Reliable Multicast Transport J. Lacan
Internet-Draft ENSICA/LAAS-CNRS
Expires: December 25, 2006 V. Roca
INRIA
J. Peltotalo
S. Peltotalo
Tampere University of Technology
June 23, 2006
Reed-Solomon Forward Error Correction (FEC)
draft-ietf-rmt-bb-fec-rs-01.txt
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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.
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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 . . . . . . . . . . . . . . . . . . . . . . . . . 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 ID . . . . . . . . . . . . . . . . . . . . . . 7
4.2. FEC Object Transmission Information . . . . . . . . . . . 8
4.2.1. Mandatory Elements . . . . . . . . . . . . . . . . . . 8
4.2.2. Common Elements . . . . . . . . . . . . . . . . . . . 8
4.2.3. Scheme-Specific Elements . . . . . . . . . . . . . . . 8
4.2.4. Encoding Format . . . . . . . . . . . . . . . . . . . 9
5. Procedures . . . . . . . . . . . . . . . . . . . . . . . . . . 11
5.1. Determining the Maximum Source Block Length (B) . . . . . 11
5.2. Determining the Number of Encoding Symbols of a Block . . 11
6. Reed-Solomon Codes Specification for the Erasure Channel . . . 13
6.1. Finite Field . . . . . . . . . . . . . . . . . . . . . . . 13
6.2. Reed-Solomon Encoding Algorithm . . . . . . . . . . . . . 14
6.2.1. Encoding Principles . . . . . . . . . . . . . . . . . 14
6.2.2. Encoding Complexity . . . . . . . . . . . . . . . . . 15
6.3. Reed-Solomon Decoding Algorithm . . . . . . . . . . . . . 15
6.3.1. Decoding Principles . . . . . . . . . . . . . . . . . 15
6.3.2. Decoding Complexity . . . . . . . . . . . . . . . . . 16
6.4. Implementation for the Packet Erasure Channel . . . . . . 16
7. Security Considerations . . . . . . . . . . . . . . . . . . . 18
8. IANA Considerations . . . . . . . . . . . . . . . . . . . . . 19
9. Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . 20
10. References . . . . . . . . . . . . . . . . . . . . . . . . . . 21
10.1. Normative References . . . . . . . . . . . . . . . . . . . 21
10.2. Informative References . . . . . . . . . . . . . . . . . . 21
Authors' Addresses . . . . . . . . . . . . . . . . . . . . . . . . 23
Intellectual Property and Copyright Statements . . . . . . . . . . 24
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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 (32-m bit field) 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 (m bit field) 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 XX.
4.2.2. Common Elements
The following elements MUST be defined with the present FEC scheme:
o Transfer-Length (L): a non-negative integer indicating the length
of the object in bytes. There are some restrictions on the
maximum Transfer-Length that can be supported :
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-Transfer-Length (L)
o FEC-OTI-Encoding-Symbol-Length (E)
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o FEC-OTI-Maximum-Source-Block-Length (B)
o FEC-OTI-Max-Number-of-Encoding-Symbols (max_n)
o FEC-OTI-Number-of-Encoding-Symbols-per-Group (optional) (G)
o FEC-OTI-Finite-Field-Parameter (optional) (m)
When no G parameter is to be carried in the FEC OTI, the sender
simply omits the 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.
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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 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
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 value of n 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 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 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)
m = 8, "101110001", (1+x^^2+x^^3+x^^4+x^^8)
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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
alpha the primitive element of GF(2^^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
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
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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
corresponding to the received elements. Indeed, since any encoding
element is obtained by multiplying the source vector by one column of
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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 received correctly 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 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 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.
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------------ --------------- -------------------
0 | | | | | | | | | | | |
| | | | | * | generator | = | | | | |
| | | | | | matrix | | | | | |
| | | | | | GM | | | | | |
source |--------------| | | |---------------------|
vector | | | | | | | --------------- ->| | | | | | |
j |--------------| / |---------------------|
| | | | | / | | | | |
| | | | | encoding | | | | |
| | | | | vector | | | | |
| | | | | j | | | | |
| | | | | | | | | |
S-1 | | | | | | | | | |
------------ -------------------
k source symbols n encoding symbols
(source + repair)
Figure 4: 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 XX 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-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.
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-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.
[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-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),
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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 .
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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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