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Versions: 00 01 02 03 04 05 RFC 5510

Reliable Multicast Transport                                    J. Lacan
Internet-Draft                                          ENSICA/LAAS-CNRS
Expires: August 27, 2006                                         V. Roca
                                                                   INRIA
                                                            J. Peltotalo
                                                            S. Peltotalo
                                        Tampere University of Technology
                                                       February 23, 2006


              Reed-Solomon Forward Error Correction (FEC)
                    draft-ietf-rmt-bb-fec-rs-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
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   This Internet-Draft will expire on August 27, 2006.

Copyright Notice

   Copyright (C) The Internet Society (2006).

Abstract

   This document describes a Fully-Specified FEC scheme for the Reed-
   Solomon forward error correction code and its application to reliable
   delivery of data objects on the packet erasure channel.




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   The Reed-Solomon codes belong to the class of Maximum Distance
   Separable (MDS) codes, i.e, they enable a receiver to recover the k
   source symbols from any set of k received symbols.

   The implementation described here is compatible with the IPR-free
   implementation from Luigi Rizzo.


Table of Contents

   1.  Introduction . . . . . . . . . . . . . . . . . . . . . . . . .  3
   2.  Terminology  . . . . . . . . . . . . . . . . . . . . . . . . .  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.  Determining the Maximum Source Block Length (B)  . . . . . 10
     5.2.  Determining the Number of Encoding Symbols of a Block  . . 10
   6.  Reed-Solomon Codes . . . . . . . . . . . . . . . . . . . . . . 12
     6.1.  Finite field . . . . . . . . . . . . . . . . . . . . . . . 12
     6.2.  Reed-Solomon Encoding Algorithm  . . . . . . . . . . . . . 13
       6.2.1.  Encoding Complexity  . . . . . . . . . . . . . . . . . 14
     6.3.  Reed-Solomon Decoding Algorithm for the Erasure Channel  . 14
       6.3.1.  Decoding Complexity  . . . . . . . . . . . . . . . . . 14
     6.4.  Implementation . . . . . . . . . . . . . . . . . . . . . . 15
       6.4.1.  Implementation for the Packet Erasure Channel  . . . . 15
   7.  Security Considerations  . . . . . . . . . . . . . . . . . . . 17
   8.  Intellectual Property  . . . . . . . . . . . . . . . . . . . . 18
   9.  IANA Considerations  . . . . . . . . . . . . . . . . . . . . . 19
   10. Acknowledgments  . . . . . . . . . . . . . . . . . . . . . . . 20
   11. References . . . . . . . . . . . . . . . . . . . . . . . . . . 21
     11.1. Normative References . . . . . . . . . . . . . . . . . . . 21
     11.2. Informative References . . . . . . . . . . . . . . . . . . 21
   Authors' Addresses . . . . . . . . . . . . . . . . . . . . . . . . 22
   Intellectual Property and Copyright Statements . . . . . . . . . . 23








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1.  Introduction

   The use of Forward Error Correction (FEC) codes is a classical
   solution to improve the reliability of multicast and broadcast
   transmissions.  The [RFC3452] and [draft-ietf-rmt-fec-bb-revised-03]
   documents describe a general framework to use FEC in Content Delivery
   Protocols (CDP).  The companion document [RFC3453] describes some
   applications of FEC codes for content delivery.

   Recent FEC schemes like [draft-ietf-rmt-bb-fec-raptor-object-03] and
   [draft-ietf-rmt-bb-fec-ldpc-01] proposed erasure codes based on
   sparse graphs/matrices.  These codes are efficient in terms of CPU
   but not optimal in terms of correction capabilities, at least for
   small objects.

   The FEC scheme presented in this document belongs to the class of
   Maximum-Distance Separable codes, i.e., it is optimal in terms of
   erasure correction capability.  In others words, it enables the
   receiver to recover the k source symbols from any set of k encoding
   symbols.  Even if the encoding/decoding complexity is larger than
   that of [draft-ietf-rmt-bb-fec-raptor-object-03] or
   [draft-ietf-rmt-bb-fec-ldpc-01], this family of codes is very useful
   for applications sending small objects (e.g., for video and audio
   streaming).

   Indeed many applications dealing with content transmission or content
   storage already rely on packet-based Reed-Solomon codes.  In
   particular, many of them are derived from the implementation of Luigi
   Rizzo [RS-Rizzo].  This latter is compatible with the Reed-Solomon
   codes specification of the present document.





















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2.  Terminology

   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 RFC 2119 [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 [draft-ietf-rmt-fec-bb-revised-03].  Additionally, it uses the
   following definitions:

      Source symbol: unit of data used during the encoding process.

      Encoding symbol: unit of data generated by the encoding process.

      Repair symbol: encoding symbols that are not source symbols.

      Systematic code: a code in which the source symbols are part of
      the encoding symbols

      Source block: a block of k source symbols that are considered
      together for the encoding.

      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 number of source symbols in a source block

      n_r denotes the number of repair symbols generated for a source
      block

      n denotes the encoding block length, i.e., the number of encoding
      symbols generated for a source block.  Then n = k+ n_r

      max_n Maximum Number of Encoding Symbols generated for any source
      block

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




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

      E denotes the encoding symbol length in bytes

      sz denotes the symbol size in units of m bit elements

      m defines the number of elements in the finite field, namely q       2^^m.

      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

      a^^b denotes a raised to the power b

      a^^-1 denotes the inverse of a

      I_k denotes the k*k identity matrix

3.3.  Abbreviations

   This document uses the following abbreviations:

      ESI Encoding Symbol ID

      RS Reed-Solomon

      MDS Maximum Distance Separable code

      GF(q) finite field (A.K.A. Galois Field) with q elements



















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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:

   o  The Source Block Number (16 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^^16 blocks per
      object.

   o  The Encoding Symbol ID (16 bit field) identifies which specific
      encoding symbol generated from the source block is carried in the
      packet payload.  There is a maximum of 2^^16 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 an
   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 by incrementing sequentially the ESI.

    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 (16 bits) |  Encoding Symbol ID (16 bits) |
   +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+

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

4.2.  FEC Object Transmission Information

4.2.1.  Mandatory Elements

   o  FEC Encoding ID: the Fully-Specified FEC Scheme described in this
      document use the FEC Encoding ID XX.

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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         max_transfer_length = 2^^16 * B * E

      For instance, if B = 2^^8-1 (because the codec operates on a
      finite field with 2^^8 elements), and if E = 1024 bytes, then the
      maximum transfer length is 2^^34 bytes (i.e., a bit more than 17
      Giga Bytes).  For larger objects, it is expected that other FEC
      codes (e.g., LDPC codes) or another Reed-Solomon FEC Scheme with a
      larger Source Block Number field in the FEC Payload ID be used.

   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

   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.  When no G
      parameter is communicated to the decoder, then this latter MUST
      assume that G = 1.

   o  Finite Field size parameter, m: The m parameter defines the finite
      field size equal to q = p^^m elements.  The default value is m       8.  When no finite field size parameter is communicated to the
      decoder, then this latter MUST assume that m = 8.

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)                      |
   +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
   |       m       |       G       |   Encoding Symbol Length (E)  |
   +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
   |  Max Source Block Length (B)  |  Max Nb Enc. Symbols (max_n)  |
   +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+

   Figure 2: EXT_FTI Header Format

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 (optional)

   o  FEC-OTI-Finite-Field-Size-Parameter (optional)

   When no finite field size parameter is to be carried in the FEC OTI,
   the sender simply omits the FEC-OTI-Finite-Field-Size-Parameter
   element.
















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5.  Procedures

   This section defines procedures for FEC Encoding ID XX.

5.1.  Determining the Maximum Source Block Length (B)

   The finite field size parameter, m, defines the number of non zero
   elements in this field, q = 2^^m-1.  Note that q is also the
   theoretical maximum number of encoding symbols that can be produced
   for a source block.  For instance, when m = 8 (default):

      max1_B = 2^^8-1

   Additionally, a codec MAY impose other limitations on the maximum
   block size.  Yet it is not expected that such limits exist when using
   m = 8 (default).  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.

5.2.  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:

   Input:

      B: Maximum source block length, for any source block.  Section 5.1
      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:





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      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^^m) 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);

   Note that a Reed-Solomon decoder does not need to know the n value.
   Therefore the receiver part of the "n-algorithm" is not necessary
   from the Reed-Solomon decoder point of view.  Yet a receiving
   application using the Reed-Solomon FEC scheme will sometimes need to
   know the value of n used by the sender, for instance for memory
   management optimizations.  To that purpose, all the needed
   information is carried in the FEC OTI.













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6.  Reed-Solomon Codes

   Reed-Solomon (RS) codes form a special class of linear block codes,
   which offer maximum erasure correction capability.  A [n,k]-RS code
   encodes a sequence of k source elements defined over a finite field
   GF(q) into a sequence of n encoding elements, where n is upperbounded
   by q-1.  The implementation described in this document is based on a
   generator matrix built from a Vandermonde matrix put into systematic
   form.

6.1.  Finite field

   A finite field GF(q) is defined as a finite set of q elements which
   have a structure of field.  It contains necessarily q = p^^m
   elements, where p is a prime number.  With packet erasure channels, p
   is always set to 2.  The elements of the field GF(2^^m) can be
   represented by polynomials with binary coefficients (i.e., over
   GF(2)) of degree less than m.  The polynomials can be associated to
   binary vectors of length m.  For example, the vector (11001)
   represents the polynomial 1 + x + x^^4.  This representation is often
   called polynomial representation.  The addition between two elements
   is defined as the addition of binary polynomials in GF(2) and the
   multiplication is the multiplication modulo a given irreducible
   (i.e., non-factorizable) polynomial of degree m with coefficients in
   GF(2).

   Since a finite field GF(2^^m) is completely characterized by the
   irreducible polynomial, we propose the following polynomials to
   represent the field GF(2^^m), for m varying from 2 to 16:

      m = 2, "111" (1+x+x^^2)

      m = 3, "1101", (1+x+x^^3)

      m = 4, "11001", (1+x+x^^4)

      m = 5, "101001", (1+x^^2+x^^5)

      m = 6, "1100001", (1+x+x^^6)

      m = 7, "10010001", (1+x^^3+x^^7)

      m = 8, "101110001", (1+x^^2+x^^3+x^^4+x^^8)

      m = 9, "1000100001", (1+x^^4+x^^9)

      m = 10, "10010000001", (1+x^^3+x^^10)




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      m = 11, "101000000001", (1+x^^2+x^^11)

      m = 12, "1100101000001", (1+x+x^^4+x^^6+x^^12)

      m = 13, "11011000000001", (1+x+x^^3+x^^4+x^^13)

      m = 14, "110000100010001", (1+x+x^^6+x^^10+x^^14)

      m = 15, "1100000000000001", (1+x+x^^15)

      m = 16, "11010000000010001", (1+x+x^^3+x^^12+x^^16)

   For implementation issues, these polynomials are also primitive
   elements of GF(2^^m), i.e., any element of GF(2^^m) can be expressed
   as a power of a root of this polynomial.  These polynomials also
   contain the minimum number of monomials.

6.2.  Reed-Solomon Encoding Algorithm

   The encoding algorithm produces a vector of n encoding elements
   e=(e_0, ..., e_(n-1)) over GF(2^^m) from a source vector of k
   elements s=(s_0, ..., s_(k-1) ) over GF(2^^m).

   The linear codes can be encoded by multiplying the source vector by a
   generator matrix Gm of k rows and n columns over GF(2^^m).  Thus: e    s * Gm. The definition of the generator matrix completely
   characterizes the code.

   Let us consider that: n = 2^^m - 1 and: 0 < k <= n.  Let us denote
   alpha a primitive element of GF(2^^m) (i.e., any element of GF(2^^m)
   can be expressed as a power of alpha).

   The generator matrix is build from a k*n-Vandermonde matrix denoted
   by V_{k,n}. The entries of V_{k,n} are v_{i,j} = alpha^^(i*j), where
   0 <= i <= k - 1 and 0 <= j <= n - 1.  This matrix generates a MDS
   code.  However, it is not systematic as required by most of network
   applications.  To obtain a systematic matrix, the simplest solution
   is to consider the matrix V_{k,k} formed by the first k columns of
   V_{k,n} then to invert it and to multiply this inverse by V_{k,n}.
   Clearly, the product V_{k,k}^^-1 * V_{k,n} contains the identity
   matrix I_k on its first k columns and generates a MDS code.

   The product V_{k,k}^^-1 * V_{k,n} is denoted by Gm and is the
   generator matrix of the code considered in this document.

   Note that, for practical applications, the length of the code can be
   shortened to k <= n' < n by considering the sub-matrix formed by the
   n' first columns of Gm.



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6.2.1.  Encoding Complexity

   The encoding process can be done by first pre-computing G and by
   multiplying the source vector by Gm. The complexity is one
   multiplication s*Gm, where Gm is a k*(n-k) matrix. The complexity of
   the vector-matrix multiplication is then k*(n-k) (i.e., k operations
   per repair element).

   The encoding can also be processed by first computing the product s*
   V_{k,k}^^-1 and then by multiplying the result by V_{k,n}.  The
   multiplication by the inverse of a square Vandermonde matrix is known
   as the interpolation problem and its complexity is O(k log^^2 (k)).
   The multiplication by a Vandermonde matrix, known as the multipoint
   evaluation problem, requires O((n-k) log(k)) by using Fast Fourier
   Transform, as explained in [fastMatrix-vectorMultiplication].  The
   total complexity of this encoding algorithm is then O(k/(n-k) log^^2
   (k)+ log(k)) operations per repair symbol.

6.3.  Reed-Solomon Decoding Algorithm for the Erasure Channel

   The Reed-Solomon decoding algorithm for the erasure channel allows
   the recovery of the k source elements from any set of k received
   elements.  It is based on the fundamental property of the generator
   matrix which is such that any k*k-submatrix is invertible (see
   [MWS]).  The first step of the decoding consists in extracting the
   k*k submatrix of the generator matrix obtained by considering the
   columns corresponding to the received symbols.  Indeed, since any
   encoding element is obtained by multiplying the source vector by one
   column of the generator matrix, the received vector of k encoding
   symbols can be considered as the result of the multiplication of the
   source vector by a k*k submatrix of the generator matrix.  Since this
   submatrix is invertible, the second step of the algorithm is to
   invert this matrix and to multiply the received vector by the
   obtained matrix to recover the source vector.

6.3.1.  Decoding Complexity

   The decoding algorithm described previously includes the matrix
   inversion and the vector-matrix multiplication.  With the classical
   Gauss-Jordan algorithm, the matrix inversion requires O(k^^3)
   operations and the vector-matrix multiplication is performed in
   O(k^^2) operations.

   This complexity can be improved by considering that the received
   submatrix of Gm is the product between the inverse of a Vandermonde
   matrix (V_(k,k)^^-1) and another Vandermonde matrix (denoted by V'
   which is a submatrix of V_(k,n)).  The decoding can be done by
   multiplying the received vector by V'^^-1 (interpolation problem with



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   complexity O( k log^^2(k)) ) then by V_{k,k} (multipoint evaluation
   with complexity O( k log(k)) ).  The global decoding complexity is
   then O(log^^2(k)) operations per source symbol.

6.4.  Implementation

6.4.1.  Implementation for the Packet Erasure Channel

   In a packet erasure channel, each packet is either received correctly
   or erased.  The location of the erased packets in the sequence of
   packets must be known.  The following specification describes the use
   of Reed-Solomon codes for generating redundant packets from k source
   packets and to recover the source packets from k received packets.

   The k source symbols of a source block are assumed to be composed of
   sz m-bit elements.  Each m-bit element is associated to an element of
   the finite field GF(2^^m) through the polynomial representation
   (Section 6.1).  If some of the source symbols contain less than sz
   elements, they are virtually padded with zero elements (it can be the
   case for the last symbol of the last block of the object).

   The encoding processing produces n-k repair symbols of sz elements by
   encoding each of the sz encoding vectors from the sz source vectors
   (Figure 3).  The j-th source vector is composed of the j-th element
   of each of the source symbols.  Similarly, the j-th encoding vector
   is composed of the j-th element of each encoding symbol.

            ------------     ---------------       -------------------
            | | |    | |    |               |      | | |           | |
            | | |    | |  * |  generator    |  =   | | |           | |
            | | |    | |    |   matrix      |      | | |           | |
            | | |    | |    |    Gm         |      | | |           | |
   source |--------------|  |               |    |---------------------|
   vector | | | |    | | |   ---------------   ->| | | |           | | |
      j   |--------------|                    /  |---------------------|
            | | |    | |                     /     | | |           | |
            | | |    | |                encoding   | | |           | |
            | | |    | |                 vector    | | |           | |
            | | |    | |                    j      | | |           | |
            | | |    | |                           | | |           | |
            | | |    | |                           | | |           | |
            ------------                           -------------------
          k source symbols                         n encoding symbols

   Figure 3: Packet encoding scheme

   An asset of this scheme is that the loss of some of encoding symbols
   produce the same erasure pattern for each of the sz encoding vectors.



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   It follows that the matrix inversion must be done only once and will
   be used by all the sz encoding vectors.  For large sz, this
   complexity cost of the inversion becomes negligible compared to the
   sz matrix-vector multiplications.

   Another asset is that repair symbols can be produced on demand, e.g.,
   depending on the observed erasures on the channel.  The only
   constraint is the finite field size (see Section 6.1)











































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7.  Security Considerations

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















































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8.  Intellectual Property

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













































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9.  IANA Considerations

   Values of FEC Encoding IDs and FEC Instance IDs are subject to IANA
   registration.  For general guidelines on IANA considerations as they
   apply to this document, see [draft-ietf-rmt-fec-bb-revised-03].  This
   document assigns the Fully-Specified FEC Encoding ID XX under the
   ietf:rmt:fec:encoding name-space to "Reed-Solomon Codes".












































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10.  Acknowledgments


















































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11.  References

11.1.  Normative References

   [RFC3452]  Luby, M., "Forward Error Correction (FEC) Building Block",
              RFC 3452, December 2002.

   [RFC3453]  Luby, M., "The Use of Forward Error Correction (FEC) in
              Reliable Multicast", RFC 3453, December 2002.

   [draft-ietf-rmt-fec-bb-revised-03]
              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.

   [rfc2119]  Bradner, S., "Key words for use in RFCs to Indicate
              Requirement Levels", RFC 2119.

11.2.  Informative References

   [MWS]      Mac Williams, F. and N. Sloane, "The Theory of Error
              Correcting Codes", North Holland, 1977 .

   [RS-Rizzo]
              Rizzo, L., "New version of the FEC code (revised 2 july
              98), available at
              http://info.iet.unipi.it/~luigi/vdm98/vdm980702.tgz",
              July 1998.

   [draft-ietf-rmt-bb-fec-ldpc-01]
              Roca, V., Neumann, C., and D. Furodet, "Low Density Parity
              Check (LDPC) Forward Error Correction",
               draft-ietf-rmt-bb-fec-ldpc-01.txt (work in progress),
              March 2006.

   [draft-ietf-rmt-bb-fec-raptor-object-03]
              Luby, M., "Raptor Forward Error Correction Scheme",
              Internet Draft (draft-ietf-rmt-bb-fec-raptor-object-03 :
              work in progress), October 2005.

   [fastMatrix-vectorMultiplication]
              Gohberg, I. and V. Olshevsky, "Fast algorithms with
              preprocessing for matrix-vector multiplication problems",
              Journal of Complexity, pp. 411-427, vol. 10, 1994 .






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Authors' Addresses

   Jerome Lacan
   ENSICA/LAAS-CNRS
   1, place Emile Blouin
   Toulouse  31056
   France

   Email: jerome.lacan@ensica.fr
   URI:


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

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


   Jani Peltotalo
   Tampere University of Technology
   P.O. Box 553 (Korkeakoulunkatu 1)
   Tampere  FIN-33101
   Finland

   Email: jani.peltotalo@tut.fi
   URI:


   Sami Peltotalo
   Tampere University of Technology
   P.O. Box 553 (Korkeakoulunkatu 1)
   Tampere  FIN-33101
   Finland

   Email: sami.peltotalo@tut.fi
   URI:










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Intellectual Property Statement

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   http://www.ietf.org/ipr.

   The IETF invites any interested party to bring to its attention any
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   ietf-ipr@ietf.org.


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Acknowledgment

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   Internet Society.




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