 1/draftietfrmtbbfecldpc02.txt 20060718 22:12:22.000000000 +0200
+++ 2/draftietfrmtbbfecldpc03.txt 20060718 22:12:22.000000000 +0200
@@ 1,22 +1,22 @@
RMT V. Roca
InternetDraft INRIA
Expires: December 23, 2006 C. Neumann
+Expires: January 19, 2007 C. Neumann
Thomson Research
D. Furodet
STMicroelectronics
 June 21, 2006
+ July 18, 2006
Low Density Parity Check (LDPC) Staircase and Triangle Forward Error
Correction (FEC) Schemes
 draftietfrmtbbfecldpc02.txt
+ draftietfrmtbbfecldpc03.txt
Status of this Memo
By submitting this InternetDraft, 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.
InternetDrafts are working documents of the Internet Engineering
Task Force (IETF), its areas, and its working groups. Note that
@@ 27,21 +27,21 @@
and may be updated, replaced, or obsoleted by other documents at any
time. It is inappropriate to use InternetDrafts as reference
material or to cite them other than as "work in progress."
The list of current InternetDrafts can be accessed at
http://www.ietf.org/ietf/1idabstracts.txt.
The list of InternetDraft Shadow Directories can be accessed at
http://www.ietf.org/shadow.html.
 This InternetDraft will expire on December 23, 2006.
+ This InternetDraft will expire on January 19, 2007.
Copyright Notice
Copyright (C) The Internet Society (2006).
Abstract
This document describes two FullySpecified FEC Schemes, LDPC
Staircase and LDPCTriangle, and their application to the reliable
delivery of objects on packet erasure channels. These systematic FEC
@@ 53,23 +53,23 @@
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
+ 4.2.1. Mandatory Element . . . . . . . . . . . . . . . . . . 7
4.2.2. Common Elements . . . . . . . . . . . . . . . . . . . 7
 4.2.3. SchemeSpecific Element . . . . . . . . . . . . . . . 8
+ 4.2.3. SchemeSpecific Elements . . . . . . . . . . . . . . . 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 LDPCStaircase Scheme . . . . . . . 20
@@ 91,41 +91,40 @@
Authors' Addresses . . . . . . . . . . . . . . . . . . . . . . . . 32
Intellectual Property and Copyright Statements . . . . . . . . . . 33
1. Introduction
RFC 3453 [3] introduces large block FEC codes as an alternative to
small block FEC codes like ReedSolomon. 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 FullySpecified FEC Encoding ID
 XX that is intended to be used with the "Low Density Parity Check"
 (LDPC) Staircase FEC codes, and the FullySpecified 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.
+ XX that is intended to be used with the LDPCStaircase FEC codes, and
+ the FullySpecified FEC Encoding ID YY that is intended to be used
+ with the LDPCTriangle FEC codes [4][7]. 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.
+ FEC Object Transmission Information.
A publicly available reference implementation of these codes is
available and distributed under a GNU/LGPL license [6].
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].
@@ 165,74 +164,81 @@
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 pseudorandom
number generator, where s is the seed (s > 0)

 rand(m) denotes a pseudorandom number generator, that returns a
+ rand(m) denotes a pseudorandom number generator that returns a
new random integer in [0; m1] each time it is called
3.3. Abbreviations
This document uses the following abbreviations:
ESI: Encoding Symbol ID
FEC OTI: FEC Object Transmission Information
+ FPI: FEC Payload ID
+
+ LDPC: Low Density Parity Check
+
+ PRNG: Pseudo Random Number Generator
+
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.
+ 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 k1) identify source symbols, the
remaining nk values (k to nk1) identify repair symbols.
 There MUST be exactly one FEC Payload ID per packet. In case of en
+ There MUST be exactly one FEC Payload ID per packet. In case of an
Encoding Symbol Group, when multiple encoding symbols are sent in the
same packet, the FEC Payload ID refers to the first symbol of the
packet. The other symbols can be deduced from the ESI of the first
symbol thanks to a dedicated function, as explained in Section 5.5
0 1 2 3
0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1
+++++++++++++++++++++++++++++++++
 Source Block Number  Encoding Symbol ID (20 bits) 
+++++++++++++++++++++++++++++++++
Figure 1: FEC Payload ID encoding format for FEC Encoding ID XX and
YY
4.2. FEC Object Transmission Information
4.2.1. Mandatory Elements
+4.2.1. Mandatory Element
 o FEC Encoding ID: the FullySpecified FEC Schemes described in this
 document use the FEC Encoding ID XX for LDPCStaircase and FEC
 Encoding ID YY for LDPCTriangle.
+ o FEC Encoding ID: the LDPCStaircase and LDPCTriangle Fully
+ Specified FEC Schemes use respectively the FEC Encoding ID XX
+ (Staircase) and YY (Triangle).
4.2.2. Common Elements
The following elements MUST be defined with the present FEC Scheme:
o TransferLength (L): a nonnegative integer indicating the length
of the object in bytes. There are some restrictions on the
maximum TransferLength that can be supported:
maximum transfer length = 2^^12 * B * E
@@ 249,53 +255,52 @@
o MaximumSourceBlockLength (B): a nonnegative 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 MaxNumberofEncodingSymbols (max_n): a nonnegative 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
+ Section 5 explains how to define the values of each of these
elements.
4.2.3. SchemeSpecific Element
+4.2.3. SchemeSpecific Elements
 The following element MUST be defined with the present FEC Scheme.
 It contains two distinct pieces of information:
+ The following elements MUST be defined with the present FEC Scheme:
o G: a nonnegative integer indicating the number of encoding
 symbols per group used for the object. The default value is 1,
+ symbols per group (i.e. per packet). The default value is 1,
meaning that each packet contains exactly one symbol. Values
greater than 1 can also be defined, as explained in Section 5.3.
 o PRNG seed: The seed is a 32 bit unsigned integer between 1 and
+ o PRNG seed: the seed is a 32 bit unsigned integer between 1 and
0x7FFFFFFE (i.e. 2^^312) 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
+ appropriate mechanism (Section 4.2.4). When the PRNG seed is not
+ carried within the FEC OTI, it is assumed that encoder and
decoders use another way to communicate the information, or use a
fixed, predefined value.
4.2.4. Encoding Format
This section shows two possible encoding formats of the above FEC
OTI. The present document does not specify when or how these
encoding formats should be used.
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.
+ mechanism is used (e.g. within the ALC [11] or NORM [13] 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)  
+++++++++++++++++ +
 TransferLength (L) 
+++++++++++++++++++++++++++++++++
 Encoding Symbol Length (E)  G  B (MSB) 
+++++++++++++++++++++++++++++++++
@@ 314,30 +319,30 @@
for field alignment purposes.
o The MaximumSourceBlockLength (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 32bit 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:
+ a FLUTE session [12], the following XML elements must be described
+ for the associated object:
o FECOTITransferlength
o FECOTIEncodingSymbolLength
o FECOTIMaximumSourceBlockLength
 o FECOTIMaxNumberofEncodingSymbols
+ o FECOTIMaxNumberofEncodingSymbols
o FECOTINumberEncodingSymbolsperGroup
o FECOTIPRNGseed (optional)
When no PRNG seed is to be carried in the FEC OTI, the sender simply
omits the FECOTIPRNGseed element.
5. Procedures
This section defines procedures that are common to FEC Encoding IDs
@@ 356,21 +361,21 @@
are numbered consecutively from 0 to N1. The first I source blocks
consist of A_large source symbols, the remaining NI 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
+ 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 k1) which describes the occurrence of
each source symbol in the equation system; and the right side (from
column k to n1) 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
LDPCStaircase and LDPCTriangle schemes is the construction of the
@@ 395,37 +400,37 @@
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 rate == 1 (no repair symbols): max1_B = 2^^20 = 1,048,576
 o 1 > rate >= 1/2: max1_B = 2^^19 = 524,288 symbols
+ o 1/2 <= rate < 1: max1_B = 2^^19 = 524,288 symbols
 o 1/2 > rate >= 1/4: max1_B = 2^^18 = 262,144 symbols
+ o 1/4 <= rate < 1/2: max1_B = 2^^18 = 262,144 symbols
 o 1/4 > rate >= 1/8: max1_B = 2^^17 = 131,072 symbols
+ o 1/8 <= rate < 1/4: max1_B = 2^^17 = 131,072 symbols
Additionally, a codec MAY impose other limitations on the maximum
block size. This is the case for instance when the codec uses
 internally 16 bit 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.
+ internally 16 bit unsigned integers to store the Encoding Symbol ID,
+ since it does not enable to store all the possible values of a 20 bit
+ field. In that case, if for instance 1/2 <= rate < 1, then the
+ maximum source block length is 2^^15. Other limitations may also
+ apply, for instance because of a limited working memory size. This
+ decision MUST be clarified at implementation time, when the target
+ use case is known. This results in a max2_B limitation.
Then, B is given by:
B = min(max1_B, max2_B)
Note that this calculation is only required at the coder, since the B
parameter is communicated to the decoder through the FEC OTI.
5.3. Determining the Encoding Symbol Length (E) and Number of Encoding
Symbols per Group (G)
@@ 440,21 +445,21 @@
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
+ 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:
@@ 503,41 +508,40 @@
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");
+ 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)
+ (NB: if B has been defined as explained in Section 5.2, this error
+ should never happen)
n = floor(k * max_n / B);
AT A RECEIVER:
Input:
B: Extracted from the received FEC OTI
max_n: Extracted from the received FEC OTI
k: Given by the source blocking algorithm
Output:
 n:

+ n: Number of encoding symbols generated for this source block
Algorithm:
n = floor(k * max_n / B);
5.5. Identifying the Symbols of an Encoding Symbol Group
When multiple encoding symbols are sent in the same packet, the FEC
Payload ID information of the packet MUST refer to the first encoding
symbol. It MUST then be possible to identify each symbol from this
single FEC Payload ID. To that purpose, the symbols of an Encoding
@@ 554,38 +558,49 @@
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.
+ o are stored in sequence in the packet, without any padding. In
+ other words, the last byte of the symbol with ESI i (where i is
+ the ith ESI returned by the function ESIs_of_group()) is
+ immediately followed by the first byte of symbol i+1.
+
The system must first be initialized by creating a random permutation
of the nk indexes. This initialization function MUST be called
immediately after creating the parity check matrix. More precisely,
since the PRNG seed is not reinitialized, 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((nk) * sizeof(int));
+ IDtoTxseq = malloc((nk) * sizeof(int));
/* initialize the two tables that map ID
* (i.e. ESIk) 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];
@@ 594,67 +609,71 @@
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 of the packet under construction
+ * 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(n/G)} range
 * ESIs[] (OUT): list of ESI of the packet
+ * PktIdx (IN): index of the packet, in
+ * {0..ceil(k/G)+ceil((nk)/G)} range
+ * ESIs[] (OUT): list of ESIs for the packet
*/
+ void
sender_find_ESIs_of_group (int PktIdx,
ESI_t ESIs[])
{
int i;
 if (is_source_packet(PktIdx) == true) {
+ if (PktIdx < nbSourcePkts) {
/* this is a source packet */
 ESIs[0] = (PktIdx * G) % k;
 for (i = 0; i < G; i++) {
 ESIs[i] = ESIs[0] + i;
+ 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 of a packet received.
+ * 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 ESI of the packet
+ * ESIs[] (OUT): list of ESIs for the packet
*/
+ void
receiver_find_ESIs_of_group (ESI_t esi0,
ESI_t ESIs[])
{
int i;
 if (is_source_packet(esi0) == true) {
+ if (esi0 < k) {
/* this is a source packet */
 for (i = 0; i < G; i++) {
+ 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)];
}
@@ 738,21 +757,21 @@
6.2. Parity Check Matrix Creation
The LDPCStaircase 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 possible choices to
+ /* initialize a list of all possible choices in order to
* guarantee a homogeneous "1" distribution */
for (h = 3*k1; h >= 0; h) {
u[h] = h % (nk);
}
/* 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 */
@@ 772,21 +791,22 @@
} else {
/* no choice left, choose one randomly */
do {
i = rand(nk);
} while (matrix_has_entry(i, j));
matrix_insert_entry(i, j);
}
}
}
 /* Add extra bits to avoid rows with less than two "1s" */
+ /* 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 < nk; 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);
@@ 811,32 +831,32 @@
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 nk linear
 equations whose variables are the source an repair symbols. Of
 course, the final goal is to recover the value of source symbols
 only.
+ 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]: 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
+ channel, the parity check matrix defines a set of linear equations
whose variables are the source symbols and repair symbols. Receiving
or decoding a symbol is equivalent to having the value of a variable.
Appendix A sketches a possible implementation of this algorithm.
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
@@ 900,24 +920,24 @@
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.
7.4. Decoding
Decoding basically consists in solving a system of nk 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.
+ equations, whose variables are the n source and repair symbols. Of
+ course, the final goal is to recover the value of the k source
+ symbols only. To that purpose, many techniques are possible, as
+ explained in Section 6.4.
Because interoperability does not depend on the decoding algorithm
used, the current document does not recommend any particular
technique. This choice is left to the codec implementer.
8. Security Considerations
The security considerations for this document are the same as that of
[2].
@@ 971,97 +991,121 @@
[9] Carta, D., "Two Fast Implementations of the Minimal Standard
Random Number Generator", Communications of the ACM, Vol. 33,
No. 1, pp.8788, January 1990.
[10] Zyablov, V. and M. Pinsker, "Decoding Complexity of LowDensity
Codes for Transmission in a Channel with Erasures", Translated
from Problemy Peredachi Informatsii, Vol.10, No. 1, pp.1528,
JanuaryMarch 1974.
+ [11] Luby, M., Watson, M., and L. Vicisano, "Asynchronous Layered
+ Coding (ALC) Protocol Instantiation",
+ draftietfrmtpialcrevised03.txt (work in progress),
+ April 2006.
+
+ [12] Paila, T., Walsh, R., Luby, M., Lehtonen, R., and V. Roca,
+ "FLUTE  File Delivery over Unidirectional Transport",
+ draftietfrmtfluterevised01.txt (work in progress),
+ January 2006.
+
+ [13] Adamson, B., Bormann, C., Handley, M., and J. Macker,
+ "Negativeacknowledgment (NACK)Oriented Reliable Multicast
+ (NORM) Protocol", draftietfrmtpinormrevised02.txt (work
+ in progress), 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[nk], one per equation;
 Reset all the buffers to 0;
+ Initialization: allocate a table partial_sum[nk] 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.
 * new_esi (IN): ESI of the new symbol, which is also the index
 * in [0; n1]
 * new_symb (IN): New symbol received or decoded
+ * 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) {
 Return; /* decoding is now finished */
+ Remember that decoding is finished;
+ Return; /* work is over now... */
}
 Create an empty list of equations having symbols decoded during
 this decoding step;
+ 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.
+ * 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 (degree of equation eq == 1) {
 /* new symbol can be decoded, remember the equation */
+ 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 symbols.
+ * 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
+ 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;
}
Authors' Addresses
Vincent Roca
INRIA
655, av. de l'Europe
Zirst; Montbonnot
ST ISMIER cedex 38334
France