212   | 41    | 10    | 619   |
                 +-------+-------+-------+-------+-------+
                 | 619   | 45    | 41    | 10    | 631   |
                 +-------+-------+-------+-------+-------+
                 | 633   | 898   | 43    | 10    | 647   |
                 +-------+-------+-------+-------+-------+
                 | 640   | 527   | 43    | 10    | 653   |
                 +-------+-------+-------+-------+-------+
                 | 648   | 558   | 43    | 10    | 661   |
                 +-------+-------+-------+-------+-------+
                 | 666   | 460   | 47    | 10    | 683   |
                 +-------+-------+-------+-------+-------+
                 | 675   | 5     | 47    | 10    | 691   |
                 +-------+-------+-------+-------+-------+
                 | 685   | 895   | 47    | 10    | 701   |
                 +-------+-------+-------+-------+-------+
                 | 693   | 996   | 47    | 10    | 709   |
                 +-------+-------+-------+-------+-------+
                 | 703   | 282   | 47    | 10    | 719   |
                 +-------+-------+-------+-------+-------+
                 | 718   | 513   | 47    | 10    | 733   |
                 +-------+-------+-------+-------+-------+
                 | 728   | 865   | 47    | 10    | 743   |
                 +-------+-------+-------+-------+-------+
                 | 736   | 870   | 47    | 10    | 751   |
                 +-------+-------+-------+-------+-------+
                 | 747   | 239   | 47    | 10    | 761   |
                 +-------+-------+-------+-------+-------+
                 | 759   | 452   | 47    | 10    | 773   |
                 +-------+-------+-------+-------+-------+
                 | 778   | 862   | 53    | 10    | 797   |
                 +-------+-------+-------+-------+-------+
                 | 792   | 852   | 53    | 10    | 811   |
                 +-------+-------+-------+-------+-------+
                 | 802   | 643   | 53    | 10    | 821   |
                 +-------+-------+-------+-------+-------+




<span class="grey">Luby, et al.                 Standards Track                   [Page 45]</span></pre>
<hr class='noprint'/><!--NewPage--><pre class='newpage'><span id="page-46" ></span>
<span class="grey"><a href="./rfc6330">RFC 6330</a>                   RaptorQ FEC Scheme                August 2011</span>


                 +-------+-------+-------+-------+-------+
                 | 811   | 543   | 53    | 10    | 829   |
                 +-------+-------+-------+-------+-------+
                 | 821   | 447   | 53    | 10    | 839   |
                 +-------+-------+-------+-------+-------+
                 | 835   | 321   | 53    | 10    | 853   |
                 +-------+-------+-------+-------+-------+
                 | 845   | 287   | 53    | 10    | 863   |
                 +-------+-------+-------+-------+-------+
                 | 860   | 12    | 53    | 10    | 877   |
                 +-------+-------+-------+-------+-------+
                 | 870   | 251   | 53    | 10    | 887   |
                 +-------+-------+-------+-------+-------+
                 | 891   | 30    | 53    | 10    | 907   |
                 +-------+-------+-------+-------+-------+
                 | 903   | 621   | 53    | 10    | 919   |
                 +-------+-------+-------+-------+-------+
                 | 913   | 555   | 53    | 10    | 929   |
                 +-------+-------+-------+-------+-------+
                 | 926   | 127   | 53    | 10    | 941   |
                 +-------+-------+-------+-------+-------+
                 | 938   | 400   | 53    | 10    | 953   |
                 +-------+-------+-------+-------+-------+
                 | 950   | 91    | 59    | 10    | 971   |
                 +-------+-------+-------+-------+-------+
                 | 963   | 916   | 59    | 10    | 983   |
                 +-------+-------+-------+-------+-------+
                 | 977   | 935   | 59    | 10    | 997   |
                 +-------+-------+-------+-------+-------+
                 | 989   | 691   | 59    | 10    | 1009  |
                 +-------+-------+-------+-------+-------+
                 | 1002  | 299   | 59    | 10    | 1021  |
                 +-------+-------+-------+-------+-------+
                 | 1020  | 282   | 59    | 10    | 1039  |
                 +-------+-------+-------+-------+-------+
                 | 1032  | 824   | 59    | 10    | 1051  |
                 +-------+-------+-------+-------+-------+
                 | 1050  | 536   | 59    | 11    | 1069  |
                 +-------+-------+-------+-------+-------+
                 | 1074  | 596   | 59    | 11    | 1093  |
                 +-------+-------+-------+-------+-------+
                 | 1085  | 28    | 59    | 11    | 1103  |
                 +-------+-------+-------+-------+-------+
                 | 1099  | 947   | 59    | 11    | 1117  |
                 +-------+-------+-------+-------+-------+
                 | 1111  | 162   | 59    | 11    | 1129  |
                 +-------+-------+-------+-------+-------+




<span class="grey">Luby, et al.                 Standards Track                   [Page 46]</span></pre>
<hr class='noprint'/><!--NewPage--><pre class='newpage'><span id="page-47" ></span>
<span class="grey"><a href="./rfc6330">RFC 6330</a>                   RaptorQ FEC Scheme                August 2011</span>


                 +-------+-------+-------+-------+-------+
                 | 1136  | 536   | 59    | 11    | 1153  |
                 +-------+-------+-------+-------+-------+
                 | 1152  | 1000  | 61    | 11    | 1171  |
                 +-------+-------+-------+-------+-------+
                 | 1169  | 251   | 61    | 11    | 1187  |
                 +-------+-------+-------+-------+-------+
                 | 1183  | 673   | 61    | 11    | 1201  |
                 +-------+-------+-------+-------+-------+
                 | 1205  | 559   | 61    | 11    | 1223  |
                 +-------+-------+-------+-------+-------+
                 | 1220  | 923   | 61    | 11    | 1237  |
                 +-------+-------+-------+-------+-------+
                 | 1236  | 81    | 67    | 11    | 1259  |
                 +-------+-------+-------+-------+-------+
                 | 1255  | 478   | 67    | 11    | 1277  |
                 +-------+-------+-------+-------+-------+
                 | 1269  | 198   | 67    | 11    | 1291  |
                 +-------+-------+-------+-------+-------+
                 | 1285  | 137   | 67    | 11    | 1307  |
                 +-------+-------+-------+-------+-------+
                 | 1306  | 75    | 67    | 11    | 1327  |
                 +-------+-------+-------+-------+-------+
                 | 1347  | 29    | 67    | 11    | 1367  |
                 +-------+-------+-------+-------+-------+
                 | 1361  | 231   | 67    | 11    | 1381  |
                 +-------+-------+-------+-------+-------+
                 | 1389  | 532   | 67    | 11    | 1409  |
                 +-------+-------+-------+-------+-------+
                 | 1404  | 58    | 67    | 11    | 1423  |
                 +-------+-------+-------+-------+-------+
                 | 1420  | 60    | 67    | 11    | 1439  |
                 +-------+-------+-------+-------+-------+
                 | 1436  | 964   | 71    | 11    | 1459  |
                 +-------+-------+-------+-------+-------+
                 | 1461  | 624   | 71    | 11    | 1483  |
                 +-------+-------+-------+-------+-------+
                 | 1477  | 502   | 71    | 11    | 1499  |
                 +-------+-------+-------+-------+-------+
                 | 1502  | 636   | 71    | 11    | 1523  |
                 +-------+-------+-------+-------+-------+
                 | 1522  | 986   | 71    | 11    | 1543  |
                 +-------+-------+-------+-------+-------+
                 | 1539  | 950   | 71    | 11    | 1559  |
                 +-------+-------+-------+-------+-------+
                 | 1561  | 735   | 73    | 11    | 1583  |
                 +-------+-------+-------+-------+-------+




<span class="grey">Luby, et al.                 Standards Track                   [Page 47]</span></pre>
<hr class='noprint'/><!--NewPage--><pre class='newpage'><span id="page-48" ></span>
<span class="grey"><a href="./rfc6330">RFC 6330</a>                   RaptorQ FEC Scheme                August 2011</span>


                 +-------+-------+-------+-------+-------+
                 | 1579  | 866   | 73    | 11    | 1601  |
                 +-------+-------+-------+-------+-------+
                 | 1600  | 203   | 73    | 11    | 1621  |
                 +-------+-------+-------+-------+-------+
                 | 1616  | 83    | 73    | 11    | 1637  |
                 +-------+-------+-------+-------+-------+
                 | 1649  | 14    | 73    | 11    | 1669  |
                 +-------+-------+-------+-------+-------+
                 | 1673  | 522   | 79    | 11    | 1699  |
                 +-------+-------+-------+-------+-------+
                 | 1698  | 226   | 79    | 11    | 1723  |
                 +-------+-------+-------+-------+-------+
                 | 1716  | 282   | 79    | 11    | 1741  |
                 +-------+-------+-------+-------+-------+
                 | 1734  | 88    | 79    | 11    | 1759  |
                 +-------+-------+-------+-------+-------+
                 | 1759  | 636   | 79    | 11    | 1783  |
                 +-------+-------+-------+-------+-------+
                 | 1777  | 860   | 79    | 11    | 1801  |
                 +-------+-------+-------+-------+-------+
                 | 1800  | 324   | 79    | 11    | 1823  |
                 +-------+-------+-------+-------+-------+
                 | 1824  | 424   | 79    | 11    | 1847  |
                 +-------+-------+-------+-------+-------+
                 | 1844  | 999   | 79    | 11    | 1867  |
                 +-------+-------+-------+-------+-------+
                 | 1863  | 682   | 83    | 11    | 1889  |
                 +-------+-------+-------+-------+-------+
                 | 1887  | 814   | 83    | 11    | 1913  |
                 +-------+-------+-------+-------+-------+
                 | 1906  | 979   | 83    | 11    | 1931  |
                 +-------+-------+-------+-------+-------+
                 | 1926  | 538   | 83    | 11    | 1951  |
                 +-------+-------+-------+-------+-------+
                 | 1954  | 278   | 83    | 11    | 1979  |
                 +-------+-------+-------+-------+-------+
                 | 1979  | 580   | 83    | 11    | 2003  |
                 +-------+-------+-------+-------+-------+
                 | 2005  | 773   | 83    | 11    | 2029  |
                 +-------+-------+-------+-------+-------+
                 | 2040  | 911   | 89    | 11    | 2069  |
                 +-------+-------+-------+-------+-------+
                 | 2070  | 506   | 89    | 11    | 2099  |
                 +-------+-------+-------+-------+-------+
                 | 2103  | 628   | 89    | 11    | 2131  |
                 +-------+-------+-------+-------+-------+




<span class="grey">Luby, et al.                 Standards Track                   [Page 48]</span></pre>
<hr class='noprint'/><!--NewPage--><pre class='newpage'><span id="page-49" ></span>
<span class="grey"><a href="./rfc6330">RFC 6330</a>                   RaptorQ FEC Scheme                August 2011</span>


                 +-------+-------+-------+-------+-------+
                 | 2125  | 282   | 89    | 11    | 2153  |
                 +-------+-------+-------+-------+-------+
                 | 2152  | 309   | 89    | 11    | 2179  |
                 +-------+-------+-------+-------+-------+
                 | 2195  | 858   | 89    | 11    | 2221  |
                 +-------+-------+-------+-------+-------+
                 | 2217  | 442   | 89    | 11    | 2243  |
                 +-------+-------+-------+-------+-------+
                 | 2247  | 654   | 89    | 11    | 2273  |
                 +-------+-------+-------+-------+-------+
                 | 2278  | 82    | 97    | 11    | 2311  |
                 +-------+-------+-------+-------+-------+
                 | 2315  | 428   | 97    | 11    | 2347  |
                 +-------+-------+-------+-------+-------+
                 | 2339  | 442   | 97    | 11    | 2371  |
                 +-------+-------+-------+-------+-------+
                 | 2367  | 283   | 97    | 11    | 2399  |
                 +-------+-------+-------+-------+-------+
                 | 2392  | 538   | 97    | 11    | 2423  |
                 +-------+-------+-------+-------+-------+
                 | 2416  | 189   | 97    | 11    | 2447  |
                 +-------+-------+-------+-------+-------+
                 | 2447  | 438   | 97    | 11    | 2477  |
                 +-------+-------+-------+-------+-------+
                 | 2473  | 912   | 97    | 11    | 2503  |
                 +-------+-------+-------+-------+-------+
                 | 2502  | 1     | 97    | 11    | 2531  |
                 +-------+-------+-------+-------+-------+
                 | 2528  | 167   | 97    | 11    | 2557  |
                 +-------+-------+-------+-------+-------+
                 | 2565  | 272   | 97    | 11    | 2593  |
                 +-------+-------+-------+-------+-------+
                 | 2601  | 209   | 101   | 11    | 2633  |
                 +-------+-------+-------+-------+-------+
                 | 2640  | 927   | 101   | 11    | 2671  |
                 +-------+-------+-------+-------+-------+
                 | 2668  | 386   | 101   | 11    | 2699  |
                 +-------+-------+-------+-------+-------+
                 | 2701  | 653   | 101   | 11    | 2731  |
                 +-------+-------+-------+-------+-------+
                 | 2737  | 669   | 101   | 11    | 2767  |
                 +-------+-------+-------+-------+-------+
                 | 2772  | 431   | 101   | 11    | 2801  |
                 +-------+-------+-------+-------+-------+
                 | 2802  | 793   | 103   | 11    | 2833  |
                 +-------+-------+-------+-------+-------+




<span class="grey">Luby, et al.                 Standards Track                   [Page 49]</span></pre>
<hr class='noprint'/><!--NewPage--><pre class='newpage'><span id="page-50" ></span>
<span class="grey"><a href="./rfc6330">RFC 6330</a>                   RaptorQ FEC Scheme                August 2011</span>


                 +-------+-------+-------+-------+-------+
                 | 2831  | 588   | 103   | 11    | 2861  |
                 +-------+-------+-------+-------+-------+
                 | 2875  | 777   | 107   | 11    | 2909  |
                 +-------+-------+-------+-------+-------+
                 | 2906  | 939   | 107   | 11    | 2939  |
                 +-------+-------+-------+-------+-------+
                 | 2938  | 864   | 107   | 11    | 2971  |
                 +-------+-------+-------+-------+-------+
                 | 2979  | 627   | 107   | 11    | 3011  |
                 +-------+-------+-------+-------+-------+
                 | 3015  | 265   | 109   | 11    | 3049  |
                 +-------+-------+-------+-------+-------+
                 | 3056  | 976   | 109   | 11    | 3089  |
                 +-------+-------+-------+-------+-------+
                 | 3101  | 988   | 113   | 11    | 3137  |
                 +-------+-------+-------+-------+-------+
                 | 3151  | 507   | 113   | 11    | 3187  |
                 +-------+-------+-------+-------+-------+
                 | 3186  | 640   | 113   | 11    | 3221  |
                 +-------+-------+-------+-------+-------+
                 | 3224  | 15    | 113   | 11    | 3259  |
                 +-------+-------+-------+-------+-------+
                 | 3265  | 667   | 113   | 11    | 3299  |
                 +-------+-------+-------+-------+-------+
                 | 3299  | 24    | 127   | 11    | 3347  |
                 +-------+-------+-------+-------+-------+
                 | 3344  | 877   | 127   | 11    | 3391  |
                 +-------+-------+-------+-------+-------+
                 | 3387  | 240   | 127   | 11    | 3433  |
                 +-------+-------+-------+-------+-------+
                 | 3423  | 720   | 127   | 11    | 3469  |
                 +-------+-------+-------+-------+-------+
                 | 3466  | 93    | 127   | 11    | 3511  |
                 +-------+-------+-------+-------+-------+
                 | 3502  | 919   | 127   | 11    | 3547  |
                 +-------+-------+-------+-------+-------+
                 | 3539  | 635   | 127   | 11    | 3583  |
                 +-------+-------+-------+-------+-------+
                 | 3579  | 174   | 127   | 11    | 3623  |
                 +-------+-------+-------+-------+-------+
                 | 3616  | 647   | 127   | 11    | 3659  |
                 +-------+-------+-------+-------+-------+
                 | 3658  | 820   | 127   | 11    | 3701  |
                 +-------+-------+-------+-------+-------+
                 | 3697  | 56    | 127   | 11    | 3739  |
                 +-------+-------+-------+-------+-------+




<span class="grey">Luby, et al.                 Standards Track                   [Page 50]</span></pre>
<hr class='noprint'/><!--NewPage--><pre class='newpage'><span id="page-51" ></span>
<span class="grey"><a href="./rfc6330">RFC 6330</a>                   RaptorQ FEC Scheme                August 2011</span>


                 +-------+-------+-------+-------+-------+
                 | 3751  | 485   | 127   | 11    | 3793  |
                 +-------+-------+-------+-------+-------+
                 | 3792  | 210   | 127   | 11    | 3833  |
                 +-------+-------+-------+-------+-------+
                 | 3840  | 124   | 127   | 11    | 3881  |
                 +-------+-------+-------+-------+-------+
                 | 3883  | 546   | 127   | 11    | 3923  |
                 +-------+-------+-------+-------+-------+
                 | 3924  | 954   | 131   | 11    | 3967  |
                 +-------+-------+-------+-------+-------+
                 | 3970  | 262   | 131   | 11    | 4013  |
                 +-------+-------+-------+-------+-------+
                 | 4015  | 927   | 131   | 11    | 4057  |
                 +-------+-------+-------+-------+-------+
                 | 4069  | 957   | 131   | 11    | 4111  |
                 +-------+-------+-------+-------+-------+
                 | 4112  | 726   | 137   | 11    | 4159  |
                 +-------+-------+-------+-------+-------+
                 | 4165  | 583   | 137   | 11    | 4211  |
                 +-------+-------+-------+-------+-------+
                 | 4207  | 782   | 137   | 11    | 4253  |
                 +-------+-------+-------+-------+-------+
                 | 4252  | 37    | 137   | 11    | 4297  |
                 +-------+-------+-------+-------+-------+
                 | 4318  | 758   | 137   | 11    | 4363  |
                 +-------+-------+-------+-------+-------+
                 | 4365  | 777   | 137   | 11    | 4409  |
                 +-------+-------+-------+-------+-------+
                 | 4418  | 104   | 139   | 11    | 4463  |
                 +-------+-------+-------+-------+-------+
                 | 4468  | 476   | 139   | 11    | 4513  |
                 +-------+-------+-------+-------+-------+
                 | 4513  | 113   | 149   | 11    | 4567  |
                 +-------+-------+-------+-------+-------+
                 | 4567  | 313   | 149   | 11    | 4621  |
                 +-------+-------+-------+-------+-------+
                 | 4626  | 102   | 149   | 11    | 4679  |
                 +-------+-------+-------+-------+-------+
                 | 4681  | 501   | 149   | 11    | 4733  |
                 +-------+-------+-------+-------+-------+
                 | 4731  | 332   | 149   | 11    | 4783  |
                 +-------+-------+-------+-------+-------+
                 | 4780  | 786   | 149   | 11    | 4831  |
                 +-------+-------+-------+-------+-------+
                 | 4838  | 99    | 149   | 11    | 4889  |
                 +-------+-------+-------+-------+-------+




<span class="grey">Luby, et al.                 Standards Track                   [Page 51]</span></pre>
<hr class='noprint'/><!--NewPage--><pre class='newpage'><span id="page-52" ></span>
<span class="grey"><a href="./rfc6330">RFC 6330</a>                   RaptorQ FEC Scheme                August 2011</span>


                 +-------+-------+-------+-------+-------+
                 | 4901  | 658   | 149   | 11    | 4951  |
                 +-------+-------+-------+-------+-------+
                 | 4954  | 794   | 149   | 11    | 5003  |
                 +-------+-------+-------+-------+-------+
                 | 5008  | 37    | 151   | 11    | 5059  |
                 +-------+-------+-------+-------+-------+
                 | 5063  | 471   | 151   | 11    | 5113  |
                 +-------+-------+-------+-------+-------+
                 | 5116  | 94    | 157   | 11    | 5171  |
                 +-------+-------+-------+-------+-------+
                 | 5172  | 873   | 157   | 11    | 5227  |
                 +-------+-------+-------+-------+-------+
                 | 5225  | 918   | 157   | 11    | 5279  |
                 +-------+-------+-------+-------+-------+
                 | 5279  | 945   | 157   | 11    | 5333  |
                 +-------+-------+-------+-------+-------+
                 | 5334  | 211   | 157   | 11    | 5387  |
                 +-------+-------+-------+-------+-------+
                 | 5391  | 341   | 157   | 11    | 5443  |
                 +-------+-------+-------+-------+-------+
                 | 5449  | 11    | 163   | 11    | 5507  |
                 +-------+-------+-------+-------+-------+
                 | 5506  | 578   | 163   | 11    | 5563  |
                 +-------+-------+-------+-------+-------+
                 | 5566  | 494   | 163   | 11    | 5623  |
                 +-------+-------+-------+-------+-------+
                 | 5637  | 694   | 163   | 11    | 5693  |
                 +-------+-------+-------+-------+-------+
                 | 5694  | 252   | 163   | 11    | 5749  |
                 +-------+-------+-------+-------+-------+
                 | 5763  | 451   | 167   | 11    | 5821  |
                 +-------+-------+-------+-------+-------+
                 | 5823  | 83    | 167   | 11    | 5881  |
                 +-------+-------+-------+-------+-------+
                 | 5896  | 689   | 167   | 11    | 5953  |
                 +-------+-------+-------+-------+-------+
                 | 5975  | 488   | 173   | 11    | 6037  |
                 +-------+-------+-------+-------+-------+
                 | 6039  | 214   | 173   | 11    | 6101  |
                 +-------+-------+-------+-------+-------+
                 | 6102  | 17    | 173   | 11    | 6163  |
                 +-------+-------+-------+-------+-------+
                 | 6169  | 469   | 173   | 11    | 6229  |
                 +-------+-------+-------+-------+-------+
                 | 6233  | 263   | 179   | 11    | 6299  |
                 +-------+-------+-------+-------+-------+




<span class="grey">Luby, et al.                 Standards Track                   [Page 52]</span></pre>
<hr class='noprint'/><!--NewPage--><pre class='newpage'><span id="page-53" ></span>
<span class="grey"><a href="./rfc6330">RFC 6330</a>                   RaptorQ FEC Scheme                August 2011</span>


                 +-------+-------+-------+-------+-------+
                 | 6296  | 309   | 179   | 11    | 6361  |
                 +-------+-------+-------+-------+-------+
                 | 6363  | 984   | 179   | 11    | 6427  |
                 +-------+-------+-------+-------+-------+
                 | 6427  | 123   | 179   | 11    | 6491  |
                 +-------+-------+-------+-------+-------+
                 | 6518  | 360   | 179   | 11    | 6581  |
                 +-------+-------+-------+-------+-------+
                 | 6589  | 863   | 181   | 11    | 6653  |
                 +-------+-------+-------+-------+-------+
                 | 6655  | 122   | 181   | 11    | 6719  |
                 +-------+-------+-------+-------+-------+
                 | 6730  | 522   | 191   | 11    | 6803  |
                 +-------+-------+-------+-------+-------+
                 | 6799  | 539   | 191   | 11    | 6871  |
                 +-------+-------+-------+-------+-------+
                 | 6878  | 181   | 191   | 11    | 6949  |
                 +-------+-------+-------+-------+-------+
                 | 6956  | 64    | 191   | 11    | 7027  |
                 +-------+-------+-------+-------+-------+
                 | 7033  | 387   | 191   | 11    | 7103  |
                 +-------+-------+-------+-------+-------+
                 | 7108  | 967   | 191   | 11    | 7177  |
                 +-------+-------+-------+-------+-------+
                 | 7185  | 843   | 191   | 11    | 7253  |
                 +-------+-------+-------+-------+-------+
                 | 7281  | 999   | 193   | 11    | 7351  |
                 +-------+-------+-------+-------+-------+
                 | 7360  | 76    | 197   | 11    | 7433  |
                 +-------+-------+-------+-------+-------+
                 | 7445  | 142   | 197   | 11    | 7517  |
                 +-------+-------+-------+-------+-------+
                 | 7520  | 599   | 197   | 11    | 7591  |
                 +-------+-------+-------+-------+-------+
                 | 7596  | 576   | 199   | 11    | 7669  |
                 +-------+-------+-------+-------+-------+
                 | 7675  | 176   | 211   | 11    | 7759  |
                 +-------+-------+-------+-------+-------+
                 | 7770  | 392   | 211   | 11    | 7853  |
                 +-------+-------+-------+-------+-------+
                 | 7855  | 332   | 211   | 11    | 7937  |
                 +-------+-------+-------+-------+-------+
                 | 7935  | 291   | 211   | 11    | 8017  |
                 +-------+-------+-------+-------+-------+
                 | 8030  | 913   | 211   | 11    | 8111  |
                 +-------+-------+-------+-------+-------+




<span class="grey">Luby, et al.                 Standards Track                   [Page 53]</span></pre>
<hr class='noprint'/><!--NewPage--><pre class='newpage'><span id="page-54" ></span>
<span class="grey"><a href="./rfc6330">RFC 6330</a>                   RaptorQ FEC Scheme                August 2011</span>


                 +-------+-------+-------+-------+-------+
                 | 8111  | 608   | 211   | 11    | 8191  |
                 +-------+-------+-------+-------+-------+
                 | 8194  | 212   | 211   | 11    | 8273  |
                 +-------+-------+-------+-------+-------+
                 | 8290  | 696   | 211   | 11    | 8369  |
                 +-------+-------+-------+-------+-------+
                 | 8377  | 931   | 223   | 11    | 8467  |
                 +-------+-------+-------+-------+-------+
                 | 8474  | 326   | 223   | 11    | 8563  |
                 +-------+-------+-------+-------+-------+
                 | 8559  | 228   | 223   | 11    | 8647  |
                 +-------+-------+-------+-------+-------+
                 | 8654  | 706   | 223   | 11    | 8741  |
                 +-------+-------+-------+-------+-------+
                 | 8744  | 144   | 223   | 11    | 8831  |
                 +-------+-------+-------+-------+-------+
                 | 8837  | 83    | 223   | 11    | 8923  |
                 +-------+-------+-------+-------+-------+
                 | 8928  | 743   | 223   | 11    | 9013  |
                 +-------+-------+-------+-------+-------+
                 | 9019  | 187   | 223   | 11    | 9103  |
                 +-------+-------+-------+-------+-------+
                 | 9111  | 654   | 227   | 11    | 9199  |
                 +-------+-------+-------+-------+-------+
                 | 9206  | 359   | 227   | 11    | 9293  |
                 +-------+-------+-------+-------+-------+
                 | 9303  | 493   | 229   | 11    | 9391  |
                 +-------+-------+-------+-------+-------+
                 | 9400  | 369   | 233   | 11    | 9491  |
                 +-------+-------+-------+-------+-------+
                 | 9497  | 981   | 233   | 11    | 9587  |
                 +-------+-------+-------+-------+-------+
                 | 9601  | 276   | 239   | 11    | 9697  |
                 +-------+-------+-------+-------+-------+
                 | 9708  | 647   | 239   | 11    | 9803  |
                 +-------+-------+-------+-------+-------+
                 | 9813  | 389   | 239   | 11    | 9907  |
                 +-------+-------+-------+-------+-------+
                 | 9916  | 80    | 239   | 11    | 10009 |
                 +-------+-------+-------+-------+-------+
                 | 10017 | 396   | 241   | 11    | 10111 |
                 +-------+-------+-------+-------+-------+
                 | 10120 | 580   | 251   | 11    | 10223 |
                 +-------+-------+-------+-------+-------+
                 | 10241 | 873   | 251   | 11    | 10343 |
                 +-------+-------+-------+-------+-------+




<span class="grey">Luby, et al.                 Standards Track                   [Page 54]</span></pre>
<hr class='noprint'/><!--NewPage--><pre class='newpage'><span id="page-55" ></span>
<span class="grey"><a href="./rfc6330">RFC 6330</a>                   RaptorQ FEC Scheme                August 2011</span>


                 +-------+-------+-------+-------+-------+
                 | 10351 | 15    | 251   | 11    | 10453 |
                 +-------+-------+-------+-------+-------+
                 | 10458 | 976   | 251   | 11    | 10559 |
                 +-------+-------+-------+-------+-------+
                 | 10567 | 584   | 251   | 11    | 10667 |
                 +-------+-------+-------+-------+-------+
                 | 10676 | 267   | 257   | 11    | 10781 |
                 +-------+-------+-------+-------+-------+
                 | 10787 | 876   | 257   | 11    | 10891 |
                 +-------+-------+-------+-------+-------+
                 | 10899 | 642   | 257   | 12    | 11003 |
                 +-------+-------+-------+-------+-------+
                 | 11015 | 794   | 257   | 12    | 11119 |
                 +-------+-------+-------+-------+-------+
                 | 11130 | 78    | 263   | 12    | 11239 |
                 +-------+-------+-------+-------+-------+
                 | 11245 | 736   | 263   | 12    | 11353 |
                 +-------+-------+-------+-------+-------+
                 | 11358 | 882   | 269   | 12    | 11471 |
                 +-------+-------+-------+-------+-------+
                 | 11475 | 251   | 269   | 12    | 11587 |
                 +-------+-------+-------+-------+-------+
                 | 11590 | 434   | 269   | 12    | 11701 |
                 +-------+-------+-------+-------+-------+
                 | 11711 | 204   | 269   | 12    | 11821 |
                 +-------+-------+-------+-------+-------+
                 | 11829 | 256   | 271   | 12    | 11941 |
                 +-------+-------+-------+-------+-------+
                 | 11956 | 106   | 277   | 12    | 12073 |
                 +-------+-------+-------+-------+-------+
                 | 12087 | 375   | 277   | 12    | 12203 |
                 +-------+-------+-------+-------+-------+
                 | 12208 | 148   | 277   | 12    | 12323 |
                 +-------+-------+-------+-------+-------+
                 | 12333 | 496   | 281   | 12    | 12451 |
                 +-------+-------+-------+-------+-------+
                 | 12460 | 88    | 281   | 12    | 12577 |
                 +-------+-------+-------+-------+-------+
                 | 12593 | 826   | 293   | 12    | 12721 |
                 +-------+-------+-------+-------+-------+
                 | 12726 | 71    | 293   | 12    | 12853 |
                 +-------+-------+-------+-------+-------+
                 | 12857 | 925   | 293   | 12    | 12983 |
                 +-------+-------+-------+-------+-------+
                 | 13002 | 760   | 293   | 12    | 13127 |
                 +-------+-------+-------+-------+-------+




<span class="grey">Luby, et al.                 Standards Track                   [Page 55]</span></pre>
<hr class='noprint'/><!--NewPage--><pre class='newpage'><span id="page-56" ></span>
<span class="grey"><a href="./rfc6330">RFC 6330</a>                   RaptorQ FEC Scheme                August 2011</span>


                 +-------+-------+-------+-------+-------+
                 | 13143 | 130   | 293   | 12    | 13267 |
                 +-------+-------+-------+-------+-------+
                 | 13284 | 641   | 307   | 12    | 13421 |
                 +-------+-------+-------+-------+-------+
                 | 13417 | 400   | 307   | 12    | 13553 |
                 +-------+-------+-------+-------+-------+
                 | 13558 | 480   | 307   | 12    | 13693 |
                 +-------+-------+-------+-------+-------+
                 | 13695 | 76    | 307   | 12    | 13829 |
                 +-------+-------+-------+-------+-------+
                 | 13833 | 665   | 307   | 12    | 13967 |
                 +-------+-------+-------+-------+-------+
                 | 13974 | 910   | 307   | 12    | 14107 |
                 +-------+-------+-------+-------+-------+
                 | 14115 | 467   | 311   | 12    | 14251 |
                 +-------+-------+-------+-------+-------+
                 | 14272 | 964   | 311   | 12    | 14407 |
                 +-------+-------+-------+-------+-------+
                 | 14415 | 625   | 313   | 12    | 14551 |
                 +-------+-------+-------+-------+-------+
                 | 14560 | 362   | 317   | 12    | 14699 |
                 +-------+-------+-------+-------+-------+
                 | 14713 | 759   | 317   | 12    | 14851 |
                 +-------+-------+-------+-------+-------+
                 | 14862 | 728   | 331   | 12    | 15013 |
                 +-------+-------+-------+-------+-------+
                 | 15011 | 343   | 331   | 12    | 15161 |
                 +-------+-------+-------+-------+-------+
                 | 15170 | 113   | 331   | 12    | 15319 |
                 +-------+-------+-------+-------+-------+
                 | 15325 | 137   | 331   | 12    | 15473 |
                 +-------+-------+-------+-------+-------+
                 | 15496 | 308   | 331   | 12    | 15643 |
                 +-------+-------+-------+-------+-------+
                 | 15651 | 800   | 337   | 12    | 15803 |
                 +-------+-------+-------+-------+-------+
                 | 15808 | 177   | 337   | 12    | 15959 |
                 +-------+-------+-------+-------+-------+
                 | 15977 | 961   | 337   | 12    | 16127 |
                 +-------+-------+-------+-------+-------+
                 | 16161 | 958   | 347   | 12    | 16319 |
                 +-------+-------+-------+-------+-------+
                 | 16336 | 72    | 347   | 12    | 16493 |
                 +-------+-------+-------+-------+-------+
                 | 16505 | 732   | 347   | 12    | 16661 |
                 +-------+-------+-------+-------+-------+




<span class="grey">Luby, et al.                 Standards Track                   [Page 56]</span></pre>
<hr class='noprint'/><!--NewPage--><pre class='newpage'><span id="page-57" ></span>
<span class="grey"><a href="./rfc6330">RFC 6330</a>                   RaptorQ FEC Scheme                August 2011</span>


                 +-------+-------+-------+-------+-------+
                 | 16674 | 145   | 349   | 12    | 16831 |
                 +-------+-------+-------+-------+-------+
                 | 16851 | 577   | 353   | 12    | 17011 |
                 +-------+-------+-------+-------+-------+
                 | 17024 | 305   | 353   | 12    | 17183 |
                 +-------+-------+-------+-------+-------+
                 | 17195 | 50    | 359   | 12    | 17359 |
                 +-------+-------+-------+-------+-------+
                 | 17376 | 351   | 359   | 12    | 17539 |
                 +-------+-------+-------+-------+-------+
                 | 17559 | 175   | 367   | 12    | 17729 |
                 +-------+-------+-------+-------+-------+
                 | 17742 | 727   | 367   | 12    | 17911 |
                 +-------+-------+-------+-------+-------+
                 | 17929 | 902   | 367   | 12    | 18097 |
                 +-------+-------+-------+-------+-------+
                 | 18116 | 409   | 373   | 12    | 18289 |
                 +-------+-------+-------+-------+-------+
                 | 18309 | 776   | 373   | 12    | 18481 |
                 +-------+-------+-------+-------+-------+
                 | 18503 | 586   | 379   | 12    | 18679 |
                 +-------+-------+-------+-------+-------+
                 | 18694 | 451   | 379   | 12    | 18869 |
                 +-------+-------+-------+-------+-------+
                 | 18909 | 287   | 383   | 12    | 19087 |
                 +-------+-------+-------+-------+-------+
                 | 19126 | 246   | 389   | 12    | 19309 |
                 +-------+-------+-------+-------+-------+
                 | 19325 | 222   | 389   | 12    | 19507 |
                 +-------+-------+-------+-------+-------+
                 | 19539 | 563   | 397   | 12    | 19727 |
                 +-------+-------+-------+-------+-------+
                 | 19740 | 839   | 397   | 12    | 19927 |
                 +-------+-------+-------+-------+-------+
                 | 19939 | 897   | 401   | 12    | 20129 |
                 +-------+-------+-------+-------+-------+
                 | 20152 | 409   | 401   | 12    | 20341 |
                 +-------+-------+-------+-------+-------+
                 | 20355 | 618   | 409   | 12    | 20551 |
                 +-------+-------+-------+-------+-------+
                 | 20564 | 439   | 409   | 12    | 20759 |
                 +-------+-------+-------+-------+-------+
                 | 20778 | 95    | 419   | 13    | 20983 |
                 +-------+-------+-------+-------+-------+
                 | 20988 | 448   | 419   | 13    | 21191 |
                 +-------+-------+-------+-------+-------+




<span class="grey">Luby, et al.                 Standards Track                   [Page 57]</span></pre>
<hr class='noprint'/><!--NewPage--><pre class='newpage'><span id="page-58" ></span>
<span class="grey"><a href="./rfc6330">RFC 6330</a>                   RaptorQ FEC Scheme                August 2011</span>


                 +-------+-------+-------+-------+-------+
                 | 21199 | 133   | 419   | 13    | 21401 |
                 +-------+-------+-------+-------+-------+
                 | 21412 | 938   | 419   | 13    | 21613 |
                 +-------+-------+-------+-------+-------+
                 | 21629 | 423   | 431   | 13    | 21841 |
                 +-------+-------+-------+-------+-------+
                 | 21852 | 90    | 431   | 13    | 22063 |
                 +-------+-------+-------+-------+-------+
                 | 22073 | 640   | 431   | 13    | 22283 |
                 +-------+-------+-------+-------+-------+
                 | 22301 | 922   | 433   | 13    | 22511 |
                 +-------+-------+-------+-------+-------+
                 | 22536 | 250   | 439   | 13    | 22751 |
                 +-------+-------+-------+-------+-------+
                 | 22779 | 367   | 439   | 13    | 22993 |
                 +-------+-------+-------+-------+-------+
                 | 23010 | 447   | 443   | 13    | 23227 |
                 +-------+-------+-------+-------+-------+
                 | 23252 | 559   | 449   | 13    | 23473 |
                 +-------+-------+-------+-------+-------+
                 | 23491 | 121   | 457   | 13    | 23719 |
                 +-------+-------+-------+-------+-------+
                 | 23730 | 623   | 457   | 13    | 23957 |
                 +-------+-------+-------+-------+-------+
                 | 23971 | 450   | 457   | 13    | 24197 |
                 +-------+-------+-------+-------+-------+
                 | 24215 | 253   | 461   | 13    | 24443 |
                 +-------+-------+-------+-------+-------+
                 | 24476 | 106   | 467   | 13    | 24709 |
                 +-------+-------+-------+-------+-------+
                 | 24721 | 863   | 467   | 13    | 24953 |
                 +-------+-------+-------+-------+-------+
                 | 24976 | 148   | 479   | 13    | 25219 |
                 +-------+-------+-------+-------+-------+
                 | 25230 | 427   | 479   | 13    | 25471 |
                 +-------+-------+-------+-------+-------+
                 | 25493 | 138   | 479   | 13    | 25733 |
                 +-------+-------+-------+-------+-------+
                 | 25756 | 794   | 487   | 13    | 26003 |
                 +-------+-------+-------+-------+-------+
                 | 26022 | 247   | 487   | 13    | 26267 |
                 +-------+-------+-------+-------+-------+
                 | 26291 | 562   | 491   | 13    | 26539 |
                 +-------+-------+-------+-------+-------+
                 | 26566 | 53    | 499   | 13    | 26821 |
                 +-------+-------+-------+-------+-------+




<span class="grey">Luby, et al.                 Standards Track                   [Page 58]</span></pre>
<hr class='noprint'/><!--NewPage--><pre class='newpage'><span id="page-59" ></span>
<span class="grey"><a href="./rfc6330">RFC 6330</a>                   RaptorQ FEC Scheme                August 2011</span>


                 +-------+-------+-------+-------+-------+
                 | 26838 | 135   | 499   | 13    | 27091 |
                 +-------+-------+-------+-------+-------+
                 | 27111 | 21    | 503   | 13    | 27367 |
                 +-------+-------+-------+-------+-------+
                 | 27392 | 201   | 509   | 13    | 27653 |
                 +-------+-------+-------+-------+-------+
                 | 27682 | 169   | 521   | 13    | 27953 |
                 +-------+-------+-------+-------+-------+
                 | 27959 | 70    | 521   | 13    | 28229 |
                 +-------+-------+-------+-------+-------+
                 | 28248 | 386   | 521   | 13    | 28517 |
                 +-------+-------+-------+-------+-------+
                 | 28548 | 226   | 523   | 13    | 28817 |
                 +-------+-------+-------+-------+-------+
                 | 28845 | 3     | 541   | 13    | 29131 |
                 +-------+-------+-------+-------+-------+
                 | 29138 | 769   | 541   | 13    | 29423 |
                 +-------+-------+-------+-------+-------+
                 | 29434 | 590   | 541   | 13    | 29717 |
                 +-------+-------+-------+-------+-------+
                 | 29731 | 672   | 541   | 13    | 30013 |
                 +-------+-------+-------+-------+-------+
                 | 30037 | 713   | 547   | 13    | 30323 |
                 +-------+-------+-------+-------+-------+
                 | 30346 | 967   | 547   | 13    | 30631 |
                 +-------+-------+-------+-------+-------+
                 | 30654 | 368   | 557   | 14    | 30949 |
                 +-------+-------+-------+-------+-------+
                 | 30974 | 348   | 557   | 14    | 31267 |
                 +-------+-------+-------+-------+-------+
                 | 31285 | 119   | 563   | 14    | 31583 |
                 +-------+-------+-------+-------+-------+
                 | 31605 | 503   | 569   | 14    | 31907 |
                 +-------+-------+-------+-------+-------+
                 | 31948 | 181   | 571   | 14    | 32251 |
                 +-------+-------+-------+-------+-------+
                 | 32272 | 394   | 577   | 14    | 32579 |
                 +-------+-------+-------+-------+-------+
                 | 32601 | 189   | 587   | 14    | 32917 |
                 +-------+-------+-------+-------+-------+
                 | 32932 | 210   | 587   | 14    | 33247 |
                 +-------+-------+-------+-------+-------+
                 | 33282 | 62    | 593   | 14    | 33601 |
                 +-------+-------+-------+-------+-------+
                 | 33623 | 273   | 593   | 14    | 33941 |
                 +-------+-------+-------+-------+-------+




<span class="grey">Luby, et al.                 Standards Track                   [Page 59]</span></pre>
<hr class='noprint'/><!--NewPage--><pre class='newpage'><span id="page-60" ></span>
<span class="grey"><a href="./rfc6330">RFC 6330</a>                   RaptorQ FEC Scheme                August 2011</span>


                 +-------+-------+-------+-------+-------+
                 | 33961 | 554   | 599   | 14    | 34283 |
                 +-------+-------+-------+-------+-------+
                 | 34302 | 936   | 607   | 14    | 34631 |
                 +-------+-------+-------+-------+-------+
                 | 34654 | 483   | 607   | 14    | 34981 |
                 +-------+-------+-------+-------+-------+
                 | 35031 | 397   | 613   | 14    | 35363 |
                 +-------+-------+-------+-------+-------+
                 | 35395 | 241   | 619   | 14    | 35731 |
                 +-------+-------+-------+-------+-------+
                 | 35750 | 500   | 631   | 14    | 36097 |
                 +-------+-------+-------+-------+-------+
                 | 36112 | 12    | 631   | 14    | 36457 |
                 +-------+-------+-------+-------+-------+
                 | 36479 | 958   | 641   | 14    | 36833 |
                 +-------+-------+-------+-------+-------+
                 | 36849 | 524   | 641   | 14    | 37201 |
                 +-------+-------+-------+-------+-------+
                 | 37227 | 8     | 643   | 14    | 37579 |
                 +-------+-------+-------+-------+-------+
                 | 37606 | 100   | 653   | 14    | 37967 |
                 +-------+-------+-------+-------+-------+
                 | 37992 | 339   | 653   | 14    | 38351 |
                 +-------+-------+-------+-------+-------+
                 | 38385 | 804   | 659   | 14    | 38749 |
                 +-------+-------+-------+-------+-------+
                 | 38787 | 510   | 673   | 14    | 39163 |
                 +-------+-------+-------+-------+-------+
                 | 39176 | 18    | 673   | 14    | 39551 |
                 +-------+-------+-------+-------+-------+
                 | 39576 | 412   | 677   | 14    | 39953 |
                 +-------+-------+-------+-------+-------+
                 | 39980 | 394   | 683   | 14    | 40361 |
                 +-------+-------+-------+-------+-------+
                 | 40398 | 830   | 691   | 15    | 40787 |
                 +-------+-------+-------+-------+-------+
                 | 40816 | 535   | 701   | 15    | 41213 |
                 +-------+-------+-------+-------+-------+
                 | 41226 | 199   | 701   | 15    | 41621 |
                 +-------+-------+-------+-------+-------+
                 | 41641 | 27    | 709   | 15    | 42043 |
                 +-------+-------+-------+-------+-------+
                 | 42067 | 298   | 709   | 15    | 42467 |
                 +-------+-------+-------+-------+-------+
                 | 42490 | 368   | 719   | 15    | 42899 |
                 +-------+-------+-------+-------+-------+




<span class="grey">Luby, et al.                 Standards Track                   [Page 60]</span></pre>
<hr class='noprint'/><!--NewPage--><pre class='newpage'><span id="page-61" ></span>
<span class="grey"><a href="./rfc6330">RFC 6330</a>                   RaptorQ FEC Scheme                August 2011</span>


                 +-------+-------+-------+-------+-------+
                 | 42916 | 755   | 727   | 15    | 43331 |
                 +-------+-------+-------+-------+-------+
                 | 43388 | 379   | 727   | 15    | 43801 |
                 +-------+-------+-------+-------+-------+
                 | 43840 | 73    | 733   | 15    | 44257 |
                 +-------+-------+-------+-------+-------+
                 | 44279 | 387   | 739   | 15    | 44701 |
                 +-------+-------+-------+-------+-------+
                 | 44729 | 457   | 751   | 15    | 45161 |
                 +-------+-------+-------+-------+-------+
                 | 45183 | 761   | 751   | 15    | 45613 |
                 +-------+-------+-------+-------+-------+
                 | 45638 | 855   | 757   | 15    | 46073 |
                 +-------+-------+-------+-------+-------+
                 | 46104 | 370   | 769   | 15    | 46549 |
                 +-------+-------+-------+-------+-------+
                 | 46574 | 261   | 769   | 15    | 47017 |
                 +-------+-------+-------+-------+-------+
                 | 47047 | 299   | 787   | 15    | 47507 |
                 +-------+-------+-------+-------+-------+
                 | 47523 | 920   | 787   | 15    | 47981 |
                 +-------+-------+-------+-------+-------+
                 | 48007 | 269   | 787   | 15    | 48463 |
                 +-------+-------+-------+-------+-------+
                 | 48489 | 862   | 797   | 15    | 48953 |
                 +-------+-------+-------+-------+-------+
                 | 48976 | 349   | 809   | 15    | 49451 |
                 +-------+-------+-------+-------+-------+
                 | 49470 | 103   | 809   | 15    | 49943 |
                 +-------+-------+-------+-------+-------+
                 | 49978 | 115   | 821   | 15    | 50461 |
                 +-------+-------+-------+-------+-------+
                 | 50511 | 93    | 821   | 16    | 50993 |
                 +-------+-------+-------+-------+-------+
                 | 51017 | 982   | 827   | 16    | 51503 |
                 +-------+-------+-------+-------+-------+
                 | 51530 | 432   | 839   | 16    | 52027 |
                 +-------+-------+-------+-------+-------+
                 | 52062 | 340   | 853   | 16    | 52571 |
                 +-------+-------+-------+-------+-------+
                 | 52586 | 173   | 853   | 16    | 53093 |
                 +-------+-------+-------+-------+-------+
                 | 53114 | 421   | 857   | 16    | 53623 |
                 +-------+-------+-------+-------+-------+
                 | 53650 | 330   | 863   | 16    | 54163 |
                 +-------+-------+-------+-------+-------+




<span class="grey">Luby, et al.                 Standards Track                   [Page 61]</span></pre>
<hr class='noprint'/><!--NewPage--><pre class='newpage'><span id="page-62" ></span>
<span class="grey"><a href="./rfc6330">RFC 6330</a>                   RaptorQ FEC Scheme                August 2011</span>


                 +-------+-------+-------+-------+-------+
                 | 54188 | 624   | 877   | 16    | 54713 |
                 +-------+-------+-------+-------+-------+
                 | 54735 | 233   | 877   | 16    | 55259 |
                 +-------+-------+-------+-------+-------+
                 | 55289 | 362   | 883   | 16    | 55817 |
                 +-------+-------+-------+-------+-------+
                 | 55843 | 963   | 907   | 16    | 56393 |
                 +-------+-------+-------+-------+-------+
                 | 56403 | 471   | 907   | 16    | 56951 |
                 +-------+-------+-------+-------+-------+

             Table 2: Systematic Indices and Other Parameters

<span class="h3"><a class="selflink" id="section-5.7" href="#section-5.7">5.7</a>.  Operating with Octets, Symbols, and Matrices</span>

<span class="h4"><a class="selflink" id="section-5.7.1" href="#section-5.7.1">5.7.1</a>.  General</span>

   The remainder of this section describes the arithmetic operations
   that are used to generate encoding symbols from source symbols and to
   generate source symbols from encoding symbols.  Mathematically,
   octets can be thought of as elements of a finite field, i.e., the
   finite field GF(256) with 256 elements, and thus the addition and
   multiplication operations and identity elements and inverses over
   both operations are defined.  Matrix operations and symbol operations
   are defined based on the arithmetic operations on octets.  This
   allows a full implementation of these arithmetic operations without
   having to understand the underlying mathematics of finite fields.

<span class="h4"><a class="selflink" id="section-5.7.2" href="#section-5.7.2">5.7.2</a>.  Arithmetic Operations on Octets</span>

   Octets are mapped to non-negative integers in the range 0 through 255
   in the usual way: A single octet of data from a symbol,
   B[7],B[6],B[5],B[4],B[3],B[2],B[1],B[0], where B[7] is the highest
   order bit and B[0] is the lowest order bit, is mapped to the integer
   i=B[7]*128+B[6]*64+B[5]*32+B[4]*16+B[3]*8+B[2]*4+B[1]*2+B[0].

   The addition of two octets u and v is defined as the exclusive-or
   operation, i.e.,

      u + v = u ^ v.

   Subtraction is defined in the same way, so we also have

      u - v = u ^ v.






<span class="grey">Luby, et al.                 Standards Track                   [Page 62]</span></pre>
<hr class='noprint'/><!--NewPage--><pre class='newpage'><span id="page-63" ></span>
<span class="grey"><a href="./rfc6330">RFC 6330</a>                   RaptorQ FEC Scheme                August 2011</span>


   The zero element (additive identity) is the octet represented by the
   integer 0.  The additive inverse of u is simply u, i.e.,

      u + u = 0.

   The multiplication of two octets is defined with the help of two
   tables OCT_EXP and OCT_LOG, which are given in <a href="#section-5.7.3">Section 5.7.3</a> and
   <a href="#section-5.7.4">Section 5.7.4</a>, respectively.  The table OCT_LOG maps octets (other
   than the zero element) to non-negative integers, and OCT_EXP maps
   non-negative integers to octets.  For two octets u and v, we define

      u * v =

         0, if either u or v are 0,

         OCT_EXP[OCT_LOG[u] + OCT_LOG[v]] otherwise.

   Note that the '+' on the right-hand side of the above is the usual
   integer addition, since its arguments are ordinary integers.

   The division u / v of two octets u and v, and where v != 0, is
   defined as follows:

      u / v =

         0, if u == 0,

         OCT_EXP[OCT_LOG[u] - OCT_LOG[v] + 255] otherwise.

   The one element (multiplicative identity) is the octet represented by
   the integer 1.  For an octet u that is not the zero element, i.e.,
   the multiplicative inverse of u is

      OCT_EXP[255 - OCT_LOG[u]].

   The octet denoted by alpha is the octet with the integer
   representation 2.  If i is a non-negative integer 0 &lt;= i &lt; 256, we
   have

      alpha^^i = OCT_EXP[i].

<span class="h4"><a class="selflink" id="section-5.7.3" href="#section-5.7.3">5.7.3</a>.  The Table OCT_EXP</span>

   The table OCT_EXP contains 510 octets.  The indexing starts at 0 and
   ranges to 509, and the entries are the octets with the following
   positive integer representation:





<span class="grey">Luby, et al.                 Standards Track                   [Page 63]</span></pre>
<hr class='noprint'/><!--NewPage--><pre class='newpage'><span id="page-64" ></span>
<span class="grey"><a href="./rfc6330">RFC 6330</a>                   RaptorQ FEC Scheme                August 2011</span>


   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, 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

<span class="h4"><a class="selflink" id="section-5.7.4" href="#section-5.7.4">5.7.4</a>.  The Table OCT_LOG</span>

   The table OCT_LOG contains 255 non-negative integers.  The table is
   indexed by octets interpreted as integers.  The octet corresponding
   to the zero element, which is represented by the integer 0, is
   excluded as an index, and thus indexing starts at 1 and ranges up to
   255, and the entries are the following:

   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,



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   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

<span class="h4"><a class="selflink" id="section-5.7.5" href="#section-5.7.5">5.7.5</a>.  Operations on Symbols</span>

   Operations on symbols have the same semantics as operations on
   vectors of octets of length T in this specification.  Thus, if U and
   V are two symbols formed by the octets u[0], ..., u[T-1] and v[0],
   ..., v[T-1], respectively, the sum of symbols U + V is defined to be
   the component-wise sum of octets, i.e., equal to the symbol D formed
   by the octets d[0], ..., d[T-1], such that

      d[i] = u[i] + v[i], 0 &lt;= i &lt; T.

   Furthermore, if beta is an octet, the product beta*U is defined to be
   the symbol D obtained by multiplying each octet of U by beta, i.e.,

      d[i] = beta*u[i], 0 &lt;= i &lt; T.

<span class="h4"><a class="selflink" id="section-5.7.6" href="#section-5.7.6">5.7.6</a>.  Operations on Matrices</span>

   All matrices in this specification have entries that are octets, and
   thus matrix operations and definitions are defined in terms of the
   underlying octet arithmetic, e.g., operations on a matrix, matrix
   rank, and matrix inversion.

<span class="h3"><a class="selflink" id="section-5.8" href="#section-5.8">5.8</a>.  Requirements for a Compliant Decoder</span>

   If a RaptorQ-compliant decoder receives a mathematically sufficient
   set of encoding symbols generated according to the encoder
   specification in <a href="#section-5.3">Section 5.3</a> for reconstruction of a source block,
   then such a decoder SHOULD recover the entire source block.

   A RaptorQ-compliant decoder SHALL have the following recovery
   properties for source blocks with K' source symbols for all values of
   K' in Table 2 of <a href="#section-5.6">Section 5.6</a>.



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   1.  If the decoder receives K' encoding symbols generated according
       to the encoder specification in <a href="#section-5.3">Section 5.3</a> with corresponding
       ESIs chosen independently and uniformly at random from the range
       of possible ESIs, then on average the decoder will fail to
       recover the entire source block at most 1 out of 100 times.

   2.  If the decoder receives K'+1 encoding symbols generated according
       to the encoder specification in <a href="#section-5.3">Section 5.3</a> with corresponding
       ESIs chosen independently and uniformly at random from the range
       of possible ESIs, then on average the decoder will fail to
       recover the entire source block at most 1 out of 10,000 times.

   3.  If the decoder receives K'+2 encoding symbols generated according
       to the encoder specification in <a href="#section-5.3">Section 5.3</a> with corresponding
       ESIs chosen independently and uniformly at random from the range
       of possible ESIs, then on average the decoder will fail to
       recover the entire source block at most 1 out of 1,000,000 times.

   Note that the Example FEC Decoder specified in <a href="#section-5.4">Section 5.4</a> fulfills
   both requirements, i.e.,

   1.  it can reconstruct a source block as long as it receives a
       mathematically sufficient set of encoding symbols generated
       according to the encoder specification in <a href="#section-5.3">Section 5.3</a>, and

   2.  it fulfills the mandatory recovery properties from above.

<span class="h2"><a class="selflink" id="section-6" href="#section-6">6</a>.  Security Considerations</span>

   Data delivery can be subject to denial-of-service attacks by
   attackers that send corrupted packets that are accepted as legitimate
   by receivers.  This is particularly a concern for multicast delivery
   because a corrupted packet may be injected into the session close to
   the root of the multicast tree, in which case the corrupted packet
   will arrive at many receivers.  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-256 hash [<a href="#ref-FIPS.180-3.2008">FIPS.180-3.2008</a>] of an object may be appended before
   transmission, and the SHA-256 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 [<a href="./rfc4082" title="&quot;Timed Efficient Stream Loss-Tolerant Authentication (TESLA): Multicast Source Authentication Transform Introduction&quot;">RFC4082</a>] 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



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   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.

<span class="h2"><a class="selflink" id="section-7" href="#section-7">7</a>.  IANA Considerations</span>

   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 [<a href="./rfc5052" title="&quot;Forward Error Correction (FEC) Building Block&quot;">RFC5052</a>].  IANA has assigned the value 6
   under the ietf:rmt:fec:encoding registry to "RaptorQ Code" as the
   Fully-Specified FEC Encoding ID value associated with this
   specification.

<span class="h2"><a class="selflink" id="section-8" href="#section-8">8</a>.  Acknowledgements</span>

   Thanks are due to Ranganathan (Ranga) Krishnan.  Ranga Krishnan has
   been very supportive in finding and resolving implementation details
   and in finding the systematic indices.  In addition, Habeeb Mohiuddin
   Mohammed and Antonios Pitarokoilis, both from the Munich University
   of Technology (TUM), and Alan Shinsato have done two independent
   implementations of the RaptorQ encoder/decoder that have helped to
   clarify and to resolve issues with this specification.

<span class="h2"><a class="selflink" id="section-9" href="#section-9">9</a>.  References</span>

<span class="h3"><a class="selflink" id="section-9.1" href="#section-9.1">9.1</a>.  Normative References</span>

   [<a id="ref-FIPS.180-3.2008">FIPS.180-3.2008</a>]
              National Institute of Standards and Technology, "Secure
              Hash Standard", FIPS PUB 180-3, October 2008.

   [<a id="ref-RFC2119">RFC2119</a>]  Bradner, S., "Key words for use in RFCs to Indicate
              Requirement Levels", <a href="https://www.rfc-editor.org/bcp/bcp14">BCP 14</a>, <a href="./rfc2119">RFC 2119</a>, March 1997.

   [<a id="ref-RFC4082">RFC4082</a>]  Perrig, A., Song, D., Canetti, R., Tygar, J., and B.
              Briscoe, "Timed Efficient Stream Loss-Tolerant
              Authentication (TESLA): Multicast Source Authentication
              Transform Introduction", <a href="./rfc4082">RFC 4082</a>, June 2005.



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   [<a id="ref-RFC5052">RFC5052</a>]  Watson, M., Luby, M., and L. Vicisano, "Forward Error
              Correction (FEC) Building Block", <a href="./rfc5052">RFC 5052</a>, August 2007.

<span class="h3"><a class="selflink" id="section-9.2" href="#section-9.2">9.2</a>.  Informative References</span>

   [<a id="ref-LTCodes">LTCodes</a>]  Luby, M., "LT codes", Annual IEEE Symposium on Foundations
              of Computer Science, pp. 271-280, November 2002.

   [<a id="ref-RFC3453">RFC3453</a>]  Luby, M., Vicisano, L., Gemmell, J., Rizzo, L., Handley,
              M., and J. Crowcroft, "The Use of Forward Error Correction
              (FEC) in Reliable Multicast", <a href="./rfc3453">RFC 3453</a>, December 2002.

   [<a id="ref-RFC5053">RFC5053</a>]  Luby, M., Shokrollahi, A., Watson, M., and T. Stockhammer,
              "Raptor Forward Error Correction Scheme for Object
              Delivery", <a href="./rfc5053">RFC 5053</a>, October 2007.

   [<a id="ref-RaptorCodes">RaptorCodes</a>]
              Shokrollahi, A. and M. Luby, "Raptor Codes", Foundations
              and Trends in Communications and Information Theory: Vol.
              6: No. 3-4, pp. 213-322, 2011.































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Authors' Addresses

   Michael Luby
   Qualcomm Incorporated
   3165 Kifer Road
   Santa Clara, CA  95051
   U.S.A.

   EMail: luby@qualcomm.com


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

   EMail: amin.shokrollahi@epfl.ch


   Mark Watson
   Netflix Inc.
   100 Winchester Circle
   Los Gatos, CA  95032
   U.S.A.

   EMail: watsonm@netflix.com


   Thomas Stockhammer
   Nomor Research
   Brecherspitzstrasse 8
   Munich  81541
   Germany

   EMail: stockhammer@nomor.de


   Lorenz Minder
   Qualcomm Incorporated
   3165 Kifer Road
   Santa Clara, CA  95051
   U.S.A.

   EMail: lminder@qualcomm.com




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