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NWCRG                                                           J. Heide
Internet-Draft                                             Steinwurf Aps
Intended status: Experimental                                     S. Shi
Expires: January 3, 2020                                        K. Fouli
                                                               M. Medard
                                              Code On Network Coding LLC
                                                                V. Chook
                                                            Inmarsat PLC
                                                           July 02, 2019


    Random Linear Network Coding (RLNC)-Based Symbol Representation
                       draft-heide-nwcrg-rlnc-02

Abstract

   This document describes a symbol representation for Random Linear
   Network Coding (RLNC) schemes used for reliable data transfer.
   Specifically, the following features are discussed and incorporated:
   both block RLNC and a sliding window RLNC, varying data frame sizes,
   and one or multiple symbols associated with a single symbol
   representation header.

Requirements Language

   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].

Status of This Memo

   This Internet-Draft is submitted in full conformance with the
   provisions of BCP 78 and BCP 79.

   Internet-Drafts are working documents of the Internet Engineering
   Task Force (IETF).  Note that other groups may also distribute
   working documents as Internet-Drafts.  The list of current Internet-
   Drafts is at https://datatracker.ietf.org/drafts/current/.

   Internet-Drafts are draft documents valid for a maximum of six months
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   time.  It is inappropriate to use Internet-Drafts as reference
   material or to cite them other than as "work in progress."

   This Internet-Draft will expire on January 3, 2020.






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Copyright Notice

   Copyright (c) 2019 IETF Trust and the persons identified as the
   document authors.  All rights reserved.

   This document is subject to BCP 78 and the IETF Trust's Legal
   Provisions Relating to IETF Documents
   (https://trustee.ietf.org/license-info) in effect on the date of
   publication of this document.  Please review these documents
   carefully, as they describe your rights and restrictions with respect
   to this document.  Code Components extracted from this document must
   include Simplified BSD License text as described in Section 4.e of
   the Trust Legal Provisions and are provided without warranty as
   described in the Simplified BSD License.

Table of Contents

   1.  Introduction  . . . . . . . . . . . . . . . . . . . . . . . .   2
   2.  Symbol Representation . . . . . . . . . . . . . . . . . . . .   3
     2.1.  Representation Setup  . . . . . . . . . . . . . . . . . .   4
     2.2.  Field Types and Formats . . . . . . . . . . . . . . . . .   4
     2.3.  Externally Specified Parameters Required  . . . . . . . .   7
     2.4.  Small Encoding Window . . . . . . . . . . . . . . . . . .   7
       2.4.1.  Examples  . . . . . . . . . . . . . . . . . . . . . .   9
     2.5.  Large Encoding Window . . . . . . . . . . . . . . . . . .  10
   3.  Security Considerations . . . . . . . . . . . . . . . . . . .  11
   4.  IANA Considerations . . . . . . . . . . . . . . . . . . . . .  11
   5.  References  . . . . . . . . . . . . . . . . . . . . . . . . .  11
     5.1.  Normative References  . . . . . . . . . . . . . . . . . .  11
     5.2.  Informative References  . . . . . . . . . . . . . . . . .  12
   Authors' Addresses  . . . . . . . . . . . . . . . . . . . . . . .  12

1.  Introduction

   Symbol representation specifies the format of the symbol-carrying
   data unit that is to be used in network coding operations, including
   header format and symbol concatenation.  This document describes a
   symbol representation format intended to be used for Network Coding
   in general, and for Random Linear Network Coding (RLNC) in particular
   [HK03].

   Owing to its dynamic structure, network coding has requirements that
   are distinct from conventional point-to-point codes, leading to a
   highly reconfigurable symbol set.  Consequently, the design choices
   related to symbol representation are particularly important in
   network coding as they have a direct impact on the viability of
   network protocols, topologies, and architecture [RLNC-Background].
   For example, recoding [RLNC-Background] requires the coefficients to



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   be accessible at the recoding nodes.  Hence, architectures and
   protocols requiring recoding must specify coefficient location in
   their symbol representation.

   In addition to providing background on RLNC, [RLNC-Background] argues
   that careful design and specification of a symbol representation is a
   requirement for any viable network coding protocol, architecture, or
   topology.

2.  Symbol Representation

   This section provides a symbol representation design for implementing
   RLNC-based erasure correction schemes.  In the decribed symbol
   representation design, multiple symbols are concatenated and
   associated with a single symbol representation header.

   The symbol representation design is provided for constructing a data
   payload portion of a data packet for a protocol that utilizes a
   generation-based or sliding-window RLNC, where recoding can be used
   at intermediate nodes.  A data packet data payload comprises one or
   more symbol representations.  Each symbol representation in turn
   comprises one or more symbols that can be systematic, coded or
   recoded.  The use of this symbol representation design is not limited
   by transmission schemes.  It can be applied to unicast, multiple-
   unicast, multicast, multi-path, and multi-source settings and the
   like.

   Coding coefficient vectors must be implicitly or explicitly
   transmitted from the sender to the receiver, generally along with the
   coded data for successful decoding of the original data.  One option
   is to attach each coding coefficient vector to the corresponding
   coded symbol as a header, thus also enabling recoding at intermediate
   nodes.  Another option is to attach the current state of a pseudo-
   random generator for generating the coding coefficient vector, to
   reduce the size of the header.  Adding a header to each symbol may
   result in a high overhead when the symbol size is small or when
   generation or sliding window size is large.  Adding a joint header to
   the beginning of each generation may also cause synchronization to be
   re-initiated only at the beginning of each generation instead of
   every symbol.  In what follows, a symbol representation is provided
   that allow for both of these options such that both a general
   representation with coding coefficients and a compact representation
   with a seed for generating the coding coefficients can be used, in
   order to reduce the header overhead.







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2.1.  Representation Setup

   This section specifies a symbol representation that enables both a
   general form with coding vectors attached, and a compact form where a
   seed is attached instead for the first symbol in the symbol
   representation, and where subsequent coding vectors are deduced from
   the first one.  Different maximum generation and window size are
   supported for RLNC encoding, recoding, and decoding.

   To encode over a set of data symbols, a coding vector is first
   generated, comprising a number of finite field elements as specified
   by a GENERATION SIZE or WINDOW SIZE variable.  For a generation based
   code the GENERATION SIZE defines the number of original symbols in
   each generation.  For a window based code the window size specifies
   the maximal number of symbols in the window over which coding can be
   performed.  In the case of systematic codes, systematic symbols
   correspond to unit coding vectors.

   Figure 1 illustrates the general symbol representation design.  Four
   header fields precede the symbol data: TYPE flag (T), SYMBOLS,
   ENCODER RANK, and SEED or CODING COEFFICIENTS.  The TYPE Flag (T)
   indicates if the symbol is systematic, coded, or recoded.  SYMBOLS
   indicates the number of symbols in the SYMBOLS(S) DATA field.
   ENCODER RANK represents the current rank of the encoder, which is the
   number of symbols being linearly combined.  SEED is used to generate
   the coding coefficient vector(s) using a pseudo-random number
   generator, for a compact form of the symbol representation.  The
   CODING COEFFICIENTS field is a list of SYMBOLS number of coding
   vectors used to generate the ensuing SYMBOL(S) DATA.

    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
   +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
   | T |SYMBOLS|   ENCODER RANK    |  SEED or CODING COEFFICIENTS  |
   +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
   |                                                               |
   |                        SYMBOL(S) DATA                         |
   |                              ...                              |
   |                                                               |
   +---------------------------------------------------------------+

   Figure 1: A general symbol representation design.

2.2.  Field Types and Formats

   The TYPE Flag (T) indicates if the symbol is systematic, coded, or
   recoded, and has the following properties:




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   o  2 bits long.

   o  If the TYPE flag is '1', all symbols included in this symbol
      representation are systematic or uncoded, with symbol index
      starting from ENCODER RANK.  This option allows for efficient
      representation of systematic symbols.

   o  If the TYPE is '2', all symbols included in this symbol
      representation are coded, with coding vectors generated using the
      included SEED and the ENCODER RANK.  Consequently, only the first
      ENCODER RANK elements in the coding vector can be non-zero,
      whereas the remaing elements (e.g.  GENERATION SIZE - ENCODER
      RANK) in the coding vector are zeros.  This option allows for
      compact and efficient representation of coded symbols, which may
      also subsequently be recoded.

   o  If the TYPE is '3', all symbols included in this symbol
      representation are either uncoded, coded or recoded.  Each coding
      vector included is composed of GENERATION SIZE or WINDOW SIZE
      coefficients.

   SYMBOLS indicates the number of symbols in the 'Symbol(s) Data'
   field, and has the following properties:

   o  4 bits long.  A maximum number of 15 symbols are concatenated
      within each symbol representation.

   o  The special case of SYMBOLS = 0 indicates that zero symbols are
      included, and consequently the size of SYMBOLS(S) DATA is 0 bytes.
      This can, for example, be used to implement a flush functionality
      or ensure that protocol operations do not stop in certain case for
      purely event-driven protocols.

   ENCODER RANK represents the current rank of the encoder, and has the
   following properties:

   o  MUST be no larger than generation/window size.

   o  If TYPE flag is '1', ENCODER RANK is the symbol index of the first
      data symbol in this symbol representation.

   o  If TYPE flag is '2' or '3', ENCODER RANK is the number of data
      symbols over which coding was performed for all coded symbols in
      this symbol representation.

   o  Coded symbols can be generated before a generation or window is
      filled.  ENCODER RANK describes the number of original symbols
      included in the coded symbol(s).



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   SEED is used to generate the coding coefficient vector(s) using a
   pseudo-random number generator, for a compact form of the symbol
   representation, and has the following properties:

   o  The SEED field is only present when TYPE flag is '2'.  If TYPE is
      '1' or '3', this field is absent.

   o  The pseudo-random generator MUST be seeded with this value and all
      coding coefficient vectors are produced by the same generator.
      For example, if ENCODER RANK is 12, then the coding vector for the
      first symbol in this symbol representation is coefficients 0
      through 11 generated by the pseudo-random generator seeded by
      SEED, and coding vector for the second symbol in this symbol
      representation is coefficients 12 through 23 generated by the
      pseudo-random generator seeded by SEED.  If generation/window size
      is larger than ENCODER RANK, the remaining coefficients in the
      coding vector are zero.

   o  To ensure that SEED can be interpreted correctly at the receiver,
      the same pseudo-random number generator MUST be used by the sender
      and a recoding or receiving node.  Otherwise, more than one SEED
      field would need to be used.

   o  8 bits long.  Thus, 256 different seed values can be served.  One
      SEED is used per symbol representation, each of which can contain
      up to 15 symbols, all derived using the same SEED.  For distinct
      ENCODER RANKs, different coding vectors would be generated from
      the same SEED, since only an ENCODER RANK number of coefficients
      from the random generator is grouped as a coding coefficient
      vector, before progressing to the next coding vector for the next
      symbol in the symbol representation.  Consequently, the maximal
      number of coded symbols that can be generated for a generation
      is |SEED| * |SYMBOLS| * |ENCODER RANK| which in the best case is
      (2^8)*(2^4-1)*(2^10) ~ 2^22, which for all practical
      considerations can be considered as an infinite number of coded
      symbols.  If all coded symbols that can be represented using a
      SEED is exhausted, symbols where the coding vectors is included
      can be sent instead.

   o  In the case where no random number generator is available, or
      where its use is not desired, the coding coefficients can be
      produced by other means, such as functions of the data, state of
      the network, or the like, and transmitted explicitly by setting
      the TYPE flag to '3'

   CODING COEFFICIENTS field is a list of SYMBOLS number of coding
   vectors used to generate the ensuing SYMBOL(S) DATA, and has the
   following properties:



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   o  The CODING COEFFICIENT field is only present when TYPE flag is
      '3'.  If TYPE is '1' or '2', this field is absent.

   o  Each coding vector includes ENCODER RANK number of coding
      coefficients, each coding coefficient having a predetermined field
      size.

2.3.  Externally Specified Parameters Required

   This section specifies parameters that are REQUIRED for the use of
   this symbol representation but which are not included in the symbol
   representation and therefore MUST be communicated by means of some
   outer mecanism.  Typically these parameters will be static throughout
   the instantitaion of a protocol and can therefore be globally
   defined.  Consequently, there is little to gain by incorperating
   these parameters into the representation but conversely it would add
   additional overhead.

   o  Field polynomial, the underlying field over which coding is
      performed.

   o  Pseudo Random Generator, used to generate coding vectors.

   o  Symbol Size, used to divide the original data into symbols.

   o  Generation Size or Window Size, for block and sliding window
      codes, respectively.

   o  Small or large encoding window, this symbol representation
      supports both a small and a large coding window, but the variant
      used is not communicated.

2.4.  Small Encoding Window

   In a first small encoding window symbol representation, ENCODER RANK
   is 10 bits long, and the maximum generation/window size is 2^10.

   Figures 2 to 4 below illustrate systematic, coded, and recoded symbol
   representations within an encoding window of size 2^10.  Systematic
   symbols are uncoded.  Coded symbols are compact in form and comprise
   a seed for coding coefficient generation.  Recoded symbols are
   general in form and comprise the coding coefficient vectors
   explicitly.








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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
   +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
   | 1 |SYMBOLS|   ENCODER RANK    |        SYMBOL(S) DATA         |
   +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
   |                                                               |
   |                   SYMBOLS(S) DATA continued                   |
   |                              ...                              |
   |                                                               |
   +---------------------------------------------------------------+

   Figure 2: A systematic symbol representation.

    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
   +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
   | 2 |SYMBOLS|   ENCODER RANK    |     SEED      |SYMBOL(S) DATA |
   +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
   |                                                               |
   |                    SYMBOL(S) DATA continued                   |
   |                              ...                              |
   |                                                               |
   +---------------------------------------------------------------+

   Figure 3: A compact, coded symbol representation.

    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
   +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
   | 3 |SYMBOLS|   ENCODER RANK    |      CODING COEFFICIENTS      |
   +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
   |                                                               |
   |                 CODING COEFFICIENTS continued                 |
   |                              ...                              |
   |                                                               |
   +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
   |                                                               |
   |                         SYMBOL(S) DATA                        |
   |                               ...                             |
   |                                                               |
   +---------------------------------------------------------------+

   Figure 4: A recoded symbol representation.








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2.4.1.  Examples

   The following examples show different symbol representations for an
   illustrative case where the symbol size is 2 bytes, generation/window
   size is 8, and field size is 2^8.

   Example 1: Three systematic symbols with ID 0, 1 and 2.  As the TYPE
   flag is '1' , SEED/CODING COEFFICIENTS is absent, and ENCODER RANK is
   the symbol index of the first data symbol with ID 0 in this compact
   symbol representation.

    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
   +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
   | 1 |   3   |         0         |   Systematic Symbol 0 Data    |
   +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
   |   Systematic Symbol 1 Data    |   Systematic Symbol 2 Data    |
   +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+

   Figure 5: A symbol representation with 3 systematic, uncoded symbols.

   Example 2: Two coded symbols using a compact representation.  In this
   example, TYPE is '2', the SEED to the pseudo-random number generator
   shared by the sender and receiver is 4.  The coding vector for Symbol
   A is coefficients 0 to 7 generated by the pseudo-random number
   generator, the coding vector for symbol B is coefficients 8 to 15
   generated by the pseudo-random number generator.

    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
   +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
   | 2 |   2   |         8         |       4       | Coded Symbol A
   +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
    Data           |      Coded Symbol B Data      |
   +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+

   Figure 6: A symbol representation with 2 coded symbols.

   Example 3: Two recoded symbols.  Coefficients A0 to A7 constitute the
   coding vector for Symbol A, coefficients B0 to B7 constitute the
   coding vector for symbol B.  In practical implementations, symbol
   sizes are much larger than 2, leading to amortization of the coding
   coefficient overheads.








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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
   +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
   | 3 |   2   |         8         |      A0       |      A1       |
   +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
   |      A2       |      A3       |      A4       |      A5       |
   +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
   |      A6       |      A7       |      B0       |      B1       |
   +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
   |      B2       |      B3       |      B4       |      B5       |
   +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
   |      B6       |      B7       |      Coded Symbol A Data      |
   +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
   |      Coded Symbol B Data      |
   +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+

   Figure 7: A symbol representation with 2 recoded symbols having
   coding coefficients attached.

2.5.  Large Encoding Window

   In a second large encoding window symbol representation, ENCODER RANK
   is 18-bit long, and the maximum generation/window size is 2^18.

   Figures 8 to 10 below illustrate systematic, coded, and recoded
   symbol representations within an encoding window of size 2^18.
   Systematic symbols are uncoded.  Coded symbols are compact in form
   and comprise a seed for coding coefficient generation.  Recoded
   symbols are general in form and comprise the coding coefficient
   vectors explicitly (CODING COEFFICIENTS or CODING COEFFS).

    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
   +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
   | 1 |SYMBOLS|           ENCODER RANK            |SYMBOL(S) DATA |
   +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
   |                                                               |
   |                    SYMBOL(S) DATA continued                   |
   |                              ...                              |
   |                                                               |
   +---------------------------------------------------------------+

   Figure 8: A systematic symbol representation.








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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
   +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
   | 2 |SYMBOLS|           ENCODER RANK            |     SEED      |
   +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
   |                                                               |
   |                         SYMBOL(S) DATA                        |
   |                               ...                             |
   |                                                               |
   +---------------------------------------------------------------+

   Figure 9: A coded symbol representation.

    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
   +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
   | 3 |SYMBOLS|           ENCODER RANK            | CODING COEFFS |
   +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
   |                                                               |
   |                 CODING COEFFICIENTS continued                 |
   |                               ...                             |
   |                                                               |
   +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
   |                                                               |
   |                         SYMBOL(S) DATA                        |
   |                               ...                             |
   |                                                               |
   +---------------------------------------------------------------+

   Figure 10: A recoded symbol representation.

3.  Security Considerations

   This document does not present new security considerations.

4.  IANA Considerations

   This document has no actions for IANA.

5.  References

5.1.  Normative References

   [RFC2119]  Bradner, S., "Key words for use in RFCs to Indicate
              Requirement Levels", BCP 14, RFC 2119,
              DOI 10.17487/RFC2119, March 1997,
              <https://www.rfc-editor.org/info/rfc2119>.




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   [RLNC-Background]
              Heide, J., Shi, S., Fouli, K., Medard, M., and V. Chook,
              "Random Linear Network Coding (RLNC): Background and
              Practical Considerations", February 2018,
              <https://datatracker.ietf.org/doc/
              draft-heide-nwcrg-rlnc-background/>.

5.2.  Informative References

   [HK03]     Ho, T., Koetter, R., Medard, M., Karger, D., and M.
              Effros, "The Benefits of Coding over Routing in a
              Randomized Setting", July 2003,
              <http://ieeexplore.ieee.org/document/1228459/>.

Authors' Addresses

   Janus Heide
   Steinwurf Aps
   Aalborg
   Denmark

   Email: janus@steinwurf.com


   Shirley Shi
   Code On Network Coding LLC
   Cambridge
   USA

   Email: xshi@alum.mit.edu


   Kerim Fouli
   Code On Network Coding LLC
   Cambridge
   USA

   Email: fouli@codeontechnologies.com


   Muriel Medard
   Code On Network Coding LLC
   Cambridge
   USA

   Email: muriel.medard@codeontechnologies.com





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   Vince Chook
   Inmarsat PLC
   London
   United Kingdom

   Email: Vince.Chook@inmarsat.com













































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