RMT V. Roca
Internet-Draft C. Neumann
Expires: December 30, 2005 INRIA
D. Furodet
STMicroelectronics
June 28, 2005
Low Density Parity Check (LDPC) Forward Error Correction
draft-roca-rmt-ldpc-00.txt
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Copyright (C) The Internet Society (2005).
Abstract
This document describes an Under-Specified FEC Scheme that can be
used with the broad class of Low Density Parity Check (LDPC) codes
and their application to the reliable delivery of objects on packet
erasure channels. Additionally, this document describes the LDPC-
Staircase and LDPC-Triangle Forward Error Correction codes, two
instances of the LDPC FEC Scheme, in a way that enables fully inter-
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operable implementations. The LDPC codes belong to the class of
large block FEC codes, as defined in RFC3453, which enables them to
efficiently encode/decode large objects, in a single block. They
also enable a receiver to recover the k source symbols from any set
of a little bit more than k encoding symbols.
Table of Contents
1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 3
2. Requirements notation . . . . . . . . . . . . . . . . . . . . 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 . . . . . . . . . . . . . . . . . . . . . . . . . . 9
5.1 General . . . . . . . . . . . . . . . . . . . . . . . . . 9
5.2 Parity Check Matrix . . . . . . . . . . . . . . . . . . . 10
5.3 Derivations and Interpretation of the Fields Provided
in the FPI and FEC OTI . . . . . . . . . . . . . . . . . . 10
5.4 Pseudo Random Number Generator . . . . . . . . . . . . . . 11
6. Full Specification of the LDPC-Staircase Scheme . . . . . . . 12
6.1 Instance Specific Parameters . . . . . . . . . . . . . . . 12
6.2 Parity Check Matrix Creation . . . . . . . . . . . . . . . 12
6.3 Encoding . . . . . . . . . . . . . . . . . . . . . . . . . 14
6.4 Decoding . . . . . . . . . . . . . . . . . . . . . . . . . 14
7. Full Specification of the LDPC-Triangle Scheme . . . . . . . . 15
7.1 Instance Specific Parameters . . . . . . . . . . . . . . . 15
7.2 Parity Check Matrix Creation . . . . . . . . . . . . . . . 15
7.3 Encoding . . . . . . . . . . . . . . . . . . . . . . . . . 15
7.4 Decoding . . . . . . . . . . . . . . . . . . . . . . . . . 16
8. Security Considerations . . . . . . . . . . . . . . . . . . . 17
9. Intellectual Property . . . . . . . . . . . . . . . . . . . . 18
10. Acknowledgments . . . . . . . . . . . . . . . . . . . . . . 19
11. References . . . . . . . . . . . . . . . . . . . . . . . . . 20
11.1 Normative References . . . . . . . . . . . . . . . . . . . 20
11.2 Informative References . . . . . . . . . . . . . . . . . . 20
Authors' Addresses . . . . . . . . . . . . . . . . . . . . . . 21
A. Iterative Decoding Algorithm (Informative) . . . . . . . . . . 22
Intellectual Property and Copyright Statements . . . . . . . . 24
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1. Introduction
RFC 3453 [RFC3453] introduces large block FEC codes as an alternative
to small block FEC codes like Reed-Solomon. The main advantage of
such large block codes is the possibility to operate efficiently on
source blocks of several tens of thousands (or more) source symbols
of size.
The present document introduces the Under-Specified FEC Encoding ID
132 that is intended to be used with the "Low Density Parity Check"
(LDPC) FEC codes, that belong the class of large block codes. LDPC
codes rely on a dedicated matrix, called a "Parity Check Matrix", at
the encoding and decoding ends. The parity check matrix defines
relationships (or constraints) between the various encoding symbols
(i.e. source symbols and repair symbols), that are later used by the
decoder to reconstruct the original k source symbols if some of them
are missing. These codes are systematic, in the sense that the
encoding symbols include the source symbols in addition to the
redundant symbols.
-- editor's note: This document makes use of the FEC Encoding ID
value 132, but this may change after IANA assignment --
Since the encoder and decoder must operate on the same parity check
matrix, some information must be communicated between them, as part
of the FEC Object Transmission Information. Its content and the
associated EXT_FTI are fully described in Section 4.2.
The two variants specified in this document belong to this broad
class of LDPC codes. But other codes, existing or forthcoming, may
also be added in the future, taking advantage of the framework
provided by the Under-Specified FEC Encoding ID 132. More
specifically, this document reserves the FEC Instance ID value 0 for
the LDPC-Staircase codes [Roca04][Mac03] and reserves the FEC
Instance ID value 1 for the LDPC-Triangle codes [Roca04]. A publicly
available reference implementation of these codes is available and
distributed under a GNU/LGPL license [LDPCrefimpl].
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2. Requirements notation
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 [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 [fec-bb-revised]. Additionally, it uses the following
definitions:
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 denotes the number of encoding symbols
E denotes the encoding symbol length in bytes
B denotes the maximum source block length in terms of symbols
N denotes the number of source blocks into which the object shall
be partitioned
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
max_n Maximum Number of Encoding Symbols per encoding block. This
depends on FEC code rate.
rand(m) denotes a pseudo-random number generator, that returns a
new random integer in [0; m-1] each time it is called.
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3.3 Abbreviations
This document uses the following abbreviations:
ESI Encoding Symbol ID
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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:
The Source Block Number identifies from which source block of the
object the encoding symbol(s) in the payload is(are) generated.
The Encoding Symbol ID identifies which specific encoding symbol
generated from the source block is carried in the packet payload.
Each encoding symbol is either an original source symbol or a
redundant symbol generated by the encoder.
There MUST be exactly one FEC Payload ID per packet. 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 as explained in Section 5.1
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 |
+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
| Encoding Symbol ID |
+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
Figure 1: FEC Payload ID encoding format for FEC Encoding ID 132
4.2 FEC Object Transmission Information
4.2.1 Mandatory Elements
FEC Encoding ID: the Under-Specified FEC Scheme described in this
document uses the FEC Encoding ID 132.
FEC Instance ID: this document reserves the FEC Instance ID value 0
for the LDPC-Staircase codes (Section 6) and the FEC Instance ID
value 1 for the LDPC-Triangle codes (Section 7).
4.2.2 Common Elements
The following elements MUST be used with the present FEC Scheme:
Transfer-Length: a non-negative integer indicating the length of the
object in bytes.
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Encoding-Symbol-Length: a non-negative integer indicating the length
of each encoding symbol in bytes.
Maximum-Source-Block-Length: a non-negative integer indicating the
maximum number of source symbols in a source block.
Max-Number-of-Encoding-Symbols: a non-negative integer indicating the
maximum number of encoding symbols (i.e. source plus repair symbols
in the case of a systematic code).
Section 5.3 describes how to derive the values of each of these
elements.
4.2.3 Scheme-Specific Elements
PRNG seed: Seed (a 32 bit value) used to initiate the Pseudo Random
Generator (defined in Section 5.4). This element is optional and may
be used by some specific Instance IDs.
Other elements MAY be defined for Instance-Specific needs.
4.2.4 Encoding Format
This section shows possible encoding formats of the above FEC OTI.
4.2.4.1 Using the General EXT_FTI Format
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) |
+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
| FEC Instance ID | Encoding Symbol Length (E) |
+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
| Max Source Block Length (B) |
+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
| Max Nb of Enc. Symbols (max_n) |
+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
.. Scheme Specific optional elements .
+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
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5. Procedures
This section defines procedures that are common to all FEC Instance
IDs scoped by FEC Encoding ID 132.
5.1 General
The source object is first partitioned into blocks, using the block
partitioning algorithm specified in [fec-bb-revised]. To that
purpose, the B (maximum source block length in symbols), L (object
transfer length in bytes), and E (encoding symbol length in bytes)
arguments are provided. As an output, the object is partitioned into
N source blocks. These blocks are numbered consecutively from 0 to
N-1. The first I source blocks consist of A_large source symbols,
the remaining N-I source blocks consist of A_small source symbols.
Each source symbol is E bytes in length, except perhaps the last
symbol which may be shorter as explained in [fec-bb-revised].
FEC encoding and decoding is done block per block, independently.
When multiple encoding symbols are sent in the same packet, it MUST
be possible to identify each symbol from this single FEC Payload ID.
To that purpose, the symbols of an Encoding Symbol Group (i.e.
packet):
o MUST be in sequence, from ESI i to ESI i+G-1 (inclusive),
o MUST all be either source symbols, or repair symbols. Therefore,
only source packets and repair packets are permitted, not mixed
ones.
The FEC Payload ID information MUST refer to the first encoding
symbol of the packet.
This specification does not specify what value for B should be used.
This decision SHOULD be clarified either at implementation time, when
the target use case is known, or in the specification of a FEC
Instance ID, for instance to take into account some specificities of
a FEC scheme.
Similarly, this specification does not specify if and when Encoding
Symbol Groups should be used or not, i.e. if and when we have G>1.
This decision SHOULD be clarified either at implementation time, when
the target use case is known, or in the specification of a FEC
Instance ID, for instance to take into account some specificities of
a FEC scheme.
In both cases, a receiver can derive the B and G values from the
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information it receives.
5.2 Parity Check Matrix
LDPC codes rely on a parity check matrix, which represents a linear
equation system between repair symbols and source symbols of a given
block. The basic operator is XOR and the matrix can only be filled
with 1s and 0s.
The parity check matrix is logically divided into two parts: the left
side (from column 0 to k-1) which describes the occurrence of each
source symbol in the equation system; and the right side (from column
k to n-1) which describes the occurrence of each repair symbol in the
equation system.
An entry (a "1") in the matrix at position (i,j), i.e. at row i and
column j, means that the symbol with ESI i appears in equation j.
5.3 Derivations and Interpretation of the Fields Provided in the FPI
and FEC OTI
The fields provided in the FEC OTI are derived using the
"n-algorithm", described below:
AT A SENDER:
Input:
B Maximum Source Block Length, i.e., the maximum number of source
symbols per source block. This is given by the FEC codec
specifications and/or the execution environment limitations.
k Source Block Length, i.e., the number of source symbols per
source block. This is given by source blocking algorithm.
rate or (k,n) FEC code rate, which is given by the user (e.g. when
starting a FLUTE sending application). It is expressed either as
a floating point value, R, or as a quotient k/n. The latter
option is RECOMMENDED for the integer math version of the
algorithm.
Output:
max_n Maximum Number of Encoding Symbols per encoding block. This
depends on FEC code rate.
n Encoding Block Length, i.e., the number of encoding symbols
generated for the source block.
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Algorithm:
a. max_n = B / R rounded down to the nearest integer (max_n = (B *
b) div a)
b. n = k * max_n / B rounded down to the nearest integer (n = (k *
max_n) div B)
AT A RECEIVER:
Input: B, max_n, k
Output: n
Algorithm:
a. n = k * max_n / B rounded down to the nearest integer (n = (k *
max_n) div B)
Notes: (1) X div Y denotes the integer quotient of the division X/Y
The use of floating point arithmetic in the algorithm might lead to
erroneous results caused by rounding problems, depending on the
mathematical library used. These problems can be avoided by using
only integer math in all algorithm calculations. It is strongly
recommended not to use rounding functions, and how to do that is
presented in brackets
5.4 Pseudo Random Number Generator
The present FEC Encoding ID relies on a pseudo-random number
generator that must be fully specified in order to enable the
receivers and the senders to build the same parity check matrix.
-- editor's note: The PRNG to use is TBD. Current implementation
relies on the GNU C Library lrand48() function, but this may not
be the most appropriate choice. --
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6. Full Specification of the LDPC-Staircase Scheme
6.1 Instance Specific Parameters
LDPC-Staircase is identified by th Under-Specified FEC Encoding ID
132 and the the FEC Instance ID 0.
LDPC-Staircase is based on a pseudo-random number generator as
specified in Section 5.4. Therefore the seed used to initiate the
PRNG is an instance-specific FEC Object Transmission Information
element and MUST be transmitted within the FEC OTI, as specified in
Section 4.2.
6.2 Parity Check Matrix Creation
The matrix creation algorithm for LDPC Staircase is described in the
following. The algorithm can be divided into two parts: The left
side of the matrix where the occurrence of the source symbols in the
equations is described, and the right side of the matrix where repair
symbols are described. The left side is generated with the following
algorithm:
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/* initialize a list of possible choices to
* guarantee a homogeneous "1" distribution */
for(h = 3*k-1; h >= 0; h--) {
u[h] = h % (n-k);
}
/* left limit within the list of possible choices, u[] */
t = 0;
for(j = 0; j < k; j++) { /* for each source symbol column */
for(h = 0; h < 3; h++) { /* add 3 "1s" */
/* check that valid available choices remain */
for(i = t; i < 3*k && matrix_has_entry(u[i],j); i++);
if(i < 3*k) {
/* choose one index within the
* list of possible choices */
do {
i = t + rand() % (3*k-t);
} while (matrix_has_entry(u[i],j));
matrix_insert_entry(u[i],j);
/* replace with u[t] which has never been chosen */
u[i] = u[t];
t++;
} else {
/* no choice left, choose one randomly */
do {
i = rand() % (n-k);
} while (matrix_has_entry(i,j));
matrix_insert_entry(i,j);
}
}
}
/* Add extra bits to avoid rows with less than two checks. */
for(i = 0; i < n-k; i++) { /* for each row */
if(degree_of_row(i) == 0) {
j = rand() % k;
e = matrix_insert_entry(i,j);
}
if(degree_of_row(i) == 1) {
do {
j = rand()% k;
} while (matrix_has_entry(i,j));
matrix_insert_entry(i,j);
}
}
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The right side (the staircase) is generated with the following
algorithm:
for(i = 0; i < n-k; i++) { /* for each row */
matrix_insert_entry(i,k+i);
if (i > 0)
matrix_insert_entry(i,k+i-1);
}
6.3 Encoding
Thanks to the staircase matrix, repair symbol creation is
straightforward: each repair symbol is equal to the sum of all source
symbols in the associated equation, plus the previous repair packet.
Therefore encoding should follow the natural repair symbol order,
i.e. generate repair symbol with ESI i before symbol ESI i+1.
6.4 Decoding
Decoding can be done using the general LDPC iterative decoding
algorithm as described in Appendix A.
Other techniques can be used, for instance solving th system of n-k
linear equations whose variables are the source an repair symbols
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7. Full Specification of the LDPC-Triangle Scheme
7.1 Instance Specific Parameters
LDPC-Triangle is identified by th Under-Specified FEC Encoding ID 132
and the the FEC Instance ID 1.
LDPC-Triangle is based on a pseudo-random number generator as
specified in Section 5.4. Therefore the seed used to initiate the
PRNG is an instance-specific FEC Object Transmission Information
element, and MUST be transmitted within the FEC OTI, as specified in
Section 4.2.
7.2 Parity Check Matrix Creation
The matrix creation algorithm for LDPC Triangle is the following.
The left side is the same as for LDPC Staircase (see Section 6.2).
The right side (the triangle) is generated with the following
algorithm:
for(i = 0; i < n-k; i++) { /* for each row */
/* create the identity */
matrix_insert_entry(i,k+i);
if (i > 0) {
/* create the staircase */
matrix_insert_entry(i,k+i-1);
/* fill the triangle */
int j = i;
for (l = 0; l < j; l++) {
if (j != 0) {
temp = rand() % j;
matrix_insert_entry(pchkMatrix, i, k+j);
}
}
}
}
7.3 Encoding
Just like LDPC-Triangle repair symbol creation is straightforward:
each repair symbol is equal to the sum of all source symbols in the
associated equation, plus some previous repair packets specified in
the triangle. Encoding should follow the natural repair symbol
order, i.e. generate repair symbol with ESI i before symbol ESI i+1.
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7.4 Decoding
Decoding can be done using the general LDPC iterative decoding
algorithm as described in Appendix A.
Other techniques can be used, for instance solving th system of n-k
linear equation whose variables are the source an repair symbols
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8. Security Considerations
The security considerations for this document are the same as they
are for RFC 3452 [RFC3452].
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9. Intellectual Property
The authors are not aware of any intellectual property rights
associated to the two LDPC codes specified within this document. Yet
other LDPC codes and associated techniques MAY be covered by IPR.
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10. Acknowledgments
Section 5.3 is derived from a previous Internet-Draft, and we would
like to thank S. Peltotalo and J. Peltotalo for their contribution.
We would also like to thank Pascal Moniot from STMicroelectronics for
his comments.
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11. References
11.1 Normative References
[RFC2119] Bradner, S., "Key words for use in RFCs to Indicate
Requirement Levels", RFC 2119, BCP 14, March 1997.
[RFC3452] Luby, M., Vicisano, L., Gemmell, J., Rizzo, L., Handley,
M., and J. Crowcroft, "Forward Error Correction (FEC)
Building Block", RFC 3452, December 2002.
[RFC3453] 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.
[fec-bb-revised]
Watson, M., Luby, M., and L. Vicisano, "Forward Error
Correction (FEC) Building Block (revised)", draft-ietf-
rmt-fec-bb-revised-00.txt draft-ietf-rmt-fec-bb-revised-
00.txt, April 2005.
11.2 Informative References
[LDPCrefimpl]
Roca, V., Neumann, C., and J. Laboure, "LDPC-Staircase/
LDPC-Triangle Codec Reference Implementation", MCLv3
project PLANETE Research Team, INRIA Rhone-Alpes,
June 2005.
[Mac03] MacKay, D., "Information Theory, Inference and Learning
Algorithms", Cambridge University Press, ISBN: 0521642981,
2003.
[Roca04] Roca, V. and C. Neumann, "Design, Evaluation and
Comparison of Four Large Block FEC Codecs: LDPC, LDGM,
LDGM Staircase and LDGM Triangle, Plus a Reed-Solomon
Small Block FEC Codec", INRIA Research Report RR-5225,
June 2004.
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Authors' Addresses
Vincent Roca
INRIA
655, av. de l'Europe
Zirst; Montbonnot
ST ISMIER cedex 38334
France
Phone:
Email: vincent.roca@inrialpes.fr
URI:
Christoph Neumann
INRIA
655, av. de l'Europe
Zirst; Montbonnot
ST ISMIER cedex 38334
France
Phone:
Email: christoph.neumann@inrialpes.fr
URI:
David Furodet
STMicroelectronics
12, Rue Jules Horowitz
BP217
Grenoble Cedex 38019
France
Phone:
Email: david.furodet@st.com
URI:
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Appendix A. Iterative Decoding Algorithm (Informative)
LDPC decoding over a packet erasure channel can be achieved through a
trivial iterative decoding algorithm. The underlying idea is the
following:
Given a set of linear equations, if one of them has only one
remaining unknown variable, then the value of this variable is
that of the constant term. So, replace this variable by its value
in all remaining linear equations, and reiterate. The value of
several variables can therefore be found by this recursive
algorithm.
Applied to LDPC FEC codes working over an erasure packet, the parity
check matrix defines a set of linear equations. The variables are
the source symbols and repair symbols. Of course, from a decoding
point of view, finding (i.e. decoding) all source symbols is the
target. Finding repair symbols is often required to that purpose,
but this is not the final goal. The iterative decoding algorithm is
the following:
Initialization: allocate a partial sum buffer partial_sum_i for
each line i: set it to 0.
For each newly received or decoded symbol s_i with ESI i:
1. If s_i is an already decoded or received symbol, return
immediately and do nothing.
2. If s_i is a source symbol, it is permanently stored in memory.
3. For each equation j having a degree greater than one (i.e.
more than one unknown variable), with an entry in column i
(i.e. having s_i as a variable), do the following:
+ add s_i to partial_sum_i;
+ remove the entry (j, i) of the H matrix.
+ If the new degree of equation j is one, we have decoded a
new packet and have to remember the index of the equation
in a list of indexes for newly decoded packets for step 4.
4. For all newly generated packets in step 3:
+ remove the last entry in equation j,
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Internet-Draft LDPC Forward Error Correction June 2005
+ move partial_sum_j into th buffer of symbol s_l,
+ goto step 1 with the newly created symbol s_l
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Internet-Draft LDPC Forward Error Correction June 2005
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