Reliable Multicast Transport                                    J. Lacan
Internet-Draft                                          ENSICA/LAAS-CNRS
Expires: August 27, December 25, 2006                                       V. Roca
                                                                   INRIA
                                                            J. Peltotalo
                                                            S. Peltotalo
                                        Tampere University of Technology
                                                       February
                                                           June 23, 2006

              Reed-Solomon Forward Error Correction (FEC)
                    draft-ietf-rmt-bb-fec-rs-00.txt
                    draft-ietf-rmt-bb-fec-rs-01.txt

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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 the
   reliable delivery of data objects on the packet erasure channel.

   The

   Reed-Solomon codes belong to the class of Maximum Distance Separable
   (MDS) codes, i.e, 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 ID . . . . . . . . . . . . . . . . . . . . . .  7
     4.2.  FEC Object Transmission Information  . . . . . . . . . . .  7  8
       4.2.1.  Mandatory Elements . . . . . . . . . . . . . . . . . .  7  8
       4.2.2.  Common Elements  . . . . . . . . . . . . . . . . . . .  7  8
       4.2.3.  Scheme-Specific Elements . . . . . . . . . . . . . . .  8
       4.2.4.  Encoding Format  . . . . . . . . . . . . . . . . . . .  8  9
   5.  Procedures . . . . . . . . . . . . . . . . . . . . . . . . . . 10 11
     5.1.  Determining the Maximum Source Block Length (B)  . . . . . 10 11
     5.2.  Determining the Number of Encoding Symbols of a Block  . . 10 11
   6.  Reed-Solomon Codes Specification for the Erasure Channel . . . 13
     6.1.  Finite Field . . . . . . . . . . . . . . . . . . . 12
     6.1.  Finite field . . . . . . 13
     6.2.  Reed-Solomon Encoding Algorithm  . . . . . . . . . . . . . 14
       6.2.1.  Encoding Principles  . . . . 12
     6.2.  Reed-Solomon Encoding Algorithm . . . . . . . . . . . . . 13
       6.2.1. 14
       6.2.2.  Encoding Complexity  . . . . . . . . . . . . . . . . . 14 15
     6.3.  Reed-Solomon Decoding Algorithm for the Erasure Channel  . 14
       6.3.1.  Decoding Complexity  . . . . . . . . .  . . . . . . . . 14
     6.4.  Implementation . . . . . 15
       6.3.1.  Decoding Principles  . . . . . . . . . . . . . . . . . 15
       6.4.1.  Implementation for the Packet Erasure Channel  .
       6.3.2.  Decoding Complexity  . . . 15
   7.  Security Considerations . . . . . . . . . . . . . . 16
     6.4.  Implementation for the Packet Erasure Channel  . . . . . 17
   8.  Intellectual Property . 16
   7.  Security Considerations  . . . . . . . . . . . . . . . . . . . 18
   9.
   8.  IANA Considerations  . . . . . . . . . . . . . . . . . . . . . 19
   10.
   9.  Acknowledgments  . . . . . . . . . . . . . . . . . . . . . . . 20
   11.
   10. References . . . . . . . . . . . . . . . . . . . . . . . . . . 21
     11.1.
     10.1. Normative References . . . . . . . . . . . . . . . . . . . 21
     11.2.
     10.2. Informative References . . . . . . . . . . . . . . . . . . 21
   Authors' Addresses . . . . . . . . . . . . . . . . . . . . . . . . 22 23
   Intellectual Property and Copyright Statements . . . . . . . . . . 23 24

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 [2] document describes a general framework to use
   FEC in Content Delivery Protocols (CDP).  The companion document [RFC3453] [3]
   describes some applications of FEC codes for content delivery.

   Recent FEC schemes like [draft-ietf-rmt-bb-fec-raptor-object-03] [6] and
   [draft-ietf-rmt-bb-fec-ldpc-01] [7] proposed erasure codes based on
   sparse graphs/matrices.  These codes are efficient in terms of CPU
   processing but not optimal in terms of correction capabilities, at least for
   small capabilities when
   dealing with "small" objects.

   The FEC scheme presented described in this document belongs to the class of
   Maximum-Distance
   Maximum Distance Separable codes, i.e., it is codes that are optimal in terms of erasure
   correction capability.  In others words, it enables the 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 [draft-ietf-rmt-bb-fec-raptor-object-03] [6] or
   [draft-ietf-rmt-bb-fec-ldpc-01], [7], this family of codes is very useful
   for applications sending small objects (e.g., for video and audio
   streaming).

   Indeed many useful.

   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 use the implementation Reed-Solomon codec of Luigi Rizzo [RS-Rizzo].  This latter is compatible with the Reed-Solomon
   codes specification
   [4].  The goal of the present document. document to specify an implementation
   of Reed-Solomon codes that is compatible with this codec.

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]. [1].

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]. [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 symbols symbol that are is not a source symbols. symbol.

      Systematic code: a FEC code in which the source symbols are part of
      the encoding symbols 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,
      together within the same packet, and whose relationships to the
      source object block can be derived from a single Encoding Symbol ID.

      Source Packet Packet: a data packet containing only source symbols.

      Repair Packet 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 bytes.

      k denotes the number of source symbols in a source block block.

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

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

      max_n Maximum Number denotes the maximum number of Encoding Symbols encoding symbols generated for
      any source
      block block.

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

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

      E denotes the encoding symbol length in bytes

      sz bytes.

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

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

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

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

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

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

      a^^b denotes a raised to the power b b.

      a^^-1 denotes the inverse of a a.

      I_k denotes the k*k identity matrix matrix.

3.3.  Abbreviations

   This document uses the following abbreviations:

      ESI stands for Encoding Symbol ID ID.

      FEC OTI stands for FEC Object Transmission Information.

      RS Reed-Solomon stands for Reed-Solomon.

      MDS stands for Maximum Distance Separable code code.

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

4.  Formats and Codes

4.1.  FEC Payload IDs ID

   The FEC Payload ID is composed of the Source Block Number and the
   Encoding Symbol ID: 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 (16 (32-m bit field) identifies from which
      source block of the object the encoding symbol(s) in the payload
      is (are) generated.  There is are a maximum of 2^^16 2^^(32-m) blocks per
      object.

   o  The Encoding Symbol ID (16 (m bit field) identifies which specific
      encoding symbol symbol(s) generated from the source block is is(are) carried
      in the packet payload.  There is are a maximum of 2^^16 2^^m encoding
      symbols per block.  The first k values (0 to k-1) 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 (16 (32-8=24 bits)        |  Encoding Symbol Enc. Symb. ID (16 bits) |
   +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+

   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 Encoding Payload ID XX encoding format for  m = 16

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 uses FEC Encoding ID XX.

4.2.2.  Common Elements

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

         max_transfer_length = 2^^16 2^^(32-m) * B * E

      For instance, if for m = 8, for B = 2^^8-1 2^^8 - 1 (because the codec
      operates on a finite field with 2^^8 elements), elements) and if E = 1024
      bytes, then the maximum transfer length is 2^^34 approximately equal to
      2^^42 bytes (i.e., a bit more than 17
      Giga (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, it is expected that other FEC
      codes (e.g., LDPC codes) or
      another Reed-Solomon FEC Scheme scheme, with a larger Source Block Number field in the
      FEC Payload ID ID, could be used. 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:

   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 is the length of the
      finite field size equal to elements, in bits.  It also characterizes the number
      of elements in the finite field: q = p^^m 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 or how these one encoding formats 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. 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 2: 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, session [10], the following XML elements attributes must be described
   for the associated object:

   o  FEC-OTI-Transfer-length  FEC-OTI-Transfer-Length (L)

   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-Number-Encoding-Symbols-per-Group  FEC-OTI-Number-of-Encoding-Symbols-per-Group (optional) (G)

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

   When no finite field size G parameter is to be carried in the FEC OTI, the sender
   simply omits the FEC-OTI-Finite-Field-Size-Parameter
   element. FEC-OTI-Number-of-Encoding-Symbols-per-Group
   attribute.  When no Finite Field parameter is to be carried in the
   FEC OTI, the sender simply omits the FEC-OTI-Finite-Field-Parameter
   attribute.

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, field which is equal to: q - 1 = 2^^m-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 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 (default). value.  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
      block partitioning algorithm.

      rate: FEC code rate, which is given by the user (e.g., (e.g. when
      starting a FLUTE sending application) for a given use case. application).  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:

      max_n = floor(B / rate);

      if (max_n >= 2^^m) > 2^^m - 1) then return an error ("invalid code rate");

      n = floor(k * max_n / B);

   AT A RECEIVER:

   Input:

      B

      B: Extracted from the received FEC OTI

      max_n

      max_n: Extracted from the received FEC OTI

      k

      k: Given by the source blocking 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 value of n used by the sender, for instance for memory
   management optimizations.  To that purpose, the FEC OTI carries all
   the parameters needed
   information is carried in for a receiver to execute the FEC OTI. above algorithm.

6.  Reed-Solomon Codes Specification for the Erasure Channel

   Reed-Solomon (RS) codes form a special class of are linear block codes,
   which offer maximum erasure correction capability. 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 upperbounded upper bounded by q-1. 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 Field

   A finite field GF(q) is defined as a finite set of q elements which
   have
   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., (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 over GF(2) of
   degree m with coefficients in GF(2).

   Since a  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, we propose the 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)

      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)

   For implementation issues,

   In order to facilitate the implementation, these polynomials are also primitive
   elements of GF(2^^m), i.e.,
   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

   The encoding algorithm produces a vector of n encoding elements
   e=(e_0,

6.2.1.  Encoding Principles

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

   The  Being a linear codes can be encoded code, encoding is
   performed by multiplying the source vector by a generator matrix Gm matrix, GM,
   of k rows and n columns over GF(2^^m).  Thus:

      e = s * Gm. 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
   alpha a the primitive element of GF(2^^m) (i.e., any element chosen in the list of GF(2^^m)
   can be expressed as a power
   Section 6.1 for the corresponding value of alpha).

   The generator matrix is build from m.  Let us consider a k*n-Vandermonde
   Vandermonde matrix of k rows and n columns, denoted by V_{k,n}. The entries V_{k,n}, and
   built as follows: the {i, j} entry of V_{k,n} are is v_{i,j} =
   alpha^^(i*j), where 0 <= i <= k - 1 and 0 <= j <= n - 1.  This matrix
   generates a MDS code.  However, it this MDS code is not systematic as required by most of network systematic,
   which is a problem for many networking applications.  To obtain a
   systematic matrix, matrix (and code), the simplest solution
   is to consider consists in
   considering the matrix V_{k,k} formed by the first k columns of
   V_{k,n}
   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 columns, meaning that the first k encoding
   elements are equal to source elements.  Besides the associated code
   keeps the MDS code.

   The product V_{k,k}^^-1 * V_{k,n} is denoted by Gm and is property.

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

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

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

6.2.1. GM.

6.2.2.  Encoding Complexity

   The encoding process

   Encoding can be done performed by first pre-computing G GM and by
   multiplying the source vector (k elements) by Gm. GM (k rows and n
   columns).  The complexity is one of the pre-computation of the generator
   matrix can be estimated as the complexity of the multiplication s*Gm, where Gm is of
   the inverse of a k*(n-k) matrix. The Vandermonde matrix by n-k vectors (i.e. the last n-k
   columns of V_{k,n}).  Since the complexity of the vector-matrix multiplication inverse of a k*k-
   Vandermonde matrix by a vector is then k*(n-k) (i.e., 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).

   The encoding element (vector-matrix multiplication).

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

6.3.  Reed-Solomon Decoding Algorithm for the Erasure Channel

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
   [MWS]). [5]).
   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. 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
   symbols 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.

6.3.1.

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 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)) ). O(k * log(k))).  The global decoding complexity is
   then O(log^^2(k)) operations per source symbol. element.

6.4.  Implementation

6.4.1.  Implementation for the Packet Erasure Channel

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

   The k source symbols of a source block are assumed to be composed of
   sz
   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 sz 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 processing process produces n-k repair symbols of sz elements by
   encoding each size S m-bit
   elements, the k source symbols being also part of the sz n encoding vectors from the sz source vectors
   symbols (Figure 3).  The 4).  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
   of each encoding symbol.

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

   Figure 3: 4: Packet encoding scheme

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

   Another asset is that the n-k repair symbols can be produced on demand, e.g.,
   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.  The only
   constraint 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 finite field size (see Section 6.1) S source vectors
   with column j of GM.

7.  Security Considerations

   The security considerations for this document are the same as that of
   [RFC3452].
   [2].

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.

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

10.

9.  Acknowledgments

11.

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

10.  References

11.1.

10.1.  Normative References

   [RFC3452]

   [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",
              RFC 3452, December 2002.

   [RFC3453]  draft-ietf-rmt-fec-bb-revised-03.txt
        (work in progress), January 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.

   [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.

10.2.  Informative References

   [MWS]

   [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 .

   [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]

   [6]   Luby, M., Shokrollahi, A., Watson, M., and T. Stockhammer,
         "Raptor Forward Error Correction Scheme", Internet
         Draft draft-ietf-rmt-bb-fec-raptor-object-03 (work in
         progress), October 2005.

   [7]   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]

   [8]   Luby, M., "Raptor Forward Error Correction Scheme",
              Internet Draft (draft-ietf-rmt-bb-fec-raptor-object-03 :
              work Watson, M., and L. Vicisano, "Asynchronous Layered
         Coding (ALC) Protocol Instantiation",
          draft-ietf-rmt-pi-alc-revised-03.txt (work in progress), October 2005.

   [fastMatrix-vectorMultiplication]
         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-01.txt (work
         in progress), March 2006.

   [10]  Paila, T., Walsh, R., Luby, M., Lehtonen, R., and V. Roca,
         "FLUTE - File Delivery over Unidirectional Transport",
          draft-ietf-rmt-flute-revised-01.txt (work in progress),
         January 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 .

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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