RMT V. Roca
Internet-Draft INRIA
Expires: December 23, 2006 C. Neumann
Thomson Research
D. Furodet
STMicroelectronics
June 21, 2006
Low Density Parity Check (LDPC) Staircase and Triangle Forward Error
Correction (FEC) Schemes
draft-ietf-rmt-bb-fec-ldpc-02.txt
Status of this Memo
By submitting this Internet-Draft, each author represents that any
applicable patent or other IPR claims of which he or she is aware
have been or will be disclosed, and any of which he or she becomes
aware will be disclosed, in accordance with Section 6 of BCP 79.
Internet-Drafts are working documents of the Internet Engineering
Task Force (IETF), its areas, and its working groups. Note that
other groups may also distribute working documents as Internet-
Drafts.
Internet-Drafts are draft documents valid for a maximum of six months
and may be updated, replaced, or obsoleted by other documents at any
time. It is inappropriate to use Internet-Drafts as reference
material or to cite them other than as "work in progress."
The list of current Internet-Drafts can be accessed at
http://www.ietf.org/ietf/1id-abstracts.txt.
The list of Internet-Draft Shadow Directories can be accessed at
http://www.ietf.org/shadow.html.
This Internet-Draft will expire on December 23, 2006.
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''
Roca, et al. Expires December 23, 2006 [Page 1]
Internet-Draft LDPC Staircase and Triangle FEC June 2006
(LDPC) codes, and are large block FEC codes in these sense of
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. Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . 27
10. References . . . . . . . . . . . . . . . . . . . . . . . . . . 28
10.1. Normative References . . . . . . . . . . . . . . . . . . . 28
10.2. Informative References . . . . . . . . . . . . . . . . . . 28
Appendix A. Trivial Decoding Algorithm (Informative Only) . . . . 30
Authors' Addresses . . . . . . . . . . . . . . . . . . . . . . . . 32
Intellectual Property and Copyright Statements . . . . . . . . . . 33
Roca, et al. Expires December 23, 2006 [Page 2]
Internet-Draft LDPC Staircase and Triangle FEC June 2006
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
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 [4][7]. 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
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 [6].
Roca, et al. Expires December 23, 2006 [Page 3]
Internet-Draft LDPC Staircase and Triangle FEC June 2006
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].
Roca, et al. Expires December 23, 2006 [Page 4]
Internet-Draft LDPC Staircase and Triangle FEC June 2006
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
srand(s) denotes the initialization function of the pseudo-random
number generator, where s is the seed (s > 0)
rand(m) denotes a pseudo-random number generator, that returns a
new random integer in [0; m-1] each time it is called
Roca, et al. Expires December 23, 2006 [Page 5]
Internet-Draft LDPC Staircase and Triangle FEC June 2006
3.3. Abbreviations
This document uses the following abbreviations:
ESI: Encoding Symbol ID
FEC OTI: FEC Object Transmission Information
Roca, et al. Expires December 23, 2006 [Page 6]
Internet-Draft LDPC Staircase and Triangle FEC June 2006
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:
Roca, et al. Expires December 23, 2006 [Page 7]
Internet-Draft LDPC Staircase and Triangle FEC June 2006
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 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 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 (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
Roca, et al. Expires December 23, 2006 [Page 8]
Internet-Draft LDPC Staircase and Triangle FEC June 2006
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.
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 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 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
Roca, et al. Expires December 23, 2006 [Page 9]
Internet-Draft LDPC Staircase and Triangle FEC June 2006
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
omits the FEC-OTI-PRNG-seed element.
Roca, et al. Expires December 23, 2006 [Page 10]
Internet-Draft LDPC Staircase and Triangle FEC June 2006
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 [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 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 [5].
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
Roca, et al. Expires December 23, 2006 [Page 11]
Internet-Draft LDPC Staircase and Triangle FEC June 2006
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.
In that case, if for instance 1 > rate >= 1/2, then the maximum block
size 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
Roca, et al. Expires December 23, 2006 [Page 12]
Internet-Draft LDPC Staircase and Triangle FEC June 2006
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.
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 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: nb_pkts = ceil(L / pkt_sz). Finally 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.
Roca, et al. Expires December 23, 2006 [Page 13]
Internet-Draft LDPC Staircase and Triangle FEC June 2006
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 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");
(NB: if max_n 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
k: Given by the source blocking algorithm
Output:
n:
Roca, et al. Expires December 23, 2006 [Page 14]
Internet-Draft LDPC Staircase and Triangle FEC June 2006
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, 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.
Roca, et al. Expires December 23, 2006 [Page 15]
Internet-Draft LDPC Staircase and Triangle FEC June 2006
/*
* 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.
Roca, et al. Expires December 23, 2006 [Page 16]
Internet-Draft LDPC Staircase and Triangle FEC June 2006
/*
* 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.
Roca, et al. Expires December 23, 2006 [Page 17]
Internet-Draft LDPC Staircase and Triangle FEC June 2006
/*
* 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 [8] 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 [9].
Roca, et al. Expires December 23, 2006 [Page 18]
Internet-Draft LDPC Staircase and Triangle FEC June 2006
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 Recipies in C] */
return ((unsigned long)
((double)seed * (double)maxv / (double)0x7FFFFFFF));
}
Roca, et al. Expires December 23, 2006 [Page 19]
Internet-Draft LDPC Staircase and Triangle FEC June 2006
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:
Roca, et al. Expires December 23, 2006 [Page 20]
Internet-Draft LDPC Staircase and Triangle FEC June 2006
/* 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);
}
}
Roca, et al. Expires December 23, 2006 [Page 21]
Internet-Draft LDPC Staircase and Triangle FEC June 2006
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 [10]: 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 optimized derivative) is
another possible decoding technique. Hybrid solutions that start by
using the trivial algorithm above and finish with a Gauss elimination
are also possible.
Because interoperability does not depend on the decoding algorithm
Roca, et al. Expires December 23, 2006 [Page 22]
Internet-Draft LDPC Staircase and Triangle FEC June 2006
used, the current document does not recommend any particular
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.
Roca, et al. Expires December 23, 2006 [Page 23]
Internet-Draft LDPC Staircase and Triangle FEC June 2006
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
Roca, et al. Expires December 23, 2006 [Page 24]
Internet-Draft LDPC Staircase and Triangle FEC June 2006
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.
Roca, et al. Expires December 23, 2006 [Page 25]
Internet-Draft LDPC Staircase and Triangle FEC June 2006
8. Security Considerations
The security considerations for this document are the same as that of
[2].
Roca, et al. Expires December 23, 2006 [Page 26]
Internet-Draft LDPC Staircase and Triangle FEC June 2006
9. 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.
Roca, et al. Expires December 23, 2006 [Page 27]
Internet-Draft LDPC Staircase and Triangle FEC June 2006
10. References
10.1. 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-03.txt
(work in progress), January 2006.
[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.
10.2. Informative References
[4] 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.
[5] 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.
[6] Roca, V., Neumann, C., and J. Laboure, "LDPC-Staircase/
LDPC-Triangle Codec Reference Implementation", INRIA Rhone-
Alpes and STMicroelectronics,
http://planete-bcast.inrialpes.fr/.
[7] MacKay, D., "Information Theory, Inference and Learning
Algorithms", Cambridge University Press, ISBN: 0521642981,
2003.
[8] 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.
[9] 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.
[10] 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,
Roca, et al. Expires December 23, 2006 [Page 28]
Internet-Draft LDPC Staircase and Triangle FEC June 2006
January-March 1974.
Roca, et al. Expires December 23, 2006 [Page 29]
Internet-Draft LDPC Staircase and Triangle FEC June 2006
Appendix A. Trivial Decoding Algorithm (Informative Only)
A trivial decoding algorithm is sketched below (please see [6] for
the details omitted here):
Initialization: allocate a table of partial sum buffers:
partial_sum[n-k], one per equation;
Reset all the buffers to 0;
/*
* For each newly received or decoded symbol, try to make progress
* in the decoding of the associated source block.
* new_esi (IN): ESI of the new symbol, which is also the index
* in [0; n-1]
* new_symb (IN): 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) {
Return; /* decoding is now finished */
}
Create an empty list of equations having symbols decoded during
this decoding step;
/*
* First add this new symbol to all partial sums of the
* associated equations.
*/
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;
If (degree of equation eq == 1) {
/* new symbol can be decoded, remember the equation */
Append eq to the list of equations having symbols
decoded during this decoding step;
}
}
Roca, et al. Expires December 23, 2006 [Page 30]
Internet-Draft LDPC Staircase and Triangle FEC June 2006
/*
* Then finish with recursive calls to decoding_step() for each
* newly decoded symbols.
*/
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;
/* finally, call this function recursively */
decoding_step(dec_esi, dec_symb);
}
}
}
Roca, et al. Expires December 23, 2006 [Page 31]
Internet-Draft LDPC Staircase and Triangle FEC June 2006
Authors' Addresses
Vincent Roca
INRIA
655, av. de l'Europe
Zirst; Montbonnot
ST ISMIER cedex 38334
France
Email: vincent.roca@inrialpes.fr
URI: http://planete.inrialpes.fr/~roca/
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/
Roca, et al. Expires December 23, 2006 [Page 32]
Internet-Draft LDPC Staircase and Triangle FEC June 2006
Intellectual Property Statement
The IETF takes no position regarding the validity or scope of any
Intellectual Property Rights or other rights that might be claimed to
pertain to the implementation or use of the technology described in
this document or the extent to which any license under such rights
might or might not be available; nor does it represent that it has
made any independent effort to identify any such rights. Information
on the procedures with respect to rights in RFC documents can be
found in BCP 78 and BCP 79.
Copies of IPR disclosures made to the IETF Secretariat and any
assurances of licenses to be made available, or the result of an
attempt made to obtain a general license or permission for the use of
such proprietary rights by implementers or users of this
specification can be obtained from the IETF on-line IPR repository at
http://www.ietf.org/ipr.
The IETF invites any interested party to bring to its attention any
copyrights, patents or patent applications, or other proprietary
rights that may cover technology that may be required to implement
this standard. Please address the information to the IETF at
ietf-ipr@ietf.org.
Disclaimer of Validity
This document and the information contained herein are provided on an
"AS IS" basis and THE CONTRIBUTOR, THE ORGANIZATION HE/SHE REPRESENTS
OR IS SPONSORED BY (IF ANY), THE INTERNET SOCIETY AND THE INTERNET
ENGINEERING TASK FORCE DISCLAIM ALL WARRANTIES, EXPRESS OR IMPLIED,
INCLUDING BUT NOT LIMITED TO ANY WARRANTY THAT THE USE OF THE
INFORMATION HEREIN WILL NOT INFRINGE ANY RIGHTS OR ANY IMPLIED
WARRANTIES OF MERCHANTABILITY OR FITNESS FOR A PARTICULAR PURPOSE.
Copyright Statement
Copyright (C) The Internet Society (2006). This document is subject
to the rights, licenses and restrictions contained in BCP 78, and
except as set forth therein, the authors retain all their rights.
Acknowledgment
Funding for the RFC Editor function is currently provided by the
Internet Society.
Roca, et al. Expires December 23, 2006 [Page 33]