RMT V. Roca
Internet-Draft C. Neumann
Expires: April 13, 2006 INRIA
D. Furodet
STMicroelectronics
October 10, 2005
Low Density Parity Check (LDPC) Forward Error Correction
draft-ietf-rmt-bb-fec-ldpc-00.txt
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Copyright Notice
Copyright (C) The Internet Society (2005).
Abstract
This document describes two Fully-Specified FEC Schemes, LDPC-
Staircase and LDPC-Triangle, and their application to the reliable
delivery of objects on packet erasure channels. These systematic FEC
codes belong to the well known class of ``Low Density Parity Check''
(LDPC) codes, and are large block FEC codes in these sense of
RFC3453.
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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 . . . . . . . . . . . . . . . . . . . . . . . . . . 10
5.1 General . . . . . . . . . . . . . . . . . . . . . . . . . 10
5.2 Determining the Maximum Source Block Length (B) . . . . . 10
5.3 Determining the Encoding Symbol Length (E) . . . . . . . . 11
5.4 Determining the Number of Encoding Symbols of a Block . . 11
5.5 Identifying the Symbols of an Encoding Symbol Group . . . 13
5.6 Pseudo Random Number Generator . . . . . . . . . . . . . . 16
6. Full Specification of the LDPC-Staircase Scheme . . . . . . . 18
6.1 General . . . . . . . . . . . . . . . . . . . . . . . . . 18
6.2 Parity Check Matrix Creation . . . . . . . . . . . . . . . 18
6.3 Encoding . . . . . . . . . . . . . . . . . . . . . . . . . 20
6.4 Decoding . . . . . . . . . . . . . . . . . . . . . . . . . 20
7. Full Specification of the LDPC-Triangle Scheme . . . . . . . . 21
7.1 General . . . . . . . . . . . . . . . . . . . . . . . . . 21
7.2 Parity Check Matrix Creation . . . . . . . . . . . . . . . 21
7.3 Encoding . . . . . . . . . . . . . . . . . . . . . . . . . 21
7.4 Decoding . . . . . . . . . . . . . . . . . . . . . . . . . 22
8. Security Considerations . . . . . . . . . . . . . . . . . . . 23
9. Intellectual Property . . . . . . . . . . . . . . . . . . . . 24
10. Acknowledgments . . . . . . . . . . . . . . . . . . . . . . 25
11. References . . . . . . . . . . . . . . . . . . . . . . . . . 26
11.1 Normative References . . . . . . . . . . . . . . . . . . . 26
11.2 Informative References . . . . . . . . . . . . . . . . . . 26
Authors' Addresses . . . . . . . . . . . . . . . . . . . . . . 27
A. Trivial Decoding Algoritm (Informative Only) . . . . . . . . . 28
Intellectual Property and Copyright Statements . . . . . . . . 29
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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 size several tens of thousands (or more) source
symbols. The present document introduces the Fully-Specified FEC
Encoding ID XX that is intended to be used with the "Low Density
Parity Check" (LDPC) Staircase FEC codes, and the Fully-Specified FEC
Encoding ID YY that is intended to be used with the "Low Density
Parity Check" (LDPC)-Triangle FEC codes [Roca04][Mac03]. Both
schemes belong the broad class of large block codes.
-- editor's note: This document makes use of the FEC Encoding ID
values XX and YY that will be specified after IANA assignment --
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.
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.
A publicly available reference implementation of these codes is
available and distributed under a GNU/LGPL license [LDPCrefimpl]. To
the best of our knowledge, there is no patent or patent application
identified as being used in the LDPC-Staircase and LDPC-Triangle FEC
schemes.
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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 source block length in symbols, i.e. the number of
source symbols of a source block
n denotes the encoding block length, i.e. the number of encoding
symbols generated for a source block
E denotes the encoding symbol length in bytes
B denotes the maximum source block length in symbols, i.e. the
maximum number of source symbols per source block
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 generated for any source
block
srand(s) denotes the initialization function of the pseudo-random
number generator, where s is the seed (s > 0)
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rand(m) denotes a pseudo-random number generator, that returns a
new random integer in [0; m-1] each time it is called
3.3 Abbreviations
This document uses the following abbreviations:
ESI Encoding Symbol ID
FEC OTI FEC Object Transmission Information
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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 (12 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^^12 blocks per object.
The Encoding Symbol ID (20 bit field) identifies which specific
encoding symbol generated from the source block is carried in the
packet payload. There is a maximum of 2^^20 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 en
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 thanks to a dedicated function, as explained in Section 5.5
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 (20 bits) |
+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
Figure 1: FEC Payload ID encoding format for FEC Encoding ID XX and
YY
4.2 FEC Object Transmission Information
4.2.1 Mandatory Elements
o FEC Encoding ID: the Fully-Specified FEC Schemes described in this
document use the FEC Encoding ID XX for LDPC-Staircase and FEC
Encoding ID YY for LDPC-Triangle.
4.2.2 Common Elements
The following elements MUST be used with the present FEC Scheme:
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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:
maximum transfer length = 2^^12 * B * E
For instance, if B=2^^19 (because of a code rate of 1/2,
Section 5.2), and if E=1024 bytes, then the maximum transfer
length is 2^^41 bytes.
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
o PRNG seed: The seed is a 32 bit value used to initialize the
Pseudo Random Number Generator (defined in Section 5.6). This
element is optional. Whether or not it is present in the FEC OTI
will be signaled in the associated encoding format through an
appropriate mechanism (see Section 4.2.4). When the PRNG seed is
not carried within the FEC OTI, it is assumed that encoder and
decoders use another way to communicate the information, or use a
fixed, predefined value.
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) |
+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
| 0 (not applicable) | Encoding Symbol Length (E) |
+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
| Max Source Block Length (B) |
+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
| Max Nb of Enc. Symbols (max_n) |
+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
. Optional PRNG seed .
+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
The HEL (Header Extension Length) indicates whether the optional PRNG
seed is present or not.
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-PRNG-seed (optional)
When no PRNG seed is to be carried in the FEC OTI, the sender simply
omits the FEC-OTI-PRNG-seed element.
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5. Procedures
This section defines procedures that are common to FEC Encoding IDs
XX and YY.
5.1 General
The B (maximum source block length in symbols) and E (encoding symbol
length in bytes) parameters are first determined, as explained in the
following sections.
The source object is then partitioned using the block partitioning
algorithm specified in [fec-bb-revised]. To that purpose, the B, L
(object transfer length in bytes), and E arguments are provided. As
a result, 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.
For each block the actual number of encoding symbols is determined,
as explained in the following section.
Then, FEC encoding and decoding can be done block per block,
independently. To that purpose, a parity check matrix is created,
that forms a system of linear equations between the repair and source
symbols of a given block, where the basic operator is XOR.
This 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 of the system. The only difference between the
LDPC-Staircase and LDPC-Triangle schemes is the construction of the
right sub-matrix.
The following sections detail how the B, E, and n parameters are
determined (respectively Section 5.2, Section 5.3 and Section 5.4),
how encoding symbol groups are created (Section 5.5), and finally
specify the PRNG (Section 5.6).
5.2 Determining the Maximum Source Block Length (B)
The B parameter (maximum source block length in symbols) depends on
several parameters: the code rate (rate), the Encoding Symbol ID
field length of the FEC Payload ID (20 bits), as well as possible
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internal codec limitations.
The B parameter cannot be larger than the following values, derived
from the FEC Payload ID limitations, for a given code rate:
max1_B = 2 ^^ (20 - ceil(Log2(1/rate)))
Some common max1_B values are:
o rate == 1 (no repair symbols): max_B = 2 ^^ 20 = 1,048,576
o 1 > rate >= 1/2: max1_B = 2 ^^ 19 = 524,288 symbols
o 1/2 > rate >= 1/4: max1_B = 2 ^^ 18 = 262,144 symbols
o 1/4 > rate >= 1/8: max1_B = 2 ^^ 17 = 131,072 symbols
Additionally, a codec MAY impose other limitations on the maximum
block size. This is the case for instance when the codec uses
internally 16 bit integers to store the Encoding Symbol ID, since it
does not enable to store all the possible values of a 20 bit field.
Other limitations (e.g. available working memory) may also apply.
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.
5.3 Determining the Encoding Symbol Length (E)
-- editor's note: the E parameter is a function of the object
transfer length. Since LDPC codes are known to offer better
protection for large blocks, the smaller the object, the smaller E
should be in order to increase the number of symbols the object is
composed of. The optimal values that should be used for E as a
function of the object transfer length are under study. --
Note that this step is only required at the coder, since the E
parameter is communicated to the decoder through the FEC OTI.
5.4 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.
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AT A SENDER:
Input:
B Maximum source block length, for any source block. Section 5.2
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:
max_n Maximum number of encoding symbols generated for any source
block
n Number of encoding symbols generated for this source block
Algorithm:
a. max_n = floor(B / R)
b. 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:
a. n = floor(k * max_n / B)
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5.5 Identifying the Symbols of an Encoding Symbol Group
When multiple encoding symbols are sent in the same packet, the FEC
Payload ID information of the packet MUST refer to the first encoding
symbol. It MUST then 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 all be either source symbols, or repair symbols. Therefore
only source packets and repair packets are permitted, not mixed
ones.
o are identified by a function, ESIs_of_group(), that takes as
argument:
* for a sender, the index of the Encoding Symbol Group (i.e.
packet) that the application wants to create,
* for a receiver, the ESI information contained in the FEC
Payload ID.
and returns the list of G Encoding Symbol IDs that will be packed
together. In case of a source packet, the G source symbols are
taken consecutively. In case of a repair packet, the G repair
symbols are chosen randomly, as explained below.
The system must first be initialized by creating a random permutation
of the n-k indexes. This initialization function MUST be called
immediately after creating the parity check matrix. More precisely,
since the PRNG seed is not re-initialized, no call to the PRNG
function must have happened between the time the parity check matrix
has been initialized and the time the following initialization
function is called. This is true both at a sender and at a receiver.
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/*
* Use only in case G > 1, i.e. when encoding symbol
* groups are actually needed.
*/
initialize_tables ()
{
int i;
int randInd;
/* initialize the two tables that map ID
* (i.e. ESI-k) to/from TxSequence:
* - IDtoTxseq
* - txseqToID
*/
for (i = 0; i < n - k; i++) {
IDtoTxseq[i]=i;
txseqToID[i]=i;
}
/* now randomize everything */
for (i = 0; i < n - k; i++) {
randInd = rand(n - k);
backup = IDtoTxseq[i];
IDtoTxseq[i] = IDtoTxseq[randInd];
IDtoTxseq[randInd] = backup;
txseqToID[IDtoTxseq[i]] = i;
txseqToID[IDtoTxseq[randInd]] = randInd;
}
return;
}
It is then possible, at the sender, to determine the sequence of G
Encoding Symbol IDs that will be part of the group.
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/*
* Use only in case G > 1, i.e. when encoding symbol
* groups are actually needed.
* PktIdx (IN): index of the packet, in {0..ceil(T/G)} range
* ESIs[] (OUT): list of ESI of the packet
*/
ESIs_of_group (int PktIdx,
ESI_t ESIs[])
{
int i;
if (is_source_packet(PktIdx) == true) {
/* this is a source packet */
ESIs[0] = (PktIdx * G) % k;
for (i = 0; i < G; i++) {
ESIs[i] = ESIs[0] + i;
}
} else {
/* this is a repair packet */
for (i = 0; i < G; i++) {
ESIs[i] =
k +
txseqToID[(i + (PktIdx - nbSourcePkts) * G)
% (n - k)];
}
}
return;
}
Similarly, upon receiving an Encoding Symbol Group (i.e. packet), a
receiver can determine the sequence of G Encoding Symbol IDs from the
first ESI, esi0, that is contained in the FEC Payload ID.
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/*
* Use only in case G > 1, i.e. when encoding symbol
* groups are actually needed.
* esi0 (IN): : ESI contained in the FEC Payload ID
* ESIs[] (OUT): list of ESI of the packet
*/
ESIs_of_group (ESI_t esi0,
ESI_t ESIs[])
{
int i;
if (is_source_packet(esi0) == true) {
/* this is a source packet */
for (i = 0; i < G; i++) {
ESIs[i] = (esi0 + i) % k;
}
} else {
/* this is a repair packet */
for (i = 0; i < G; i++) {
ESIs[i] =
k +
txseqToID[(i + IDtoTxseq[esi0 - k])
% (n - k)];
}
}
}
5.6 Pseudo Random Number Generator
The present FEC Encoding ID relies on a pseudo-random number
generator (PRNG) that must be fully specified, in particular in order
to enable the receivers and the senders to build the same parity
check matrix. The minimal standard generator [Park88] is used. It
defines a simple multiplicative congruential algorithm: Ij+1 = A * Ij
(modulo M), with the following choices: A = 7^^5 = 16807 and M =
2^^31 - 1 = 2147483647. The PRNG must be initialized with a seed
that is strictly greater than 0.
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double seed; /* assumed initialized with a seed > 0 */
#define A 16807.0
#define M 2147483647.0
#define Q 127773.0 /* M div A */
#define R 2836.0 /* M mod A */
/*
* Initialize the PRNG with a seed > 0.
*/
void srand (int s)
{
if (s > 0) seed = s;
else exit(-1);
}
/*
* Returns a random integer in [0; maxv-1]
* derived from [Park88].
*/
int rand (int maxv)
{
double lo, hi, test;
double rand;
hi = (int)(seed / Q);
lo = seed - Q*hi;
test = A*lo - R*hi;
if (test > 0.0)
seed = test;
else
seed = test + M;
rand = seed / M;
if (rand == 1.0)
return 0;
else
return ((int)(rand * (double)maxv));
}
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6. Full Specification of the LDPC-Staircase Scheme
6.1 General
LDPC-Staircase is identified by the Fully-Specified FEC Encoding ID
XX.
LDPC-Staircase is based on the pseudo-random number generator
specified in Section 5.6. Therefore the seed used to initiate the
PRNG is an instance-specific FEC Object Transmission Information
optional element. When the PRNG seed is not carried within the FEC
OTI, it is assumed that encoder and decoders use another way to
communicate the information, or use a fixed, predefined value.
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);
}
Note that just after creating this parity check matrix, when encoding
symbol groups are used, the function initializing the two random
permutation tables (Section 5.5) MUST be called. This is true both
at a sender and at a receiver.
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 MUST follow the natural repair symbol order, i.e.
generate repair symbol with ESI i before symbol ESI i+1 and MUST
start with the first repair symbol.
6.4 Decoding
Decoding basically consists in solving a system of n-k linear
equations whose variables are the source an repair symbols. Of
course, the final goal is to recover the value of source symbols
only.
To that purpose, many techniques are possible. One of them is the
following trivial algorithm: 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 the remaining linear equations and
reiterate. The value of several variables can therefore be found
recursively. Applied to LDPC FEC codes working over an erasure
packet, the parity check matrix defines a set of linear equations
whose variables are the source symbols and repair symbols. Receiving
or decoding a symbol is equivalent to having the value of a variable.
Appendix A sketches a possible implementation of this algorithm.
The pivot of Gauss technique, as well as derived versions, is another
possibility.
Because interoperability does not depend on the decoding algorithm
used, the current document does not recommand any particular
technique. This choice is left to the codec implementer.
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7. Full Specification of the LDPC-Triangle Scheme
7.1 General
LDPC-Triangle is identified by the Fully-Specified FEC Encoding ID
YY.
LDPC-Triangle is based on the pseudo-random number generator
specified in Section 5.6. Therefore the seed used to initiate the
PRNG is an instance-specific FEC Object Transmission Information
optional element. When the PRNG seed is not carried within the FEC
OTI, it is assumed that encoder and decoders use another way to
communicate the information, or use a fixed, predefined value.
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);
}
}
}
}
Note that just after creating this parity check matrix, when encoding
symbol groups are used, the function initializing the two random
permutation tables (Section 5.5) MUST be called. This is true both
at a sender and at a receiver.
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
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associated equation, plus the (previously built) repair packets
specified in the triangle. Therefore encoding MUST follow the
natural repair symbol order, i.e. generate repair symbol with ESI i
before symbol ESI i+1 and MUST start with the first repair symbol.
7.4 Decoding
Decoding basically consists in solving a system of n-k linear
equations, whose variables are the source an repair symbols. Of
course, the final goal is to recover the value of source symbols
only. To that purpose, many techniques are possible, as explained in
Section 6.4.
Because interoperability does not depend on the decoding algorithm
used, the current document does not recommand any particular
technique. This choice is left to the codec implementer.
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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
To the best of our knowledge, there is no patent or patent
application identified as being used in the LDPC-Staircase and LDPC-
Triangle FEC schemes. Yet other LDPC codes and associated techniques
MAY be covered by Intellectual Property Rights.
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10. Acknowledgments
Section 5.4 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-01.txt draft-ietf-rmt-fec-bb-revised-
01.txt, September 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.
[Park88] Park, S. and K. Miller, "Random Number Generators: Good
Ones are Hard to Find", Communications of the ACM Vol 31,
No 10, pp.1192-1201, 1988.
[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. Trivial Decoding Algoritm (Informative Only)
A trivial decoding algorithm is the following:
Initialization: allocate a partial sum buffer, partial_sum_i, for
each line i, and reset 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 s_l in step 3:
+ remove the last entry in equation j,
+ copy partial_sum_j to the buffer associate with symbol s_l,
+ goto step 1 with the newly created symbol s_l
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