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Copyright © The Internet Society (2006).
This document describes two Fully-Specified FEC Schemes, LDPC-Staircase and LDPC-Triangle, and their application to the reliable delivery of objects on packet erasure channels. These systematic FEC codes belong to the well known class of ``Low Density Parity Check'' (LDPC) codes, and are large block FEC codes in these sense of RFC3453.
1.
Introduction
2.
Requirements notation
3.
Definitions, Notations and Abbreviations
3.1.
Definitions
3.2.
Notations
3.3.
Abbreviations
4.
Formats and Codes
4.1.
FEC Payload IDs
4.2.
FEC Object Transmission Information
4.2.1.
Mandatory Element
4.2.2.
Common Elements
4.2.3.
Scheme-Specific Elements
4.2.4.
Encoding Format
5.
Procedures
5.1.
General
5.2.
Determining the Maximum Source Block Length (B)
5.3.
Determining the Encoding Symbol Length (E) and Number of
Encoding Symbols per Group (G)
5.4.
Determining the Number of Encoding Symbols of a Block
5.5.
Identifying the Symbols of an Encoding Symbol Group
5.6.
Pseudo Random Number Generator
6.
Full Specification of the LDPC-Staircase Scheme
6.1.
General
6.2.
Parity Check Matrix Creation
6.3.
Encoding
6.4.
Decoding
7.
Full Specification of the LDPC-Triangle Scheme
7.1.
General
7.2.
Parity Check Matrix Creation
7.3.
Encoding
7.4.
Decoding
8.
Security Considerations
9.
Acknowledgments
10.
References
10.1.
Normative References
10.2.
Informative References
Appendix A.
Trivial Decoding Algorithm (Informative Only)
§
Authors' Addresses
§
Intellectual Property and Copyright Statements
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RFC 3453 (Luby, M., Vicisano, L., Gemmell, J., Rizzo, L., Handley, M., and J. Crowcroft, “The Use of Forward Error Correction (FEC) in Reliable Multicast,” December 2002.) [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 LDPC-Staircase FEC codes, and the Fully-Specified FEC Encoding ID YY that is intended to be used with the LDPC-Triangle FEC codes [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,” June 2004.)[7] (MacKay, D., “Information Theory, Inference and Learning Algorithms,” 2003.). Both schemes belong to 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, V., Neumann, C., and J. Laboure, “LDPC-Staircase/LDPC-Triangle Codec Reference Implementation,” .).
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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] (Bradner, S., “Key words for use in RFCs to Indicate Requirement Levels,” March 1997.).
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This document uses the same terms and definitions as those specified in [2] (Watson, M., Luby, M., and L. Vicisano, “Forward Error Correction (FEC) Building Block,” January 2006.). 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.
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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)
rand(m) denotes a pseudo-random number generator that returns a new random integer in [0; m-1] each time it is called
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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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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.
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 (Identifying the Symbols of an Encoding Symbol Group)
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 |
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The following elements MUST be defined with the present FEC Scheme:
For instance, if B=2^^19 (because of a code rate of 1/2, Section 5.2 (Determining the Maximum Source Block Length (B))), 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).maximum transfer length = 2^^12 * B * E
Section 5 (Procedures) explains how to define the values of each of these elements.
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The following elements MUST be defined with the present FEC Scheme:
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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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The FEC OTI binary format is the following, when the EXT_FTI mechanism is used (e.g. within the ALC [11] (Luby, M., Watson, M., and L. Vicisano, “Asynchronous Layered Coding (ALC) Protocol Instantiation,” April 2006.) or NORM [13] (Adamson, B., Bormann, C., Handley, M., and J. Macker, “Negative-acknowledgment (NACK)-Oriented Reliable Multicast (NORM) Protocol,” June 2006.) 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 . +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
| EXT_FTI Header for FEC Encoding ID 132 |
In particular:
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When it is desired that the FEC OTI be carried in the FDT Instance of a FLUTE session [12] (Paila, T., Walsh, R., Luby, M., Lehtonen, R., and V. Roca, “FLUTE - File Delivery over Unidirectional Transport,” January 2006.), the following XML elements must be described for the associated object:
When no PRNG seed is to be carried in the FEC OTI, the sender simply omits the FEC-OTI-PRNG-seed element.
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This section defines procedures that are common to FEC Encoding IDs XX and YY.
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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] (Watson, M., Luby, M., and L. Vicisano, “Forward Error Correction (FEC) Building Block,” January 2006.). 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 [5] (Neumann, C., Roca, V., Francillon, A., and D. Furodet, “Impacts of Packet Scheduling and Packet Loss Distribution on FEC Performances: Observations and Recommendations,” October 2005.).
The following sections detail how the B, E, and n parameters are determined (respectively Section 5.2 (Determining the Maximum Source Block Length (B)), Section 5.3 (Determining the Encoding Symbol Length (E) and Number of Encoding Symbols per Group (G)) and Section 5.4 (Determining the Number of Encoding Symbols of a Block)), how encoding symbol groups are created (Section 5.5 (Identifying the Symbols of an Encoding Symbol Group)), and finally specify the PRNG (Section 5.6 (Pseudo Random Number Generator)).
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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:
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.
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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 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 (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 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 |
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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 (Determining the Maximum Source Block Length (B)) 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 (FEC Payload IDs)).
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 (Determining the Maximum Source Block Length (B)), 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: Number of encoding symbols generated for this source block
Algorithm:
n = floor(k * max_n / B);
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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):
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.
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.
/*
* 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.
/*
* 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)];
}
}
}
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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] (Park, S. and K. Miller, “Random Number Generators: Good Ones are Hard to Find,” 1988.) 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] (Carta, D., “Two Fast Implementations of the Minimal Standard Random Number Generator,” January 1990.).
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));
}
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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 (Scheme-Specific Elements)). 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.
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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:
/* 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));
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 (Identifying the Symbols of an Encoding Symbol Group)) MUST be called. This is true both at a sender and at a receiver.
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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.
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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 [10] (Zyablov, V. and M. Pinsker, “Decoding Complexity of Low-Density Codes for Transmission in a Channel with Erasures,” January-March 1974.): 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 (Trivial Decoding Algorithm (Informative Only)) 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 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.
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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 (Scheme-Specific Elements)). 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.
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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 (Parity Check Matrix Creation)).
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 (Identifying the Symbols of an Encoding Symbol Group)) MUST be called. This is true both at a sender and at a receiver.
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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 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.
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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 (Decoding).
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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The security considerations for this document are the same as that of [2] (Watson, M., Luby, M., and L. Vicisano, “Forward Error Correction (FEC) Building Block,” January 2006.).
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Section 5.4 (Determining the Number of Encoding Symbols of a Block) 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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| [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. |
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| [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, January-March 1974. |
| [11] | Luby, M., Watson, M., and L. Vicisano, “Asynchronous Layered Coding (ALC) Protocol Instantiation,” draft-ietf-rmt-pi-alc-revised-03.txt (work in progress), April 2006. |
| [12] | Paila, T., Walsh, R., Luby, M., Lehtonen, R., and V. Roca, “FLUTE - File Delivery over Unidirectional Transport,” draft-ietf-rmt-flute-revised-01.txt (work in progress), January 2006. |
| [13] | Adamson, B., Bormann, C., Handley, M., and J. Macker, “Negative-acknowledgment (NACK)-Oriented Reliable Multicast (NORM) Protocol,” draft-ietf-rmt-pi-norm-revised-02.txt (work in progress), June 2006. |
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A trivial decoding algorithm is sketched below (please see [6] (Roca, V., Neumann, C., and J. Laboure, “LDPC-Staircase/LDPC-Triangle Codec Reference Implementation,” .) 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.
* 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;
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;
/* finally, call this function recursively */
decoding_step(dec_esi, dec_symb);
}
}
Free the list of equations having symbols decoded;
Return;
}
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| 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/ |
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