Reliable Multicast Transport J. Lacan
Internet-Draft ENSICA/LAAS-CNRS
Expires: August 27, 2006 V. Roca
INRIA
J. Peltotalo
S. Peltotalo
Tampere University of Technology
February 23, 2006
Reed-Solomon Forward Error Correction (FEC)
draft-ietf-rmt-bb-fec-rs-00.txt
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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 reliable
delivery of data objects on the packet erasure channel.
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The Reed-Solomon codes belong to the class of Maximum Distance
Separable (MDS) codes, i.e, they enable a receiver to recover the k
source symbols from any set of k received symbols.
The implementation described here is compatible with the IPR-free
implementation from Luigi Rizzo.
Table of Contents
1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 3
2. Terminology . . . . . . . . . . . . . . . . . . . . . . . . . 4
3. Definitions Notations and Abbreviations . . . . . . . . . . . 5
3.1. Definitions . . . . . . . . . . . . . . . . . . . . . . . 5
3.2. Notations . . . . . . . . . . . . . . . . . . . . . . . . 5
3.3. Abbreviations . . . . . . . . . . . . . . . . . . . . . . 6
4. Formats and Codes . . . . . . . . . . . . . . . . . . . . . . 7
4.1. FEC Payload IDs . . . . . . . . . . . . . . . . . . . . . 7
4.2. FEC Object Transmission Information . . . . . . . . . . . 7
4.2.1. Mandatory Elements . . . . . . . . . . . . . . . . . . 7
4.2.2. Common Elements . . . . . . . . . . . . . . . . . . . 7
4.2.3. Scheme-Specific Elements . . . . . . . . . . . . . . . 8
4.2.4. Encoding Format . . . . . . . . . . . . . . . . . . . 8
5. Procedures . . . . . . . . . . . . . . . . . . . . . . . . . . 10
5.1. Determining the Maximum Source Block Length (B) . . . . . 10
5.2. Determining the Number of Encoding Symbols of a Block . . 10
6. Reed-Solomon Codes . . . . . . . . . . . . . . . . . . . . . . 12
6.1. Finite field . . . . . . . . . . . . . . . . . . . . . . . 12
6.2. Reed-Solomon Encoding Algorithm . . . . . . . . . . . . . 13
6.2.1. Encoding Complexity . . . . . . . . . . . . . . . . . 14
6.3. Reed-Solomon Decoding Algorithm for the Erasure Channel . 14
6.3.1. Decoding Complexity . . . . . . . . . . . . . . . . . 14
6.4. Implementation . . . . . . . . . . . . . . . . . . . . . . 15
6.4.1. Implementation for the Packet Erasure Channel . . . . 15
7. Security Considerations . . . . . . . . . . . . . . . . . . . 17
8. Intellectual Property . . . . . . . . . . . . . . . . . . . . 18
9. IANA Considerations . . . . . . . . . . . . . . . . . . . . . 19
10. Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . 20
11. References . . . . . . . . . . . . . . . . . . . . . . . . . . 21
11.1. Normative References . . . . . . . . . . . . . . . . . . . 21
11.2. Informative References . . . . . . . . . . . . . . . . . . 21
Authors' Addresses . . . . . . . . . . . . . . . . . . . . . . . . 22
Intellectual Property and Copyright Statements . . . . . . . . . . 23
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1. Introduction
The use of Forward Error Correction (FEC) codes is a classical
solution to improve the reliability of multicast and broadcast
transmissions. The [RFC3452] and [draft-ietf-rmt-fec-bb-revised-03]
documents describe a general framework to use FEC in Content Delivery
Protocols (CDP). The companion document [RFC3453] describes some
applications of FEC codes for content delivery.
Recent FEC schemes like [draft-ietf-rmt-bb-fec-raptor-object-03] and
[draft-ietf-rmt-bb-fec-ldpc-01] proposed erasure codes based on
sparse graphs/matrices. These codes are efficient in terms of CPU
but not optimal in terms of correction capabilities, at least for
small objects.
The FEC scheme presented in this document belongs to the class of
Maximum-Distance Separable codes, i.e., it is optimal in terms of
erasure correction capability. In others words, it enables the
receiver to recover the k source symbols from any set of k encoding
symbols. Even if the encoding/decoding complexity is larger than
that of [draft-ietf-rmt-bb-fec-raptor-object-03] or
[draft-ietf-rmt-bb-fec-ldpc-01], this family of codes is very useful
for applications sending small objects (e.g., for video and audio
streaming).
Indeed many applications dealing with content transmission or content
storage already rely on packet-based Reed-Solomon codes. In
particular, many of them are derived from the implementation of Luigi
Rizzo [RS-Rizzo]. This latter is compatible with the Reed-Solomon
codes specification of the present document.
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2. Terminology
The key words "MUST", "MUST NOT", "REQUIRED", "SHALL", "SHALL NOT",
"SHOULD", "SHOULD NOT", "RECOMMENDED", "MAY", and "OPTIONAL" in this
document are to be interpreted as described in RFC 2119 [rfc2119].
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3. Definitions Notations and Abbreviations
3.1. Definitions
This document uses the same terms and definitions as those specified
in [draft-ietf-rmt-fec-bb-revised-03]. Additionally, it uses the
following definitions:
Source symbol: unit of data used during the encoding process.
Encoding symbol: unit of data generated by the encoding process.
Repair symbol: encoding symbols that are not source symbols.
Systematic code: a code in which the source symbols are part of
the encoding symbols
Source block: a block of k source symbols that are considered
together for the encoding.
Encoding Symbol Group: a group of encoding symbols that are sent
together, within the same packet, and whose relationships to the
source object can be derived from a single Encoding Symbol ID.
Source Packet a data packet containing only source symbols.
Repair Packet a data packet containing only repair symbols.
3.2. Notations
This document uses the following notations:
L denotes the object transfer length in bytes
k denotes the number of source symbols in a source block
n_r denotes the number of repair symbols generated for a source
block
n denotes the encoding block length, i.e., the number of encoding
symbols generated for a source block. Then n = k+ n_r
max_n Maximum Number of Encoding Symbols generated for any source
block
B denotes the maximum source block length in symbols, i.e., the
maximum number of source symbols per source block
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N denotes the number of source blocks into which the object shall
be partitioned
E denotes the encoding symbol length in bytes
sz denotes the symbol size in units of m bit elements
m defines the number of elements in the finite field, namely q 2^^m.
G denotes the number of encoding symbols per group, i.e., the
number of symbols sent in the same packet
rate denotes the so-called "code rate", i.e. the k/n ratio
a^^b denotes a raised to the power b
a^^-1 denotes the inverse of a
I_k denotes the k*k identity matrix
3.3. Abbreviations
This document uses the following abbreviations:
ESI Encoding Symbol ID
RS Reed-Solomon
MDS Maximum Distance Separable code
GF(q) finite field (A.K.A. Galois Field) with q elements
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4. Formats and Codes
4.1. FEC Payload IDs
The FEC Payload ID is composed of the Source Block Number and the
Encoding Symbol ID:
o The Source Block Number (16 bit field) identifies from which
source block of the object the encoding symbol(s) in the payload
is (are) generated. There is a maximum of 2^^16 blocks per
object.
o The Encoding Symbol ID (16 bit field) identifies which specific
encoding symbol generated from the source block is carried in the
packet payload. There is a maximum of 2^^16 encoding symbols per
block. The first k values (0 to k-1) identify source symbols, the
remaining n-k values identify repair symbols.
There MUST be exactly one FEC Payload ID per packet. In case of an
Encoding Symbol Group, when multiple encoding symbols are sent in the
same packet, the FEC Payload ID refers to the first symbol of the
packet. The other symbols can be deduced from the ESI of the first
symbol by incrementing sequentially the ESI.
0 1 2 3
0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1
+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
| Source Block Number (16 bits) | Encoding Symbol ID (16 bits) |
+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
Figure 1: FEC Payload ID encoding format for FEC Encoding ID XX
4.2. FEC Object Transmission Information
4.2.1. Mandatory Elements
o FEC Encoding ID: the Fully-Specified FEC Scheme described in this
document use the FEC Encoding ID XX.
4.2.2. Common Elements
The following elements MUST be defined with the present FEC Scheme:
o Transfer-Length (L): a non-negative integer indicating the length
of the object in bytes. There are some restrictions on the
maximum Transfer-Length that can be supported:
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max_transfer_length = 2^^16 * B * E
For instance, if B = 2^^8-1 (because the codec operates on a
finite field with 2^^8 elements), and if E = 1024 bytes, then the
maximum transfer length is 2^^34 bytes (i.e., a bit more than 17
Giga Bytes). For larger objects, it is expected that other FEC
codes (e.g., LDPC codes) or another Reed-Solomon FEC Scheme with a
larger Source Block Number field in the FEC Payload ID be used.
o Encoding-Symbol-Length (E): a non-negative integer indicating the
length of each encoding symbol in bytes.
o Maximum-Source-Block-Length (B): a non-negative integer indicating
the maximum number of source symbols in a source block.
o Max-Number-of-Encoding-Symbols (max_n): a non-negative integer
indicating the maximum number of encoding symbols generated for
any source block.
Section 5 explains how to derive the values of each of these
elements.
4.2.3. Scheme-Specific Elements
The following element MUST be defined with the present FEC Scheme.
It contains two distinct pieces of information:
o G: a non-negative integer indicating the number of encoding
symbols per group used for the object. The default value is 1,
meaning that each packet contains exactly one symbol. When no G
parameter is communicated to the decoder, then this latter MUST
assume that G = 1.
o Finite Field size parameter, m: The m parameter defines the finite
field size equal to q = p^^m elements. The default value is m 8. When no finite field size parameter is communicated to the
decoder, then this latter MUST assume that m = 8.
4.2.4. Encoding Format
This section shows two possible encoding formats of the above FEC
OTI. The present document does not specify when or how these
encoding formats should be used.
4.2.4.1. Using the General EXT_FTI Format
The FEC OTI binary format is the following, when the EXT_FTI
mechanism is used.
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0 1 2 3
0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1
+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
| HET = 64 | HEL | |
+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+ +
| Transfer-Length (L) |
+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
| m | G | Encoding Symbol Length (E) |
+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
| Max Source Block Length (B) | Max Nb Enc. Symbols (max_n) |
+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
Figure 2: EXT_FTI Header Format
4.2.4.2. Using the FDT Instance (FLUTE specific)
When it is desired that the FEC OTI be carried in the FDT Instance of
a FLUTE session, the following XML elements must be described for the
associated object:
o FEC-OTI-Transfer-length
o FEC-OTI-Encoding-Symbol-Length
o FEC-OTI-Maximum-Source-Block-Length
o FEC-OTI-Max-Number-of-Encoding-Symbols
o FEC-OTI-Number-Encoding-Symbols-per-Group (optional)
o FEC-OTI-Finite-Field-Size-Parameter (optional)
When no finite field size parameter is to be carried in the FEC OTI,
the sender simply omits the FEC-OTI-Finite-Field-Size-Parameter
element.
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5. Procedures
This section defines procedures for FEC Encoding ID XX.
5.1. Determining the Maximum Source Block Length (B)
The finite field size parameter, m, defines the number of non zero
elements in this field, q = 2^^m-1. Note that q is also the
theoretical maximum number of encoding symbols that can be produced
for a source block. For instance, when m = 8 (default):
max1_B = 2^^8-1
Additionally, a codec MAY impose other limitations on the maximum
block size. Yet it is not expected that such limits exist when using
m = 8 (default). This decision SHOULD be clarified at implementation
time, when the target use case is known. This results in a max2_B
limitation.
Then, B is given by:
B = min(max1_B, max2_B)
Note that this calculation is only required at the coder, since the B
parameter is communicated to the decoder through the FEC OTI.
5.2. Determining the Number of Encoding Symbols of a Block
The following algorithm, also called "n-algorithm", explains how to
determine the actual number of encoding symbols for a given block.
AT A SENDER:
Input:
B: Maximum source block length, for any source block. Section 5.1
explains how to determine its value.
k: Current source block length. This parameter is given by the
source blocking algorithm.
rate: FEC code rate, which is given by the user (e.g., when
starting a FLUTE sending application) for a given use case. It is
expressed as a floating point value.
Output:
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max_n: Maximum number of encoding symbols generated for any source
block
n: Number of encoding symbols generated for this source block
Algorithm:
max_n = floor(B / rate);
if (max_n >= 2^^m) then return an error ("invalid code rate");
n = floor(k * max_n / B);
AT A RECEIVER:
Input:
B Extracted from the received FEC OTI
max_n Extracted from the received FEC OTI
k Given by the source blocking algorithm
Output:
n
Algorithm:
n = floor(k * max_n / B);
Note that a Reed-Solomon decoder does not need to know the n value.
Therefore the receiver part of the "n-algorithm" is not necessary
from the Reed-Solomon decoder point of view. Yet a receiving
application using the Reed-Solomon FEC scheme will sometimes need to
know the value of n used by the sender, for instance for memory
management optimizations. To that purpose, all the needed
information is carried in the FEC OTI.
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6. Reed-Solomon Codes
Reed-Solomon (RS) codes form a special class of linear block codes,
which offer maximum erasure correction capability. A [n,k]-RS code
encodes a sequence of k source elements defined over a finite field
GF(q) into a sequence of n encoding elements, where n is upperbounded
by q-1. The implementation described in this document is based on a
generator matrix built from a Vandermonde matrix put into systematic
form.
6.1. Finite field
A finite field GF(q) is defined as a finite set of q elements which
have a structure of field. It contains necessarily q = p^^m
elements, where p is a prime number. With packet erasure channels, p
is always set to 2. The elements of the field GF(2^^m) can be
represented by polynomials with binary coefficients (i.e., over
GF(2)) of degree less than m. The polynomials can be associated to
binary vectors of length m. For example, the vector (11001)
represents the polynomial 1 + x + x^^4. This representation is often
called polynomial representation. The addition between two elements
is defined as the addition of binary polynomials in GF(2) and the
multiplication is the multiplication modulo a given irreducible
(i.e., non-factorizable) polynomial of degree m with coefficients in
GF(2).
Since a finite field GF(2^^m) is completely characterized by the
irreducible polynomial, we propose the following polynomials to
represent the field GF(2^^m), for m varying from 2 to 16:
m = 2, "111" (1+x+x^^2)
m = 3, "1101", (1+x+x^^3)
m = 4, "11001", (1+x+x^^4)
m = 5, "101001", (1+x^^2+x^^5)
m = 6, "1100001", (1+x+x^^6)
m = 7, "10010001", (1+x^^3+x^^7)
m = 8, "101110001", (1+x^^2+x^^3+x^^4+x^^8)
m = 9, "1000100001", (1+x^^4+x^^9)
m = 10, "10010000001", (1+x^^3+x^^10)
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m = 11, "101000000001", (1+x^^2+x^^11)
m = 12, "1100101000001", (1+x+x^^4+x^^6+x^^12)
m = 13, "11011000000001", (1+x+x^^3+x^^4+x^^13)
m = 14, "110000100010001", (1+x+x^^6+x^^10+x^^14)
m = 15, "1100000000000001", (1+x+x^^15)
m = 16, "11010000000010001", (1+x+x^^3+x^^12+x^^16)
For implementation issues, these polynomials are also primitive
elements of GF(2^^m), i.e., any element of GF(2^^m) can be expressed
as a power of a root of this polynomial. These polynomials also
contain the minimum number of monomials.
6.2. Reed-Solomon Encoding Algorithm
The encoding algorithm produces a vector of n encoding elements
e=(e_0, ..., e_(n-1)) over GF(2^^m) from a source vector of k
elements s=(s_0, ..., s_(k-1) ) over GF(2^^m).
The linear codes can be encoded by multiplying the source vector by a
generator matrix Gm of k rows and n columns over GF(2^^m). Thus: e s * Gm. The definition of the generator matrix completely
characterizes the code.
Let us consider that: n = 2^^m - 1 and: 0 < k <= n. Let us denote
alpha a primitive element of GF(2^^m) (i.e., any element of GF(2^^m)
can be expressed as a power of alpha).
The generator matrix is build from a k*n-Vandermonde matrix denoted
by V_{k,n}. The entries of V_{k,n} are v_{i,j} = alpha^^(i*j), where
0 <= i <= k - 1 and 0 <= j <= n - 1. This matrix generates a MDS
code. However, it is not systematic as required by most of network
applications. To obtain a systematic matrix, the simplest solution
is to consider the matrix V_{k,k} formed by the first k columns of
V_{k,n} then to invert it and to multiply this inverse by V_{k,n}.
Clearly, the product V_{k,k}^^-1 * V_{k,n} contains the identity
matrix I_k on its first k columns and generates a MDS code.
The product V_{k,k}^^-1 * V_{k,n} is denoted by Gm and is the
generator matrix of the code considered in this document.
Note that, for practical applications, the length of the code can be
shortened to k <= n' < n by considering the sub-matrix formed by the
n' first columns of Gm.
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6.2.1. Encoding Complexity
The encoding process can be done by first pre-computing G and by
multiplying the source vector by Gm. The complexity is one
multiplication s*Gm, where Gm is a k*(n-k) matrix. The complexity of
the vector-matrix multiplication is then k*(n-k) (i.e., k operations
per repair element).
The encoding can also be processed by first computing the product s*
V_{k,k}^^-1 and then by multiplying the result by V_{k,n}. The
multiplication by the inverse of a square Vandermonde matrix is known
as the interpolation problem and its complexity is O(k log^^2 (k)).
The multiplication by a Vandermonde matrix, known as the multipoint
evaluation problem, requires O((n-k) log(k)) by using Fast Fourier
Transform, as explained in [fastMatrix-vectorMultiplication]. The
total complexity of this encoding algorithm is then O(k/(n-k) log^^2
(k)+ log(k)) operations per repair symbol.
6.3. Reed-Solomon Decoding Algorithm for the Erasure Channel
The Reed-Solomon decoding algorithm for the erasure channel allows
the recovery of the k source elements from any set of k received
elements. It is based on the fundamental property of the generator
matrix which is such that any k*k-submatrix is invertible (see
[MWS]). The first step of the decoding consists in extracting the
k*k submatrix of the generator matrix obtained by considering the
columns corresponding to the received symbols. Indeed, since any
encoding element is obtained by multiplying the source vector by one
column of the generator matrix, the received vector of k encoding
symbols can be considered as the result of the multiplication of the
source vector by a k*k submatrix of the generator matrix. Since this
submatrix is invertible, the second step of the algorithm is to
invert this matrix and to multiply the received vector by the
obtained matrix to recover the source vector.
6.3.1. Decoding Complexity
The decoding algorithm described previously includes the matrix
inversion and the vector-matrix multiplication. With the classical
Gauss-Jordan algorithm, the matrix inversion requires O(k^^3)
operations and the vector-matrix multiplication is performed in
O(k^^2) operations.
This complexity can be improved by considering that the received
submatrix of Gm is the product between the inverse of a Vandermonde
matrix (V_(k,k)^^-1) and another Vandermonde matrix (denoted by V'
which is a submatrix of V_(k,n)). The decoding can be done by
multiplying the received vector by V'^^-1 (interpolation problem with
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complexity O( k log^^2(k)) ) then by V_{k,k} (multipoint evaluation
with complexity O( k log(k)) ). The global decoding complexity is
then O(log^^2(k)) operations per source symbol.
6.4. Implementation
6.4.1. Implementation for the Packet Erasure Channel
In a packet erasure channel, each packet is either received correctly
or erased. The location of the erased packets in the sequence of
packets must be known. The following specification describes the use
of Reed-Solomon codes for generating redundant packets from k source
packets and to recover the source packets from k received packets.
The k source symbols of a source block are assumed to be composed of
sz m-bit elements. Each m-bit element is associated to an element of
the finite field GF(2^^m) through the polynomial representation
(Section 6.1). If some of the source symbols contain less than sz
elements, they are virtually padded with zero elements (it can be the
case for the last symbol of the last block of the object).
The encoding processing produces n-k repair symbols of sz elements by
encoding each of the sz encoding vectors from the sz source vectors
(Figure 3). The j-th source vector is composed of the j-th element
of each of the source symbols. Similarly, the j-th encoding vector
is composed of the j-th element of each encoding symbol.
------------ --------------- -------------------
| | | | | | | | | | | |
| | | | | * | generator | = | | | | |
| | | | | | matrix | | | | | |
| | | | | | Gm | | | | | |
source |--------------| | | |---------------------|
vector | | | | | | | --------------- ->| | | | | | |
j |--------------| / |---------------------|
| | | | | / | | | | |
| | | | | encoding | | | | |
| | | | | vector | | | | |
| | | | | j | | | | |
| | | | | | | | | |
| | | | | | | | | |
------------ -------------------
k source symbols n encoding symbols
Figure 3: Packet encoding scheme
An asset of this scheme is that the loss of some of encoding symbols
produce the same erasure pattern for each of the sz encoding vectors.
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It follows that the matrix inversion must be done only once and will
be used by all the sz encoding vectors. For large sz, this
complexity cost of the inversion becomes negligible compared to the
sz matrix-vector multiplications.
Another asset is that repair symbols can be produced on demand, e.g.,
depending on the observed erasures on the channel. The only
constraint is the finite field size (see Section 6.1)
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7. Security Considerations
The security considerations for this document are the same as that of
[RFC3452].
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8. Intellectual Property
To the best of our knowledge, there is no patent or patent
application identified as being used in the Reed-Solomon FEC scheme.
Yet other flavors of Reed-Solomon codes and associated techniques MAY
be covered by Intellectual Property Rights.
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9. IANA Considerations
Values of FEC Encoding IDs and FEC Instance IDs are subject to IANA
registration. For general guidelines on IANA considerations as they
apply to this document, see [draft-ietf-rmt-fec-bb-revised-03]. This
document assigns the Fully-Specified FEC Encoding ID XX under the
ietf:rmt:fec:encoding name-space to "Reed-Solomon Codes".
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10. Acknowledgments
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11. References
11.1. Normative References
[RFC3452] Luby, M., "Forward Error Correction (FEC) Building Block",
RFC 3452, December 2002.
[RFC3453] Luby, M., "The Use of Forward Error Correction (FEC) in
Reliable Multicast", RFC 3453, December 2002.
[draft-ietf-rmt-fec-bb-revised-03]
Watson, M., Luby, M., and L. Vicisano, "Forward Error
Correction (FEC) Building Block",
draft-ietf-rmt-fec-bb-revised-03.txt (work in progress),
January 2006.
[rfc2119] Bradner, S., "Key words for use in RFCs to Indicate
Requirement Levels", RFC 2119.
11.2. Informative References
[MWS] Mac Williams, F. and N. Sloane, "The Theory of Error
Correcting Codes", North Holland, 1977 .
[RS-Rizzo]
Rizzo, L., "New version of the FEC code (revised 2 july
98), available at
http://info.iet.unipi.it/~luigi/vdm98/vdm980702.tgz",
July 1998.
[draft-ietf-rmt-bb-fec-ldpc-01]
Roca, V., Neumann, C., and D. Furodet, "Low Density Parity
Check (LDPC) Forward Error Correction",
draft-ietf-rmt-bb-fec-ldpc-01.txt (work in progress),
March 2006.
[draft-ietf-rmt-bb-fec-raptor-object-03]
Luby, M., "Raptor Forward Error Correction Scheme",
Internet Draft (draft-ietf-rmt-bb-fec-raptor-object-03 :
work in progress), October 2005.
[fastMatrix-vectorMultiplication]
Gohberg, I. and V. Olshevsky, "Fast algorithms with
preprocessing for matrix-vector multiplication problems",
Journal of Complexity, pp. 411-427, vol. 10, 1994 .
Lacan, et al. Expires August 27, 2006 [Page 21]
Internet-Draft Reed-Solomon Forward Error Correction February 2006
Authors' Addresses
Jerome Lacan
ENSICA/LAAS-CNRS
1, place Emile Blouin
Toulouse 31056
France
Email: jerome.lacan@ensica.fr
URI:
Vincent Roca
INRIA
655, av. de l'Europe
Zirst; Montbonnot
ST ISMIER cedex 38334
France
Email: vincent.roca@inrialpes.fr
URI: http://planete.inrialpes.fr/~roca/
Jani Peltotalo
Tampere University of Technology
P.O. Box 553 (Korkeakoulunkatu 1)
Tampere FIN-33101
Finland
Email: jani.peltotalo@tut.fi
URI:
Sami Peltotalo
Tampere University of Technology
P.O. Box 553 (Korkeakoulunkatu 1)
Tampere FIN-33101
Finland
Email: sami.peltotalo@tut.fi
URI:
Lacan, et al. Expires August 27, 2006 [Page 22]
Internet-Draft Reed-Solomon Forward Error Correction February 2006
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Lacan, et al. Expires August 27, 2006 [Page 23]
