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

RMT                                                              V. Roca
Internet-Draft                                                C. Neumann
Expires: April 13, 2006                                            INRIA
                                                              D. Furodet
                                                      STMicroelectronics
                                                        October 10, 2005


        Low Density Parity Check (LDPC) Forward Error Correction
                   draft-ietf-rmt-bb-fec-ldpc-00.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
   aware will be disclosed, in accordance with Section 6 of BCP 79.

   Internet-Drafts are working documents of the Internet Engineering
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   This Internet-Draft will expire on April 13, 2006.

Copyright Notice

   Copyright (C) The Internet Society (2005).

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
   RFC3453.



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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 Elements . . . . . . . . . . . . . . .  8
       4.2.4   Encoding Format  . . . . . . . . . . . . . . . . . . .  8
   5.  Procedures . . . . . . . . . . . . . . . . . . . . . . . . . . 10
     5.1   General  . . . . . . . . . . . . . . . . . . . . . . . . . 10
     5.2   Determining the Maximum Source Block Length (B)  . . . . . 10
     5.3   Determining the Encoding Symbol Length (E) . . . . . . . . 11
     5.4   Determining the Number of Encoding Symbols of a Block  . . 11
     5.5   Identifying the Symbols of an Encoding Symbol Group  . . . 13
     5.6   Pseudo Random Number Generator . . . . . . . . . . . . . . 16
   6.  Full Specification of the LDPC-Staircase Scheme  . . . . . . . 18
     6.1   General  . . . . . . . . . . . . . . . . . . . . . . . . . 18
     6.2   Parity Check Matrix Creation . . . . . . . . . . . . . . . 18
     6.3   Encoding . . . . . . . . . . . . . . . . . . . . . . . . . 20
     6.4   Decoding . . . . . . . . . . . . . . . . . . . . . . . . . 20
   7.  Full Specification of the LDPC-Triangle Scheme . . . . . . . . 21
     7.1   General  . . . . . . . . . . . . . . . . . . . . . . . . . 21
     7.2   Parity Check Matrix Creation . . . . . . . . . . . . . . . 21
     7.3   Encoding . . . . . . . . . . . . . . . . . . . . . . . . . 21
     7.4   Decoding . . . . . . . . . . . . . . . . . . . . . . . . . 22
   8.  Security Considerations  . . . . . . . . . . . . . . . . . . . 23
   9.  Intellectual Property  . . . . . . . . . . . . . . . . . . . . 24
   10.   Acknowledgments  . . . . . . . . . . . . . . . . . . . . . . 25
   11.   References . . . . . . . . . . . . . . . . . . . . . . . . . 26
     11.1  Normative References . . . . . . . . . . . . . . . . . . . 26
     11.2  Informative References . . . . . . . . . . . . . . . . . . 26
       Authors' Addresses . . . . . . . . . . . . . . . . . . . . . . 27
   A.  Trivial Decoding Algoritm (Informative Only) . . . . . . . . . 28
       Intellectual Property and Copyright Statements . . . . . . . . 29










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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 is a maximum of 2^^12 blocks per object.

      The Encoding Symbol ID (20 bit field) identifies which specific
      encoding symbol generated from the source block is carried in the
      packet payload.  There is 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 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 used with the present FEC Scheme:






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   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:

         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.

   o  Max-Number-of-Encoding-Symbols (max_n): a non-negative integer
      indicating the maximum number of encoding symbols generated for
      any source block.

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

4.2.3  Scheme-Specific Elements

   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.

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.






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    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       |                               |
   +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+                               +
   |                      Transfer-Length (L)                      |
   +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
   |      0 (not applicable)       |   Encoding Symbol Length (E)  |
   +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
   |                   Max Source Block Length (B)                 |
   +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
   |                 Max Nb of Enc. Symbols  (max_n)               |
   +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
   .                       Optional PRNG seed                      .
   +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+

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

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-PRNG-seed (optional)

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

   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
   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



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   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 (e.g. available working memory) may also apply.
   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)

      -- editor's note: the E parameter is a function of the object
      transfer length.  Since LDPC codes are known to offer better
      protection for large blocks, the smaller the object, the smaller E
      should be in order to increase the number of symbols the object is
      composed of.  The optimal values that should be used for E as a
      function of the object transfer length are under study. --

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

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.



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   AT A SENDER:

   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 given by the user (e.g. when starting
      a FLUTE sending application) for a given use case.  It is
      expressed as a floating point value.

   Output:

      max_n     Maximum number of encoding symbols generated for any source
      block

      n         Number of encoding symbols generated for this source block

   Algorithm:

   a.  max_n = floor(B / R)

   b.  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:

   a.  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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   /*
    * Use only in case G > 1, i.e. when encoding symbol
    * groups are actually needed.
    */
   initialize_tables ()
   {
       int i;
       int randInd;

       /* initialize the two tables that map ID
        * (i.e. ESI-k) to/from TxSequence:
        *    - IDtoTxseq
        *    - txseqToID
        */
       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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   /*
    * Use only in case G > 1, i.e. when encoding symbol
    * groups are actually needed.
    * PktIdx (IN):  index of the packet, in {0..ceil(T/G)} range
    * ESIs[] (OUT): list of ESI of the packet
    */
   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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   /*
    * Use only in case G > 1, i.e. when encoding symbol
    * groups are actually needed.
    * esi0 (IN):  : ESI contained in the FEC Payload ID
    * ESIs[] (OUT): list of ESI of the packet
    */
   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.  The PRNG must be initialized with a seed
   that is strictly greater than 0.













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      double seed; /* assumed initialized with a seed > 0 */
      #define A 16807.0
      #define M 2147483647.0
      #define Q 127773.0 /* M div A */
      #define R 2836.0 /* M mod A */

      /*
       * Initialize the PRNG with a seed > 0.
       */
      void srand (int s)
      {
         if (s > 0) seed = s;
         else exit(-1);
      }

      /*
       * Returns a random integer in [0; maxv-1]
       * derived from [Park88].
       */
      int rand (int maxv)
      {
         double lo, hi, test;
         double rand;

         hi = (int)(seed / Q);
         lo = seed - Q*hi;
         test = A*lo - R*hi;
         if (test > 0.0)
            seed = test;
         else
            seed = test + M;
         rand = seed / M;
         if (rand == 1.0)
            return 0;
         else
            return ((int)(rand * (double)maxv));
      }














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6.  Full Specification of the LDPC-Staircase Scheme

6.1  General

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

   LDPC-Staircase is based on the pseudo-random number generator
   specified in Section 5.6.  Therefore the seed used to initiate the
   PRNG is an instance-specific FEC Object Transmission Information
   optional element.  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.

6.2  Parity Check Matrix Creation

   The matrix creation algorithm for LDPC-Staircase is described in the
   following.  The algorithm can be divided into two parts: The left
   side of the matrix where the occurrence of the source symbols in the
   equations is described, and the right side of the matrix where repair
   symbols are described.  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 checks. */
      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 with the following
   algorithm:

      for (i = 0; i < n-k; i++) { /* for each row */
          matrix_insert_entry(i,k+i);
          if (i > 0)
              matrix_insert_entry(i,k+i-1);
      }

   Note that just after creating this parity check matrix, when encoding
   symbol groups are used, 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 packet.
   Therefore encoding MUST follow the natural repair symbol order, i.e.
   generate repair symbol with ESI i before symbol ESI i+1 and MUST
   start with the first repair symbol.

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: 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 pivot of Gauss technique, as well as derived versions, is another
   possibility.

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



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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.

   LDPC-Triangle is based on the pseudo-random number generator
   specified in Section 5.6.  Therefore the seed used to initiate the
   PRNG is an instance-specific FEC Object Transmission Information
   optional element.  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.

7.2  Parity Check Matrix Creation

   The matrix creation algorithm for LDPC-Triangle is the following.
   The left side is the same as for LDPC-Staircase (see Section 6.2).
   The right side (the triangle) is generated with the following
   algorithm:

      for (i = 0; i < n-k; i++) { /* for each row */
          /* create the identity */
          matrix_insert_entry(i,k+i);
          if (i > 0) {
              /* create the staircase */
              matrix_insert_entry(i,k+i-1);

              /* fill the triangle */
              int j = i;
              for (l = 0; l < j; l++) {
                  if (j != 0) {
                      temp = rand(j);
                      matrix_insert_entry(pchkMatrix, i, k+j);
                  }
              }
          }
      }

   Note that just after creating this parity check matrix, when encoding
   symbol groups are used, 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

   Just like LDPC-Triangle repair symbol creation is straightforward:
   each repair symbol is equal to the sum of all source symbols in the



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   associated equation, plus the (previously built) repair packets
   specified in the triangle.  Therefore encoding MUST follow the
   natural repair symbol order, i.e. generate repair symbol with ESI i
   before symbol ESI i+1 and MUST start with the first repair symbol.

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 recommand 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 they
   are for RFC 3452 [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 from STMicroelectronics for
   his 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 (revised)", draft-ietf-
              rmt-fec-bb-revised-01.txt draft-ietf-rmt-fec-bb-revised-
              01.txt, September 2005.

11.2  Informative References

   [LDPCrefimpl]
              Roca, V., Neumann, C., and J. Laboure, "LDPC-Staircase/
              LDPC-Triangle Codec Reference Implementation", MCLv3
              project PLANETE Research Team, INRIA Rhone-Alpes,
              June 2005.

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

   [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.

   [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.









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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:


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

   Phone:
   Email: christoph.neumann@inrialpes.fr
   URI:


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

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















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Appendix A.  Trivial Decoding Algoritm (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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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
   assurances of licenses to be made available, or the result of an
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   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 (2005).  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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