RMT V. Roca
Internet-Draft INRIA
Intended status: Experimental C. Neumann
Expires: November 8, 2007 Thomson Research
D. Furodet
STMicroelectronics
May 7, 2007
Low Density Parity Check (LDPC) Staircase and Triangle Forward Error
Correction (FEC) Schemes
draft-ietf-rmt-bb-fec-ldpc-06.txt
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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 the sense of RFC3453.
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Table of Contents
1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 4
2. Requirements notation . . . . . . . . . . . . . . . . . . . . 5
3. Definitions, Notations and Abbreviations . . . . . . . . . . . 6
3.1. Definitions . . . . . . . . . . . . . . . . . . . . . . . 6
3.2. Notations . . . . . . . . . . . . . . . . . . . . . . . . 6
3.3. Abbreviations . . . . . . . . . . . . . . . . . . . . . . 7
4. Formats and Codes . . . . . . . . . . . . . . . . . . . . . . 8
4.1. FEC Payload IDs . . . . . . . . . . . . . . . . . . . . . 8
4.2. FEC Object Transmission Information . . . . . . . . . . . 8
4.2.1. Mandatory Element . . . . . . . . . . . . . . . . . . 8
4.2.2. Common Elements . . . . . . . . . . . . . . . . . . . 8
4.2.3. Scheme-Specific Elements . . . . . . . . . . . . . . . 9
4.2.4. Encoding Format . . . . . . . . . . . . . . . . . . . 9
5. Procedures . . . . . . . . . . . . . . . . . . . . . . . . . . 12
5.1. General . . . . . . . . . . . . . . . . . . . . . . . . . 12
5.2. Determining the Maximum Source Block Length (B) . . . . . 13
5.3. Determining the Encoding Symbol Length (E) and Number
of Encoding Symbols per Group (G) . . . . . . . . . . . . 13
5.4. Determining the Number of Encoding Symbols of a Block . . 15
5.5. Identifying the Symbols of an Encoding Symbol Group . . . 16
5.6. Pseudo Random Number Generator . . . . . . . . . . . . . . 19
6. Full Specification of the LDPC-Staircase Scheme . . . . . . . 21
6.1. General . . . . . . . . . . . . . . . . . . . . . . . . . 21
6.2. Parity Check Matrix Creation . . . . . . . . . . . . . . . 21
6.3. Encoding . . . . . . . . . . . . . . . . . . . . . . . . . 22
6.4. Decoding . . . . . . . . . . . . . . . . . . . . . . . . . 23
7. Full Specification of the LDPC-Triangle Scheme . . . . . . . . 24
7.1. General . . . . . . . . . . . . . . . . . . . . . . . . . 24
7.2. Parity Check Matrix Creation . . . . . . . . . . . . . . . 24
7.3. Encoding . . . . . . . . . . . . . . . . . . . . . . . . . 24
7.4. Decoding . . . . . . . . . . . . . . . . . . . . . . . . . 25
8. Security Considerations . . . . . . . . . . . . . . . . . . . 26
9. IANA Considerations . . . . . . . . . . . . . . . . . . . . . 27
10. Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . 28
11. References . . . . . . . . . . . . . . . . . . . . . . . . . . 29
11.1. Normative References . . . . . . . . . . . . . . . . . . . 29
11.2. Informative References . . . . . . . . . . . . . . . . . . 29
Appendix A. Trivial Decoding Algorithm (Informative Only) . . . . 31
Authors' Addresses . . . . . . . . . . . . . . . . . . . . . . . . 33
Intellectual Property and Copyright Statements . . . . . . . . . . 34
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1. Introduction
RFC 3453 [3] 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 3
that is intended to be used with the LDPC-Staircase FEC codes, and
the Fully-Specified FEC Encoding ID 4 that is intended to be used
with the LDPC-Triangle FEC codes [6][9]. Both schemes belong to the
broad 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
repair symbols.
Since the encoder and decoder must operate on the same parity check
matrix, 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 [8].
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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 [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:
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 "code rate", i.e. the k/n ratio
max_n denotes the maximum number of encoding symbols generated for
any source block
H denotes the parity check matrix
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
FPI: FEC Payload ID
LDPC: Low Density Parity Check
PRNG: Pseudo Random Number Generator
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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 packet
payload is(are) generated. There are a maximum of 2^^12 blocks
per object. Source block numbering starts at 0.
The Encoding Symbol ID (20 bit field) identifies which encoding
symbol(s) generated from the source block is(are) carried in the
packet payload. There are 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 (k to n-k-1) 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 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 3 and 4
4.2. FEC Object Transmission Information
4.2.1. Mandatory Element
o FEC Encoding ID: the LDPC-Staircase and LDPC-Triangle Fully-
Specified FEC Schemes use respectively the FEC Encoding ID 3
(Staircase) and 4 (Triangle).
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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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 (or 2 TB). The upper limit, with symbols of
size 2^^16-1 bytes and a code rate larger or equal to 1/2, amounts
to 2^^47 bytes (or 128 TB).
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. There are
some restrictions on the maximum B value, as explained in
Section 5.2.
o Max-Number-of-Encoding-Symbols (max_n): a non-negative integer
indicating the maximum number of encoding symbols generated for
any source block. There are some restrictions on the maximum
max_n value. In particular max_n is at most equal to 2^^20.
Section 5 explains how to define the values of each of these
elements.
4.2.3. Scheme-Specific Elements
The following elements MUST be defined with the present FEC Scheme:
o G: a non-negative integer indicating the number of encoding
symbols per group (i.e. per packet). The default value is 1,
meaning that each packet contains exactly one symbol. Values
greater than 1 can also be defined, as explained in Section 5.3.
o PRNG seed: the seed is a 32 bit unsigned integer between 1 and
0x7FFFFFFE (i.e. 2^^31-2) inclusive. This value is used to
initialize the Pseudo Random Number Generator (Section 5.6). This
element is optional. Whether or not it is present in the FEC OTI
is signaled in the associated encoding format through an
appropriate mechanism (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.
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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 [13] or NORM [15] 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 (=4 or 5) | |
+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+ +
| Transfer-Length (L) |
+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
| Encoding Symbol Length (E) | G | B (MSB) |
+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
| B (LSB) | Max Nb of Enc. Symbols (max_n) |
+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
. Optional PRNG seed .
+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
Figure 2: EXT_FTI Header for FEC Encoding ID 3 and 4.
In particular:
o The HEL (Header Extension Length) indicates whether the optional
PRNG seed is present (HEL=5) or not (HEL=4).
o The Transfer-Length (L) field size (48 bits) is larger than the
size required to store the maximum transfer length (Section 4.2.2)
for field alignment purposes.
o The Maximum-Source-Block-Length (B) field (20 bits) is split into
two parts: the 8 most significant bits (MSB) are in the third 32-
bit word of the EXT_FTI, and the remaining 12 least significant
bits (LSB) are in the fourth 32-bit word.
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 [14], the following XML attributes must be described
for the associated object:
o FEC-OTI-FEC-Encoding-ID
o FEC-OTI-Transfer-length
o FEC-OTI-Encoding-Symbol-Length
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o FEC-OTI-Maximum-Source-Block-Length
o FEC-OTI-Max-Number-of-Encoding-Symbols
o FEC-OTI-Scheme-Specific-Info
The FEC-OTI-Scheme-Specific-Info contains the string resulting from
the Base64 encoding (in the XML Schema xs:base64Binary sense) of the
following value:
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
+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
| PRNG seed |
+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
| G |
+-+-+-+-+-+-+-+-+
Figure 3: FEC OTI Scheme Specific Information to be Included in the
FDT Instance for FEC Encoding ID 3 and 4.
When no PRNG seed is to be carried in the FEC OTI, the seed field
MUST be set to 0 (which is not a valid seed value). Otherwise the
seed field contains a valid value as explained in Section 4.2.3.
After Base64 encoding, the 5 bytes of the FEC OTI Scheme Specific
Information are transformed into a string of 8 printable characters
(in the 64-character alphabet) and added to the FEC-OTI-Scheme-
Specific-Info attribute.
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5. Procedures
This section defines procedures that are common to FEC Encoding IDs 3
and 4.
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 [2]. 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 source and repair
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.
When the parity symbols have been created, the sender will transmit
source and parity symbols. The way this transmission occurs can
largely impact the erasure recovery capabilities of the LDPC-* FEC.
In particular, sending parity symbols in sequence is suboptimal.
Instead it is usually recommended the shuffle these symbols. The
interested reader will find more details in [7].
The following sections detail how the B, E, and n parameters are
determined (respectively in Section 5.2, Section 5.3 and
Section 5.4), how encoding symbol groups are created (Section 5.5),
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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
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 symbol): max1_B = 2^^20 = 1,048,576
o 1/2 <= rate < 1: max1_B = 2^^19 = 524,288 symbols
o 1/4 <= rate < 1/2: max1_B = 2^^18 = 262,144 symbols
o 1/8 <= rate < 1/4: 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 unsigned integers to store the Encoding Symbol ID,
since it does not enable to store all the possible values of a 20 bit
field. In that case, if for instance 1/2 <= rate < 1, then the
maximum source block length is 2^^15. Other limitations may also
apply, for instance because of a limited working memory size. This
decision MUST 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) and Number of Encoding
Symbols per Group (G)
The E parameter usually depends on the maximum transmission unit on
the path (PMTU) from the source to the receivers. In order to
minimize the protocol header overhead (e.g. the LCT/UDP/IPv4 or IPv6
headers in case of ALC), E is chosen as large as possible. In that
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case, E is chosen so that the size of a packet composed of a single
symbol (G=1) remains below but close to the PMTU.
However other considerations can exist. For instance, the E
parameter can be made a function of the object transfer length.
Indeed, LDPC codes are known to offer better protection for large
blocks. In case of small objects, it can be advantageous to reduce
the encoding symbol length (E) in order to artificially increase the
number of symbols, and therefore the block size.
In order to minimize the protocol header overhead, several symbols
can be grouped in the same Encoding Symbol Group (i.e. G > 1).
Depending on how many symbols are grouped (G) and on the packet loss
rate (G symbols are lost for each packet erasure), this strategy
might or might not be appropriate. A balance must therefore be
found.
The current specification does not mandate any value for either E or
G. The current specification only provides an example of possible
choices for E and G. Note that this choice is done by the sender.
Then the E and G parameters are communicated to the receivers thanks
to the FEC OTI.
Example:
First define the target packet payload size, pkt_sz (at most equal to
the PMTU minus the size of the various protocol headers). The pkt_sz
must be chosen in such a way that the symbol size is an integer.
This can require that pkt_sz be a multiple of 4, 8 or 16 (see the
table below). Then calculate the number of packets in the object:
nb_pkts = ceil(L / pkt_sz). Finally, thanks to nb_pkts, use the
following table to find a possible G value.
+------------------------+----+-------------+-------------------+
| Number of packets | G | Symbol size | k |
+------------------------+----+-------------+-------------------+
| 4000 <= nb_pkts | 1 | pkt_sz | 4000 <= k |
| | | | |
| 1000 <= nb_pkts < 4000 | 4 | pkt_sz / 4 | 4000 <= k < 16000 |
| | | | |
| 500 <= nb_pkts < 1000 | 8 | pkt_sz / 8 | 4000 <= k < 8000 |
| | | | |
| 1 <= nb_pkts < 500 | 16 | pkt_sz / 16 | 16 <= k < 8000 |
+------------------------+----+-------------+-------------------+
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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.
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. It is provided by the user, for instance
when starting a FLUTE sending application. It is expressed as a
floating point value. The rate value must be such that the
resulting number of encoding symbols per block is at most equal to
2^^20 (Section 4.1).
Output:
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^^20) then return an error ("invalid code rate");
(NB: if B has been defined as explained in Section 5.2, this error
should never happen)
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
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k: Given by the source blocking algorithm
Output:
n: Number of encoding symbols generated for this source block
Algorithm:
n = floor(k * max_n / B);
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, sender(resp.
receiver)_find_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 a list of G Encoding Symbol IDs. In case of a source
packet, the G Encoding Symbol IDs are chosen consecutively, by
incrementing the ESI. In case of a repair packet, the G repair
symbols are chosen randomly, as explained below.
o are stored in sequence in the packet, without any padding. In
other words, the last byte of the i-th symbol is immediately
followed by the first byte of (i+1)-th symbol.
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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int *txseqToID;
int *IDtoTxseq;
/*
* Initialization function.
* Warning: use only when G > 1.
*/
void
initialize_tables ()
{
int i;
int randInd;
int backup;
txseqToID = malloc((n-k) * sizeof(int));
IDtoTxseq = malloc((n-k) * sizeof(int));
/* initialize the two tables that map ID
* (i.e. ESI-k) to/from TxSequence. */
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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/*
* Determine the sequence of ESIs for the packet under construction
* at a sender.
* Warning: use only when G > 1.
* PktIdx (IN): index of the packet, in
* {0..ceil(k/G)+ceil((n-k)/G)} range
* ESIs[] (OUT): list of ESIs for the packet
*/
void
sender_find_ESIs_of_group (int PktIdx,
ESI_t ESIs[])
{
int i;
if (PktIdx < nbSourcePkts) {
/* this is a source packet */
ESIs[0] = PktIdx * G;
for (i = 1; i < G; i++) {
ESIs[i] = (ESIs[0] + i) % k;
}
} 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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/*
* Determine the sequence of ESIs for the packet received.
* Warning: use only when G > 1.
* esi0 (IN): : ESI contained in the FEC Payload ID
* ESIs[] (OUT): list of ESIs for the packet
*/
void
receiver_find_ESIs_of_group (ESI_t esi0,
ESI_t ESIs[])
{
int i;
if (esi0 < k) {
/* this is a source packet */
ESIs[0] = esi0;
for (i = 1; 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 [10] 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. Several implementations of this PRNG are
known and discussed in the literature. All of them provide the same
sequence of pseudo random numbers. A validation criteria of such a
PRNG is the following: if seed = 1, then the 10,000th value returned
MUST be equal to 1043618065.
The following implementation uses the Park and Miller algorithm with
the optimization suggested by D. Carta in [11].
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unsigned long seed;
/*
* Initialize the PRNG with a seed between
* 1 and 0x7FFFFFFE (i.e. 2^^31-2) inclusive.
*/
void srand (unsigned long s)
{
if ((s > 0) && (s < 0x7FFFFFFF))
seed = s;
else
exit(-1);
}
/*
* Returns a random integer in [0; maxv-1]
* Derived from rand31pmc, Robin Whittle,
* September 20th, 2005.
* http://www.firstpr.com.au/dsp/rand31/
* 16807 multiplier constant (7^^5)
* 0x7FFFFFFF modulo constant (2^^31-1)
* The inner PRNG produces a value between 1 and
* 0x7FFFFFFE (2^^31-2) inclusive.
* This value is then scaled between 0 and maxv-1
* inclusive.
*/
unsigned long
rand (unsigned long maxv)
{
unsigned long hi, lo;
lo = 16807 * (seed & 0xFFFF);
hi = 16807 * (seed >> 16); /* binary shift to right */
lo += (hi & 0x7FFF) < < 16; /* binary shift to left */
lo += hi >> 15;
if (lo > 0x7FFFFFFF)
lo -= 0x7FFFFFFF;
seed = (long)lo;
/* don't use modulo, least significant bits are less random
* than most significant bits [Numerical Recipes in C] */
return ((unsigned long)
((double)seed * (double)maxv / (double)0x7FFFFFFF));
}
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6. Full Specification of the LDPC-Staircase Scheme
6.1. General
The LDPC-Staircase scheme is identified by the Fully-Specified FEC
Encoding ID 3.
The PRNG used by the LDPC-Staircase scheme must be initialized by a
seed. This PRNG seed is an optional instance-specific FEC OTI
attribute (Section 4.2.3). When this PRNG seed is not carried within
the FEC OTI, it is assumed that encoder and decoders either use
another way to communicate the seed value or use a fixed, predefined
value.
6.2. Parity Check Matrix Creation
The LDPC-Staircase matrix can be divided into two parts: the left
side of the matrix defines in which equations the source symbols are
involved; the right side of the matrix defines in which equations the
repair symbols are involved.
The left side is generated with the following algorithm:
/*
* Derived from: "Software for Low Density Parity Check Codes"
* Version of 2001-11-18, Radford M. Neal, Univ. of Toronto.
* Copyright (c) 1995, 1996, 2000, 2001 by Radford M. Neal
* http://www.cs.toronto.edu/~radford/ldpc.software.html
*/
/* initialize a list of all possible choices in order 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));
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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 "1s".
* This is needed when the code rate is smaller than 2/5. */
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);
}
}
The right side (the staircase) is generated by the following
algorithm:
matrix_insert_entry(0, k); /* first row */
for (i = 1; i < n-k; i++) { /* for the following rows */
matrix_insert_entry(i, k+i); /* identity */
matrix_insert_entry(i, k+i-1); /* staircase */
}
Note that just after creating this parity check matrix, when encoding
symbol groups are used (i.e. G > 1), 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
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symbols in the associated equation, plus the previous repair symbol
(except for the first repair symbol). Therefore encoding MUST follow
the natural repair symbol order: start with the first repair symbol,
and generate repair symbol with ESI i before symbol ESI i+1.
6.4. Decoding
Decoding basically consists in solving a system of n-k linear
equations whose variables are the n source and repair symbols. Of
course, the final goal is to recover the value of the k source
symbols only.
To that purpose, many techniques are possible. One of them is the
following trivial algorithm [12]: 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
channel, 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.
A Gaussian elimination (or any optimized derivative) is another
possible decoding technique. Hybrid solutions that start by using
the trivial algorithm above and finish with a Gaussian elimination
are also possible.
Because interoperability does not depend on the decoding algorithm
used, the current document does not recommend any particular
technique. This choice is left to the codec developer.
However choosing a decoding technique will have great practical
impacts. It will impact the erasure capabilities: a Gaussian
elimination enables to solve the system with a smaller number of
known symbols compared to the trivial technique. It will also impact
the CPU load: a Gaussian elimination requires more processing than
the above trivial algorithm. Depending on the target use case, the
codec developer will favor one feature or the other.
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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 4.
The PRNG used by the LDPC-Triangle scheme must be initialized by a
seed. This PRNG seed is an optional instance-specific FEC OTI
attribute (Section 4.2.3). When this PRNG seed is not carried within
the FEC OTI, it is assumed that encoder and decoders either use
another way to communicate the seed value or use a fixed, predefined
value.
7.2. Parity Check Matrix Creation
The LDPC-Triangle matrix can be divided into two parts: the left side
of the matrix defines in which equations the source symbols are
involved; the right side of the matrix defines in which equations the
repair symbols are involved.
The left side is generated with the same algorithm as that of LDPC-
Staircase (Section 6.2).
The right side (the triangle) is generated with the following
algorithm:
matrix_insert_entry(0, k); /* first row */
for (i = 1; i < n-k; i++) { /* for the following rows */
matrix_insert_entry(i, k+i); /* identity */
matrix_insert_entry(i, k+i-1); /* staircase */
/* now fill the triangle */
j = i-1;
for (l = 0; l < j; l++) { /* limit the # of "1s" added */
j = rand(j);
matrix_insert_entry(i, k+j);
}
}
Note that just after creating this parity check matrix, when encoding
symbol groups are used (i.e. G > 1), 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
Here also repair symbol creation is straightforward: each repair
symbol is equal to the sum of all source symbols in the associated
equation, plus the repair symbols in the triangle. Therefore
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encoding MUST follow the natural repair symbol order: start with the
first repair symbol, and generate repair symbol with ESI i before
symbol ESI i+1.
7.4. Decoding
Decoding basically consists in solving a system of n-k linear
equations, whose variables are the n source and repair symbols. Of
course, the final goal is to recover the value of the k 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 recommend any particular
technique. This choice is left to the codec implementer.
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8. Security Considerations
Data delivery can be subject to denial-of-service attacks by
attackers which send corrupted packets that are accepted as
legitimate by receivers. This is particularly a concern for
multicast delivery because a corrupted packet may be injected into
the session close to the root of the multicast tree, in which case
the corrupted packet will arrive at many receivers. This is
particularly a concern for the FEC building block because the use of
even one corrupted packet containing encoding data may result in the
decoding of an object that is completely corrupted and unusable. It
is thus RECOMMENDED that source authentication and integrity checking
are applied to decoded objects before delivering objects to an
application. For example, a SHA-1 hash [4] of an object may be
appended before transmission, and the SHA-1 hash is computed and
checked after the object is decoded but before it is delivered to an
application. Source authentication SHOULD be provided, for example
by including a digital signature verifiable by the receiver computed
on top of the hash value. It is also RECOMMENDED that a packet
authentication protocol such as TESLA [5] be used to detect and
discard corrupted packets upon arrival. Furthermore, it is
RECOMMENDED that Reverse Path Forwarding checks be enabled in all
network routers and switches along the path from the sender to
receivers to limit the possibility of a bad agent successfully
injecting a corrupted packet into the multicast tree data path.
Another security concern is that some FEC information may be obtained
by receivers out-of-band in a session description, and if the session
description is forged or corrupted then the receivers will not use
the correct protocol for decoding content from received packets. To
avoid these problems, it is RECOMMENDED that measures be taken to
prevent receivers from accepting incorrect session descriptions,
e.g., by using source authentication to ensure that receivers only
accept legitimate session descriptions from authorized senders.
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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 [2].
This document assigns the Fully-Specified FEC Encoding ID 3 under the
"ietf:rmt:fec:encoding" name-space to "LDPC Staircase Codes".
This document assigns the Fully-Specified FEC Encoding ID 4 under the
"ietf:rmt:fec:encoding" name-space to "LDPC Triangle Codes".
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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, Laurent Fazio, Aurelien
Francillon and Shao Wenjian for their comments.
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11. References
11.1. Normative References
[1] Bradner, S., "Key words for use in RFCs to Indicate Requirement
Levels", RFC 2119, BCP 14, March 1997.
[2] Watson, M., Luby, M., and L. Vicisano, "Forward Error
Correction (FEC) Building Block",
draft-ietf-rmt-fec-bb-revised-07.txt (work in progress),
April 2007.
[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.
[4] "HMAC: Keyed-Hashing for Message Authentication", RFC 2104,
February 1997.
[5] "Timed Efficient Stream Loss-Tolerant Authentication (TESLA):
Multicast Source Authentication Transform Introduction",
RFC 4082, June 2005.
11.2. Informative References
[6] 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.
[7] Neumann, C., Roca, V., Francillon, A., and D. Furodet, "Impacts
of Packet Scheduling and Packet Loss Distribution on FEC
Performances: Observations and Recommendations", ACM CoNEXT'05
Conference, Toulouse, France (an extended version is available
as INRIA Research Report RR-5578), October 2005.
[8] Roca, V., Neumann, C., and J. Laboure, "LDPC-Staircase/
LDPC-Triangle Codec Reference Implementation", INRIA Rhone-
Alpes and STMicroelectronics,
http://planete-bcast.inrialpes.fr/.
[9] MacKay, D., "Information Theory, Inference and Learning
Algorithms", Cambridge University Press, ISBN: 0521642981,
2003.
[10] 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.
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[11] Carta, D., "Two Fast Implementations of the Minimal Standard
Random Number Generator", Communications of the ACM, Vol. 33,
No. 1, pp.87-88, January 1990.
[12] Zyablov, V. and M. Pinsker, "Decoding Complexity of Low-Density
Codes for Transmission in a Channel with Erasures", Translated
from Problemy Peredachi Informatsii, Vol.10, No. 1, pp.15-28,
January-March 1974.
[13] Luby, M., Watson, M., and L. Vicisano, "Asynchronous Layered
Coding (ALC) Protocol Instantiation",
draft-ietf-rmt-pi-alc-revised-04.txt (work in progress),
February 2007.
[14] Paila, T., Walsh, R., Luby, M., Lehtonen, R., and V. Roca,
"FLUTE - File Delivery over Unidirectional Transport",
draft-ietf-rmt-flute-revised-03.txt (work in progress),
January 2007.
[15] Adamson, B., Bormann, C., Handley, M., and J. Macker,
"Negative-acknowledgment (NACK)-Oriented Reliable Multicast
(NORM) Protocol", draft-ietf-rmt-pi-norm-revised-04.txt (work
in progress), March 2007.
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Appendix A. Trivial Decoding Algorithm (Informative Only)
A trivial decoding algorithm is sketched below (please see [8] for
the details omitted here):
Initialization: allocate a table partial_sum[n-k] of buffers, each
buffer being of size the symbol size. There's one
entry per equation since the buffers are meant to
store the partial sum of each equation; Reset all
the buffers to zero;
/*
* For each newly received or decoded symbol, try to make progress
* in the decoding of the associated source block.
* NB: in case of a symbol group (G>1), this function is called for
* each symbol of the received packet.
* NB: a callback function indicates to the caller that new symbol(s)
* has(have) been decoded.
* new_esi (IN): ESI of the new symbol received or decoded
* new_symb (IN): Buffer of the new symbol received or decoded
*/
void
decoding_step(ESI_t new_esi,
symbol_t *new_symb)
{
If (new_symb is an already decoded or received symbol) {
Return; /* don't waste time with this symbol */
}
If (new_symb is the last missing source symbol) {
Remember that decoding is finished;
Return; /* work is over now... */
}
Create an empty list of equations having symbols decoded
during this decoding step;
/*
* First add this new symbol to the partial sum of all the
* equations where the symbol appears.
*/
For (each equation eq in which new_symb is a variable and
having more than one unknown variable) {
Add new_symb to partial_sum[eq];
Remove entry(eq, new_esi) from the H matrix;
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If (the new degree of equation eq == 1) {
/* a new symbol can be decoded, remember the
* equation */
Append eq to the list of equations having symbols
decoded during this decoding step;
}
}
/*
* Then finish with recursive calls to decoding_step() for each
* newly decoded symbol.
*/
For (each equation eq in the list of equations having symbols
decoded during this decoding step) {
/*
* Because of the recursion below, we need to check that
* decoding is not finished, and that the equation is
* __still__ of degree 1
*/
If (decoding is finished) {
break; /* exit from the loop */
}
If ((degree of equation eq == 1) {
Let dec_esi be the ESI of the newly decoded symbol in
equation eq;
Remove entry(eq, dec_esi);
Allocate a buffer, dec_symb, for this symbol and
copy partial_sum[eq] to dec_symb;
Inform the caller that a new symbol has been
decoded via a callback function;
/* finally, call this function recursively */
decoding_step(dec_esi, dec_symb);
}
}
Free the list of equations having symbols decoded;
Return;
}
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Authors' Addresses
Vincent Roca
INRIA
655, av. de l'Europe
Inovallee; Montbonnot
ST ISMIER cedex 38334
France
Email: vincent.roca@inrialpes.fr
URI: http://planete.inrialpes.fr/~roca/
Christoph Neumann
Thomson Research
46, Quai A. Le Gallo
Boulogne Cedex 92648
France
Email: christoph.neumann@thomson.net
URI: http://planete.inrialpes.fr/~chneuman/
David Furodet
STMicroelectronics
12, Rue Jules Horowitz
BP217
Grenoble Cedex 38019
France
Email: david.furodet@st.com
URI: http://www.st.com/
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