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Versions: 00 01 02 03 04 05 RFC 5510

Reliable Multicast Transport                                    J. Lacan
Internet-Draft                                          ENSICA/LAAS-CNRS
Intended status: Experimental                                    V. Roca
Expires: June 25, 2007                                             INRIA
                                                            J. Peltotalo
                                                            S. Peltotalo
                                        Tampere University of Technology
                                                       December 22, 2006


              Reed-Solomon Forward Error Correction (FEC)
                    draft-ietf-rmt-bb-fec-rs-02.txt

Status of this Memo

   By submitting this Internet-Draft, each author represents that any
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   This Internet-Draft will expire on June 25, 2007.

Copyright Notice

   Copyright (C) The IETF Trust (2006).










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Abstract

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

   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
   implementation from Luigi Rizzo.


Table of Contents

   1.  Introduction . . . . . . . . . . . . . . . . . . . . . . . . .  4
   2.  Terminology  . . . . . . . . . . . . . . . . . . . . . . . . .  5
   3.  Definitions Notations and Abbreviations  . . . . . . . . . . .  6
     3.1.  Definitions  . . . . . . . . . . . . . . . . . . . . . . .  6
     3.2.  Notations  . . . . . . . . . . . . . . . . . . . . . . . .  6
     3.3.  Abbreviations  . . . . . . . . . . . . . . . . . . . . . .  7
   4.  Formats and Codes  . . . . . . . . . . . . . . . . . . . . . .  8
     4.1.  FEC Payload ID . . . . . . . . . . . . . . . . . . . . . .  8
     4.2.  FEC Object Transmission Information  . . . . . . . . . . .  9
       4.2.1.  Mandatory Elements . . . . . . . . . . . . . . . . . .  9
       4.2.2.  Common Elements  . . . . . . . . . . . . . . . . . . .  9
       4.2.3.  Scheme-Specific Elements . . . . . . . . . . . . . . .  9
       4.2.4.  Encoding Format  . . . . . . . . . . . . . . . . . . . 10
   5.  Procedures . . . . . . . . . . . . . . . . . . . . . . . . . . 12
     5.1.  Determining the Maximum Source Block Length (B)  . . . . . 12
     5.2.  Determining the Number of Encoding Symbols of a Block  . . 12
   6.  Reed-Solomon Codes Specification for the Erasure Channel . . . 14
     6.1.  Finite Field . . . . . . . . . . . . . . . . . . . . . . . 14
     6.2.  Reed-Solomon Encoding Algorithm  . . . . . . . . . . . . . 15
       6.2.1.  Encoding Principles  . . . . . . . . . . . . . . . . . 15
       6.2.2.  Encoding Complexity  . . . . . . . . . . . . . . . . . 16
     6.3.  Reed-Solomon Decoding Algorithm  . . . . . . . . . . . . . 16
       6.3.1.  Decoding Principles  . . . . . . . . . . . . . . . . . 16
       6.3.2.  Decoding Complexity  . . . . . . . . . . . . . . . . . 17
     6.4.  Implementation for the Packet Erasure Channel  . . . . . . 17
   7.  Security Considerations  . . . . . . . . . . . . . . . . . . . 19
   8.  IANA Considerations  . . . . . . . . . . . . . . . . . . . . . 20
   9.  Acknowledgments  . . . . . . . . . . . . . . . . . . . . . . . 21
   10. References . . . . . . . . . . . . . . . . . . . . . . . . . . 22
     10.1. Normative References . . . . . . . . . . . . . . . . . . . 22
     10.2. Informative References . . . . . . . . . . . . . . . . . . 22
   Authors' Addresses . . . . . . . . . . . . . . . . . . . . . . . . 24



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   Intellectual Property and Copyright Statements . . . . . . . . . . 25


















































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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 [2] document describes a general framework to use
   FEC in Content Delivery Protocols (CDP).  The companion document [3]
   describes some applications of FEC codes for content delivery.

   Recent FEC schemes like [6] and [7] proposed erasure codes based on
   sparse graphs/matrices.  These codes are efficient in terms of
   processing but not optimal in terms of correction capabilities when
   dealing with "small" objects.

   The FEC scheme described in this document belongs to the class of
   Maximum Distance Separable codes that are optimal in terms of erasure
   correction capability.  In others words, it enables a receiver to
   recover the k source symbols from any set of exactly k encoding
   symbols.  Even if the encoding/decoding complexity is larger than
   that of [6] or [7], this family of codes is very useful.

   Many applications dealing with content transmission or content
   storage already rely on packet-based Reed-Solomon codes.  In
   particular, many of them use the Reed-Solomon codec of Luigi Rizzo
   [4].  The goal of the present document to specify an implementation
   of Reed-Solomon codes that is compatible with this codec.


























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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 [1].














































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3.  Definitions Notations and Abbreviations

3.1.  Definitions

   This document uses the same terms and definitions as those specified
   in [2].  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 symbol that is not a source symbol.

      Systematic code: FEC 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 block 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.  Therefore: n = k + n_r.

      max_n denotes the 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.

      S denotes the symbol size in units of m-bit elements.  When m = 8,
      then S and E are equal.

      m defines the length of the elements in the finite field, in bits.

      q defines the number of elements in the finite field.  We have: q
      = 2^^m in this specification.

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

      GM denotes the Generator Matrix of a Reed-Solomon code.

      rate denotes the "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 stands for Encoding Symbol ID.

      FEC OTI stands for FEC Object Transmission Information.

      RS stands for Reed-Solomon.

      MDS stands for Maximum Distance Separable code.

      GF(q) denotes a finite field (A.K.A. Galois Field) with q
      elements.  We assume that q = 2^^m in this document.











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4.  Formats and Codes

4.1.  FEC Payload ID

   The FEC Payload ID is composed of the Source Block Number and the
   Encoding Symbol ID.  The length of these two fields depends on the
   parameter m (which is transmitted in the FEC OTI) as follows :

   o  The Source Block Number (field of size 32-m bits) identifies from
      which source block of the object the encoding symbol(s) in the
      payload is (are) generated.  There are a maximum of 2^^(32-m)
      blocks per object.

   o  The Encoding Symbol ID (field of size m bits) identifies which
      specific encoding symbol(s) generated from the source block
      is(are) carried in the packet payload.  There are a maximum of
      2^^m 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 source or repair 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.

   The format of the FEC Payload ID for m = 8 and m = 16 is illustrated
   in Figure 1 and Figure 2 respectively.

    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 (32-8=24 bits)        | Enc. Symb. ID |
   +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+

       Figure 1: FEC Payload ID encoding format for m = 8 (default)


    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
   +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
   | Src Block Nb (32-16=16 bits)  |  Enc. Symbol ID (m=16 bits)   |
   +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+

            Figure 2: FEC Payload ID encoding format for m = 16






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4.2.  FEC Object Transmission Information

4.2.1.  Mandatory Elements

   o  FEC Encoding ID: the Fully-Specified FEC Scheme described in this
      document uses FEC Encoding ID 2.

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 :

         max_transfer_length = 2^^(32-m) * B * E

      For instance, for m = 8, for 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 approximately equal to
      2^^42 bytes (i.e. 4 Tera Bytes).  Similarly, for m = 16, for B =
      2^^16 - 1 and if E = 1024 bytes, then the maximum transfer length
      is also approximately equal to 2^^42 bytes.  For larger objects,
      another FEC scheme, with a larger Source Block Number field in the
      FEC Payload ID, could be defined.  Another solution consists in
      fragmenting large objects into smaller objects, each of them
      complying with the above limits.

   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:






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   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 parameter, m: The m parameter is the length of the
      finite field elements, in bits.  It also characterizes the number
      of elements in the finite field: q = 2^^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 one encoding format
   or the other 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 (e.g. within the ALC [8] or NORM [9] protocols).

    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 3: 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 [10], the following XML attributes must be described
   for the associated object:

   o  FEC-OTI-FEC-Encoding-ID

   o  FEC-OTI-Transfer-Length (L)




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   o  FEC-OTI-Encoding-Symbol-Length (E)

   o  FEC-OTI-Maximum-Source-Block-Length (B)

   o  FEC-OTI-Max-Number-of-Encoding-Symbols (max_n)

   o  FEC-OTI-Scheme-Specific-Info

   The FEC-OTI-Scheme-Specific-Info contains the string resulting from
   the Base64 encoding (in the XML Schema xs:base64Binary sense) of the
   following value:

    0                   1
    0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5
   +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
   |       m       |       G       |
   +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+

    Figure 4: FEC OTI Scheme Specific Information to be Included in the
                               FDT Instance

   When no m parameter is to be carried in the FEC OTI, the m field is
   set to 0 (which is not a valid seed value).  Otherwise the m field
   contains a valid value as explained in Section 4.2.3.  Similarly,
   when no G parameter is to be carried in the FEC OTI, the G field is
   set to 0 (which is not a valid seed value).  Otherwise the G field
   contains a valid value as explained in Section 4.2.3.  When neither m
   nor G are to be carried in the FEC OTI, then the sender simply omits
   the FEC-OTI-Scheme-Specific-Info attribute.

   After Base64 encoding, the 2 bytes of the FEC OTI Scheme Specific
   Information are transformed into a string of 4 printable characters
   (in the 64-character alphabet) and added to the FEC-OTI-Scheme-
   Specific-Info attribute.

















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5.  Procedures

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 which is equal to: q - 1 = 2^^m - 1.  Note
   that q - 1 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 = 255

   Additionally, a codec MAY impose other limitations on the maximum
   block size.  Yet it is not expected that such limits exist when using
   the default m = 8 value.  This decision MUST 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
      block partitioning algorithm.

      rate: FEC code rate, which is given by the user (e.g. when
      starting a FLUTE sending application).  It is expressed as a
      floating point value.

   Output:

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



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      n: Number of encoding symbols generated for this source block

   Algorithm:

      max_n = floor(B / rate);

      if (max_n > 2^^m - 1) 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 block partitioning 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 n value used by the sender, for instance for memory
   management optimizations.  To that purpose, the FEC OTI carries all
   the parameters needed for a receiver to execute the above algorithm.
















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6.  Reed-Solomon Codes Specification for the Erasure Channel

   Reed-Solomon (RS) codes are linear block codes.  They also belong to
   the class of MDS codes.  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 upper bounded by q - 1.  The
   implementation described in this document is based on a generator
   matrix built from a Vandermonde matrix put into systematic form.

   Section 6.1 to Section 6.3 specify the [n,k]-RS codes when applied to
   m-bit elements, and Section 6.4 the use of [n,k]-RS codes when
   applied to symbols composed of several m-bit elements, which is the
   target of this specification.

6.1.  Finite Field

   A finite field GF(q) is defined as a finite set of q elements which
   has 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
   lower or equal than m-1.  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
   polynomial over GF(2) of degree m with coefficients in GF(2).  Note
   that all the roots of this polynomial are in GF(2^^m) but not in
   GF(2).

   A finite field GF(2^^m) is completely characterized by the
   irreducible polynomial.  The following polynomials are chosen 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)





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      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)

      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)

   In order to facilitate the implementation, these polynomials are also
   primitive.  This means that any element of GF(2^^m) can be expressed
   as a power of a given root of this polynomial.  These polynomials are
   also chosen so that they contain the minimum number of monomials.

6.2.  Reed-Solomon Encoding Algorithm

6.2.1.  Encoding Principles

   Let s = (s_0, ..., s_{k-1}) be a source vector of k elements over
   GF(2^^m).  Let e = (e_0, ..., e_{n-1}) be the corresponding encoding
   vector of n elements over GF(2^^m).  Being a linear code, encoding is
   performed 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
   RS code.

   Let us consider that: n = 2^^m - 1 and: 0 < k <= n.  Let us denote by
   alpha the root of the primitive polynomial of degree m chosen in the
   list of Section 6.1 for the corresponding value of m.  Let us
   consider a Vandermonde matrix of k rows and n columns, denoted by
   V_{k,n}, and built as follows: the {i, j} entry of V_{k,n} is v_{i,j}
   = alpha^^(i*j), where 0 <= i <= k - 1 and 0 <= j <= n - 1.  This
   matrix generates a MDS code.  However, this MDS code is not
   systematic, which is a problem for many networking applications.  To
   obtain a systematic matrix (and code), the simplest solution consists
   in considering the matrix V_{k,k} formed by the first k columns of



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   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, meaning that the first k encoding
   elements are equal to source elements.  Besides the associated code
   keeps the MDS property.

   Therefore, the generator matrix of the code considered in this
   document is:

      GM = (V_{k,k}^^-1) * V_{k,n}

   Note that, in practice, the [n,k]-RS code can be shortened to a
   [n',k]-RS code, where k <= n' < n, by considering the sub-matrix
   formed by the n' first columns of GM.

6.2.2.  Encoding Complexity

   Encoding can be performed by first pre-computing GM and by
   multiplying the source vector (k elements) by GM (k rows and n
   columns).  The complexity of the pre-computation of the generator
   matrix can be estimated as the complexity of the multiplication of
   the inverse of a Vandermonde matrix by n-k vectors (i.e. the last n-k
   columns of V_{k,n}).  Since the complexity of the inverse of a k*k-
   Vandermonde matrix by a vector is O(k * log^^2(k)), the generator
   matrix can be computed in 0((n-k)* k * log^^2(k)) operations.  When
   the genarator matrix is pre-computed, the encoding needs k operations
   per repair element (vector-matrix multiplication).

   Encoding can also be performed by first computing the product s *
   V_{k,k}^^-1 and then by multiplying the result with 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 [11].  The total complexity of this
   encoding algorithm is then O((k/(n-k)) * log^^2(k) + log(k))
   operations per repair element.

6.3.  Reed-Solomon Decoding Algorithm

6.3.1.  Decoding Principles

   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 [5]).
   The first step of the decoding consists in extracting the k*k
   submatrix of the generator matrix obtained by considering the columns



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   corresponding to the received elements.  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 elements 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.

6.3.2.  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
   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 element.

6.4.  Implementation for the Packet Erasure Channel

   In a packet erasure channel, each packet (and symbol(s) since packets
   contain G >= 1 symbols) is either correctly received or erased.  The
   location of the erased symbols in the sequence of symbols must be
   known.  The following specification describes the use of Reed-Solomon
   codes for generating redundant symbols from the k source symbols and
   to recover the source symbols from any set of k received symbols.

   The k source symbols of a source block are assumed to be composed of
   S 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 S
   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 process produces n-k repair symbols of size S m-bit
   elements, the k source symbols being also part of the n encoding
   symbols (Figure 5).  These repair symbols are created m-bit element
   per m-bit element.  More specifically, 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



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   of each encoding symbol.

                                    n
            +-+-+----+-+    +---------------+      +-+-+-----------+-+
      0     | | |    | |    |               |      | | |           | |
            | | |    | |  * |   generator   |  =   | | |           | |
            | | |    | |    |    matrix     |      | | |           | |
            | | |    | |    |      GM       |      | | |           | |
   source +--------------+  |   (k x n)     |    +---------------------+
   vector | | | |    | | |  +---------------+  ->| | | |           | | |
      j   +--------------+                    /  +---------------------+
            | | |    | |                     /     | | |           | |
            | | |    | |                encoding   | | |           | |
            | | |    | |                 vector    | | |           | |
            | | |    | |                    j      | | |           | |
            | | |    | |                           | | |           | |
     S-1    | | |    | |                           | | |           | |
            +-+-+----+-+                           +-+-+-----------+-+
          k source symbols                         n encoding symbols
                                                    (source + repair)

                     Figure 5: Packet encoding scheme

   An asset of this scheme is that the loss of some encoding symbols
   produces the same erasure pattern for each of the S encoding vectors.
   It follows that the matrix inversion must be done only once and will
   be used by all the S encoding vectors.  For large S, this matrix
   inversion cost becomes negligible in front of the S matrix-vector
   multiplications.

   Another asset is that the n-k repair symbols can be produced on
   demand.  For instance, a sender can start by producing a limited
   number of repair symbols and later on, depending on the observed
   erasures on the channel, decide to produce additional repair symbols,
   up to the n-k upper limit.  Indeed, to produce the repair symbol e_j,
   where k <= j < n, it is sufficient to multiply the S source vectors
   with column j of GM.














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7.  Security Considerations

   The security considerations for this document are the same as that of
   [2].















































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8.  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 [2].  This document assigns the Fully-
   Specified FEC Encoding ID 2 under the ietf:rmt:fec:encoding name-
   space to "Reed-Solomon Codes".












































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9.  Acknowledgments

   The authors want to thank Luigi Rizzo for comments on the subject and
   for the design of the reference Reed-Solomon codec.















































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10.  References

10.1.  Normative References

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

   [2]   Watson, M., Luby, M., and L. Vicisano, "Forward Error
         Correction (FEC) Building Block",
          draft-ietf-rmt-fec-bb-revised-04.txt (work in progress),
         September 2006.

   [3]   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.

10.2.  Informative References

   [4]   Rizzo, L., "Reed-Solomon FEC codec (revised version of July
         2nd, 1998), available at
         http://info.iet.unipi.it/~luigi/vdm98/vdm980702.tgz",
         July 1998.

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

   [6]   Luby, M., Shokrollahi, A., Watson, M., and T. Stockhammer,
         "Raptor Forward Error Correction Scheme", Internet
         Draft draft-ietf-rmt-bb-fec-raptor-object-04 (work in
         progress), June 2006.

   [7]   Roca, V., Neumann, C., and D. Furodet, "Low Density Parity
         Check (LDPC) Forward Error Correction",
          draft-ietf-rmt-bb-fec-ldpc-04.txt (work in progress),
         December 2006.

   [8]   Luby, M., Watson, M., and L. Vicisano, "Asynchronous Layered
         Coding (ALC) Protocol Instantiation",
          draft-ietf-rmt-pi-alc-revised-03.txt (work in progress),
         April 2006.

   [9]   Adamson, B., Bormann, C., Handley, M., and J. Macker,
         "Negative-acknowledgment (NACK)-Oriented Reliable Multicast
         (NORM) Protocol",  draft-ietf-rmt-pi-norm-revised-03.txt (work
         in progress), September 2006.

   [10]  Paila, T., Walsh, R., Luby, M., Lehtonen, R., and V. Roca,
         "FLUTE - File Delivery over Unidirectional Transport",



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          draft-ietf-rmt-flute-revised-02.txt (work in progress),
         August 2006.

   [11]  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:   http://dmi.ensica.fr/auteur.php3?id_auteur=5


   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:   http://atm.tut.fi/mad


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

   Email: sami.peltotalo@tut.fi
   URI:   http://atm.tut.fi/mad










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Full Copyright Statement

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