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Versions: 00 01 02 03 04 05 06 07 08 RFC 5170

RMT                                                              V. Roca
Internet-Draft                                                     INRIA
Expires: August 5, 2006                                       C. Neumann
                                                        Thomson Research
                                                              D. Furodet
                                                      STMicroelectronics
                                                           February 2006


        Low Density Parity Check (LDPC) Forward Error Correction
                   draft-ietf-rmt-bb-fec-ldpc-01.txt

Status of this Memo

   By submitting this Internet-Draft, each author represents that any
   applicable patent or other IPR claims of which he or she is aware
   have been or will be disclosed, and any of which he or she becomes
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   This Internet-Draft will expire on August 5, 2006.

Copyright Notice

   Copyright (C) The Internet Society (2006).

Abstract

   This document describes two Fully-Specified FEC Schemes, LDPC-
   Staircase and LDPC-Triangle, and their application to the reliable
   delivery of objects on packet erasure channels.  These systematic FEC
   codes belong to the well known class of ``Low Density Parity Check''
   (LDPC) codes, and are large block FEC codes in these sense of



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   RFC3453.


Table of Contents

   1.  Introduction . . . . . . . . . . . . . . . . . . . . . . . . .  3
   2.  Requirements notation  . . . . . . . . . . . . . . . . . . . .  4
   3.  Definitions, Notations and Abbreviations . . . . . . . . . . .  5
     3.1.  Definitions  . . . . . . . . . . . . . . . . . . . . . . .  5
     3.2.  Notations  . . . . . . . . . . . . . . . . . . . . . . . .  5
     3.3.  Abbreviations  . . . . . . . . . . . . . . . . . . . . . .  6
   4.  Formats and Codes  . . . . . . . . . . . . . . . . . . . . . .  7
     4.1.  FEC Payload IDs  . . . . . . . . . . . . . . . . . . . . .  7
     4.2.  FEC Object Transmission Information  . . . . . . . . . . .  7
       4.2.1.  Mandatory Elements . . . . . . . . . . . . . . . . . .  7
       4.2.2.  Common Elements  . . . . . . . . . . . . . . . . . . .  7
       4.2.3.  Scheme-Specific Element  . . . . . . . . . . . . . . .  8
       4.2.4.  Encoding Format  . . . . . . . . . . . . . . . . . . .  8
   5.  Procedures . . . . . . . . . . . . . . . . . . . . . . . . . . 11
     5.1.  General  . . . . . . . . . . . . . . . . . . . . . . . . . 11
     5.2.  Determining the Maximum Source Block Length (B)  . . . . . 12
     5.3.  Determining the Encoding Symbol Length (E) and Number
           of Encoding Symbols per Group (G)  . . . . . . . . . . . . 12
     5.4.  Determining the Number of Encoding Symbols of a Block  . . 13
     5.5.  Identifying the Symbols of an Encoding Symbol Group  . . . 15
     5.6.  Pseudo Random Number Generator . . . . . . . . . . . . . . 18
   6.  Full Specification of the LDPC-Staircase Scheme  . . . . . . . 20
     6.1.  General  . . . . . . . . . . . . . . . . . . . . . . . . . 20
     6.2.  Parity Check Matrix Creation . . . . . . . . . . . . . . . 20
     6.3.  Encoding . . . . . . . . . . . . . . . . . . . . . . . . . 22
     6.4.  Decoding . . . . . . . . . . . . . . . . . . . . . . . . . 22
   7.  Full Specification of the LDPC-Triangle Scheme . . . . . . . . 24
     7.1.  General  . . . . . . . . . . . . . . . . . . . . . . . . . 24
     7.2.  Parity Check Matrix Creation . . . . . . . . . . . . . . . 24
     7.3.  Encoding . . . . . . . . . . . . . . . . . . . . . . . . . 24
     7.4.  Decoding . . . . . . . . . . . . . . . . . . . . . . . . . 25
   8.  Security Considerations  . . . . . . . . . . . . . . . . . . . 26
   9.  Intellectual Property  . . . . . . . . . . . . . . . . . . . . 27
   10. Acknowledgments  . . . . . . . . . . . . . . . . . . . . . . . 28
   11. References . . . . . . . . . . . . . . . . . . . . . . . . . . 29
     11.1. Normative References . . . . . . . . . . . . . . . . . . . 29
     11.2. Informative References . . . . . . . . . . . . . . . . . . 29
   Appendix A.  Trivial Decoding Algorithm (Informative Only) . . . . 31
   Authors' Addresses . . . . . . . . . . . . . . . . . . . . . . . . 32
   Intellectual Property and Copyright Statements . . . . . . . . . . 33






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1.  Introduction

   RFC 3453 [RFC3453] introduces large block FEC codes as an alternative
   to small block FEC codes like Reed-Solomon.  The main advantage of
   such large block codes is the possibility to operate efficiently on
   source blocks of size several tens of thousands (or more) source
   symbols.  The present document introduces the Fully-Specified FEC
   Encoding ID XX that is intended to be used with the "Low Density
   Parity Check" (LDPC) Staircase FEC codes, and the Fully-Specified FEC
   Encoding ID YY that is intended to be used with the "Low Density
   Parity Check" (LDPC)-Triangle FEC codes [Roca04][Mac03].  Both
   schemes belong the broad class of large block codes.

      -- editor's note: This document makes use of the FEC Encoding ID
      values XX and YY that will be specified after IANA assignment --

   LDPC codes rely on a dedicated matrix, called a "Parity Check
   Matrix", at the encoding and decoding ends.  The parity check matrix
   defines relationships (or constraints) between the various encoding
   symbols (i.e. source symbols and repair symbols), that are later used
   by the decoder to reconstruct the original k source symbols if some
   of them are missing.  These codes are systematic, in the sense that
   the encoding symbols include the source symbols in addition to the
   redundant symbols.

   Since the encoder and decoder must operate on the same parity check
   matrix, some information must be communicated between them, as part
   of the FEC Object Transmission information.

   A publicly available reference implementation of these codes is
   available and distributed under a GNU/LGPL license [LDPCrefimpl].  To
   the best of our knowledge, there is no patent or patent application
   identified as being used in the LDPC-Staircase and LDPC-Triangle FEC
   schemes.

















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2.  Requirements notation

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














































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3.  Definitions, Notations and Abbreviations

3.1.  Definitions

   This document uses the same terms and definitions as those specified
   in [fec-bb-revised].  Additionally, it uses the following
   definitions:

      Encoding Symbol Group: a group of encoding symbols that are sent
      together, within the same packet, and whose relationships to the
      source object can be derived from a single Encoding Symbol ID.

      Source Packet: a data packet containing only source symbols.

      Repair Packet: a data packet containing only repair symbols.

3.2.  Notations

   This document uses the following notations:

      L denotes the object transfer length in bytes

      k denotes the source block length in symbols, i.e. the number of
      source symbols of a source block

      n denotes the encoding block length, i.e. the number of encoding
      symbols generated for a source block

      E denotes the encoding symbol length in bytes

      B denotes the maximum source block length in symbols, i.e. the
      maximum number of source symbols per source block

      N denotes the number of source blocks into which the object shall
      be partitioned

      G denotes the number of encoding symbols per group, i.e. the
      number of symbols sent in the same packet

      rate denotes the so-called "code rate", i.e. the k/n ratio

      max_n Maximum Number of Encoding Symbols generated for any source
      block

      srand(s) denotes the initialization function of the pseudo-random
      number generator, where s is the seed (s > 0)





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      rand(m) denotes a pseudo-random number generator, that returns a
      new random integer in [0; m-1] each time it is called

3.3.  Abbreviations

   This document uses the following abbreviations:

      ESI: Encoding Symbol ID

      FEC OTI: FEC Object Transmission Information









































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4.  Formats and Codes

4.1.  FEC Payload IDs

   The FEC Payload ID is composed of the Source Block Number and the
   Encoding Symbol ID:

      The Source Block Number (12 bit field) identifies from which
      source block of the object the encoding symbol(s) in the payload
      is(are) generated.  There are a maximum of 2^^12 blocks per
      object.

      The Encoding Symbol ID (20 bit field) identifies which encoding
      symbol(s) generated from the source block is(are) carried in the
      packet payload.  There are a maximum of 2^^20 encoding symbols per
      block.  The first k values (0 to k-1) identify source symbols, the
      remaining n-k values (k to n-k-1) identify repair symbols.

   There MUST be exactly one FEC Payload ID per packet.  In case of en
   Encoding Symbol Group, when multiple encoding symbols are sent in the
   same packet, the FEC Payload ID refers to the first symbol of the
   packet.  The other symbols can be deduced from the ESI of the first
   symbol thanks to a dedicated function, as explained in Section 5.5

    0                   1                   2                   3
    0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1
   +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
   |  Source Block Number  |      Encoding Symbol ID (20 bits)     |
   +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+

   Figure 1: FEC Payload ID encoding format for FEC Encoding ID XX and
   YY

4.2.  FEC Object Transmission Information

4.2.1.  Mandatory Elements

   o  FEC Encoding ID: the Fully-Specified FEC Schemes described in this
      document use the FEC Encoding ID XX for LDPC-Staircase and FEC
      Encoding ID YY for LDPC-Triangle.

4.2.2.  Common Elements

   The following elements MUST be defined with the present FEC Scheme:

   o  Transfer-Length (L): a non-negative integer indicating the length
      of the object in bytes.  There are some restrictions on the
      maximum Transfer-Length that can be supported:



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         maximum transfer length = 2^^12 * B * E

      For instance, if B=2^^19 (because of a code rate of 1/2,
      Section 5.2), and if E=1024 bytes, then the maximum transfer
      length is 2^^41 bytes.

   o  Encoding-Symbol-Length (E): a non-negative integer indicating the
      length of each encoding symbol in bytes.

   o  Maximum-Source-Block-Length (B): a non-negative integer indicating
      the maximum number of source symbols in a source block.  There are
      some restrictions on the maximum B value, as explained in
      Section 5.2.

   o  Max-Number-of-Encoding-Symbols (max_n): a non-negative integer
      indicating the maximum number of encoding symbols generated for
      any source block.  There are some restrictions on the maximum
      max_n value.  In particular max_n is at most equal to 2^^20.

   Section 5 explains how to derive the values of each of these
   elements.

4.2.3.  Scheme-Specific Element

   The following element MUST be defined with the present FEC Scheme.
   It contains two distinct pieces of information:

   o  G: a non-negative integer indicating the number of encoding
      symbols per group used for the object.  The default value is 1,
      meaning that each packet contains exactly one symbol.  Values
      greater than 1 can also be defined, as explained in Section 5.3.

   o  PRNG seed: The seed is a 32 bit value used to initialize the
      Pseudo Random Number Generator (defined in Section 5.6).  This
      element is optional.  Whether or not it is present in the FEC OTI
      will be signaled in the associated encoding format through an
      appropriate mechanism (see Section 4.2.4).  When the PRNG seed is
      not carried within the FEC OTI, it is assumed that encoder and
      decoders use another way to communicate the information, or use a
      fixed, predefined value.

4.2.4.  Encoding Format

   This section shows two possible encoding formats of the above FEC
   OTI.  The present document does not specify when or how these
   encoding formats should be used.





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4.2.4.1.  Using the General EXT_FTI Format

   The FEC OTI binary format is the following, when the EXT_FTI
   mechanism is used.

    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
   +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
   |   HET = 64    | HEL (=4 or 5) |                               |
   +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+                               +
   |                      Transfer-Length (L)                      |
   +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
   |   Encoding Symbol Length (E)  |       G       |   B (MSB)     |
   +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
   |        B (LSB)        |   Max Nb of Enc. Symbols  (max_n)     |
   +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
   .                       Optional PRNG seed                      .
   +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+

   In particular:

   o  The HEL (Header Extension Length) indicates whether the optional
      PRNG seed is present (HEL=5) or not (HEL=4).

   o  The Maximum-Source-Block-Length (B) is split into two parts: the 8
      most significant bits (MSB) are in the third 32-bit word of the
      EXT_FTI, and the remaining 12 least significant bits (LSB) are in
      fourth 32-bit word.

4.2.4.2.  Using the FDT Instance (FLUTE specific)

   When it is desired that the FEC OTI be carried in the FDT Instance of
   a FLUTE session, the following XML elements must be described for the
   associated object:

   o  FEC-OTI-Transfer-length

   o  FEC-OTI-Encoding-Symbol-Length

   o  FEC-OTI-Maximum-Source-Block-Length

   o  FEC-OTI-Max-Number-of-Encoding-Symbols

   o  FEC-OTI-Number-Encoding-Symbols-per-Group

   o  FEC-OTI-PRNG-seed (optional)

   When no PRNG seed is to be carried in the FEC OTI, the sender simply



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   omits the FEC-OTI-PRNG-seed element.


















































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5.  Procedures

   This section defines procedures that are common to FEC Encoding IDs
   XX and YY.

5.1.  General

   The B (maximum source block length in symbols) and E (encoding symbol
   length in bytes) parameters are first determined, as explained in the
   following sections.

   The source object is then partitioned using the block partitioning
   algorithm specified in [fec-bb-revised].  To that purpose, the B, L
   (object transfer length in bytes), and E arguments are provided.  As
   a result, the object is partitioned into N source blocks.  These
   blocks are numbered consecutively from 0 to N-1.  The first I source
   blocks consist of A_large source symbols, the remaining N-I source
   blocks consist of A_small source symbols.  Each source symbol is E
   bytes in length, except perhaps the last symbol which may be shorter.

   For each block the actual number of encoding symbols is determined,
   as explained in the following section.

   Then, FEC encoding and decoding can be done block per block,
   independently.  To that purpose, a parity check matrix is created,
   that forms a system of linear equations between the repair and source
   symbols of a given block, where the basic operator is XOR.

   This parity check matrix is logically divided into two parts: the
   left side (from column 0 to k-1) which describes the occurrence of
   each source symbol in the equation system; and the right side (from
   column k to n-1) which describes the occurrence of each repair symbol
   in the equation system.  An entry (a "1") in the matrix at position
   (i,j) (i.e. at row i and column j) means that the symbol with ESI i
   appears in equation j of the system.  The only difference between the
   LDPC-Staircase and LDPC-Triangle schemes is the construction of the
   right sub-matrix.

   When the parity symbols have been created, the sender will transmit
   source and parity symbols.  The way this transmission occurs can
   largely impact the erasure recovery capabilities of the LDPC-* FEC.
   In particular, sending parity symbols in sequence is suboptimal.
   Instead it is usually recommended the shuffle these symbols.  The
   interested reader will find more details in [Neumann05].

   The following sections detail how the B, E, and n parameters are
   determined (respectively Section 5.2, Section 5.3 and Section 5.4),
   how encoding symbol groups are created (Section 5.5), and finally



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   specify the PRNG (Section 5.6).

5.2.  Determining the Maximum Source Block Length (B)

   The B parameter (maximum source block length in symbols) depends on
   several parameters: the code rate (rate), the Encoding Symbol ID
   field length of the FEC Payload ID (20 bits), as well as possible
   internal codec limitations.

   The B parameter cannot be larger than the following values, derived
   from the FEC Payload ID limitations, for a given code rate:

      max1_B = 2 ^^ (20 - ceil(Log2(1/rate)))

   Some common max1_B values are:

   o  rate == 1 (no repair symbols): max_B = 2 ^^ 20 = 1,048,576

   o  1 > rate >= 1/2: max1_B = 2 ^^ 19 = 524,288 symbols

   o  1/2 > rate >= 1/4: max1_B = 2 ^^ 18 = 262,144 symbols

   o  1/4 > rate >= 1/8: max1_B = 2 ^^ 17 = 131,072 symbols

   Additionally, a codec MAY impose other limitations on the maximum
   block size.  This is the case for instance when the codec uses
   internally 16 bit integers to store the Encoding Symbol ID, since it
   does not enable to store all the possible values of a 20 bit field.
   Other limitations may also apply, for instance because of a limited
   working memory size.  This decision SHOULD be clarified at
   implementation time, when the target use case is known.  This results
   in a max2_B limitation.

   Then, B is given by:

      B = min(max1_B, max2_B)

   Note that this calculation is only required at the coder, since the B
   parameter is communicated to the decoder through the FEC OTI.

5.3.  Determining the Encoding Symbol Length (E) and Number of Encoding
      Symbols per Group (G)

   The E parameter usually depends on the maximum transmission unit on
   the path (PMTU) from the source to the receivers.  In order to
   minimize the protocol header overhead (e.g. the LCT/UDP/IPv4 or IPv6
   headers in case of ALC), E is chosen as large as possible.  In that
   case, E is chosen so that the size of a packet composed of a single



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   symbol (G=1) remains below but close to the PMTU.

   Yet other considerations can exist.  For instance, the E parameter
   can be made a function of the object transfer length.  Indeed, LDPC
   codes are known to offer better protection for large blocks.  In case
   of small objects, it can be a good practice to reduce the encoding
   symbol length (E) in order to artificially increase the number of
   symbols, and therefore the block size.

   In order to minimize the protocol header overhead, several symbols
   can be grouped in the same Encoding Symbol Group (i.e.  G > 1).
   Depending on how many symbols are grouped (G) and on the packet loss
   rate (which leads to loosing G symbols at a time), this strategy
   might or might not be appropriate.  A balance must therefore be
   found.

   The current specification does not mandate any value for either E or
   G. The current specification only provides an example of possible
   choices for E and G. Note that this choice is done by the sender.
   Then the E and G parameters are communicated to the receivers thanks
   to the FEC OTI.

   Example:

   First define the target packet size, pkt_sz (usually the PMTU minus
   the various protocol headers).  The pkt_sz must be chosen in such a
   way it is a multiple of G. Calculate the number of packets: nb_pkts =
   ceil(L / pkt_sz).  Then, use the following table to find a possible G
   value.

     +------------------------+----+-------------+-------------------+
     |    Number of packets   |  G | Symbol size |         k         |
     +------------------------+----+-------------+-------------------+
     |     4000 <= nb_pkts    |  1 |    pkt_sz   |     4000 <= k     |
     |                        |    |             |                   |
     | 1000 <= nb_pkts < 4000 |  4 |  pkt_sz / 4 | 4000 <= k < 16000 |
     |                        |    |             |                   |
     |  500 <= nb_pkts < 1000 |  8 |  pkt_sz / 8 |  4000 <= k < 8000 |
     |                        |    |             |                   |
     |   1 <= nb_pkts < 500   | 16 | pkt_sz / 16 |   16 <= k < 8000  |
     +------------------------+----+-------------+-------------------+

5.4.  Determining the Number of Encoding Symbols of a Block

   The following algorithm, also called "n-algorithm", explains how to
   determine the actual number of encoding symbols for a given block.

   AT A SENDER:



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   Input:

      B: Maximum source block length, for any source block.  Section 5.2
      explains how to determine its value.

      k: Current source block length.  This parameter is given by the
      source blocking algorithm.

      rate: FEC code rate, which is provided by the user (e.g. when
      starting a FLUTE sending application).  It is expressed as a
      floating point value.  The rate value must be such that the
      resulting number of encoding symbols per block is at most equal to
      2^^20 (Section 4.1).

   Output:

      max_n: Maximum number of encoding symbols generated for any source
      block

      n: Number of encoding symbols generated for this source block

   Algorithm:

      max_n = floor(B / rate);

      if (max_n >= 2^^20) then return an error ("invalid code rate");

      n = floor(k * max_n / B);

   AT A RECEIVER:

   Input:

      B: Extracted from the received FEC OTI

      max_n: Extracted from the received FEC OTI

      k: Given by the source blocking algorithm

   Output:

      n:

   Algorithm:

      n = floor(k * max_n / B);





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5.5.  Identifying the Symbols of an Encoding Symbol Group

   When multiple encoding symbols are sent in the same packet, the FEC
   Payload ID information of the packet MUST refer to the first encoding
   symbol.  It MUST then be possible to identify each symbol from this
   single FEC Payload ID.  To that purpose, the symbols of an Encoding
   Symbol Group (i.e. packet):

   o  MUST all be either source symbols, or repair symbols.  Therefore
      only source packets and repair packets are permitted, not mixed
      ones.

   o  are identified by a function, ESIs_of_group(), that takes as
      argument:

      *  for a sender, the index of the Encoding Symbol Group (i.e.
         packet) that the application wants to create,

      *  for a receiver, the ESI information contained in the FEC
         Payload ID.

      and returns the list of G Encoding Symbol IDs that will be packed
      together.  In case of a source packet, the G source symbols are
      taken consecutively.  In case of a repair packet, the G repair
      symbols are chosen randomly, as explained below.

   The system must first be initialized by creating a random permutation
   of the n-k indexes.  This initialization function MUST be called
   immediately after creating the parity check matrix.  More precisely,
   since the PRNG seed is not re-initialized, no call to the PRNG
   function must have happened between the time the parity check matrix
   has been initialized and the time the following initialization
   function is called.  This is true both at a sender and at a receiver.


















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   /*
    * Initialization function.
    * Warning: use only when G > 1.
    */
   initialize_tables ()
   {
       int i;
       int randInd;
       int backup;

       /* initialize the two tables that map ID
        * (i.e. ESI-k) to/from TxSequence. */
       for (i = 0; i < n - k; i++) {
           IDtoTxseq[i] = i;
           txseqToID[i] = i;
       }
       /* now randomize everything */
       for (i = 0; i < n - k; i++) {
           randInd = rand(n - k);
           backup  = IDtoTxseq[i];
           IDtoTxseq[i] = IDtoTxseq[randInd];
           IDtoTxseq[randInd] = backup;
           txseqToID[IDtoTxseq[i]] =  i;
           txseqToID[IDtoTxseq[randInd]] = randInd;
       }
       return;
   }

   It is then possible, at the sender, to determine the sequence of G
   Encoding Symbol IDs that will be part of the group.





















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   /*
    * Determine the sequence of ESIs of the packet under construction
    * at a sender.
    * Warning: use only when G > 1.
    * PktIdx (IN):  index of the packet, in {0..ceil(n/G)} range
    * ESIs[] (OUT): list of ESI of the packet
    */
   sender_find_ESIs_of_group (int      PktIdx,
                              ESI_t    ESIs[])
   {
       int i;

       if (is_source_packet(PktIdx) == true) {
           /* this is a source packet */
           ESIs[0] = (PktIdx * G) % k;
           for (i = 0; i < G; i++) {
                   ESIs[i] = ESIs[0] + i;
           }
       } else {
           /* this is a repair packet */
           for (i = 0; i < G; i++) {
               ESIs[i] =
                   k +
                   txseqToID[(i + (PktIdx - nbSourcePkts) * G)
                             % (n - k)];
           }
       }
       return;
   }

   Similarly, upon receiving an Encoding Symbol Group (i.e. packet), a
   receiver can determine the sequence of G Encoding Symbol IDs from the
   first ESI, esi0, that is contained in the FEC Payload ID.


















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   /*
    * Determine the sequence of ESIs of a packet received.
    * Warning: use only when G > 1.
    * esi0 (IN):  : ESI contained in the FEC Payload ID
    * ESIs[] (OUT): list of ESI of the packet
    */
   receiver_find_ESIs_of_group (ESI_t    esi0,
                                ESI_t    ESIs[])
   {
       int i;

       if (is_source_packet(esi0) == true) {
           /* this is a source packet */
           for (i = 0; i < G; i++) {
               ESIs[i] = (esi0 + i) % k;
           }
       } else {
           /* this is a repair packet */
           for (i = 0; i < G; i++) {
               ESIs[i] =
                   k +
                   txseqToID[(i + IDtoTxseq[esi0 - k])
                             % (n - k)];
           }
       }
   }

5.6.  Pseudo Random Number Generator

   The present FEC Encoding ID relies on a pseudo-random number
   generator (PRNG) that must be fully specified, in particular in order
   to enable the receivers and the senders to build the same parity
   check matrix.  The minimal standard generator [Park88] is used.  It
   defines a simple multiplicative congruential algorithm: Ij+1 = A * Ij
   (modulo M), with the following choices: A = 7^^5 = 16807 and M =
   2^^31 - 1 = 2147483647.  Several implementations of this PRNG are
   known and discussed in the literature.  Yet all of them provide the
   same sequence of pseudo random numbers.  For instance, if seed = 1,
   then the 10,000th value returned MUST be equal to 1043618065.  The
   following implementation uses the Park and Miller algorithm with the
   optimization suggested by D. Carta in [Carta90].










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   unsigned long           seed;


   /*
    * Initialize the PRNG with a seed between
    * 1 and 0x7FFFFFFE (i.e. 2^^31-2) inclusive.
    */
   void srand (unsigned long s)
   {
           if ((s > 0) && (s < 0x7FFFFFFF))
                   seed = s;
           else
                   exit(-1);
   }

   /*
    * Returns a random integer in [0; maxv-1]
    * Derived from rand31pmc, Robin Whittle,
    * September 20th, 2005.
    * http://www.firstpr.com.au/dsp/rand31/
    *      16807           multiplier constant (7^^5)
    *      0x7FFFFFFF      modulo constant (2^^31-1)
    * The inner PRNG produces a value between 1 and
    * 0x7FFFFFFE (2^^31-2) inclusive.
    * This value is then scaled between 0 and maxv-1
    * inclusive.
    */
   unsigned long
   rand (unsigned long maxv)
   {
           unsigned long   hi, lo;

           lo = 16807 * (seed & 0xFFFF);
           hi = 16807 * (seed >> 16);
           lo += (hi & 0x7FFF) << 16;
           lo += hi >> 15;
           if (lo > 0x7FFFFFFF)
                   lo -= 0x7FFFFFFF;
           seed = (long)lo;
           /* don't use modulo, least significant bits are less random
            * than most significant bits [Numerical Recipies in C] */
           return ((unsigned long)
                   ((double)seed * (double)maxv / (double)0x7FFFFFFF));
   }







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6.  Full Specification of the LDPC-Staircase Scheme

6.1.  General

   The LDPC-Staircase scheme is identified by the Fully-Specified FEC
   Encoding ID XX.

   The PRNG used by the LDPC-Staircase scheme must be initialized by a
   seed.  This PRNG seed is an optional instance-specific FEC OTI
   element (Section 4.2.3).  When this PRNG seed is not carried within
   the FEC OTI, it is assumed that encoder and decoders either use
   another way to communicate the seed value or use a fixed, predefined
   value.

6.2.  Parity Check Matrix Creation

   The LDPC-Staircase matrix can be divided into two parts: the left
   side of the matrix defines in which equations the source symbols are
   involved; the right side of the matrix defines in which equations the
   repair symbols are involved.

   The left side is generated with the following algorithm:





























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      /* initialize a list of possible choices to
       * guarantee a homogeneous "1" distribution */
      for (h = 3*k-1; h >= 0; h--) {
          u[h] = h % (n-k);
      }
      /* left limit within the list of possible choices, u[] */
      t = 0;

      for (j = 0; j < k; j++) { /* for each source symbol column */
          for (h = 0; h < 3; h++) { /* add 3 "1s" */
              /* check that valid available choices remain */
              for (i = t; i < 3*k && matrix_has_entry(u[i], j); i++);

              if (i < 3*k) {
                  /* choose one index within the list of possible
                   * choices */
                  do {
                      i = t + rand(3*k-t);
                  } while (matrix_has_entry(u[i], j));
                  matrix_insert_entry(u[i], j);

                  /* replace with u[t] which has never been chosen */
                  u[i] = u[t];
                  t++;
              } else {
                  /* no choice left, choose one randomly */
                  do {
                      i = rand(n-k);
                  } while (matrix_has_entry(i, j));
                  matrix_insert_entry(i, j);
              }
          }
      }

      /* Add extra bits to avoid rows with less than two "1s" */
      for (i = 0; i < n-k; i++) { /* for each row */
          if (degree_of_row(i) == 0) {
              j = rand(k);
              e = matrix_insert_entry(i, j);
          }
          if (degree_of_row(i) == 1) {
              do {
                  j = rand(k);
              } while (matrix_has_entry(i, j));
              matrix_insert_entry(i, j);
          }
      }




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   The right side (the staircase) is generated by the following
   algorithm:

      matrix_insert_entry(0, k);    /* first row */
      for (i = 1; i < n-k; i++) {   /* for the following rows */
          matrix_insert_entry(i, k+i);   /* identity */
          matrix_insert_entry(i, k+i-1); /* staircase */
      }

   Note that just after creating this parity check matrix, when encoding
   symbol groups are used (i.e.  G > 1), the function initializing the
   two random permutation tables (Section 5.5) MUST be called.  This is
   true both at a sender and at a receiver.

6.3.  Encoding

   Thanks to the staircase matrix, repair symbol creation is
   straightforward: each repair symbol is equal to the sum of all source
   symbols in the associated equation, plus the previous repair symbol
   (except for the first repair symbol).  Therefore encoding MUST follow
   the natural repair symbol order: start with the first repair symbol,
   and generate repair symbol with ESI i before symbol ESI i+1.

6.4.  Decoding

   Decoding basically consists in solving a system of n-k linear
   equations whose variables are the source an repair symbols.  Of
   course, the final goal is to recover the value of source symbols
   only.

   To that purpose, many techniques are possible.  One of them is the
   following trivial algorithm [Zyablov74]: given a set of linear
   equations, if one of them has only one remaining unknown variable,
   then the value of this variable is that of the constant term.  So,
   replace this variable by its value in all the remaining linear
   equations and reiterate.  The value of several variables can
   therefore be found recursively.  Applied to LDPC FEC codes working
   over an erasure packet, the parity check matrix defines a set of
   linear equations whose variables are the source symbols and repair
   symbols.  Receiving or decoding a symbol is equivalent to having the
   value of a variable.  Appendix A sketches a possible implementation
   of this algorithm.

   The Gauss elimination technique (or any derivative) is another
   possible decoding technique.

   Because interoperability does not depend on the decoding algorithm
   used, the current document does not recommend any particular



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   technique.  This choice is left to the codec developer.

   Yet choosing a decoding technique will have great practical impacts.
   It will impact the erasure capabilities: a Gauss elimination
   technique enables to solve the system with a smaller number of
   symbols compared to the trivial technique.  It will also impact the
   CPU load: a Gauss elimination technique requires much more processing
   than the trivial technique.  Depending on the target use case, the
   codec developer will favor one feature or the other.










































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7.   Full Specification of the LDPC-Triangle Scheme

7.1.  General

   LDPC-Triangle is identified by the Fully-Specified FEC Encoding ID
   YY.

   The PRNG used by the LDPC-Triangle scheme must be initialized by a
   seed.  This PRNG seed is an optional instance-specific FEC OTI
   element (Section 4.2.3).  When this PRNG seed is not carried within
   the FEC OTI, it is assumed that encoder and decoders either use
   another way to communicate the seed value or use a fixed, predefined
   value.

7.2.  Parity Check Matrix Creation

   The LDPC-Triangle matrix can be divided into two parts: the left side
   of the matrix defines in which equations the source symbols are
   involved; the right side of the matrix defines in which equations the
   repair symbols are involved.

   The left side is generated with the same algorithm as that of LDPC-
   Staircase (Section 6.2).

   The right side (the triangle) is generated with the following
   algorithm:

      matrix_insert_entry(0, k);    /* first row */
      for (i = 1; i < n-k; i++) {   /* for the following rows */
          matrix_insert_entry(i, k+i);   /* identity */
          matrix_insert_entry(i, k+i-1); /* staircase */
          /* now fill the triangle */
          j = i-1;
          for (l = 0; l < j; l++) { /* limit the # of "1s" added */
              j = rand(j);
              matrix_insert_entry(i, k+j);
          }
      }

   Note that just after creating this parity check matrix, when encoding
   symbol groups are used (i.e.  G > 1), the function initializing the
   two random permutation tables (Section 5.5) MUST be called.  This is
   true both at a sender and at a receiver.

7.3.  Encoding

   Here also repair symbol creation is straightforward: each repair
   symbol is equal to the sum of all source symbols in the associated



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   equation, plus the repair symbols in the triangle.  Therefore
   encoding MUST follow the natural repair symbol order: start with the
   first repair symbol, and generate repair symbol with ESI i before
   symbol ESI i+1.

7.4.  Decoding

   Decoding basically consists in solving a system of n-k linear
   equations, whose variables are the source an repair symbols.  Of
   course, the final goal is to recover the value of source symbols
   only.  To that purpose, many techniques are possible, as explained in
   Section 6.4.

   Because interoperability does not depend on the decoding algorithm
   used, the current document does not recommend any particular
   technique.  This choice is left to the codec implementer.



































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8.  Security Considerations

   The security considerations for this document are the same as that of
   [RFC3452].















































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9.  Intellectual Property

   To the best of our knowledge, there is no patent or patent
   application identified as being used in the LDPC-Staircase and LDPC-
   Triangle FEC schemes.  Yet other LDPC codes and associated techniques
   MAY be covered by Intellectual Property Rights.













































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10.  Acknowledgments

   Section 5.4 is derived from a previous Internet-Draft, and we would
   like to thank S. Peltotalo and J. Peltotalo for their contribution.
   We would also like to thank Pascal Moniot, Laurent Fazio, Aurelien
   Francillon and Shao Wenjian for their comments.













































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11.  References

11.1.  Normative References

   [RFC2119]  Bradner, S., "Key words for use in RFCs to Indicate
              Requirement Levels", RFC 2119, BCP 14, March 1997.

   [RFC3452]  Luby, M., Vicisano, L., Gemmell, J., Rizzo, L., Handley,
              M., and J. Crowcroft, "Forward Error Correction (FEC)
              Building Block", RFC 3452, December 2002.

   [RFC3453]  Luby, M., Vicisano, L., Gemmell, J., Rizzo, L., Handley,
              M., and J. Crowcroft, "The Use of Forward Error Correction
              (FEC) in Reliable Multicast", RFC 3453, December 2002.

   [fec-bb-revised]
              Watson, M., Luby, M., and L. Vicisano, "Forward Error
              Correction (FEC) Building Block",
               draft-ietf-rmt-fec-bb-revised-03.txt (work in progress),
              January 2006.

11.2.  Informative References

   [Carta90]  Carta, D., "Two Fast Implementations of the Minimal
              Standard Random Number Generator",  Communications of the
              ACM, Vol. 33, No. 1, pp.87-88, January 1990.

   [LDPCrefimpl]
              Roca, V., Neumann, C., and J. Laboure, "LDPC-Staircase/
              LDPC-Triangle Codec Reference Implementation",  PLANETE
              Research Team, INRIA Rhone-Alpes,
              http://planete.inrialpes.fr/~roca/mcl/.

   [Mac03]    MacKay, D., "Information Theory, Inference and Learning
              Algorithms", Cambridge University Press, ISBN: 0521642981,
              2003.

   [Neumann05]
              Neumann, C., Roca, V., Francillon, A., and D. Furodet,
              "Impacts of Packet Scheduling and Packet Loss Distribution
              on FEC Performances: Observations and Recommendations",
               ACM CoNEXT'05 Conference, Toulouse, France (an extended
              version is available as INRIA Research Report RR-5578),
              October 2005.

   [Park88]   Park, S. and K. Miller, "Random Number Generators: Good
              Ones are Hard to Find",  Communications of the ACM, Vol.
              31, No. 10, pp.1192-1201, 1988.



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   [Roca04]   Roca, V. and C. Neumann, "Design, Evaluation and
              Comparison of Four Large Block FEC Codecs: LDPC, LDGM,
              LDGM-Staircase and LDGM-Triangle, Plus a Reed-Solomon
              Small Block FEC Codec",  INRIA Research Report RR-5225,
              June 2004.

   [Zyablov74]
              Zyablov, V. and M. Pinsker, "Decoding Complexity of Low-
              Density Codes for Tranmission in a Channel with Erasures",
               Translated from Problemy Peredachi Informatsii, Vol.10,
              No. 1, pp.15-28, January-March 1974.








































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Appendix A.  Trivial Decoding Algorithm (Informative Only)

   A trivial decoding algorithm is the following:

      Initialization: allocate a partial sum buffer, partial_sum_i, for
      each line i, and reset it to 0.

      For each newly received or decoded symbol s_i with ESI i:

      1.  If s_i is an already decoded or received symbol, return
          immediately and do nothing.

      2.  If s_i is a source symbol, it is permanently stored in memory.

      3.  For each equation j having a degree greater than one (i.e.
          more than one unknown variable), with an entry in column i
          (i.e. having s_i as a variable), do the following:

          +  add s_i to partial_sum_i;

          +  remove the entry (j, i) of the H matrix.

          +  If the new degree of equation j is one, we have decoded a
             new packet and have to remember the index of the equation
             in a list of indexes for newly decoded packets for step 4.

      4.  For all newly generated packets s_l in step 3:

          +  remove the last entry in equation j,

          +  copy partial_sum_j to the buffer associate with symbol s_l,

          +  goto step 1 with the newly created symbol s_l


















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Authors' Addresses

   Vincent Roca
   INRIA
   655, av. de l'Europe
   Zirst; Montbonnot
   ST ISMIER cedex  38334
   France

   Phone:
   Email: vincent.roca@inrialpes.fr
   URI:   http://planete.inrialpes.fr/~roca/


   Christoph Neumann
   Thomson Research
   46, Quai A. Le Gallo
   Boulogne Cedex  92648
   France

   Phone:
   Email: christoph.neumann@thomson.net
   URI:   http://planete.inrialpes.fr/~chneuman/


   David Furodet
   STMicroelectronics
   12, Rue Jules Horowitz
   BP217
   Grenoble Cedex  38019
   France

   Phone:
   Email: david.furodet@st.com
   URI:
















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Intellectual Property Statement

   The IETF takes no position regarding the validity or scope of any
   Intellectual Property Rights or other rights that might be claimed to
   pertain to the implementation or use of the technology described in
   this document or the extent to which any license under such rights
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   made any independent effort to identify any such rights.  Information
   on the procedures with respect to rights in RFC documents can be
   found in BCP 78 and BCP 79.

   Copies of IPR disclosures made to the IETF Secretariat and any
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   such proprietary rights by implementers or users of this
   specification can be obtained from the IETF on-line IPR repository at
   http://www.ietf.org/ipr.

   The IETF invites any interested party to bring to its attention any
   copyrights, patents or patent applications, or other proprietary
   rights that may cover technology that may be required to implement
   this standard.  Please address the information to the IETF at
   ietf-ipr@ietf.org.


Disclaimer of Validity

   This document and the information contained herein are provided on an
   "AS IS" basis and THE CONTRIBUTOR, THE ORGANIZATION HE/SHE REPRESENTS
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   ENGINEERING TASK FORCE DISCLAIM ALL WARRANTIES, EXPRESS OR IMPLIED,
   INCLUDING BUT NOT LIMITED TO ANY WARRANTY THAT THE USE OF THE
   INFORMATION HEREIN WILL NOT INFRINGE ANY RIGHTS OR ANY IMPLIED
   WARRANTIES OF MERCHANTABILITY OR FITNESS FOR A PARTICULAR PURPOSE.


Copyright Statement

   Copyright (C) The Internet Society (2006).  This document is subject
   to the rights, licenses and restrictions contained in BCP 78, and
   except as set forth therein, the authors retain all their rights.


Acknowledgment

   Funding for the RFC Editor function is currently provided by the
   Internet Society.




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