RMT V. Roca
Internet-Draft INRIA
Expires: August 5, 2006 C. Neumann
Thomson Research
D. Furodet
STMicroelectronics
February 2006
Low Density Parity Check (LDPC) Forward Error Correction
draft-ietf-rmt-bb-fec-ldpc-01.txt
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Copyright Notice
Copyright (C) The Internet Society (2006).
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
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RFC3453.
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 Element . . . . . . . . . . . . . . . 8
4.2.4. Encoding Format . . . . . . . . . . . . . . . . . . . 8
5. Procedures . . . . . . . . . . . . . . . . . . . . . . . . . . 11
5.1. General . . . . . . . . . . . . . . . . . . . . . . . . . 11
5.2. Determining the Maximum Source Block Length (B) . . . . . 12
5.3. Determining the Encoding Symbol Length (E) and Number
of Encoding Symbols per Group (G) . . . . . . . . . . . . 12
5.4. Determining the Number of Encoding Symbols of a Block . . 13
5.5. Identifying the Symbols of an Encoding Symbol Group . . . 15
5.6. Pseudo Random Number Generator . . . . . . . . . . . . . . 18
6. Full Specification of the LDPC-Staircase Scheme . . . . . . . 20
6.1. General . . . . . . . . . . . . . . . . . . . . . . . . . 20
6.2. Parity Check Matrix Creation . . . . . . . . . . . . . . . 20
6.3. Encoding . . . . . . . . . . . . . . . . . . . . . . . . . 22
6.4. Decoding . . . . . . . . . . . . . . . . . . . . . . . . . 22
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. Intellectual Property . . . . . . . . . . . . . . . . . . . . 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 . . . . . . . . . . . . . . . . . . . . . . . . 32
Intellectual Property and Copyright Statements . . . . . . . . . . 33
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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 are a maximum of 2^^12 blocks per
object.
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 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 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.
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 derive the values of each of these
elements.
4.2.3. Scheme-Specific Element
The following element MUST be defined with the present FEC Scheme.
It contains two distinct pieces of information:
o G: a non-negative integer indicating the number of encoding
symbols per group used for the object. The default value is 1,
meaning that each packet contains exactly one symbol. Values
greater than 1 can also be defined, as explained in Section 5.3.
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.
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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.
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 .
+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
In particular:
o The HEL (Header Extension Length) indicates whether the optional
PRNG seed is present (HEL=5) or not (HEL=4).
o The Maximum-Source-Block-Length (B) 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
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, the following XML elements must be described for the
associated object:
o FEC-OTI-Transfer-length
o FEC-OTI-Encoding-Symbol-Length
o FEC-OTI-Maximum-Source-Block-Length
o FEC-OTI-Max-Number-of-Encoding-Symbols
o FEC-OTI-Number-Encoding-Symbols-per-Group
o FEC-OTI-PRNG-seed (optional)
When no PRNG seed is to be carried in the FEC OTI, the sender simply
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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.
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 [Neumann05].
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
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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 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 may also apply, for instance because of a limited
working memory size. 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) 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
case, E is chosen so that the size of a packet composed of a single
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symbol (G=1) remains below but close to the PMTU.
Yet 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 a good practice 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 (which leads to loosing G symbols at a time), 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 size, pkt_sz (usually the PMTU minus
the various protocol headers). The pkt_sz must be chosen in such a
way it is a multiple of G. Calculate the number of packets: nb_pkts =
ceil(L / pkt_sz). Then, 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 |
+------------------------+----+-------------+-------------------+
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:
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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 provided by the user (e.g. 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");
n = floor(k * max_n / B);
AT A RECEIVER:
Input:
B: Extracted from the received FEC OTI
max_n: Extracted from the received FEC OTI
k: Given by the source blocking algorithm
Output:
n:
Algorithm:
n = floor(k * max_n / B);
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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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/*
* Initialization function.
* Warning: use only when G > 1.
*/
initialize_tables ()
{
int i;
int randInd;
int backup;
/* 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 of the packet under construction
* at a sender.
* Warning: use only when G > 1.
* PktIdx (IN): index of the packet, in {0..ceil(n/G)} range
* ESIs[] (OUT): list of ESI of the packet
*/
sender_find_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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/*
* Determine the sequence of ESIs of a packet received.
* Warning: use only when G > 1.
* esi0 (IN): : ESI contained in the FEC Payload ID
* ESIs[] (OUT): list of ESI of the packet
*/
receiver_find_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. Several implementations of this PRNG are
known and discussed in the literature. Yet all of them provide the
same sequence of pseudo random numbers. For instance, 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 [Carta90].
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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);
lo += (hi & 0x7FFF) << 16;
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 Recipies 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 XX.
The PRNG used by the LDPC-Staircase scheme must be initialized by a
seed. This PRNG seed is an optional instance-specific FEC OTI
element (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:
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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 "1s" */
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 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
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 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 [Zyablov74]: 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 Gauss elimination technique (or any derivative) is another
possible decoding technique.
Because interoperability does not depend on the decoding algorithm
used, the current document does not recommend any particular
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technique. This choice is left to the codec developer.
Yet choosing a decoding technique will have great practical impacts.
It will impact the erasure capabilities: a Gauss elimination
technique enables to solve the system with a smaller number of
symbols compared to the trivial technique. It will also impact the
CPU load: a Gauss elimination technique requires much more processing
than the trivial technique. 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
YY.
The PRNG used by the LDPC-Triangle scheme must be initialized by a
seed. This PRNG seed is an optional instance-specific FEC OTI
element (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
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equation, plus the repair symbols in the triangle. 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.
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 recommend 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 that of
[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, Laurent Fazio, Aurelien
Francillon and Shao Wenjian for their 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",
draft-ietf-rmt-fec-bb-revised-03.txt (work in progress),
January 2006.
11.2. Informative References
[Carta90] 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.
[LDPCrefimpl]
Roca, V., Neumann, C., and J. Laboure, "LDPC-Staircase/
LDPC-Triangle Codec Reference Implementation", PLANETE
Research Team, INRIA Rhone-Alpes,
http://planete.inrialpes.fr/~roca/mcl/.
[Mac03] MacKay, D., "Information Theory, Inference and Learning
Algorithms", Cambridge University Press, ISBN: 0521642981,
2003.
[Neumann05]
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.
[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.
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[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.
[Zyablov74]
Zyablov, V. and M. Pinsker, "Decoding Complexity of Low-
Density Codes for Tranmission in a Channel with Erasures",
Translated from Problemy Peredachi Informatsii, Vol.10,
No. 1, pp.15-28, January-March 1974.
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Appendix A. Trivial Decoding Algorithm (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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Authors' Addresses
Vincent Roca
INRIA
655, av. de l'Europe
Zirst; Montbonnot
ST ISMIER cedex 38334
France
Phone:
Email: vincent.roca@inrialpes.fr
URI: http://planete.inrialpes.fr/~roca/
Christoph Neumann
Thomson Research
46, Quai A. Le Gallo
Boulogne Cedex 92648
France
Phone:
Email: christoph.neumann@thomson.net
URI: http://planete.inrialpes.fr/~chneuman/
David Furodet
STMicroelectronics
12, Rue Jules Horowitz
BP217
Grenoble Cedex 38019
France
Phone:
Email: david.furodet@st.com
URI:
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Acknowledgment
Funding for the RFC Editor function is currently provided by the
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