This is a purely informative rendering of an RFC that includes verified errata. This rendering may not be used as a reference.
The following 'Verified' errata have been incorporated in this document:
EID 5548
Internet Engineering Task Force (IETF) M. Luby
Request for Comments: 6330 Qualcomm Incorporated
Category: Standards Track A. Shokrollahi
ISSN: 2070-1721 EPFL
M. Watson
Netflix Inc.
T. Stockhammer
Nomor Research
L. Minder
Qualcomm Incorporated
August 2011
RaptorQ Forward Error Correction Scheme for Object Delivery
Abstract
This document describes a Fully-Specified Forward Error Correction
(FEC) scheme, corresponding to FEC Encoding ID 6, for the RaptorQ FEC
code and its application to reliable delivery of data objects.
RaptorQ codes are a new family of codes that provide superior
flexibility, support for larger source block sizes, and better coding
efficiency than Raptor codes in RFC 5053. RaptorQ is also a fountain
code, i.e., as many encoding symbols as needed can be generated on
the fly by the encoder from the source symbols of a source block of
data. The decoder is able to recover the source block from almost
any set of encoding symbols of sufficient cardinality -- in most
cases, a set of cardinality equal to the number of source symbols is
sufficient; in rare cases, a set of cardinality slightly more than
the number of source symbols is required.
The RaptorQ code described here is a systematic code, meaning that
all the source symbols are among the encoding symbols that can be
generated.
Status of This Memo
This is an Internet Standards Track document.
This document is a product of the Internet Engineering Task Force
(IETF). It represents the consensus of the IETF community. It has
received public review and has been approved for publication by the
Internet Engineering Steering Group (IESG). Further information on
Internet Standards is available in Section 2 of RFC 5741.
Information about the current status of this document, any errata,
and how to provide feedback on it may be obtained at
http://www.rfc-editor.org/info/rfc6330.
Copyright Notice
Copyright (c) 2011 IETF Trust and the persons identified as the
document authors. All rights reserved.
This document is subject to BCP 78 and the IETF Trust's Legal
Provisions Relating to IETF Documents
(http://trustee.ietf.org/license-info) in effect on the date of
publication of this document. Please review these documents
carefully, as they describe your rights and restrictions with respect
to this document. Code Components extracted from this document must
include Simplified BSD License text as described in Section 4.e of
the Trust Legal Provisions and are provided without warranty as
described in the Simplified BSD License.
Table of Contents
1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 4
2. Requirements Notation . . . . . . . . . . . . . . . . . . . . 4
3. Formats and Codes . . . . . . . . . . . . . . . . . . . . . . 5
3.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . 5
3.2. FEC Payload IDs . . . . . . . . . . . . . . . . . . . . . 5
3.3. FEC Object Transmission Information . . . . . . . . . . . 5
3.3.1. Mandatory . . . . . . . . . . . . . . . . . . . . . . 5
3.3.2. Common . . . . . . . . . . . . . . . . . . . . . . . . 5
3.3.3. Scheme-Specific . . . . . . . . . . . . . . . . . . . 6
4. Procedures . . . . . . . . . . . . . . . . . . . . . . . . . . 7
4.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . 7
4.2. Content Delivery Protocol Requirements . . . . . . . . . . 7
4.3. Example Parameter Derivation Algorithm . . . . . . . . . . 7
4.4. Object Delivery . . . . . . . . . . . . . . . . . . . . . 9
4.4.1. Source Block Construction . . . . . . . . . . . . . . 9
4.4.2. Encoding Packet Construction . . . . . . . . . . . . . 11
4.4.3. Example Receiver Recovery Strategies . . . . . . . . . 12
5. RaptorQ FEC Code Specification . . . . . . . . . . . . . . . . 12
5.1. Background . . . . . . . . . . . . . . . . . . . . . . . . 12
5.1.1. Definitions . . . . . . . . . . . . . . . . . . . . . 13
5.1.2. Symbols . . . . . . . . . . . . . . . . . . . . . . . 14
5.2. Overview . . . . . . . . . . . . . . . . . . . . . . . . . 17
5.3. Systematic RaptorQ Encoder . . . . . . . . . . . . . . . . 18
5.3.1. Introduction . . . . . . . . . . . . . . . . . . . . . 18
5.3.2. Encoding Overview . . . . . . . . . . . . . . . . . . 19
5.3.3. First Encoding Step: Intermediate Symbol Generation . 21
5.3.4. Second Encoding Step: Encoding . . . . . . . . . . . . 27
5.3.5. Generators . . . . . . . . . . . . . . . . . . . . . . 27
5.4. Example FEC Decoder . . . . . . . . . . . . . . . . . . . 30
5.4.1. General . . . . . . . . . . . . . . . . . . . . . . . 30
5.4.2. Decoding an Extended Source Block . . . . . . . . . . 31
5.5. Random Numbers . . . . . . . . . . . . . . . . . . . . . . 36
5.5.1. The Table V0 . . . . . . . . . . . . . . . . . . . . . 36
5.5.2. The Table V1 . . . . . . . . . . . . . . . . . . . . . 37
5.5.3. The Table V2 . . . . . . . . . . . . . . . . . . . . . 38
5.5.4. The Table V3 . . . . . . . . . . . . . . . . . . . . . 40
5.6. Systematic Indices and Other Parameters . . . . . . . . . 41
5.7. Operating with Octets, Symbols, and Matrices . . . . . . . 62
5.7.1. General . . . . . . . . . . . . . . . . . . . . . . . 62
5.7.2. Arithmetic Operations on Octets . . . . . . . . . . . 62
5.7.3. The Table OCT_EXP . . . . . . . . . . . . . . . . . . 63
5.7.4. The Table OCT_LOG . . . . . . . . . . . . . . . . . . 64
5.7.5. Operations on Symbols . . . . . . . . . . . . . . . . 65
5.7.6. Operations on Matrices . . . . . . . . . . . . . . . . 65
5.8. Requirements for a Compliant Decoder . . . . . . . . . . . 65
6. Security Considerations . . . . . . . . . . . . . . . . . . . 66
7. IANA Considerations . . . . . . . . . . . . . . . . . . . . . 67
8. Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . 67
9. References . . . . . . . . . . . . . . . . . . . . . . . . . . 67
9.1. Normative References . . . . . . . . . . . . . . . . . . . 67
9.2. Informative References . . . . . . . . . . . . . . . . . . 68
1. Introduction
This document specifies an FEC scheme for the RaptorQ forward error
correction code for object delivery applications. The concept of an
FEC scheme is defined in RFC 5052 [RFC5052], and this document
follows the format prescribed there and uses the terminology of that
document. The RaptorQ code described herein is a next generation of
the Raptor code described in RFC 5053 [RFC5053]. The RaptorQ code
provides superior reliability, better coding efficiency, and support
for larger source block sizes than the Raptor code of RFC 5053
[RFC5053]. These improvements simplify the usage of the RaptorQ code
in an object delivery Content Delivery Protocol compared to RFC 5053
RFC 5053 [RFC5053]. A detailed mathematical design and analysis of
the RaptorQ code together with extensive simulation results are
provided in [RaptorCodes].
The RaptorQ FEC scheme is a Fully-Specified FEC scheme corresponding
to FEC Encoding ID 6.
RaptorQ is a fountain code, i.e., as many encoding symbols as needed
can be generated on the fly by the encoder from the source symbols of
a block. The decoder is able to recover the source block from almost
any set of encoding symbols of cardinality only slightly larger than
the number of source symbols.
The code described in this document is a systematic code; that is,
the original unmodified source symbols, as well as a number of repair
symbols, can be sent from sender to receiver. For more background on
the use of Forward Error Correction codes in reliable multicast, see
[RFC3453].
2. Requirements Notation
The key words "MUST", "MUST NOT", "REQUIRED", "SHALL", "SHALL NOT",
"SHOULD", "SHOULD NOT", "RECOMMENDED", "MAY", and "OPTIONAL" in this
document are to be interpreted as described in [RFC2119].
3. Formats and Codes
3.1. Introduction
The octet order of all fields is network byte order, i.e., big-
endian.
3.2. FEC Payload IDs
The FEC Payload ID MUST be a 4-octet field defined as follows:
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
+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
| SBN | Encoding Symbol ID |
+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
Figure 1: FEC Payload ID Format
o Source Block Number (SBN): 8-bit unsigned integer. A non-negative
integer identifier for the source block that the encoding symbols
within the packet relate to.
o Encoding Symbol ID (ESI): 24-bit unsigned integer. A non-negative
integer identifier for the encoding symbols within the packet.
The interpretation of the Source Block Number and Encoding Symbol
Identifier is defined in Section 4.
3.3. FEC Object Transmission Information
3.3.1. Mandatory
The value of the FEC Encoding ID MUST be 6, as assigned by IANA (see
Section 7).
3.3.2. Common
EID 5548 (Verified) is as follows:
Section: 3.3.2
Original Text:
o Transfer Length (F): 40-bit unsigned integer. A non-negative
integer that is at most 946270874880. This is the transfer length
of the object in units of octets.
...
NOTE: The limit of 946270874880 on the transfer length is a
consequence of the limitation on the symbol size to 2^^16-1, the
limitation on the number of symbols in a source block to 56403,
and the limitation on the number of source blocks to 2^^8.
Corrected Text:
o Transfer Length (F): 40-bit unsigned integer. A non-negative
integer that is at most 942574504275. This is the transfer length
of the object in units of octets.
...
NOTE: The limit of 942574504275 on the transfer length is a
consequence of the limitation on the symbol size to 2^^16-1, the
limitation on the number of symbols in a source block to 56403,
and the limitation on the number of source blocks to 2^^8-1.
Notes:
Section 3.3.3 defines the number of source blocks (Z) as an unsigned 8-bit integer, whose maximum value is 2^^8-1 (255), and not 2^^8. This in turn changes the limit on the transfer length, which is now 56403*(2^^16-1)*(2^^8-1) = 942574504275.
The Common FEC Object Transmission Information elements used by this
FEC scheme are:
o Transfer Length (F): 40-bit unsigned integer. A non-negative
integer that is at most 946270874880. This is the transfer length
of the object in units of octets.
o Symbol Size (T): 16-bit unsigned integer. A positive integer that
is less than 2^^16. This is the size of a symbol in units of
octets.
The encoded Common FEC Object Transmission Information (OTI) format
is shown in Figure 2.
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
+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
| Transfer Length (F) |
+ +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
| | Reserved | Symbol Size (T) |
+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
Figure 2: Encoded Common FEC OTI for RaptorQ FEC Scheme
NOTE: The limit of 946270874880 on the transfer length is a
consequence of the limitation on the symbol size to 2^^16-1, the
limitation on the number of symbols in a source block to 56403,
and the limitation on the number of source blocks to 2^^8.
3.3.3. Scheme-Specific
The following parameters are carried in the Scheme-Specific FEC
Object Transmission Information element for this FEC scheme:
o The number of source blocks (Z): 8-bit unsigned integer.
o The number of sub-blocks (N): 16-bit unsigned integer.
o A symbol alignment parameter (Al): 8-bit unsigned integer.
These parameters are all positive integers. The encoded Scheme-
specific Object Transmission Information is a 4-octet field
consisting of the parameters Z, N, and Al as shown in Figure 3.
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
+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
| Z | N | Al |
+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
Figure 3: Encoded Scheme-Specific FEC Object Transmission Information
The encoded FEC Object Transmission Information is a 12-octet field
consisting of the concatenation of the encoded Common FEC Object
Transmission Information and the encoded Scheme-specific FEC Object
Transmission Information.
These three parameters define the source block partitioning as
described in Section 4.4.1.2.
4. Procedures
4.1. Introduction
For any undefined symbols or functions used in this section, in
particular the functions "ceil" and "floor", refer to Section 5.1.
4.2. Content Delivery Protocol Requirements
This section describes the information exchange between the RaptorQ
FEC scheme and any Content Delivery Protocol (CDP) that makes use of
the RaptorQ FEC scheme for object delivery.
The RaptorQ encoder scheme and RaptorQ decoder scheme for object
delivery require the following information from the CDP:
o F: the transfer length of the object, in octets
o Al: the symbol alignment parameter
o T: the symbol size in octets, which MUST be a multiple of Al
o Z: the number of source blocks
o N: the number of sub-blocks in each source block
The RaptorQ encoder scheme for object delivery additionally requires:
- the object to be encoded, which is F octets long
The RaptorQ encoder scheme supplies the CDP with the following
information for each packet to be sent:
o Source Block Number (SBN)
o Encoding Symbol ID (ESI)
o Encoding symbol(s)
The CDP MUST communicate this information to the receiver.
4.3. Example Parameter Derivation Algorithm
This section provides recommendations for the derivation of the three
transport parameters, T, Z, and N. This recommendation is based on
the following input parameters:
o F: the transfer length of the object, in octets
o WS: the maximum size block that is decodable in working memory, in
octets
o P': the maximum payload size in octets, which is assumed to be a
multiple of Al
o Al: the symbol alignment parameter, in octets
o SS: a parameter where the desired lower bound on the sub-symbol
size is SS*Al
o K'_max: the maximum number of source symbols per source block.
Note: Section 5.1.2 defines K'_max to be 56403.
Based on the above inputs, the transport parameters T, Z, and N are
calculated as follows:
Let
o T = P'
o Kt = ceil(F/T)
o N_max = floor(T/(SS*Al))
o for all n=1, ..., N_max
* KL(n) is the maximum K' value in Table 2 in Section 5.6 such
that
K' <= WS/(Al*(ceil(T/(Al*n))))
o Z = ceil(Kt/KL(N_max))
o N is the minimum n=1, ..., N_max such that ceil(Kt/Z) <= KL(n)
It is RECOMMENDED that each packet contains exactly one symbol.
However, receivers SHALL support the reception of packets that
contain multiple symbols.
The value Kt is the total number of symbols required to represent the
source data of the object.
The algorithm above and that defined in Section 4.4.1.2 ensure that
the sub-symbol sizes are a multiple of the symbol alignment
parameter, Al. This is useful because the sum operations used for
encoding and decoding are generally performed several octets at a
time, for example, at least 4 octets at a time on a 32-bit processor.
Thus, the encoding and decoding can be performed faster if the sub-
symbol sizes are a multiple of this number of octets.
The recommended setting for the input parameter Al is 4.
The parameter WS can be used to generate encoded data that can be
decoded efficiently with limited working memory at the decoder. Note
that the actual maximum decoder memory requirement for a given value
of WS depends on the implementation, but it is possible to implement
decoding using working memory only slightly larger than WS.
4.4. Object Delivery
4.4.1. Source Block Construction
4.4.1.1. General
In order to apply the RaptorQ encoder to a source object, the object
may be broken into Z >= 1 blocks, known as source blocks. The
RaptorQ encoder is applied independently to each source block. Each
source block is identified by a unique Source Block Number (SBN),
where the first source block has SBN zero, the second has SBN one,
etc. Each source block is divided into a number, K, of source
symbols of size T octets each. Each source symbol is identified by a
unique Encoding Symbol Identifier (ESI), where the first source
symbol of a source block has ESI zero, the second has ESI one, etc.
Each source block with K source symbols is divided into N >= 1 sub-
blocks, which are small enough to be decoded in the working memory.
Each sub-block is divided into K sub-symbols of size T'.
Note that the value of K is not necessarily the same for each source
block of an object, and the value of T' may not necessarily be the
same for each sub-block of a source block. However, the symbol size
T is the same for all source blocks of an object, and the number of
symbols K is the same for every sub-block of a source block. Exact
partitioning of the object into source blocks and sub-blocks is
described in Section 4.4.1.2 below.
4.4.1.2. Source Block and Sub-Block Partitioning
The construction of source blocks and sub-blocks is determined based
on five input parameters -- F, Al, T, Z, and N -- and a function
Partition[]. The five input parameters are defined as follows:
o F: the transfer length of the object, in octets
o Al: a symbol alignment parameter, in octets
o T: the symbol size, in octets, which MUST be a multiple of Al
o Z: the number of source blocks
o N: the number of sub-blocks in each source block
These parameters MUST be set so that ceil(ceil(F/T)/Z) <= K'_max.
Recommendations for derivation of these parameters are provided in
Section 4.3.
The function Partition[I,J] derives parameters for partitioning a
block of size I into J approximately equal-sized blocks. More
specifically, it partitions I into JL blocks of length IL and JS
blocks of length IS. The output of Partition[I, J] is the sequence
(IL, IS, JL, JS), where IL = ceil(I/J), IS = floor(I/J), JL = I - IS
* J, and JS = J - JL.
The source object MUST be partitioned into source blocks and sub-
blocks as follows:
Let
o Kt = ceil(F/T),
o (KL, KS, ZL, ZS) = Partition[Kt, Z],
o (TL, TS, NL, NS) = Partition[T/Al, N].
Then, the object MUST be partitioned into Z = ZL + ZS contiguous
source blocks, the first ZL source blocks each having KL*T octets,
i.e., KL source symbols of T octets each, and the remaining ZS source
blocks each having KS*T octets, i.e., KS source symbols of T octets
each.
If Kt*T > F, then, for encoding purposes, the last symbol of the last
source block MUST be padded at the end with Kt*T-F zero octets.
Next, each source block with K source symbols MUST be divided into N
= NL + NS contiguous sub-blocks, the first NL sub-blocks each
consisting of K contiguous sub-symbols of size of TL*Al octets and
the remaining NS sub-blocks each consisting of K contiguous sub-
symbols of size of TS*Al octets. The symbol alignment parameter Al
ensures that sub-symbols are always a multiple of Al octets.
Finally, the mth symbol of a source block consists of the
concatenation of the mth sub-symbol from each of the N sub-blocks.
Note that this implies that when N > 1, a symbol is NOT a contiguous
portion of the object.
4.4.2. Encoding Packet Construction
Each encoding packet contains the following information:
o Source Block Number (SBN)
o Encoding Symbol ID (ESI)
o encoding symbol(s)
Each source block is encoded independently of the others. Each
encoding packet contains encoding symbols generated from the one
source block identified by the SBN carried in the encoding packet.
Source blocks are numbered consecutively from zero.
Encoding Symbol ID values from 0 to K-1 identify the source symbols
of a source block in sequential order, where K is the number of
source symbols in the source block. Encoding Symbol IDs K onwards
identify repair symbols generated from the source symbols using the
RaptorQ encoder.
Each encoding packet either contains only source symbols (source
packet) or contains only repair symbols (repair packet). A packet
may contain any number of symbols from the same source block. In the
case that the last source symbol in a source packet includes padding
octets added for FEC encoding purposes, then these octets need not be
included in the packet. Otherwise, each packet MUST contain only
whole symbols.
The Encoding Symbol ID, X, carried in each source packet is the
Encoding Symbol ID of the first source symbol carried in that packet.
The subsequent source symbols in the packet have Encoding Symbol IDs
X+1 to X+G-1 in sequential order, where G is the number of symbols in
the packet.
Similarly, the Encoding Symbol ID, X, placed into a repair packet is
the Encoding Symbol ID of the first repair symbol in the repair
packet, and the subsequent repair symbols in the packet have Encoding
Symbol IDs X+1 to X+G-1 in sequential order, where G is the number of
symbols in the packet.
Note that it is not necessary for the receiver to know the total
number of repair packets.
4.4.3. Example Receiver Recovery Strategies
A receiver can use the received encoding symbols for each source
block of an object to recover the source symbols for that source
block independently of all other source blocks.
If there is one sub-block per source block, i.e., N = 1, then the
portion of the data in the original object in its original order
associated with a source block consists of the concatenation of the
source symbols of a source block in consecutive ESI order.
If there are multiple sub-blocks per source block, i.e., if N > 1,
then the portion of the data in the original object in its original
order associated with a source block consists of the concatenation of
the sub-blocks associated with the source block, where sub-symbols
within each sub-block are in consecutive ESI order. In this case,
there are different receiver source block recovery strategies worth
considering depending on the available amount of Random Access Memory
(RAM) at the receiver, as outlined below.
One strategy is to recover the source symbols of a source block using
the decoding procedures applied to the received symbols for the
source block to recover the source symbols as described in Section 5,
and then to reorder the sub-symbols of the source symbols so that all
consecutive sub-symbols of the first sub-block are first, followed by
all consecutive sub-symbols of the second sub-block, etc., followed
by all consecutive sub-symbols of the Nth sub-block. This strategy
is especially applicable if the receiver has enough RAM to decode an
entire source block.
Another strategy is to separately recover the sub-blocks of a source
block. For example, a receiver may demultiplex and store sub-symbols
associated with each sub-block separately as packets containing
encoding symbols arrive, and then use the stored sub-symbols received
for a sub-block to recover that sub-block using the decoding
procedures described in Section 5. This strategy is especially
applicable if the receiver has enough RAM to decode only one sub-
block at a time.
5. RaptorQ FEC Code Specification
5.1. Background
For the purpose of the RaptorQ FEC code specification in this
section, the following definitions, symbols, and abbreviations apply.
A basic understanding of linear algebra, matrix operations, and
finite fields is assumed in this section. In particular, matrix
multiplication and matrix inversion operations over a mixture of the
finite fields GF[2] and GF[256] are used. A basic familiarity with
sparse linear equations, and efficient implementations of algorithms
that take advantage of sparse linear equations, is also quite
beneficial to an implementer of this specification.
5.1.1. Definitions
o Source block: a block of K source symbols that are considered
together for RaptorQ encoding and decoding purposes.
o Extended Source Block: a block of K' source symbols, where K' >=
K, constructed from a source block and zero or more padding
symbols.
o Symbol: a unit of data. The size, in octets, of a symbol is known
as the symbol size. The symbol size is always a positive integer.
o Source symbol: the smallest unit of data used during the encoding
process. All source symbols within a source block have the same
size.
o Padding symbol: a symbol with all zero bits that is added to the
source block to form the extended source block.
o Encoding symbol: a symbol that can be sent as part of the encoding
of a source block. The encoding symbols of a source block consist
of the source symbols of the source block and the repair symbols
generated from the source block. Repair symbols generated from a
source block have the same size as the source symbols of that
source block.
o Repair symbol: the encoding symbols of a source block that are not
source symbols. The repair symbols are generated based on the
source symbols of a source block.
o Intermediate symbols: symbols generated from the source symbols
using an inverse encoding process based on pre-coding
relationships. The repair symbols are then generated directly
from the intermediate symbols. The encoding symbols do not
include the intermediate symbols, i.e., intermediate symbols are
not sent as part of the encoding of a source block. The
intermediate symbols are partitioned into LT symbols and PI
symbols for the purposes of the encoding process.
o LT symbols: a process similar to that described in [LTCodes] is
used to generate part of the contribution to each generated
encoding symbol from the portion of the intermediate symbols
designated as LT symbols.
o PI symbols: a process even simpler than that described in
[LTCodes] is used to generate the other part of the contribution
to each generated encoding symbol from the portion of the
intermediate symbols designated as PI symbols. In the decoding
algorithm suggested in Section 5.4, the PI symbols are inactivated
at the start, i.e., are placed into the matrix U at the beginning
of the first phase of the decoding algorithm. Because the symbols
corresponding to the columns of U are sometimes called the
"inactivated" symbols, and since the PI symbols are inactivated at
the beginning, they are considered "permanently inactivated".
o HDPC symbols: there is a small subset of the intermediate symbols
that are HDPC symbols. Each HDPC symbol has a pre-coding
relationship with a large fraction of the other intermediate
symbols. HDPC means "High Density Parity Check".
o LDPC symbols: there is a moderate-sized subset of the intermediate
symbols that are LDPC symbols. Each LDPC symbol has a pre-coding
relationship with a small fraction of the other intermediate
symbols. LDPC means "Low Density Parity Check".
o Systematic code: a code in which all source symbols are included
as part of the encoding symbols of a source block. The RaptorQ
code as described herein is a systematic code.
o Encoding Symbol ID (ESI): information that uniquely identifies
each encoding symbol associated with a source block for sending
and receiving purposes.
o Internal Symbol ID (ISI): information that uniquely identifies
each symbol associated with an extended source block for encoding
and decoding purposes.
o Arithmetic operations on octets and symbols and matrices: the
operations that are used to produce encoding symbols from source
symbols and vice versa. See Section 5.7.
5.1.2. Symbols
i, j, u, v, h, d, a, b, d1, a1, b1, v, m, x, y represent values or
variables of one type or another, depending on the context.
X denotes a non-negative integer value that is either an ISI value
or an ESI value, depending on the context.
ceil(x) denotes the smallest integer that is greater than or equal
to x, where x is a real value.
floor(x) denotes the largest integer that is less than or equal to
x, where x is a real value.
min(x,y) denotes the minimum value of the values x and y, and in
general the minimum value of all the argument values.
max(x,y) denotes the maximum value of the values x and y, and in
general the maximum value of all the argument values.
i % j denotes i modulo j.
i + j denotes the sum of i and j. If i and j are octets or symbols,
this designates the arithmetic on octets or symbols,
respectively, as defined in Section 5.7. If i and j are
integers, then it denotes the usual integer addition.
i * j denotes the product of i and j. If i and j are octets, this
designates the arithmetic on octets, as defined in Section 5.7.
If i is an octet and j is a symbol, this denotes the
multiplication of a symbol by an octet, as also defined in
Section 5.7. Finally, if i and j are integers, i * j denotes
the usual product of integers.
a ^^ b denotes the operation a raised to the power b. If a is an
octet and b is a non-negative integer, this is understood to
mean a*a*...*a (b terms), with '*' being the octet product as
defined in Section 5.7.
u ^ v denotes, for equal-length bit strings u and v, the bitwise
exclusive-or of u and v.
Transpose[A] denotes the transposed matrix of matrix A. In this
specification, all matrices have entries that are octets.
A^^-1 denotes the inverse matrix of matrix A. In this
specification, all the matrices have octets as entries, so it is
understood that the operations of the matrix entries are to be
done as stated in Section 5.7 and A^^-1 is the matrix inverse of
A with respect to octet arithmetic.
K denotes the number of symbols in a single source block.
K' denotes the number of source plus padding symbols in an extended
source block. For the majority of this specification, the
padding symbols are considered to be additional source symbols.
K'_max denotes the maximum number of source symbols that can be in a
single source block. Set to 56403.
L denotes the number of intermediate symbols for a single extended
source block.
S denotes the number of LDPC symbols for a single extended source
block. These are LT symbols. For each value of K' shown in
Table 2 in Section 5.6, the corresponding value of S is a prime
number.
H denotes the number of HDPC symbols for a single extended source
block. These are PI symbols.
B denotes the number of intermediate symbols that are LT symbols
excluding the LDPC symbols.
W denotes the number of intermediate symbols that are LT symbols.
For each value of K' in Table 2 shown in Section 5.6, the
corresponding value of W is a prime number.
P denotes the number of intermediate symbols that are PI symbols.
These contain all HDPC symbols.
P1 denotes the smallest prime number greater than or equal to P.
U denotes the number of non-HDPC intermediate symbols that are PI
symbols.
C denotes an array of intermediate symbols, C[0], C[1], C[2], ...,
C[L-1].
C' denotes an array of the symbols of the extended source block,
where C'[0], C'[1], C'[2], ..., C'[K-1] are the source symbols
of the source block and C'[K], C'[K+1], ..., C'[K'-1] are
padding symbols.
V0, V1, V2, V3 denote four arrays of 32-bit unsigned integers,
V0[0], V0[1], ..., V0[255]; V1[0], V1[1], ..., V1[255]; V2[0],
V2[1], ..., V2[255]; and V3[0], V3[1], ..., V3[255] as shown in
Section 5.5.
Rand[y, i, m] denotes a pseudo-random number generator.
Deg[v] denotes a degree generator.
Enc[K', C ,(d, a, b, d1, a1, b1)] denotes an encoding symbol
generator.
Tuple[K', X] denotes a tuple generator function.
T denotes the symbol size in octets.
J(K') denotes the systematic index associated with K'.
G denotes any generator matrix.
I_S denotes the S x S identity matrix.
5.2. Overview
This section defines the systematic RaptorQ FEC code.
Symbols are the fundamental data units of the encoding and decoding
process. For each source block, all symbols are the same size,
referred to as the symbol size T. The atomic operations performed on
symbols for both encoding and decoding are the arithmetic operations
defined in Section 5.7.
The basic encoder is described in Section 5.3. The encoder first
derives a block of intermediate symbols from the source symbols of a
source block. This intermediate block has the property that both
source and repair symbols can be generated from it using the same
process. The encoder produces repair symbols from the intermediate
block using an efficient process, where each such repair symbol is
the exclusive-or of a small number of intermediate symbols from the
block. Source symbols can also be reproduced from the intermediate
block using the same process. The encoding symbols are the
combination of the source and repair symbols.
An example of a decoder is described in Section 5.4. The process for
producing source and repair symbols from the intermediate block is
designed so that the intermediate block can be recovered from any
sufficiently large set of encoding symbols, independent of the mix of
source and repair symbols in the set. Once the intermediate block is
recovered, missing source symbols of the source block can be
recovered using the encoding process.
Requirements for a RaptorQ-compliant decoder are provided in
Section 5.8. A number of decoding algorithms are possible to achieve
these requirements. An efficient decoding algorithm to achieve these
requirements is provided in Section 5.4.
The construction of the intermediate and repair symbols is based in
part on a pseudo-random number generator described in Section 5.3.
This generator is based on a fixed set of 1024 random numbers that
must be available to both sender and receiver. These numbers are
provided in Section 5.5. Encoding and decoding operations for
RaptorQ use operations on octets. Section 5.7 describes how to
perform these operations.
Finally, the construction of the intermediate symbols from the source
symbols is governed by "systematic indices", values of which are
provided in Section 5.6 for specific extended source block sizes
between 6 and K'_max = 56403 source symbols. Thus, the RaptorQ code
supports source blocks with between 1 and 56403 source symbols.
5.3. Systematic RaptorQ Encoder
5.3.1. Introduction
For a given source block of K source symbols, for encoding and
decoding purposes, the source block is augmented with K'-K additional
padding symbols, where K' is the smallest value that is at least K in
the systematic index Table 2 of Section 5.6. The reason for padding
out a source block to a multiple of K' is to enable faster encoding
and decoding and to minimize the amount of table information that
needs to be stored in the encoder and decoder.
For purposes of transmitting and receiving data, the value of K is
used to determine the number of source symbols in a source block, and
thus K needs to be known at the sender and the receiver. In this
case, the sender and receiver can compute K' from K and the K'-K
padding symbols can be automatically added to the source block
without any additional communication. The encoding symbol ID (ESI)
is used by a sender and receiver to identify the encoding symbols of
a source block, where the encoding symbols of a source block consist
of the source symbols and the repair symbols associated with the
source block. For a source block with K source symbols, the ESIs for
the source symbols are 0, 1, 2, ..., K-1, and the ESIs for the repair
symbols are K, K+1, K+2, .... Using the ESI for identifying encoding
symbols in transport ensures that the ESI values continue
consecutively between the source and repair symbols.
For purposes of encoding and decoding data, the value of K' derived
from K is used as the number of source symbols of the extended source
block upon which encoding and decoding operations are performed,
where the K' source symbols consist of the original K source symbols
and an additional K'-K padding symbols. The Internal Symbol ID (ISI)
is used by the encoder and decoder to identify the symbols associated
with the extended source block, i.e., for generating encoding symbols
and for decoding. For a source block with K original source symbols,
the ISIs for the original source symbols are 0, 1, 2, ..., K-1, the
ISIs for the K'-K padding symbols are K, K+1, K+2, ..., K'-1, and the
ISIs for the repair symbols are K', K'+1, K'+2, .... Using the ISI
for encoding and decoding allows the padding symbols of the extended
source block to be treated the same way as other source symbols of
the extended source block. Also, it ensures that a given prefix of
repair symbols are generated in a consistent way for a given number
K' of source symbols in the extended source block, independent of K.
The relationship between the ESIs and the ISIs is simple: the ESIs
and the ISIs for the original K source symbols are the same, the K'-K
padding symbols have an ISI but do not have a corresponding ESI
(since they are symbols that are neither sent nor received), and a
repair symbol ISI is simply the repair symbol ESI plus K'-K. The
translation between ESIs (used to identify encoding symbols sent and
received) and the corresponding ISIs (used for encoding and
decoding), as well as determining the proper padding of the extended
source block with padding symbols (used for encoding and decoding),
is the internal responsibility of the RaptorQ encoder/decoder.
5.3.2. Encoding Overview
The systematic RaptorQ encoder is used to generate any number of
repair symbols from a source block that consists of K source symbols
placed into an extended source block C'. Figure 4 shows the encoding
overview.
The first step of encoding is to construct an extended source block
by adding zero or more padding symbols such that the total number of
symbols, K', is one of the values listed in Section 5.6. Each
padding symbol consists of T octets where the value of each octet is
zero. K' MUST be selected as the smallest value of K' from the table
of Section 5.6 that is greater than or equal to K.
-----------------------------------------------------------+
| |
| +-----------+ +--------------+ +-------------+ |
C' | | | C' | Intermediate | C | | |
----+--->| Padding |--->| Symbol |--->| Encoding |--+-->
K | | | K' | Generation | L | | |
| +-----------+ +--------------+ +-------------+ |
| | (d,a,b, ^ |
| | d1,a1,b1)| |
| | +------------+ |
| | K' | Tuple | |
| +----------------------------->| | |
| | Generation | |
| +------------+ |
| ^ |
+-------------------------------------------------+--------+
|
ISI X
Figure 4: Encoding Overview
Let C'[0], ..., C'[K-1] denote the K source symbols.
Let C'[K], ..., C'[K'-1] denote the K'-K padding symbols, which are
all set to zero bits. Then, C'[0], ..., C'[K'-1] are the symbols of
the extended source block upon which encoding and decoding are
performed.
In the remainder of this description, these padding symbols will be
considered as additional source symbols and referred to as such.
However, these padding symbols are not part of the encoding symbols,
i.e., they are not sent as part of the encoding. At a receiver, the
value of K' can be computed based on K, then the receiver can insert
K'-K padding symbols at the end of a source block of K' source
symbols and recover the remaining K source symbols of the source
block from received encoding symbols.
The second step of encoding is to generate a number, L > K', of
intermediate symbols from the K' source symbols. In this step, K'
source tuples (d[0], a[0], b[0], d1[0], a1[0], b1[0]), ..., (d[K'-1],
a[K'-1], b[K'-1], d1[K'-1], a1[K'-1], b1[K'-1]) are generated using
the Tuple[] generator as described in Section 5.3.5.4. The K' source
tuples and the ISIs associated with the K' source symbols are used to
determine L intermediate symbols C[0], ..., C[L-1] from the source
symbols using an inverse encoding process. This process can be
realized by a RaptorQ decoding process.
Certain "pre-coding relationships" must hold within the L
intermediate symbols. Section 5.3.3.3 describes these relationships.
Section 5.3.3.4 describes how the intermediate symbols are generated
from the source symbols.
Once the intermediate symbols have been generated, repair symbols can
be produced. For a repair symbol with ISI X > K', the tuple of non-
negative integers (d, a, b, d1, a1, b1) can be generated, using the
Tuple[] generator as described in Section 5.3.5.4. Then, the (d, a,
b, d1, a1, b1) tuple and the ISI X are used to generate the
corresponding repair symbol from the intermediate symbols using the
Enc[] generator described in Section 5.3.5.3. The corresponding ESI
for this repair symbol is then X-(K'-K). Note that source symbols of
the extended source block can also be generated using the same
process, i.e., for any X < K', the symbol generated using this
process has the same value as C'[X].
5.3.3. First Encoding Step: Intermediate Symbol Generation
5.3.3.1. General
This encoding step is a pre-coding step to generate the L
intermediate symbols C[0], ..., C[L-1] from the source symbols C'[0],
..., C'[K'-1], where L > K' is defined in Section 5.3.3.3. The
intermediate symbols are uniquely defined by two sets of constraints:
1. The intermediate symbols are related to the source symbols by a
set of source symbol tuples and by the ISIs of the source
symbols. The generation of the source symbol tuples is defined
in Section 5.3.3.2 using the Tuple[] generator as described in
Section 5.3.5.4.
2. A number of pre-coding relationships hold within the intermediate
symbols themselves. These are defined in Section 5.3.3.3.
The generation of the L intermediate symbols is then defined in
Section 5.3.3.4.
5.3.3.2. Source Symbol Tuples
Each of the K' source symbols is associated with a source symbol
tuple (d[X], a[X], b[X], d1[X], a1[X], b1[X]) for 0 <= X < K'. The
source symbol tuples are determined using the Tuple[] generator
defined in Section 5.3.5.4 as:
For each X, 0 <= X < K'
(d[X], a[X], b[X], d1[X], a1[X], b1[X]) = Tuple[K, X]
5.3.3.3. Pre-Coding Relationships
The pre-coding relationships amongst the L intermediate symbols are
defined by requiring that a set of S+H linear combinations of the
intermediate symbols evaluate to zero. There are S LDPC and H HDPC
symbols, and thus L = K'+S+H. Another partition of the L
intermediate symbols is into two sets, one set of W LT symbols and
another set of P PI symbols, and thus it is also the case that L =
W+P. The P PI symbols are treated differently than the W LT symbols
in the encoding process. The P PI symbols consist of the H HDPC
symbols together with a set of U = P-H of the other K' intermediate
symbols. The W LT symbols consist of the S LDPC symbols together
with W-S of the other K' intermediate symbols. The values of these
parameters are determined from K' as described below, where H(K'),
S(K'), and W(K') are derived from Table 2 in Section 5.6.
Let
o S = S(K')
o H = H(K')
o W = W(K')
o L = K' + S + H
o P = L - W
o P1 denote the smallest prime number greater than or equal to P.
o U = P - H
o B = W - S
o C[0], ..., C[B-1] denote the intermediate symbols that are LT
symbols but not LDPC symbols.
o C[B], ..., C[B+S-1] denote the S LDPC symbols that are also LT
symbols.
o C[W], ..., C[W+U-1] denote the intermediate symbols that are PI
symbols but not HDPC symbols.
o C[L-H], ..., C[L-1] denote the H HDPC symbols that are also PI
symbols.
The first set of pre-coding relations, called LDPC relations, is
described below and requires that at the end of this process the set
of symbols D[0] , ..., D[S-1] are all zero:
o Initialize the symbols D[0] = C[B], ..., D[S-1] = C[B+S-1].
o For i = 0, ..., B-1 do
* a = 1 + floor(i/S)
* b = i % S
* D[b] = D[b] + C[i]
* b = (b + a) % S
* D[b] = D[b] + C[i]
* b = (b + a) % S
* D[b] = D[b] + C[i]
o For i = 0, ..., S-1 do
* a = i % P
* b = (i+1) % P
* D[i] = D[i] + C[W+a] + C[W+b]
Recall that the addition of symbols is to be carried out as specified
in Section 5.7.
Note that the LDPC relations as defined in the algorithm above are
linear, so there exists an S x B matrix G_LDPC,1 and an S x P matrix
G_LDPC,2 such that
G_LDPC,1 * Transpose[(C[0], ..., C[B-1])] + G_LDPC,2 *
Transpose(C[W], ..., C[W+P-1]) + Transpose[(C[B], ..., C[B+S-1])]
= 0
(The matrix G_LDPC,1 is defined by the first loop in the above
algorithm, and G_LDPC,2 can be deduced from the second loop.)
The second set of relations among the intermediate symbols C[0], ...,
C[L-1] are the HDPC relations and they are defined as follows:
Let
o alpha denote the octet represented by integer 2 as defined in
Section 5.7.
o MT denote an H x (K' + S) matrix of octets, where for j=0, ...,
K'+S-2, the entry MT[i,j] is the octet represented by the integer
1 if i= Rand[j+1,6,H] or i = (Rand[j+1,6,H] + Rand[j+1,7,H-1] + 1)
% H, and MT[i,j] is the zero element for all other values of i,
and for j=K'+S-1, MT[i,j] = alpha^^i for i=0, ..., H-1.
o GAMMA denote a (K'+S) x (K'+S) matrix of octets, where
GAMMA[i,j] =
alpha ^^ (i-j) for i >= j,
0 otherwise.
Then, the relationship between the first K'+S intermediate symbols
C[0], ..., C[K'+S-1] and the H HDPC symbols C[K'+S], ..., C[K'+S+H-1]
is given by:
Transpose[C[K'+S], ..., C[K'+S+H-1]] + MT * GAMMA *
Transpose[C[0], ..., C[K'+S-1]] = 0,
where '*' represents standard matrix multiplication utilizing the
octet multiplication to define the multiplication between a matrix of
octets and a matrix of symbols (in particular, the column vector of
symbols), and '+' denotes addition over octet vectors.
5.3.3.4. Intermediate Symbols
5.3.3.4.1. Definition
Given the K' source symbols C'[0], C'[1], ..., C'[K'-1] the L
intermediate symbols C[0], C[1], ..., C[L-1] are the uniquely defined
symbol values that satisfy the following conditions:
1. The K' source symbols C'[0], C'[1], ..., C'[K'-1] satisfy the K'
constraints
C'[X] = Enc[K', (C[0], ..., C[L-1]), (d[X], a[X], b[X], d1[X],
a1[X], b1[X])], for all X, 0 <= X < K',
where (d[X], a[X], b[X], d1[X], a1[X], b1[X])) = Tuple[K',X],
Tuple[] is defined in Section 5.3.5.4, and Enc[] is described in
Section 5.3.5.3.
2. The L intermediate symbols C[0], C[1], ..., C[L-1] satisfy the
pre-coding relationships defined in Section 5.3.3.3.
5.3.3.4.2. Example Method for Calculation of Intermediate Symbols
This section describes a possible method for calculation of the L
intermediate symbols C[0], C[1], ..., C[L-1] satisfying the
constraints in Section 5.3.3.4.1.
The L intermediate symbols can be calculated as follows:
Let
o C denote the column vector of the L intermediate symbols, C[0],
C[1], ..., C[L-1].
o D denote the column vector consisting of S+H zero symbols followed
by the K' source symbols C'[0], C'[1], ..., C'[K'-1].
Then, the above constraints define an L x L matrix A of octets such
that:
A*C = D
The matrix A can be constructed as follows:
Let
o G_LDPC,1 and G_LDPC,2 be S x B and S x P matrices as defined in
Section 5.3.3.3.
o G_HDPC be the H x (K'+S) matrix such that
G_HDPC * Transpose(C[0], ..., C[K'+S-1]) = Transpose(C[K'+S],
..., C[L-1]),
i.e., G_HDPC = MT*GAMMA
o I_S be the S x S identity matrix
o I_H be the H x H identity matrix
o G_ENC be the K' x L matrix such that
G_ENC * Transpose[(C[0], ..., C[L-1])] =
Transpose[(C'[0],C'[1], ...,C'[K'-1])],
i.e., G_ENC[i,j] = 1 if and only if C[j] is included in the
symbols that are summed to produce Enc[K', (C[0], ..., C[L-1]),
(d[i], a[i], b[i], d1[i], a1[i], b1[i])] and G_ENC[i,j] = 0
otherwise.
Then
o The first S rows of A are equal to G_LDPC,1 | I_S | G_LDPC,2.
o The next H rows of A are equal to G_HDPC | I_H.
o The remaining K' rows of A are equal to G_ENC.
The matrix A is depicted in Figure 5 below:
B S U H
+-----------------------+-------+------------------+
| | | |
S | G_LDPC,1 | I_S | G_LDPC,2 |
| | | |
+-----------------------+-------+----------+-------+
| | |
H | G_HDPC | I_H |
| | |
+------------------------------------------+-------+
| |
| |
K' | G_ENC |
| |
| |
+--------------------------------------------------+
Figure 5: The Matrix A
The intermediate symbols can then be calculated as:
C = (A^^-1)*D
The source tuples are generated such that for any K' matrix A has
full rank and is therefore invertible. This calculation can be
realized by applying a RaptorQ decoding process to the K' source
symbols C'[0], C'[1], ..., C'[K'-1] to produce the L intermediate
symbols C[0], C[1], ..., C[L-1].
To efficiently generate the intermediate symbols from the source
symbols, it is recommended that an efficient decoder implementation
such as that described in Section 5.4 be used.
5.3.4. Second Encoding Step: Encoding
In the second encoding step, the repair symbol with ISI X (X >= K')
is generated by applying the generator Enc[K', (C[0], C[1], ...,
C[L-1]), (d, a, b, d1, a1, b1)] defined in Section 5.3.5.3 to the L
intermediate symbols C[0], C[1], ..., C[L-1] using the tuple (d, a,
b, d1, a1, b1)=Tuple[K',X].
5.3.5. Generators
5.3.5.1. Random Number Generator
The random number generator Rand[y, i, m] is defined as follows,
where y is a non-negative integer, i is a non-negative integer less
than 256, and m is a positive integer, and the value produced is an
integer between 0 and m-1. Let V0, V1, V2, and V3 be the arrays
provided in Section 5.5.
Let
o x0 = (y + i) mod 2^^8
o x1 = (floor(y / 2^^8) + i) mod 2^^8
o x2 = (floor(y / 2^^16) + i) mod 2^^8
o x3 = (floor(y / 2^^24) + i) mod 2^^8
Then
Rand[y, i, m] = (V0[x0] ^ V1[x1] ^ V2[x2] ^ V3[x3]) % m
5.3.5.2. Degree Generator
The degree generator Deg[v] is defined as follows, where v is a non-
negative integer that is less than 2^^20 = 1048576. Given v, find
index d in Table 1 such that f[d-1] <= v < f[d], and set Deg[v] =
min(d, W-2). Recall that W is derived from K' as described in
Section 5.3.3.3.
+---------+---------+---------+---------+
| Index d | f[d] | Index d | f[d] |
+---------+---------+---------+---------+
| 0 | 0 | 1 | 5243 |
+---------+---------+---------+---------+
| 2 | 529531 | 3 | 704294 |
+---------+---------+---------+---------+
| 4 | 791675 | 5 | 844104 |
+---------+---------+---------+---------+
| 6 | 879057 | 7 | 904023 |
+---------+---------+---------+---------+
| 8 | 922747 | 9 | 937311 |
+---------+---------+---------+---------+
| 10 | 948962 | 11 | 958494 |
+---------+---------+---------+---------+
| 12 | 966438 | 13 | 973160 |
+---------+---------+---------+---------+
| 14 | 978921 | 15 | 983914 |
+---------+---------+---------+---------+
| 16 | 988283 | 17 | 992138 |
+---------+---------+---------+---------+
| 18 | 995565 | 19 | 998631 |
+---------+---------+---------+---------+
| 20 | 1001391 | 21 | 1003887 |
+---------+---------+---------+---------+
| 22 | 1006157 | 23 | 1008229 |
+---------+---------+---------+---------+
| 24 | 1010129 | 25 | 1011876 |
+---------+---------+---------+---------+
| 26 | 1013490 | 27 | 1014983 |
+---------+---------+---------+---------+
| 28 | 1016370 | 29 | 1017662 |
+---------+---------+---------+---------+
| 30 | 1048576 | | |
+---------+---------+---------+---------+
Table 1: Defines the Degree Distribution for Encoding Symbols
5.3.5.3. Encoding Symbol Generator
The encoding symbol generator Enc[K', (C[0], C[1], ..., C[L-1]), (d,
a, b, d1, a1, b1)] takes the following inputs:
o K' is the number of source symbols for the extended source block.
Let L, W, B, S, P, and P1 be derived from K' as described in
Section 5.3.3.3.
o (C[0], C[1], ..., C[L-1]) is the array of L intermediate symbols
(sub-symbols) generated as described in Section 5.3.3.4.
o (d, a, b, d1, a1, b1) is a source tuple determined from ISI X
using the Tuple[] generator defined in Section 5.3.5.4, whereby
* d is a positive integer denoting an encoding symbol LT degree
* a is a positive integer between 1 and W-1 inclusive
* b is a non-negative integer between 0 and W-1 inclusive
* d1 is a positive integer that has value either 2 or 3 denoting
an encoding symbol PI degree
* a1 is a positive integer between 1 and P1-1 inclusive
* b1 is a non-negative integer between 0 and P1-1 inclusive
The encoding symbol generator produces a single encoding symbol as
output (referred to as result), according to the following algorithm:
o result = C[b]
o For j = 1, ..., d-1 do
* b = (b + a) % W
* result = result + C[b]
o While (b1 >= P) do b1 = (b1+a1) % P1
o result = result + C[W+b1]
o For j = 1, ..., d1-1 do
* b1 = (b1 + a1) % P1
* While (b1 >= P) do b1 = (b1+a1) % P1
* result = result + C[W+b1]
o Return result
5.3.5.4. Tuple Generator
The tuple generator Tuple[K',X] takes the following inputs:
o K': the number of source symbols in the extended source block
o X: an ISI
Let
o L be determined from K' as described in Section 5.3.3.3
o J = J(K') be the systematic index associated with K', as defined
in Table 2 in Section 5.6
The output of the tuple generator is a tuple, (d, a, b, d1, a1, b1),
determined as follows:
o A = 53591 + J*997
o if (A % 2 == 0) { A = A + 1 }
o B = 10267*(J+1)
o y = (B + X*A) % 2^^32
o v = Rand[y, 0, 2^^20]
o d = Deg[v]
o a = 1 + Rand[y, 1, W-1]
o b = Rand[y, 2, W]
o If (d < 4) { d1 = 2 + Rand[X, 3, 2] } else { d1 = 2 }
o a1 = 1 + Rand[X, 4, P1-1]
o b1 = Rand[X, 5, P1]
5.4. Example FEC Decoder
5.4.1. General
This section describes an efficient decoding algorithm for the
RaptorQ code introduced in this specification. Note that each
received encoding symbol is a known linear combination of the
intermediate symbols. So, each received encoding symbol provides a
linear equation among the intermediate symbols, which, together with
the known linear pre-coding relationships amongst the intermediate
symbols, gives a system of linear equations. Thus, any algorithm for
solving systems of linear equations can successfully decode the
intermediate symbols and hence the source symbols. However, the
algorithm chosen has a major effect on the computational efficiency
of the decoding.
5.4.2. Decoding an Extended Source Block
5.4.2.1. General
It is assumed that the decoder knows the structure of the source
block it is to decode, including the symbol size, T, and the number K
of symbols in the source block and the number K' of source symbols in
the extended source block.
From the algorithms described in Section 5.3, the RaptorQ decoder can
calculate the total number L = K'+S+H of intermediate symbols and
determine how they were generated from the extended source block to
be decoded. In this description, it is assumed that the received
encoding symbols for the extended source block to be decoded are
passed to the decoder. Furthermore, for each such encoding symbol,
it is assumed that the number and set of intermediate symbols whose
sum is equal to the encoding symbol are passed to the decoder. In
the case of source symbols, including padding symbols, the source
symbol tuples described in Section 5.3.3.2 indicate the number and
set of intermediate symbols that sum to give each source symbol.
Let N >= K' be the number of received encoding symbols to be used for
decoding, including padding symbols for an extended source block, and
let M = S+H+N. Then, with the notation of Section 5.3.3.4.2, we have
A*C = D.
Decoding an extended source block is equivalent to decoding C from
known A and D. It is clear that C can be decoded if and only if the
rank of A is L. Once C has been decoded, missing source symbols can
be obtained by using the source symbol tuples to determine the number
and set of intermediate symbols that must be summed to obtain each
missing source symbol.
The first step in decoding C is to form a decoding schedule. In this
step, A is converted using Gaussian elimination (using row operations
and row and column reorderings) and after discarding M - L rows, into
the L x L identity matrix. The decoding schedule consists of the
sequence of row operations and row and column reorderings during the
Gaussian elimination process, and it only depends on A and not on D.
The decoding of C from D can take place concurrently with the forming
of the decoding schedule, or the decoding can take place afterwards
based on the decoding schedule.
The correspondence between the decoding schedule and the decoding of
C is as follows. Let c[0] = 0, c[1] = 1, ..., c[L-1] = L-1 and d[0]
= 0, d[1] = 1, ..., d[M-1] = M-1 initially.
o Each time a multiple, beta, of row i of A is added to row i' in
the decoding schedule, then in the decoding process the symbol
beta*D[d[i]] is added to symbol D[d[i']].
o Each time a row i of A is multiplied by an octet beta, then in the
decoding process the symbol D[d[i]] is also multiplied by beta.
o Each time row i is exchanged with row i' in the decoding schedule,
then in the decoding process the value of d[i] is exchanged with
the value of d[i'].
o Each time column j is exchanged with column j' in the decoding
schedule, then in the decoding process the value of c[j] is
exchanged with the value of c[j'].
From this correspondence, it is clear that the total number of
operations on symbols in the decoding of the extended source block is
the number of row operations (not exchanges) in the Gaussian
elimination. Since A is the L x L identity matrix after the Gaussian
elimination and after discarding the last M - L rows, it is clear at
the end of successful decoding that the L symbols D[d[0]], D[d[1]],
..., D[d[L-1]] are the values of the L symbols C[c[0]], C[c[1]], ...,
C[c[L-1]].
The order in which Gaussian elimination is performed to form the
decoding schedule has no bearing on whether or not the decoding is
successful. However, the speed of the decoding depends heavily on
the order in which Gaussian elimination is performed. (Furthermore,
maintaining a sparse representation of A is crucial, although this is
not described here.) The remainder of this section describes an
order in which Gaussian elimination could be performed that is
relatively efficient.
5.4.2.2. First Phase
In the first phase of the Gaussian elimination, the matrix A is
conceptually partitioned into submatrices and, additionally, a matrix
X is created. This matrix has as many rows and columns as A, and it
will be a lower triangular matrix throughout the first phase. At the
beginning of this phase, the matrix A is copied into the matrix X.
The submatrix sizes are parameterized by non-negative integers i and
u, which are initialized to 0 and P, the number of PI symbols,
respectively. The submatrices of A are:
1. The submatrix I defined by the intersection of the first i rows
and first i columns. This is the identity matrix at the end of
each step in the phase.
2. The submatrix defined by the intersection of the first i rows and
all but the first i columns and last u columns. All entries of
this submatrix are zero.
3. The submatrix defined by the intersection of the first i columns
and all but the first i rows. All entries of this submatrix are
zero.
4. The submatrix U defined by the intersection of all the rows and
the last u columns.
5. The submatrix V formed by the intersection of all but the first i
columns and the last u columns and all but the first i rows.
Figure 6 illustrates the submatrices of A. At the beginning of the
first phase, V consists of the first L-P columns of A, and U consists
of the last P columns corresponding to the PI symbols. In each step,
a row of A is chosen.
+-----------+-----------------+---------+
| | | |
| I | All Zeros | |
| | | |
+-----------+-----------------+ U |
| | | |
| | | |
| All Zeros | V | |
| | | |
| | | |
+-----------+-----------------+---------+
Figure 6: Submatrices of A in the First Phase
The following graph defined by the structure of V is used in
determining which row of A is chosen. The columns that intersect V
are the nodes in the graph, and the rows that have exactly 2 nonzero
entries in V and are not HDPC rows are the edges of the graph that
connect the two columns (nodes) in the positions of the two ones. A
component in this graph is a maximal set of nodes (columns) and edges
(rows) such that there is a path between each pair of nodes/edges in
the graph. The size of a component is the number of nodes (columns)
in the component.
There are at most L steps in the first phase. The phase ends
successfully when i + u = L, i.e., when V and the all zeros submatrix
above V have disappeared, and A consists of I, the all zeros
submatrix below I, and U. The phase ends unsuccessfully in decoding
failure if at some step before V disappears there is no nonzero row
in V to choose in that step. In each step, a row of A is chosen as
follows:
o If all entries of V are zero, then no row is chosen and decoding
fails.
o Let r be the minimum integer such that at least one row of A has
exactly r nonzeros in V.
* If r != 2, then choose a row with exactly r nonzeros in V with
minimum original degree among all such rows, except that HDPC
rows should not be chosen until all non-HDPC rows have been
processed.
* If r = 2 and there is a row with exactly 2 ones in V, then
choose any row with exactly 2 ones in V that is part of a
maximum size component in the graph described above that is
defined by V.
* If r = 2 and there is no row with exactly 2 ones in V, then
choose any row with exactly 2 nonzeros in V.
After the row is chosen in this step, the first row of A that
intersects V is exchanged with the chosen row so that the chosen row
is the first row that intersects V. The columns of A among those
that intersect V are reordered so that one of the r nonzeros in the
chosen row appears in the first column of V and so that the remaining
r-1 nonzeros appear in the last columns of V. The same row and
column operations are also performed on the matrix X. Then, an
appropriate multiple of the chosen row is added to all the other rows
of A below the chosen row that have a nonzero entry in the first
column of V. Specifically, if a row below the chosen row has entry
beta in the first column of V, and the chosen row has entry alpha in
the first column of V, then beta/alpha multiplied by the chosen row
is added to this row to leave a zero value in the first column of V.
Finally, i is incremented by 1 and u is incremented by r-1, which
completes the step.
Note that efficiency can be improved if the row operations identified
above are not actually performed until the affected row is itself
chosen during the decoding process. This avoids processing of row
operations for rows that are not eventually used in the decoding
process, and in particular this avoids those rows for which beta!=1
until they are actually required. Furthermore, the row operations
required for the HDPC rows may be performed for all such rows in one
process, by using the algorithm described in Section 5.3.3.3.
5.4.2.3. Second Phase
At this point, all the entries of X outside the first i rows and i
columns are discarded, so that X has lower triangular form. The last
i rows and columns of X are discarded, so that X now has i rows and i
columns. The submatrix U is further partitioned into the first i
rows, U_upper, and the remaining M - i rows, U_lower. Gaussian
elimination is performed in the second phase on U_lower either to
determine that its rank is less than u (decoding failure) or to
convert it into a matrix where the first u rows is the identity
matrix (success of the second phase). Call this u x u identity
matrix I_u. The M - L rows of A that intersect U_lower - I_u are
discarded. After this phase, A has L rows and L columns.
5.4.2.4. Third Phase
After the second phase, the only portion of A that needs to be zeroed
out to finish converting A into the L x L identity matrix is U_upper.
The number of rows i of the submatrix U_upper is generally much
larger than the number of columns u of U_upper. Moreover, at this
time, the matrix U_upper is typically dense, i.e., the number of
nonzero entries of this matrix is large. To reduce this matrix to a
sparse form, the sequence of operations performed to obtain the
matrix U_lower needs to be inverted. To this end, the matrix X is
multiplied with the submatrix of A consisting of the first i rows of
A. After this operation, the submatrix of A consisting of the
intersection of the first i rows and columns equals to X, whereas the
matrix U_upper is transformed to a sparse form.
5.4.2.5. Fourth Phase
For each of the first i rows of U_upper, do the following: if the row
has a nonzero entry at position j, and if the value of that nonzero
entry is b, then add to this row b times row j of I_u. After this
step, the submatrix of A consisting of the intersection of the first
i rows and columns is equal to X, the submatrix U_upper consists of
zeros, the submatrix consisting of the intersection of the last u
rows and the first i columns consists of zeros, and the submatrix
consisting of the last u rows and columns is the matrix I_u.
5.4.2.6. Fifth Phase
For j from 1 to i, perform the following operations:
1. If A[j,j] is not one, then divide row j of A by A[j,j].
2. For l from 1 to j-1, if A[j,l] is nonzero, then add A[j,l]
multiplied with row l of A to row j of A.
After this phase, A is the L x L identity matrix and a complete
decoding schedule has been successfully formed. Then, the
corresponding decoding consisting of summing known encoding symbols
can be executed to recover the intermediate symbols based on the
decoding schedule. The tuples associated with all source symbols are
computed according to Section 5.3.3.2. The tuples for received
source symbols are used in the decoding. The tuples for missing
source symbols are used to determine which intermediate symbols need
to be summed to recover the missing source symbols.
5.5. Random Numbers
The four arrays V0, V1, V2, and V3 used in Section 5.3.5.1 are
provided below. There are 256 entries in each of the four arrays.
The indexing into each array starts at 0, and the entries are 32-bit
unsigned integers.
5.5.1. The Table V0
251291136, 3952231631, 3370958628, 4070167936, 123631495,
3351110283, 3218676425, 2011642291, 774603218, 2402805061,
1004366930, 1843948209, 428891132, 3746331984, 1591258008,
3067016507, 1433388735, 504005498, 2032657933, 3419319784,
2805686246, 3102436986, 3808671154, 2501582075, 3978944421,
246043949, 4016898363, 649743608, 1974987508, 2651273766,
2357956801, 689605112, 715807172, 2722736134, 191939188,
3535520147, 3277019569, 1470435941, 3763101702, 3232409631,
122701163, 3920852693, 782246947, 372121310, 2995604341,
2045698575, 2332962102, 4005368743, 218596347, 3415381967,
4207612806, 861117671, 3676575285, 2581671944, 3312220480,
681232419, 307306866, 4112503940, 1158111502, 709227802,
2724140433, 4201101115, 4215970289, 4048876515, 3031661061,
1909085522, 510985033, 1361682810, 129243379, 3142379587,
2569842483, 3033268270, 1658118006, 932109358, 1982290045,
2983082771, 3007670818, 3448104768, 683749698, 778296777,
1399125101, 1939403708, 1692176003, 3868299200, 1422476658,
593093658, 1878973865, 2526292949, 1591602827, 3986158854,
3964389521, 2695031039, 1942050155, 424618399, 1347204291,
2669179716, 2434425874, 2540801947, 1384069776, 4123580443,
1523670218, 2708475297, 1046771089, 2229796016, 1255426612,
4213663089, 1521339547, 3041843489, 420130494, 10677091,
515623176, 3457502702, 2115821274, 2720124766, 3242576090,
854310108, 425973987, 325832382, 1796851292, 2462744411,
1976681690, 1408671665, 1228817808, 3917210003, 263976645,
2593736473, 2471651269, 4291353919, 650792940, 1191583883,
3046561335, 2466530435, 2545983082, 969168436, 2019348792,
2268075521, 1169345068, 3250240009, 3963499681, 2560755113,
911182396, 760842409, 3569308693, 2687243553, 381854665,
2613828404, 2761078866, 1456668111, 883760091, 3294951678,
1604598575, 1985308198, 1014570543, 2724959607, 3062518035,
3115293053, 138853680, 4160398285, 3322241130, 2068983570,
2247491078, 3669524410, 1575146607, 828029864, 3732001371,
3422026452, 3370954177, 4006626915, 543812220, 1243116171,
3928372514, 2791443445, 4081325272, 2280435605, 885616073,
616452097, 3188863436, 2780382310, 2340014831, 1208439576,
258356309, 3837963200, 2075009450, 3214181212, 3303882142,
880813252, 1355575717, 207231484, 2420803184, 358923368,
1617557768, 3272161958, 1771154147, 2842106362, 1751209208,
1421030790, 658316681, 194065839, 3241510581, 38625260,
301875395, 4176141739, 297312930, 2137802113, 1502984205,
3669376622, 3728477036, 234652930, 2213589897, 2734638932,
1129721478, 3187422815, 2859178611, 3284308411, 3819792700,
3557526733, 451874476, 1740576081, 3592838701, 1709429513,
3702918379, 3533351328, 1641660745, 179350258, 2380520112,
3936163904, 3685256204, 3156252216, 1854258901, 2861641019,
3176611298, 834787554, 331353807, 517858103, 3010168884,
4012642001, 2217188075, 3756943137, 3077882590, 2054995199,
3081443129, 3895398812, 1141097543, 2376261053, 2626898255,
2554703076, 401233789, 1460049922, 678083952, 1064990737,
940909784, 1673396780, 528881783, 1712547446, 3629685652,
1358307511
5.5.2. The Table V1
807385413, 2043073223, 3336749796, 1302105833, 2278607931,
541015020, 1684564270, 372709334, 3508252125, 1768346005,
1270451292, 2603029534, 2049387273, 3891424859, 2152948345,
4114760273, 915180310, 3754787998, 700503826, 2131559305,
1308908630, 224437350, 4065424007, 3638665944, 1679385496,
3431345226, 1779595665, 3068494238, 1424062773, 1033448464,
4050396853, 3302235057, 420600373, 2868446243, 311689386,
259047959, 4057180909, 1575367248, 4151214153, 110249784,
3006865921, 4293710613, 3501256572, 998007483, 499288295,
1205710710, 2997199489, 640417429, 3044194711, 486690751,
2686640734, 2394526209, 2521660077, 49993987, 3843885867,
4201106668, 415906198, 19296841, 2402488407, 2137119134,
1744097284, 579965637, 2037662632, 852173610, 2681403713,
1047144830, 2982173936, 910285038, 4187576520, 2589870048,
989448887, 3292758024, 506322719, 176010738, 1865471968,
2619324712, 564829442, 1996870325, 339697593, 4071072948,
3618966336, 2111320126, 1093955153, 957978696, 892010560,
1854601078, 1873407527, 2498544695, 2694156259, 1927339682,
1650555729, 183933047, 3061444337, 2067387204, 228962564,
3904109414, 1595995433, 1780701372, 2463145963, 307281463,
3237929991, 3852995239, 2398693510, 3754138664, 522074127,
146352474, 4104915256, 3029415884, 3545667983, 332038910,
976628269, 3123492423, 3041418372, 2258059298, 2139377204,
3243642973, 3226247917, 3674004636, 2698992189, 3453843574,
1963216666, 3509855005, 2358481858, 747331248, 1957348676,
1097574450, 2435697214, 3870972145, 1888833893, 2914085525,
4161315584, 1273113343, 3269644828, 3681293816, 412536684,
1156034077, 3823026442, 1066971017, 3598330293, 1979273937,
2079029895, 1195045909, 1071986421, 2712821515, 3377754595,
2184151095, 750918864, 2585729879, 4249895712, 1832579367,
1192240192, 946734366, 31230688, 3174399083, 3549375728,
1642430184, 1904857554, 861877404, 3277825584, 4267074718,
3122860549, 666423581, 644189126, 226475395, 307789415,
1196105631, 3191691839, 782852669, 1608507813, 1847685900,
4069766876, 3931548641, 2526471011, 766865139, 2115084288,
4259411376, 3323683436, 568512177, 3736601419, 1800276898,
4012458395, 1823982, 27980198, 2023839966, 869505096,
431161506, 1024804023, 1853869307, 3393537983, 1500703614,
3019471560, 1351086955, 3096933631, 3034634988, 2544598006,
1230942551, 3362230798, 159984793, 491590373, 3993872886,
3681855622, 903593547, 3535062472, 1799803217, 772984149,
895863112, 1899036275, 4187322100, 101856048, 234650315,
3183125617, 3190039692, 525584357, 1286834489, 455810374,
1869181575, 922673938, 3877430102, 3422391938, 1414347295,
1971054608, 3061798054, 830555096, 2822905141, 167033190,
1079139428, 4210126723, 3593797804, 429192890, 372093950,
1779187770, 3312189287, 204349348, 452421568, 2800540462,
3733109044, 1235082423, 1765319556, 3174729780, 3762994475,
3171962488, 442160826, 198349622, 45942637, 1324086311,
2901868599, 678860040, 3812229107, 19936821, 1119590141,
3640121682, 3545931032, 2102949142, 2828208598, 3603378023,
4135048896
5.5.3. The Table V2
1629829892, 282540176, 2794583710, 496504798, 2990494426,
3070701851, 2575963183, 4094823972, 2775723650, 4079480416,
176028725, 2246241423, 3732217647, 2196843075, 1306949278,
4170992780, 4039345809, 3209664269, 3387499533, 293063229,
3660290503, 2648440860, 2531406539, 3537879412, 773374739,
4184691853, 1804207821, 3347126643, 3479377103, 3970515774,
1891731298, 2368003842, 3537588307, 2969158410, 4230745262,
831906319, 2935838131, 264029468, 120852739, 3200326460,
355445271, 2296305141, 1566296040, 1760127056, 20073893,
3427103620, 2866979760, 2359075957, 2025314291, 1725696734,
3346087406, 2690756527, 99815156, 4248519977, 2253762642,
3274144518, 598024568, 3299672435, 556579346, 4121041856,
2896948975, 3620123492, 918453629, 3249461198, 2231414958,
3803272287, 3657597946, 2588911389, 242262274, 1725007475,
2026427718, 46776484, 2873281403, 2919275846, 3177933051,
1918859160, 2517854537, 1857818511, 3234262050, 479353687,
200201308, 2801945841, 1621715769, 483977159, 423502325,
3689396064, 1850168397, 3359959416, 3459831930, 841488699,
3570506095, 930267420, 1564520841, 2505122797, 593824107,
1116572080, 819179184, 3139123629, 1414339336, 1076360795,
512403845, 177759256, 1701060666, 2239736419, 515179302,
2935012727, 3821357612, 1376520851, 2700745271, 966853647,
1041862223, 715860553, 171592961, 1607044257, 1227236688,
3647136358, 1417559141, 4087067551, 2241705880, 4194136288,
1439041934, 20464430, 119668151, 2021257232, 2551262694,
1381539058, 4082839035, 498179069, 311508499, 3580908637,
2889149671, 142719814, 1232184754, 3356662582, 2973775623,
1469897084, 1728205304, 1415793613, 50111003, 3133413359,
4074115275, 2710540611, 2700083070, 2457757663, 2612845330,
3775943755, 2469309260, 2560142753, 3020996369, 1691667711,
4219602776, 1687672168, 1017921622, 2307642321, 368711460,
3282925988, 213208029, 4150757489, 3443211944, 2846101972,
4106826684, 4272438675, 2199416468, 3710621281, 497564971,
285138276, 765042313, 916220877, 3402623607, 2768784621,
1722849097, 3386397442, 487920061, 3569027007, 3424544196,
217781973, 2356938519, 3252429414, 145109750, 2692588106,
2454747135, 1299493354, 4120241887, 2088917094, 932304329,
1442609203, 952586974, 3509186750, 753369054, 854421006,
1954046388, 2708927882, 4047539230, 3048925996, 1667505809,
805166441, 1182069088, 4265546268, 4215029527, 3374748959,
373532666, 2454243090, 2371530493, 3651087521, 2619878153,
1651809518, 1553646893, 1227452842, 703887512, 3696674163,
2552507603, 2635912901, 895130484, 3287782244, 3098973502,
990078774, 3780326506, 2290845203, 41729428, 1949580860,
2283959805, 1036946170, 1694887523, 4880696, 466000198,
2765355283, 3318686998, 1266458025, 3919578154, 3545413527,
2627009988, 3744680394, 1696890173, 3250684705, 4142417708,
915739411, 3308488877, 1289361460, 2942552331, 1169105979,
3342228712, 698560958, 1356041230, 2401944293, 107705232,
3701895363, 903928723, 3646581385, 844950914, 1944371367,
3863894844, 2946773319, 1972431613, 1706989237, 29917467,
3497665928
5.5.4. The Table V3
1191369816, 744902811, 2539772235, 3213192037, 3286061266,
1200571165, 2463281260, 754888894, 714651270, 1968220972,
3628497775, 1277626456, 1493398934, 364289757, 2055487592,
3913468088, 2930259465, 902504567, 3967050355, 2056499403,
692132390, 186386657, 832834706, 859795816, 1283120926,
2253183716, 3003475205, 1755803552, 2239315142, 4271056352,
2184848469, 769228092, 1249230754, 1193269205, 2660094102,
642979613, 1687087994, 2726106182, 446402913, 4122186606,
3771347282, 37667136, 192775425, 3578702187, 1952659096,
3989584400, 3069013882, 2900516158, 4045316336, 3057163251,
1702104819, 4116613420, 3575472384, 2674023117, 1409126723,
3215095429, 1430726429, 2544497368, 1029565676, 1855801827,
4262184627, 1854326881, 2906728593, 3277836557, 2787697002,
2787333385, 3105430738, 2477073192, 748038573, 1088396515,
1611204853, 201964005, 3745818380, 3654683549, 3816120877,
3915783622, 2563198722, 1181149055, 33158084, 3723047845,
3790270906, 3832415204, 2959617497, 372900708, 1286738499,
1932439099, 3677748309, 2454711182, 2757856469, 2134027055,
2780052465, 3190347618, 3758510138, 3626329451, 1120743107,
1623585693, 1389834102, 2719230375, 3038609003, 462617590,
260254189, 3706349764, 2556762744, 2874272296, 2502399286,
4216263978, 2683431180, 2168560535, 3561507175, 668095726,
680412330, 3726693946, 4180630637, 3335170953, 942140968,
2711851085, 2059233412, 4265696278, 3204373534, 232855056,
881788313, 2258252172, 2043595984, 3758795150, 3615341325,
2138837681, 1351208537, 2923692473, 3402482785, 2105383425,
2346772751, 499245323, 3417846006, 2366116814, 2543090583,
1828551634, 3148696244, 3853884867, 1364737681, 2200687771,
2689775688, 232720625, 4071657318, 2671968983, 3531415031,
1212852141, 867923311, 3740109711, 1923146533, 3237071777,
3100729255, 3247856816, 906742566, 4047640575, 4007211572,
3495700105, 1171285262, 2835682655, 1634301229, 3115169925,
2289874706, 2252450179, 944880097, 371933491, 1649074501,
2208617414, 2524305981, 2496569844, 2667037160, 1257550794,
3399219045, 3194894295, 1643249887, 342911473, 891025733,
3146861835, 3789181526, 938847812, 1854580183, 2112653794,
2960702988, 1238603378, 2205280635, 1666784014, 2520274614,
3355493726, 2310872278, 3153920489, 2745882591, 1200203158,
3033612415, 2311650167, 1048129133, 4206710184, 4209176741,
2640950279, 2096382177, 4116899089, 3631017851, 4104488173,
1857650503, 3801102932, 445806934, 3055654640, 897898279,
3234007399, 1325494930, 2982247189, 1619020475, 2720040856,
885096170, 3485255499, 2983202469, 3891011124, 546522756,
1524439205, 2644317889, 2170076800, 2969618716, 961183518,
1081831074, 1037015347, 3289016286, 2331748669, 620887395,
303042654, 3990027945, 1562756376, 3413341792, 2059647769,
2823844432, 674595301, 2457639984, 4076754716, 2447737904,
1583323324, 625627134, 3076006391, 345777990, 1684954145,
879227329, 3436182180, 1522273219, 3802543817, 1456017040,
1897819847, 2970081129, 1382576028, 3820044861, 1044428167,
612252599, 3340478395, 2150613904, 3397625662, 3573635640,
3432275192
5.6. Systematic Indices and Other Parameters
Table 2 below specifies the supported values of K'. The table also
specifies for each supported value of K' the systematic index J(K'),
the number H(K') of HDPC symbols, the number S(K') of LDPC symbols,
and the number W(K') of LT symbols. For each value of K', the
corresponding values of S(K') and W(K') are prime numbers.
The systematic index J(K') is designed to have the property that the
set of source symbol tuples (d[0], a[0], b[0], d1[0], a1[0], b1[0]),
..., (d[K'-1], a[K'-1], b[K'-1], d1[K'-1], a1[K'-1], b1[K'-1]) are
such that the L intermediate symbols are uniquely defined, i.e., the
matrix A in Figure 6 has full rank and is therefore invertible.
+-------+-------+-------+-------+-------+
| K' | J(K') | S(K') | H(K') | W(K') |
+-------+-------+-------+-------+-------+
| 10 | 254 | 7 | 10 | 17 |
+-------+-------+-------+-------+-------+
| 12 | 630 | 7 | 10 | 19 |
+-------+-------+-------+-------+-------+
| 18 | 682 | 11 | 10 | 29 |
+-------+-------+-------+-------+-------+
| 20 | 293 | 11 | 10 | 31 |
+-------+-------+-------+-------+-------+
| 26 | 80 | 11 | 10 | 37 |
+-------+-------+-------+-------+-------+
| 30 | 566 | 11 | 10 | 41 |
+-------+-------+-------+-------+-------+
| 32 | 860 | 11 | 10 | 43 |
+-------+-------+-------+-------+-------+
| 36 | 267 | 11 | 10 | 47 |
+-------+-------+-------+-------+-------+
| 42 | 822 | 11 | 10 | 53 |
+-------+-------+-------+-------+-------+
| 46 | 506 | 13 | 10 | 59 |
+-------+-------+-------+-------+-------+
| 48 | 589 | 13 | 10 | 61 |
+-------+-------+-------+-------+-------+
| 49 | 87 | 13 | 10 | 61 |
+-------+-------+-------+-------+-------+
+-------+-------+-------+-------+-------+
| 55 | 520 | 13 | 10 | 67 |
+-------+-------+-------+-------+-------+
| 60 | 159 | 13 | 10 | 71 |
+-------+-------+-------+-------+-------+
| 62 | 235 | 13 | 10 | 73 |
+-------+-------+-------+-------+-------+
| 69 | 157 | 13 | 10 | 79 |
+-------+-------+-------+-------+-------+
| 75 | 502 | 17 | 10 | 89 |
+-------+-------+-------+-------+-------+
| 84 | 334 | 17 | 10 | 97 |
+-------+-------+-------+-------+-------+
| 88 | 583 | 17 | 10 | 101 |
+-------+-------+-------+-------+-------+
| 91 | 66 | 17 | 10 | 103 |
+-------+-------+-------+-------+-------+
| 95 | 352 | 17 | 10 | 107 |
+-------+-------+-------+-------+-------+
| 97 | 365 | 17 | 10 | 109 |
+-------+-------+-------+-------+-------+
| 101 | 562 | 17 | 10 | 113 |
+-------+-------+-------+-------+-------+
| 114 | 5 | 19 | 10 | 127 |
+-------+-------+-------+-------+-------+
| 119 | 603 | 19 | 10 | 131 |
+-------+-------+-------+-------+-------+
| 125 | 721 | 19 | 10 | 137 |
+-------+-------+-------+-------+-------+
| 127 | 28 | 19 | 10 | 139 |
+-------+-------+-------+-------+-------+
| 138 | 660 | 19 | 10 | 149 |
+-------+-------+-------+-------+-------+
| 140 | 829 | 19 | 10 | 151 |
+-------+-------+-------+-------+-------+
| 149 | 900 | 23 | 10 | 163 |
+-------+-------+-------+-------+-------+
| 153 | 930 | 23 | 10 | 167 |
+-------+-------+-------+-------+-------+
| 160 | 814 | 23 | 10 | 173 |
+-------+-------+-------+-------+-------+
| 166 | 661 | 23 | 10 | 179 |
+-------+-------+-------+-------+-------+
| 168 | 693 | 23 | 10 | 181 |
+-------+-------+-------+-------+-------+
| 179 | 780 | 23 | 10 | 191 |
+-------+-------+-------+-------+-------+
+-------+-------+-------+-------+-------+
| 181 | 605 | 23 | 10 | 193 |
+-------+-------+-------+-------+-------+
| 185 | 551 | 23 | 10 | 197 |
+-------+-------+-------+-------+-------+
| 187 | 777 | 23 | 10 | 199 |
+-------+-------+-------+-------+-------+
| 200 | 491 | 23 | 10 | 211 |
+-------+-------+-------+-------+-------+
| 213 | 396 | 23 | 10 | 223 |
+-------+-------+-------+-------+-------+
| 217 | 764 | 29 | 10 | 233 |
+-------+-------+-------+-------+-------+
| 225 | 843 | 29 | 10 | 241 |
+-------+-------+-------+-------+-------+
| 236 | 646 | 29 | 10 | 251 |
+-------+-------+-------+-------+-------+
| 242 | 557 | 29 | 10 | 257 |
+-------+-------+-------+-------+-------+
| 248 | 608 | 29 | 10 | 263 |
+-------+-------+-------+-------+-------+
| 257 | 265 | 29 | 10 | 271 |
+-------+-------+-------+-------+-------+
| 263 | 505 | 29 | 10 | 277 |
+-------+-------+-------+-------+-------+
| 269 | 722 | 29 | 10 | 283 |
+-------+-------+-------+-------+-------+
| 280 | 263 | 29 | 10 | 293 |
+-------+-------+-------+-------+-------+
| 295 | 999 | 29 | 10 | 307 |
+-------+-------+-------+-------+-------+
| 301 | 874 | 29 | 10 | 313 |
+-------+-------+-------+-------+-------+
| 305 | 160 | 29 | 10 | 317 |
+-------+-------+-------+-------+-------+
| 324 | 575 | 31 | 10 | 337 |
+-------+-------+-------+-------+-------+
| 337 | 210 | 31 | 10 | 349 |
+-------+-------+-------+-------+-------+
| 341 | 513 | 31 | 10 | 353 |
+-------+-------+-------+-------+-------+
| 347 | 503 | 31 | 10 | 359 |
+-------+-------+-------+-------+-------+
| 355 | 558 | 31 | 10 | 367 |
+-------+-------+-------+-------+-------+
| 362 | 932 | 31 | 10 | 373 |
+-------+-------+-------+-------+-------+
+-------+-------+-------+-------+-------+
| 368 | 404 | 31 | 10 | 379 |
+-------+-------+-------+-------+-------+
| 372 | 520 | 37 | 10 | 389 |
+-------+-------+-------+-------+-------+
| 380 | 846 | 37 | 10 | 397 |
+-------+-------+-------+-------+-------+
| 385 | 485 | 37 | 10 | 401 |
+-------+-------+-------+-------+-------+
| 393 | 728 | 37 | 10 | 409 |
+-------+-------+-------+-------+-------+
| 405 | 554 | 37 | 10 | 421 |
+-------+-------+-------+-------+-------+
| 418 | 471 | 37 | 10 | 433 |
+-------+-------+-------+-------+-------+
| 428 | 641 | 37 | 10 | 443 |
+-------+-------+-------+-------+-------+
| 434 | 732 | 37 | 10 | 449 |
+-------+-------+-------+-------+-------+
| 447 | 193 | 37 | 10 | 461 |
+-------+-------+-------+-------+-------+
| 453 | 934 | 37 | 10 | 467 |
+-------+-------+-------+-------+-------+
| 466 | 864 | 37 | 10 | 479 |
+-------+-------+-------+-------+-------+
| 478 | 790 | 37 | 10 | 491 |
+-------+-------+-------+-------+-------+
| 486 | 912 | 37 | 10 | 499 |
+-------+-------+-------+-------+-------+
| 491 | 617 | 37 | 10 | 503 |
+-------+-------+-------+-------+-------+
| 497 | 587 | 37 | 10 | 509 |
+-------+-------+-------+-------+-------+
| 511 | 800 | 37 | 10 | 523 |
+-------+-------+-------+-------+-------+
| 526 | 923 | 41 | 10 | 541 |
+-------+-------+-------+-------+-------+
| 532 | 998 | 41 | 10 | 547 |
+-------+-------+-------+-------+-------+
| 542 | 92 | 41 | 10 | 557 |
+-------+-------+-------+-------+-------+
| 549 | 497 | 41 | 10 | 563 |
+-------+-------+-------+-------+-------+
| 557 | 559 | 41 | 10 | 571 |
+-------+-------+-------+-------+-------+
| 563 | 667 | 41 | 10 | 577 |
+-------+-------+-------+-------+-------+
+-------+-------+-------+-------+-------+
| 573 | 912 | 41 | 10 | 587 |
+-------+-------+-------+-------+-------+
| 580 | 262 | 41 | 10 | 593 |
+-------+-------+-------+-------+-------+
| 588 | 152 | 41 | 10 | 601 |
+-------+-------+-------+-------+-------+
| 594 | 526 | 41 | 10 | 607 |
+-------+-------+-------+-------+-------+
| 600 | 268 | 41 | 10 | 613 |
+-------+-------+-------+-------+-------+
| 606 | 212 | 41 | 10 | 619 |
+-------+-------+-------+-------+-------+
| 619 | 45 | 41 | 10 | 631 |
+-------+-------+-------+-------+-------+
| 633 | 898 | 43 | 10 | 647 |
+-------+-------+-------+-------+-------+
| 640 | 527 | 43 | 10 | 653 |
+-------+-------+-------+-------+-------+
| 648 | 558 | 43 | 10 | 661 |
+-------+-------+-------+-------+-------+
| 666 | 460 | 47 | 10 | 683 |
+-------+-------+-------+-------+-------+
| 675 | 5 | 47 | 10 | 691 |
+-------+-------+-------+-------+-------+
| 685 | 895 | 47 | 10 | 701 |
+-------+-------+-------+-------+-------+
| 693 | 996 | 47 | 10 | 709 |
+-------+-------+-------+-------+-------+
| 703 | 282 | 47 | 10 | 719 |
+-------+-------+-------+-------+-------+
| 718 | 513 | 47 | 10 | 733 |
+-------+-------+-------+-------+-------+
| 728 | 865 | 47 | 10 | 743 |
+-------+-------+-------+-------+-------+
| 736 | 870 | 47 | 10 | 751 |
+-------+-------+-------+-------+-------+
| 747 | 239 | 47 | 10 | 761 |
+-------+-------+-------+-------+-------+
| 759 | 452 | 47 | 10 | 773 |
+-------+-------+-------+-------+-------+
| 778 | 862 | 53 | 10 | 797 |
+-------+-------+-------+-------+-------+
| 792 | 852 | 53 | 10 | 811 |
+-------+-------+-------+-------+-------+
| 802 | 643 | 53 | 10 | 821 |
+-------+-------+-------+-------+-------+
+-------+-------+-------+-------+-------+
| 811 | 543 | 53 | 10 | 829 |
+-------+-------+-------+-------+-------+
| 821 | 447 | 53 | 10 | 839 |
+-------+-------+-------+-------+-------+
| 835 | 321 | 53 | 10 | 853 |
+-------+-------+-------+-------+-------+
| 845 | 287 | 53 | 10 | 863 |
+-------+-------+-------+-------+-------+
| 860 | 12 | 53 | 10 | 877 |
+-------+-------+-------+-------+-------+
| 870 | 251 | 53 | 10 | 887 |
+-------+-------+-------+-------+-------+
| 891 | 30 | 53 | 10 | 907 |
+-------+-------+-------+-------+-------+
| 903 | 621 | 53 | 10 | 919 |
+-------+-------+-------+-------+-------+
| 913 | 555 | 53 | 10 | 929 |
+-------+-------+-------+-------+-------+
| 926 | 127 | 53 | 10 | 941 |
+-------+-------+-------+-------+-------+
| 938 | 400 | 53 | 10 | 953 |
+-------+-------+-------+-------+-------+
| 950 | 91 | 59 | 10 | 971 |
+-------+-------+-------+-------+-------+
| 963 | 916 | 59 | 10 | 983 |
+-------+-------+-------+-------+-------+
| 977 | 935 | 59 | 10 | 997 |
+-------+-------+-------+-------+-------+
| 989 | 691 | 59 | 10 | 1009 |
+-------+-------+-------+-------+-------+
| 1002 | 299 | 59 | 10 | 1021 |
+-------+-------+-------+-------+-------+
| 1020 | 282 | 59 | 10 | 1039 |
+-------+-------+-------+-------+-------+
| 1032 | 824 | 59 | 10 | 1051 |
+-------+-------+-------+-------+-------+
| 1050 | 536 | 59 | 11 | 1069 |
+-------+-------+-------+-------+-------+
| 1074 | 596 | 59 | 11 | 1093 |
+-------+-------+-------+-------+-------+
| 1085 | 28 | 59 | 11 | 1103 |
+-------+-------+-------+-------+-------+
| 1099 | 947 | 59 | 11 | 1117 |
+-------+-------+-------+-------+-------+
| 1111 | 162 | 59 | 11 | 1129 |
+-------+-------+-------+-------+-------+
+-------+-------+-------+-------+-------+
| 1136 | 536 | 59 | 11 | 1153 |
+-------+-------+-------+-------+-------+
| 1152 | 1000 | 61 | 11 | 1171 |
+-------+-------+-------+-------+-------+
| 1169 | 251 | 61 | 11 | 1187 |
+-------+-------+-------+-------+-------+
| 1183 | 673 | 61 | 11 | 1201 |
+-------+-------+-------+-------+-------+
| 1205 | 559 | 61 | 11 | 1223 |
+-------+-------+-------+-------+-------+
| 1220 | 923 | 61 | 11 | 1237 |
+-------+-------+-------+-------+-------+
| 1236 | 81 | 67 | 11 | 1259 |
+-------+-------+-------+-------+-------+
| 1255 | 478 | 67 | 11 | 1277 |
+-------+-------+-------+-------+-------+
| 1269 | 198 | 67 | 11 | 1291 |
+-------+-------+-------+-------+-------+
| 1285 | 137 | 67 | 11 | 1307 |
+-------+-------+-------+-------+-------+
| 1306 | 75 | 67 | 11 | 1327 |
+-------+-------+-------+-------+-------+
| 1347 | 29 | 67 | 11 | 1367 |
+-------+-------+-------+-------+-------+
| 1361 | 231 | 67 | 11 | 1381 |
+-------+-------+-------+-------+-------+
| 1389 | 532 | 67 | 11 | 1409 |
+-------+-------+-------+-------+-------+
| 1404 | 58 | 67 | 11 | 1423 |
+-------+-------+-------+-------+-------+
| 1420 | 60 | 67 | 11 | 1439 |
+-------+-------+-------+-------+-------+
| 1436 | 964 | 71 | 11 | 1459 |
+-------+-------+-------+-------+-------+
| 1461 | 624 | 71 | 11 | 1483 |
+-------+-------+-------+-------+-------+
| 1477 | 502 | 71 | 11 | 1499 |
+-------+-------+-------+-------+-------+
| 1502 | 636 | 71 | 11 | 1523 |
+-------+-------+-------+-------+-------+
| 1522 | 986 | 71 | 11 | 1543 |
+-------+-------+-------+-------+-------+
| 1539 | 950 | 71 | 11 | 1559 |
+-------+-------+-------+-------+-------+
| 1561 | 735 | 73 | 11 | 1583 |
+-------+-------+-------+-------+-------+
+-------+-------+-------+-------+-------+
| 1579 | 866 | 73 | 11 | 1601 |
+-------+-------+-------+-------+-------+
| 1600 | 203 | 73 | 11 | 1621 |
+-------+-------+-------+-------+-------+
| 1616 | 83 | 73 | 11 | 1637 |
+-------+-------+-------+-------+-------+
| 1649 | 14 | 73 | 11 | 1669 |
+-------+-------+-------+-------+-------+
| 1673 | 522 | 79 | 11 | 1699 |
+-------+-------+-------+-------+-------+
| 1698 | 226 | 79 | 11 | 1723 |
+-------+-------+-------+-------+-------+
| 1716 | 282 | 79 | 11 | 1741 |
+-------+-------+-------+-------+-------+
| 1734 | 88 | 79 | 11 | 1759 |
+-------+-------+-------+-------+-------+
| 1759 | 636 | 79 | 11 | 1783 |
+-------+-------+-------+-------+-------+
| 1777 | 860 | 79 | 11 | 1801 |
+-------+-------+-------+-------+-------+
| 1800 | 324 | 79 | 11 | 1823 |
+-------+-------+-------+-------+-------+
| 1824 | 424 | 79 | 11 | 1847 |
+-------+-------+-------+-------+-------+
| 1844 | 999 | 79 | 11 | 1867 |
+-------+-------+-------+-------+-------+
| 1863 | 682 | 83 | 11 | 1889 |
+-------+-------+-------+-------+-------+
| 1887 | 814 | 83 | 11 | 1913 |
+-------+-------+-------+-------+-------+
| 1906 | 979 | 83 | 11 | 1931 |
+-------+-------+-------+-------+-------+
| 1926 | 538 | 83 | 11 | 1951 |
+-------+-------+-------+-------+-------+
| 1954 | 278 | 83 | 11 | 1979 |
+-------+-------+-------+-------+-------+
| 1979 | 580 | 83 | 11 | 2003 |
+-------+-------+-------+-------+-------+
| 2005 | 773 | 83 | 11 | 2029 |
+-------+-------+-------+-------+-------+
| 2040 | 911 | 89 | 11 | 2069 |
+-------+-------+-------+-------+-------+
| 2070 | 506 | 89 | 11 | 2099 |
+-------+-------+-------+-------+-------+
| 2103 | 628 | 89 | 11 | 2131 |
+-------+-------+-------+-------+-------+
+-------+-------+-------+-------+-------+
| 2125 | 282 | 89 | 11 | 2153 |
+-------+-------+-------+-------+-------+
| 2152 | 309 | 89 | 11 | 2179 |
+-------+-------+-------+-------+-------+
| 2195 | 858 | 89 | 11 | 2221 |
+-------+-------+-------+-------+-------+
| 2217 | 442 | 89 | 11 | 2243 |
+-------+-------+-------+-------+-------+
| 2247 | 654 | 89 | 11 | 2273 |
+-------+-------+-------+-------+-------+
| 2278 | 82 | 97 | 11 | 2311 |
+-------+-------+-------+-------+-------+
| 2315 | 428 | 97 | 11 | 2347 |
+-------+-------+-------+-------+-------+
| 2339 | 442 | 97 | 11 | 2371 |
+-------+-------+-------+-------+-------+
| 2367 | 283 | 97 | 11 | 2399 |
+-------+-------+-------+-------+-------+
| 2392 | 538 | 97 | 11 | 2423 |
+-------+-------+-------+-------+-------+
| 2416 | 189 | 97 | 11 | 2447 |
+-------+-------+-------+-------+-------+
| 2447 | 438 | 97 | 11 | 2477 |
+-------+-------+-------+-------+-------+
| 2473 | 912 | 97 | 11 | 2503 |
+-------+-------+-------+-------+-------+
| 2502 | 1 | 97 | 11 | 2531 |
+-------+-------+-------+-------+-------+
| 2528 | 167 | 97 | 11 | 2557 |
+-------+-------+-------+-------+-------+
| 2565 | 272 | 97 | 11 | 2593 |
+-------+-------+-------+-------+-------+
| 2601 | 209 | 101 | 11 | 2633 |
+-------+-------+-------+-------+-------+
| 2640 | 927 | 101 | 11 | 2671 |
+-------+-------+-------+-------+-------+
| 2668 | 386 | 101 | 11 | 2699 |
+-------+-------+-------+-------+-------+
| 2701 | 653 | 101 | 11 | 2731 |
+-------+-------+-------+-------+-------+
| 2737 | 669 | 101 | 11 | 2767 |
+-------+-------+-------+-------+-------+
| 2772 | 431 | 101 | 11 | 2801 |
+-------+-------+-------+-------+-------+
| 2802 | 793 | 103 | 11 | 2833 |
+-------+-------+-------+-------+-------+
+-------+-------+-------+-------+-------+
| 2831 | 588 | 103 | 11 | 2861 |
+-------+-------+-------+-------+-------+
| 2875 | 777 | 107 | 11 | 2909 |
+-------+-------+-------+-------+-------+
| 2906 | 939 | 107 | 11 | 2939 |
+-------+-------+-------+-------+-------+
| 2938 | 864 | 107 | 11 | 2971 |
+-------+-------+-------+-------+-------+
| 2979 | 627 | 107 | 11 | 3011 |
+-------+-------+-------+-------+-------+
| 3015 | 265 | 109 | 11 | 3049 |
+-------+-------+-------+-------+-------+
| 3056 | 976 | 109 | 11 | 3089 |
+-------+-------+-------+-------+-------+
| 3101 | 988 | 113 | 11 | 3137 |
+-------+-------+-------+-------+-------+
| 3151 | 507 | 113 | 11 | 3187 |
+-------+-------+-------+-------+-------+
| 3186 | 640 | 113 | 11 | 3221 |
+-------+-------+-------+-------+-------+
| 3224 | 15 | 113 | 11 | 3259 |
+-------+-------+-------+-------+-------+
| 3265 | 667 | 113 | 11 | 3299 |
+-------+-------+-------+-------+-------+
| 3299 | 24 | 127 | 11 | 3347 |
+-------+-------+-------+-------+-------+
| 3344 | 877 | 127 | 11 | 3391 |
+-------+-------+-------+-------+-------+
| 3387 | 240 | 127 | 11 | 3433 |
+-------+-------+-------+-------+-------+
| 3423 | 720 | 127 | 11 | 3469 |
+-------+-------+-------+-------+-------+
| 3466 | 93 | 127 | 11 | 3511 |
+-------+-------+-------+-------+-------+
| 3502 | 919 | 127 | 11 | 3547 |
+-------+-------+-------+-------+-------+
| 3539 | 635 | 127 | 11 | 3583 |
+-------+-------+-------+-------+-------+
| 3579 | 174 | 127 | 11 | 3623 |
+-------+-------+-------+-------+-------+
| 3616 | 647 | 127 | 11 | 3659 |
+-------+-------+-------+-------+-------+
| 3658 | 820 | 127 | 11 | 3701 |
+-------+-------+-------+-------+-------+
| 3697 | 56 | 127 | 11 | 3739 |
+-------+-------+-------+-------+-------+
+-------+-------+-------+-------+-------+
| 3751 | 485 | 127 | 11 | 3793 |
+-------+-------+-------+-------+-------+
| 3792 | 210 | 127 | 11 | 3833 |
+-------+-------+-------+-------+-------+
| 3840 | 124 | 127 | 11 | 3881 |
+-------+-------+-------+-------+-------+
| 3883 | 546 | 127 | 11 | 3923 |
+-------+-------+-------+-------+-------+
| 3924 | 954 | 131 | 11 | 3967 |
+-------+-------+-------+-------+-------+
| 3970 | 262 | 131 | 11 | 4013 |
+-------+-------+-------+-------+-------+
| 4015 | 927 | 131 | 11 | 4057 |
+-------+-------+-------+-------+-------+
| 4069 | 957 | 131 | 11 | 4111 |
+-------+-------+-------+-------+-------+
| 4112 | 726 | 137 | 11 | 4159 |
+-------+-------+-------+-------+-------+
| 4165 | 583 | 137 | 11 | 4211 |
+-------+-------+-------+-------+-------+
| 4207 | 782 | 137 | 11 | 4253 |
+-------+-------+-------+-------+-------+
| 4252 | 37 | 137 | 11 | 4297 |
+-------+-------+-------+-------+-------+
| 4318 | 758 | 137 | 11 | 4363 |
+-------+-------+-------+-------+-------+
| 4365 | 777 | 137 | 11 | 4409 |
+-------+-------+-------+-------+-------+
| 4418 | 104 | 139 | 11 | 4463 |
+-------+-------+-------+-------+-------+
| 4468 | 476 | 139 | 11 | 4513 |
+-------+-------+-------+-------+-------+
| 4513 | 113 | 149 | 11 | 4567 |
+-------+-------+-------+-------+-------+
| 4567 | 313 | 149 | 11 | 4621 |
+-------+-------+-------+-------+-------+
| 4626 | 102 | 149 | 11 | 4679 |
+-------+-------+-------+-------+-------+
| 4681 | 501 | 149 | 11 | 4733 |
+-------+-------+-------+-------+-------+
| 4731 | 332 | 149 | 11 | 4783 |
+-------+-------+-------+-------+-------+
| 4780 | 786 | 149 | 11 | 4831 |
+-------+-------+-------+-------+-------+
| 4838 | 99 | 149 | 11 | 4889 |
+-------+-------+-------+-------+-------+
+-------+-------+-------+-------+-------+
| 4901 | 658 | 149 | 11 | 4951 |
+-------+-------+-------+-------+-------+
| 4954 | 794 | 149 | 11 | 5003 |
+-------+-------+-------+-------+-------+
| 5008 | 37 | 151 | 11 | 5059 |
+-------+-------+-------+-------+-------+
| 5063 | 471 | 151 | 11 | 5113 |
+-------+-------+-------+-------+-------+
| 5116 | 94 | 157 | 11 | 5171 |
+-------+-------+-------+-------+-------+
| 5172 | 873 | 157 | 11 | 5227 |
+-------+-------+-------+-------+-------+
| 5225 | 918 | 157 | 11 | 5279 |
+-------+-------+-------+-------+-------+
| 5279 | 945 | 157 | 11 | 5333 |
+-------+-------+-------+-------+-------+
| 5334 | 211 | 157 | 11 | 5387 |
+-------+-------+-------+-------+-------+
| 5391 | 341 | 157 | 11 | 5443 |
+-------+-------+-------+-------+-------+
| 5449 | 11 | 163 | 11 | 5507 |
+-------+-------+-------+-------+-------+
| 5506 | 578 | 163 | 11 | 5563 |
+-------+-------+-------+-------+-------+
| 5566 | 494 | 163 | 11 | 5623 |
+-------+-------+-------+-------+-------+
| 5637 | 694 | 163 | 11 | 5693 |
+-------+-------+-------+-------+-------+
| 5694 | 252 | 163 | 11 | 5749 |
+-------+-------+-------+-------+-------+
| 5763 | 451 | 167 | 11 | 5821 |
+-------+-------+-------+-------+-------+
| 5823 | 83 | 167 | 11 | 5881 |
+-------+-------+-------+-------+-------+
| 5896 | 689 | 167 | 11 | 5953 |
+-------+-------+-------+-------+-------+
| 5975 | 488 | 173 | 11 | 6037 |
+-------+-------+-------+-------+-------+
| 6039 | 214 | 173 | 11 | 6101 |
+-------+-------+-------+-------+-------+
| 6102 | 17 | 173 | 11 | 6163 |
+-------+-------+-------+-------+-------+
| 6169 | 469 | 173 | 11 | 6229 |
+-------+-------+-------+-------+-------+
| 6233 | 263 | 179 | 11 | 6299 |
+-------+-------+-------+-------+-------+
+-------+-------+-------+-------+-------+
| 6296 | 309 | 179 | 11 | 6361 |
+-------+-------+-------+-------+-------+
| 6363 | 984 | 179 | 11 | 6427 |
+-------+-------+-------+-------+-------+
| 6427 | 123 | 179 | 11 | 6491 |
+-------+-------+-------+-------+-------+
| 6518 | 360 | 179 | 11 | 6581 |
+-------+-------+-------+-------+-------+
| 6589 | 863 | 181 | 11 | 6653 |
+-------+-------+-------+-------+-------+
| 6655 | 122 | 181 | 11 | 6719 |
+-------+-------+-------+-------+-------+
| 6730 | 522 | 191 | 11 | 6803 |
+-------+-------+-------+-------+-------+
| 6799 | 539 | 191 | 11 | 6871 |
+-------+-------+-------+-------+-------+
| 6878 | 181 | 191 | 11 | 6949 |
+-------+-------+-------+-------+-------+
| 6956 | 64 | 191 | 11 | 7027 |
+-------+-------+-------+-------+-------+
| 7033 | 387 | 191 | 11 | 7103 |
+-------+-------+-------+-------+-------+
| 7108 | 967 | 191 | 11 | 7177 |
+-------+-------+-------+-------+-------+
| 7185 | 843 | 191 | 11 | 7253 |
+-------+-------+-------+-------+-------+
| 7281 | 999 | 193 | 11 | 7351 |
+-------+-------+-------+-------+-------+
| 7360 | 76 | 197 | 11 | 7433 |
+-------+-------+-------+-------+-------+
| 7445 | 142 | 197 | 11 | 7517 |
+-------+-------+-------+-------+-------+
| 7520 | 599 | 197 | 11 | 7591 |
+-------+-------+-------+-------+-------+
| 7596 | 576 | 199 | 11 | 7669 |
+-------+-------+-------+-------+-------+
| 7675 | 176 | 211 | 11 | 7759 |
+-------+-------+-------+-------+-------+
| 7770 | 392 | 211 | 11 | 7853 |
+-------+-------+-------+-------+-------+
| 7855 | 332 | 211 | 11 | 7937 |
+-------+-------+-------+-------+-------+
| 7935 | 291 | 211 | 11 | 8017 |
+-------+-------+-------+-------+-------+
| 8030 | 913 | 211 | 11 | 8111 |
+-------+-------+-------+-------+-------+
+-------+-------+-------+-------+-------+
| 8111 | 608 | 211 | 11 | 8191 |
+-------+-------+-------+-------+-------+
| 8194 | 212 | 211 | 11 | 8273 |
+-------+-------+-------+-------+-------+
| 8290 | 696 | 211 | 11 | 8369 |
+-------+-------+-------+-------+-------+
| 8377 | 931 | 223 | 11 | 8467 |
+-------+-------+-------+-------+-------+
| 8474 | 326 | 223 | 11 | 8563 |
+-------+-------+-------+-------+-------+
| 8559 | 228 | 223 | 11 | 8647 |
+-------+-------+-------+-------+-------+
| 8654 | 706 | 223 | 11 | 8741 |
+-------+-------+-------+-------+-------+
| 8744 | 144 | 223 | 11 | 8831 |
+-------+-------+-------+-------+-------+
| 8837 | 83 | 223 | 11 | 8923 |
+-------+-------+-------+-------+-------+
| 8928 | 743 | 223 | 11 | 9013 |
+-------+-------+-------+-------+-------+
| 9019 | 187 | 223 | 11 | 9103 |
+-------+-------+-------+-------+-------+
| 9111 | 654 | 227 | 11 | 9199 |
+-------+-------+-------+-------+-------+
| 9206 | 359 | 227 | 11 | 9293 |
+-------+-------+-------+-------+-------+
| 9303 | 493 | 229 | 11 | 9391 |
+-------+-------+-------+-------+-------+
| 9400 | 369 | 233 | 11 | 9491 |
+-------+-------+-------+-------+-------+
| 9497 | 981 | 233 | 11 | 9587 |
+-------+-------+-------+-------+-------+
| 9601 | 276 | 239 | 11 | 9697 |
+-------+-------+-------+-------+-------+
| 9708 | 647 | 239 | 11 | 9803 |
+-------+-------+-------+-------+-------+
| 9813 | 389 | 239 | 11 | 9907 |
+-------+-------+-------+-------+-------+
| 9916 | 80 | 239 | 11 | 10009 |
+-------+-------+-------+-------+-------+
| 10017 | 396 | 241 | 11 | 10111 |
+-------+-------+-------+-------+-------+
| 10120 | 580 | 251 | 11 | 10223 |
+-------+-------+-------+-------+-------+
| 10241 | 873 | 251 | 11 | 10343 |
+-------+-------+-------+-------+-------+
+-------+-------+-------+-------+-------+
| 10351 | 15 | 251 | 11 | 10453 |
+-------+-------+-------+-------+-------+
| 10458 | 976 | 251 | 11 | 10559 |
+-------+-------+-------+-------+-------+
| 10567 | 584 | 251 | 11 | 10667 |
+-------+-------+-------+-------+-------+
| 10676 | 267 | 257 | 11 | 10781 |
+-------+-------+-------+-------+-------+
| 10787 | 876 | 257 | 11 | 10891 |
+-------+-------+-------+-------+-------+
| 10899 | 642 | 257 | 12 | 11003 |
+-------+-------+-------+-------+-------+
| 11015 | 794 | 257 | 12 | 11119 |
+-------+-------+-------+-------+-------+
| 11130 | 78 | 263 | 12 | 11239 |
+-------+-------+-------+-------+-------+
| 11245 | 736 | 263 | 12 | 11353 |
+-------+-------+-------+-------+-------+
| 11358 | 882 | 269 | 12 | 11471 |
+-------+-------+-------+-------+-------+
| 11475 | 251 | 269 | 12 | 11587 |
+-------+-------+-------+-------+-------+
| 11590 | 434 | 269 | 12 | 11701 |
+-------+-------+-------+-------+-------+
| 11711 | 204 | 269 | 12 | 11821 |
+-------+-------+-------+-------+-------+
| 11829 | 256 | 271 | 12 | 11941 |
+-------+-------+-------+-------+-------+
| 11956 | 106 | 277 | 12 | 12073 |
+-------+-------+-------+-------+-------+
| 12087 | 375 | 277 | 12 | 12203 |
+-------+-------+-------+-------+-------+
| 12208 | 148 | 277 | 12 | 12323 |
+-------+-------+-------+-------+-------+
| 12333 | 496 | 281 | 12 | 12451 |
+-------+-------+-------+-------+-------+
| 12460 | 88 | 281 | 12 | 12577 |
+-------+-------+-------+-------+-------+
| 12593 | 826 | 293 | 12 | 12721 |
+-------+-------+-------+-------+-------+
| 12726 | 71 | 293 | 12 | 12853 |
+-------+-------+-------+-------+-------+
| 12857 | 925 | 293 | 12 | 12983 |
+-------+-------+-------+-------+-------+
| 13002 | 760 | 293 | 12 | 13127 |
+-------+-------+-------+-------+-------+
+-------+-------+-------+-------+-------+
| 13143 | 130 | 293 | 12 | 13267 |
+-------+-------+-------+-------+-------+
| 13284 | 641 | 307 | 12 | 13421 |
+-------+-------+-------+-------+-------+
| 13417 | 400 | 307 | 12 | 13553 |
+-------+-------+-------+-------+-------+
| 13558 | 480 | 307 | 12 | 13693 |
+-------+-------+-------+-------+-------+
| 13695 | 76 | 307 | 12 | 13829 |
+-------+-------+-------+-------+-------+
| 13833 | 665 | 307 | 12 | 13967 |
+-------+-------+-------+-------+-------+
| 13974 | 910 | 307 | 12 | 14107 |
+-------+-------+-------+-------+-------+
| 14115 | 467 | 311 | 12 | 14251 |
+-------+-------+-------+-------+-------+
| 14272 | 964 | 311 | 12 | 14407 |
+-------+-------+-------+-------+-------+
| 14415 | 625 | 313 | 12 | 14551 |
+-------+-------+-------+-------+-------+
| 14560 | 362 | 317 | 12 | 14699 |
+-------+-------+-------+-------+-------+
| 14713 | 759 | 317 | 12 | 14851 |
+-------+-------+-------+-------+-------+
| 14862 | 728 | 331 | 12 | 15013 |
+-------+-------+-------+-------+-------+
| 15011 | 343 | 331 | 12 | 15161 |
+-------+-------+-------+-------+-------+
| 15170 | 113 | 331 | 12 | 15319 |
+-------+-------+-------+-------+-------+
| 15325 | 137 | 331 | 12 | 15473 |
+-------+-------+-------+-------+-------+
| 15496 | 308 | 331 | 12 | 15643 |
+-------+-------+-------+-------+-------+
| 15651 | 800 | 337 | 12 | 15803 |
+-------+-------+-------+-------+-------+
| 15808 | 177 | 337 | 12 | 15959 |
+-------+-------+-------+-------+-------+
| 15977 | 961 | 337 | 12 | 16127 |
+-------+-------+-------+-------+-------+
| 16161 | 958 | 347 | 12 | 16319 |
+-------+-------+-------+-------+-------+
| 16336 | 72 | 347 | 12 | 16493 |
+-------+-------+-------+-------+-------+
| 16505 | 732 | 347 | 12 | 16661 |
+-------+-------+-------+-------+-------+
+-------+-------+-------+-------+-------+
| 16674 | 145 | 349 | 12 | 16831 |
+-------+-------+-------+-------+-------+
| 16851 | 577 | 353 | 12 | 17011 |
+-------+-------+-------+-------+-------+
| 17024 | 305 | 353 | 12 | 17183 |
+-------+-------+-------+-------+-------+
| 17195 | 50 | 359 | 12 | 17359 |
+-------+-------+-------+-------+-------+
| 17376 | 351 | 359 | 12 | 17539 |
+-------+-------+-------+-------+-------+
| 17559 | 175 | 367 | 12 | 17729 |
+-------+-------+-------+-------+-------+
| 17742 | 727 | 367 | 12 | 17911 |
+-------+-------+-------+-------+-------+
| 17929 | 902 | 367 | 12 | 18097 |
+-------+-------+-------+-------+-------+
| 18116 | 409 | 373 | 12 | 18289 |
+-------+-------+-------+-------+-------+
| 18309 | 776 | 373 | 12 | 18481 |
+-------+-------+-------+-------+-------+
| 18503 | 586 | 379 | 12 | 18679 |
+-------+-------+-------+-------+-------+
| 18694 | 451 | 379 | 12 | 18869 |
+-------+-------+-------+-------+-------+
| 18909 | 287 | 383 | 12 | 19087 |
+-------+-------+-------+-------+-------+
| 19126 | 246 | 389 | 12 | 19309 |
+-------+-------+-------+-------+-------+
| 19325 | 222 | 389 | 12 | 19507 |
+-------+-------+-------+-------+-------+
| 19539 | 563 | 397 | 12 | 19727 |
+-------+-------+-------+-------+-------+
| 19740 | 839 | 397 | 12 | 19927 |
+-------+-------+-------+-------+-------+
| 19939 | 897 | 401 | 12 | 20129 |
+-------+-------+-------+-------+-------+
| 20152 | 409 | 401 | 12 | 20341 |
+-------+-------+-------+-------+-------+
| 20355 | 618 | 409 | 12 | 20551 |
+-------+-------+-------+-------+-------+
| 20564 | 439 | 409 | 12 | 20759 |
+-------+-------+-------+-------+-------+
| 20778 | 95 | 419 | 13 | 20983 |
+-------+-------+-------+-------+-------+
| 20988 | 448 | 419 | 13 | 21191 |
+-------+-------+-------+-------+-------+
+-------+-------+-------+-------+-------+
| 21199 | 133 | 419 | 13 | 21401 |
+-------+-------+-------+-------+-------+
| 21412 | 938 | 419 | 13 | 21613 |
+-------+-------+-------+-------+-------+
| 21629 | 423 | 431 | 13 | 21841 |
+-------+-------+-------+-------+-------+
| 21852 | 90 | 431 | 13 | 22063 |
+-------+-------+-------+-------+-------+
| 22073 | 640 | 431 | 13 | 22283 |
+-------+-------+-------+-------+-------+
| 22301 | 922 | 433 | 13 | 22511 |
+-------+-------+-------+-------+-------+
| 22536 | 250 | 439 | 13 | 22751 |
+-------+-------+-------+-------+-------+
| 22779 | 367 | 439 | 13 | 22993 |
+-------+-------+-------+-------+-------+
| 23010 | 447 | 443 | 13 | 23227 |
+-------+-------+-------+-------+-------+
| 23252 | 559 | 449 | 13 | 23473 |
+-------+-------+-------+-------+-------+
| 23491 | 121 | 457 | 13 | 23719 |
+-------+-------+-------+-------+-------+
| 23730 | 623 | 457 | 13 | 23957 |
+-------+-------+-------+-------+-------+
| 23971 | 450 | 457 | 13 | 24197 |
+-------+-------+-------+-------+-------+
| 24215 | 253 | 461 | 13 | 24443 |
+-------+-------+-------+-------+-------+
| 24476 | 106 | 467 | 13 | 24709 |
+-------+-------+-------+-------+-------+
| 24721 | 863 | 467 | 13 | 24953 |
+-------+-------+-------+-------+-------+
| 24976 | 148 | 479 | 13 | 25219 |
+-------+-------+-------+-------+-------+
| 25230 | 427 | 479 | 13 | 25471 |
+-------+-------+-------+-------+-------+
| 25493 | 138 | 479 | 13 | 25733 |
+-------+-------+-------+-------+-------+
| 25756 | 794 | 487 | 13 | 26003 |
+-------+-------+-------+-------+-------+
| 26022 | 247 | 487 | 13 | 26267 |
+-------+-------+-------+-------+-------+
| 26291 | 562 | 491 | 13 | 26539 |
+-------+-------+-------+-------+-------+
| 26566 | 53 | 499 | 13 | 26821 |
+-------+-------+-------+-------+-------+
+-------+-------+-------+-------+-------+
| 26838 | 135 | 499 | 13 | 27091 |
+-------+-------+-------+-------+-------+
| 27111 | 21 | 503 | 13 | 27367 |
+-------+-------+-------+-------+-------+
| 27392 | 201 | 509 | 13 | 27653 |
+-------+-------+-------+-------+-------+
| 27682 | 169 | 521 | 13 | 27953 |
+-------+-------+-------+-------+-------+
| 27959 | 70 | 521 | 13 | 28229 |
+-------+-------+-------+-------+-------+
| 28248 | 386 | 521 | 13 | 28517 |
+-------+-------+-------+-------+-------+
| 28548 | 226 | 523 | 13 | 28817 |
+-------+-------+-------+-------+-------+
| 28845 | 3 | 541 | 13 | 29131 |
+-------+-------+-------+-------+-------+
| 29138 | 769 | 541 | 13 | 29423 |
+-------+-------+-------+-------+-------+
| 29434 | 590 | 541 | 13 | 29717 |
+-------+-------+-------+-------+-------+
| 29731 | 672 | 541 | 13 | 30013 |
+-------+-------+-------+-------+-------+
| 30037 | 713 | 547 | 13 | 30323 |
+-------+-------+-------+-------+-------+
| 30346 | 967 | 547 | 13 | 30631 |
+-------+-------+-------+-------+-------+
| 30654 | 368 | 557 | 14 | 30949 |
+-------+-------+-------+-------+-------+
| 30974 | 348 | 557 | 14 | 31267 |
+-------+-------+-------+-------+-------+
| 31285 | 119 | 563 | 14 | 31583 |
+-------+-------+-------+-------+-------+
| 31605 | 503 | 569 | 14 | 31907 |
+-------+-------+-------+-------+-------+
| 31948 | 181 | 571 | 14 | 32251 |
+-------+-------+-------+-------+-------+
| 32272 | 394 | 577 | 14 | 32579 |
+-------+-------+-------+-------+-------+
| 32601 | 189 | 587 | 14 | 32917 |
+-------+-------+-------+-------+-------+
| 32932 | 210 | 587 | 14 | 33247 |
+-------+-------+-------+-------+-------+
| 33282 | 62 | 593 | 14 | 33601 |
+-------+-------+-------+-------+-------+
| 33623 | 273 | 593 | 14 | 33941 |
+-------+-------+-------+-------+-------+
+-------+-------+-------+-------+-------+
| 33961 | 554 | 599 | 14 | 34283 |
+-------+-------+-------+-------+-------+
| 34302 | 936 | 607 | 14 | 34631 |
+-------+-------+-------+-------+-------+
| 34654 | 483 | 607 | 14 | 34981 |
+-------+-------+-------+-------+-------+
| 35031 | 397 | 613 | 14 | 35363 |
+-------+-------+-------+-------+-------+
| 35395 | 241 | 619 | 14 | 35731 |
+-------+-------+-------+-------+-------+
| 35750 | 500 | 631 | 14 | 36097 |
+-------+-------+-------+-------+-------+
| 36112 | 12 | 631 | 14 | 36457 |
+-------+-------+-------+-------+-------+
| 36479 | 958 | 641 | 14 | 36833 |
+-------+-------+-------+-------+-------+
| 36849 | 524 | 641 | 14 | 37201 |
+-------+-------+-------+-------+-------+
| 37227 | 8 | 643 | 14 | 37579 |
+-------+-------+-------+-------+-------+
| 37606 | 100 | 653 | 14 | 37967 |
+-------+-------+-------+-------+-------+
| 37992 | 339 | 653 | 14 | 38351 |
+-------+-------+-------+-------+-------+
| 38385 | 804 | 659 | 14 | 38749 |
+-------+-------+-------+-------+-------+
| 38787 | 510 | 673 | 14 | 39163 |
+-------+-------+-------+-------+-------+
| 39176 | 18 | 673 | 14 | 39551 |
+-------+-------+-------+-------+-------+
| 39576 | 412 | 677 | 14 | 39953 |
+-------+-------+-------+-------+-------+
| 39980 | 394 | 683 | 14 | 40361 |
+-------+-------+-------+-------+-------+
| 40398 | 830 | 691 | 15 | 40787 |
+-------+-------+-------+-------+-------+
| 40816 | 535 | 701 | 15 | 41213 |
+-------+-------+-------+-------+-------+
| 41226 | 199 | 701 | 15 | 41621 |
+-------+-------+-------+-------+-------+
| 41641 | 27 | 709 | 15 | 42043 |
+-------+-------+-------+-------+-------+
| 42067 | 298 | 709 | 15 | 42467 |
+-------+-------+-------+-------+-------+
| 42490 | 368 | 719 | 15 | 42899 |
+-------+-------+-------+-------+-------+
+-------+-------+-------+-------+-------+
| 42916 | 755 | 727 | 15 | 43331 |
+-------+-------+-------+-------+-------+
| 43388 | 379 | 727 | 15 | 43801 |
+-------+-------+-------+-------+-------+
| 43840 | 73 | 733 | 15 | 44257 |
+-------+-------+-------+-------+-------+
| 44279 | 387 | 739 | 15 | 44701 |
+-------+-------+-------+-------+-------+
| 44729 | 457 | 751 | 15 | 45161 |
+-------+-------+-------+-------+-------+
| 45183 | 761 | 751 | 15 | 45613 |
+-------+-------+-------+-------+-------+
| 45638 | 855 | 757 | 15 | 46073 |
+-------+-------+-------+-------+-------+
| 46104 | 370 | 769 | 15 | 46549 |
+-------+-------+-------+-------+-------+
| 46574 | 261 | 769 | 15 | 47017 |
+-------+-------+-------+-------+-------+
| 47047 | 299 | 787 | 15 | 47507 |
+-------+-------+-------+-------+-------+
| 47523 | 920 | 787 | 15 | 47981 |
+-------+-------+-------+-------+-------+
| 48007 | 269 | 787 | 15 | 48463 |
+-------+-------+-------+-------+-------+
| 48489 | 862 | 797 | 15 | 48953 |
+-------+-------+-------+-------+-------+
| 48976 | 349 | 809 | 15 | 49451 |
+-------+-------+-------+-------+-------+
| 49470 | 103 | 809 | 15 | 49943 |
+-------+-------+-------+-------+-------+
| 49978 | 115 | 821 | 15 | 50461 |
+-------+-------+-------+-------+-------+
| 50511 | 93 | 821 | 16 | 50993 |
+-------+-------+-------+-------+-------+
| 51017 | 982 | 827 | 16 | 51503 |
+-------+-------+-------+-------+-------+
| 51530 | 432 | 839 | 16 | 52027 |
+-------+-------+-------+-------+-------+
| 52062 | 340 | 853 | 16 | 52571 |
+-------+-------+-------+-------+-------+
| 52586 | 173 | 853 | 16 | 53093 |
+-------+-------+-------+-------+-------+
| 53114 | 421 | 857 | 16 | 53623 |
+-------+-------+-------+-------+-------+
| 53650 | 330 | 863 | 16 | 54163 |
+-------+-------+-------+-------+-------+
+-------+-------+-------+-------+-------+
| 54188 | 624 | 877 | 16 | 54713 |
+-------+-------+-------+-------+-------+
| 54735 | 233 | 877 | 16 | 55259 |
+-------+-------+-------+-------+-------+
| 55289 | 362 | 883 | 16 | 55817 |
+-------+-------+-------+-------+-------+
| 55843 | 963 | 907 | 16 | 56393 |
+-------+-------+-------+-------+-------+
| 56403 | 471 | 907 | 16 | 56951 |
+-------+-------+-------+-------+-------+
Table 2: Systematic Indices and Other Parameters
5.7. Operating with Octets, Symbols, and Matrices
5.7.1. General
The remainder of this section describes the arithmetic operations
that are used to generate encoding symbols from source symbols and to
generate source symbols from encoding symbols. Mathematically,
octets can be thought of as elements of a finite field, i.e., the
finite field GF(256) with 256 elements, and thus the addition and
multiplication operations and identity elements and inverses over
both operations are defined. Matrix operations and symbol operations
are defined based on the arithmetic operations on octets. This
allows a full implementation of these arithmetic operations without
having to understand the underlying mathematics of finite fields.
5.7.2. Arithmetic Operations on Octets
Octets are mapped to non-negative integers in the range 0 through 255
in the usual way: A single octet of data from a symbol,
B[7],B[6],B[5],B[4],B[3],B[2],B[1],B[0], where B[7] is the highest
order bit and B[0] is the lowest order bit, is mapped to the integer
i=B[7]*128+B[6]*64+B[5]*32+B[4]*16+B[3]*8+B[2]*4+B[1]*2+B[0].
The addition of two octets u and v is defined as the exclusive-or
operation, i.e.,
u + v = u ^ v.
Subtraction is defined in the same way, so we also have
u - v = u ^ v.
The zero element (additive identity) is the octet represented by the
integer 0. The additive inverse of u is simply u, i.e.,
u + u = 0.
The multiplication of two octets is defined with the help of two
tables OCT_EXP and OCT_LOG, which are given in Section 5.7.3 and
Section 5.7.4, respectively. The table OCT_LOG maps octets (other
than the zero element) to non-negative integers, and OCT_EXP maps
non-negative integers to octets. For two octets u and v, we define
u * v =
0, if either u or v are 0,
OCT_EXP[OCT_LOG[u] + OCT_LOG[v]] otherwise.
Note that the '+' on the right-hand side of the above is the usual
integer addition, since its arguments are ordinary integers.
The division u / v of two octets u and v, and where v != 0, is
defined as follows:
u / v =
0, if u == 0,
OCT_EXP[OCT_LOG[u] - OCT_LOG[v] + 255] otherwise.
The one element (multiplicative identity) is the octet represented by
the integer 1. For an octet u that is not the zero element, i.e.,
the multiplicative inverse of u is
OCT_EXP[255 - OCT_LOG[u]].
The octet denoted by alpha is the octet with the integer
representation 2. If i is a non-negative integer 0 <= i < 256, we
have
alpha^^i = OCT_EXP[i].
5.7.3. The Table OCT_EXP
The table OCT_EXP contains 510 octets. The indexing starts at 0 and
ranges to 509, and the entries are the octets with the following
positive integer representation:
1, 2, 4, 8, 16, 32, 64, 128, 29, 58, 116, 232, 205, 135, 19, 38, 76,
152, 45, 90, 180, 117, 234, 201, 143, 3, 6, 12, 24, 48, 96, 192, 157,
39, 78, 156, 37, 74, 148, 53, 106, 212, 181, 119, 238, 193, 159, 35,
70, 140, 5, 10, 20, 40, 80, 160, 93, 186, 105, 210, 185, 111, 222,
161, 95, 190, 97, 194, 153, 47, 94, 188, 101, 202, 137, 15, 30, 60,
120, 240, 253, 231, 211, 187, 107, 214, 177, 127, 254, 225, 223, 163,
91, 182, 113, 226, 217, 175, 67, 134, 17, 34, 68, 136, 13, 26, 52,
104, 208, 189, 103, 206, 129, 31, 62, 124, 248, 237, 199, 147, 59,
118, 236, 197, 151, 51, 102, 204, 133, 23, 46, 92, 184, 109, 218,
169, 79, 158, 33, 66, 132, 21, 42, 84, 168, 77, 154, 41, 82, 164, 85,
170, 73, 146, 57, 114, 228, 213, 183, 115, 230, 209, 191, 99, 198,
145, 63, 126, 252, 229, 215, 179, 123, 246, 241, 255, 227, 219, 171,
75, 150, 49, 98, 196, 149, 55, 110, 220, 165, 87, 174, 65, 130, 25,
50, 100, 200, 141, 7, 14, 28, 56, 112, 224, 221, 167, 83, 166, 81,
162, 89, 178, 121, 242, 249, 239, 195, 155, 43, 86, 172, 69, 138, 9,
18, 36, 72, 144, 61, 122, 244, 245, 247, 243, 251, 235, 203, 139, 11,
22, 44, 88, 176, 125, 250, 233, 207, 131, 27, 54, 108, 216, 173, 71,
142, 1, 2, 4, 8, 16, 32, 64, 128, 29, 58, 116, 232, 205, 135, 19, 38,
76, 152, 45, 90, 180, 117, 234, 201, 143, 3, 6, 12, 24, 48, 96, 192,
157, 39, 78, 156, 37, 74, 148, 53, 106, 212, 181, 119, 238, 193, 159,
35, 70, 140, 5, 10, 20, 40, 80, 160, 93, 186, 105, 210, 185, 111,
222, 161, 95, 190, 97, 194, 153, 47, 94, 188, 101, 202, 137, 15, 30,
60, 120, 240, 253, 231, 211, 187, 107, 214, 177, 127, 254, 225, 223,
163, 91, 182, 113, 226, 217, 175, 67, 134, 17, 34, 68, 136, 13, 26,
52, 104, 208, 189, 103, 206, 129, 31, 62, 124, 248, 237, 199, 147,
59, 118, 236, 197, 151, 51, 102, 204, 133, 23, 46, 92, 184, 109, 218,
169, 79, 158, 33, 66, 132, 21, 42, 84, 168, 77, 154, 41, 82, 164, 85,
170, 73, 146, 57, 114, 228, 213, 183, 115, 230, 209, 191, 99, 198,
145, 63, 126, 252, 229, 215, 179, 123, 246, 241, 255, 227, 219, 171,
75, 150, 49, 98, 196, 149, 55, 110, 220, 165, 87, 174, 65, 130, 25,
50, 100, 200, 141, 7, 14, 28, 56, 112, 224, 221, 167, 83, 166, 81,
162, 89, 178, 121, 242, 249, 239, 195, 155, 43, 86, 172, 69, 138, 9,
18, 36, 72, 144, 61, 122, 244, 245, 247, 243, 251, 235, 203, 139, 11,
22, 44, 88, 176, 125, 250, 233, 207, 131, 27, 54, 108, 216, 173, 71,
142
5.7.4. The Table OCT_LOG
The table OCT_LOG contains 255 non-negative integers. The table is
indexed by octets interpreted as integers. The octet corresponding
to the zero element, which is represented by the integer 0, is
excluded as an index, and thus indexing starts at 1 and ranges up to
255, and the entries are the following:
0, 1, 25, 2, 50, 26, 198, 3, 223, 51, 238, 27, 104, 199, 75, 4, 100,
224, 14, 52, 141, 239, 129, 28, 193, 105, 248, 200, 8, 76, 113, 5,
138, 101, 47, 225, 36, 15, 33, 53, 147, 142, 218, 240, 18, 130, 69,
29, 181, 194, 125, 106, 39, 249, 185, 201, 154, 9, 120, 77, 228, 114,
166, 6, 191, 139, 98, 102, 221, 48, 253, 226, 152, 37, 179, 16, 145,
34, 136, 54, 208, 148, 206, 143, 150, 219, 189, 241, 210, 19, 92,
131, 56, 70, 64, 30, 66, 182, 163, 195, 72, 126, 110, 107, 58, 40,
84, 250, 133, 186, 61, 202, 94, 155, 159, 10, 21, 121, 43, 78, 212,
229, 172, 115, 243, 167, 87, 7, 112, 192, 247, 140, 128, 99, 13, 103,
74, 222, 237, 49, 197, 254, 24, 227, 165, 153, 119, 38, 184, 180,
124, 17, 68, 146, 217, 35, 32, 137, 46, 55, 63, 209, 91, 149, 188,
207, 205, 144, 135, 151, 178, 220, 252, 190, 97, 242, 86, 211, 171,
20, 42, 93, 158, 132, 60, 57, 83, 71, 109, 65, 162, 31, 45, 67, 216,
183, 123, 164, 118, 196, 23, 73, 236, 127, 12, 111, 246, 108, 161,
59, 82, 41, 157, 85, 170, 251, 96, 134, 177, 187, 204, 62, 90, 203,
89, 95, 176, 156, 169, 160, 81, 11, 245, 22, 235, 122, 117, 44, 215,
79, 174, 213, 233, 230, 231, 173, 232, 116, 214, 244, 234, 168, 80,
88, 175
5.7.5. Operations on Symbols
Operations on symbols have the same semantics as operations on
vectors of octets of length T in this specification. Thus, if U and
V are two symbols formed by the octets u[0], ..., u[T-1] and v[0],
..., v[T-1], respectively, the sum of symbols U + V is defined to be
the component-wise sum of octets, i.e., equal to the symbol D formed
by the octets d[0], ..., d[T-1], such that
d[i] = u[i] + v[i], 0 <= i < T.
Furthermore, if beta is an octet, the product beta*U is defined to be
the symbol D obtained by multiplying each octet of U by beta, i.e.,
d[i] = beta*u[i], 0 <= i < T.
5.7.6. Operations on Matrices
All matrices in this specification have entries that are octets, and
thus matrix operations and definitions are defined in terms of the
underlying octet arithmetic, e.g., operations on a matrix, matrix
rank, and matrix inversion.
5.8. Requirements for a Compliant Decoder
If a RaptorQ-compliant decoder receives a mathematically sufficient
set of encoding symbols generated according to the encoder
specification in Section 5.3 for reconstruction of a source block,
then such a decoder SHOULD recover the entire source block.
A RaptorQ-compliant decoder SHALL have the following recovery
properties for source blocks with K' source symbols for all values of
K' in Table 2 of Section 5.6.
1. If the decoder receives K' encoding symbols generated according
to the encoder specification in Section 5.3 with corresponding
ESIs chosen independently and uniformly at random from the range
of possible ESIs, then on average the decoder will fail to
recover the entire source block at most 1 out of 100 times.
2. If the decoder receives K'+1 encoding symbols generated according
to the encoder specification in Section 5.3 with corresponding
ESIs chosen independently and uniformly at random from the range
of possible ESIs, then on average the decoder will fail to
recover the entire source block at most 1 out of 10,000 times.
3. If the decoder receives K'+2 encoding symbols generated according
to the encoder specification in Section 5.3 with corresponding
ESIs chosen independently and uniformly at random from the range
of possible ESIs, then on average the decoder will fail to
recover the entire source block at most 1 out of 1,000,000 times.
Note that the Example FEC Decoder specified in Section 5.4 fulfills
both requirements, i.e.,
1. it can reconstruct a source block as long as it receives a
mathematically sufficient set of encoding symbols generated
according to the encoder specification in Section 5.3, and
2. it fulfills the mandatory recovery properties from above.
6. Security Considerations
Data delivery can be subject to denial-of-service attacks by
attackers that send corrupted packets that are accepted as legitimate
by receivers. This is particularly a concern for multicast delivery
because a corrupted packet may be injected into the session close to
the root of the multicast tree, in which case the corrupted packet
will arrive at many receivers. The use of even one corrupted packet
containing encoding data may result in the decoding of an object that
is completely corrupted and unusable. It is thus RECOMMENDED that
source authentication and integrity checking are applied to decoded
objects before delivering objects to an application. For example, a
SHA-256 hash [FIPS.180-3.2008] of an object may be appended before
transmission, and the SHA-256 hash is computed and checked after the
object is decoded but before it is delivered to an application.
Source authentication SHOULD be provided, for example, by including a
digital signature verifiable by the receiver computed on top of the
hash value. It is also RECOMMENDED that a packet authentication
protocol such as TESLA [RFC4082] be used to detect and discard
corrupted packets upon arrival. This method may also be used to
provide source authentication. Furthermore, it is RECOMMENDED that
Reverse Path Forwarding checks be enabled in all network routers and
switches along the path from the sender to receivers to limit the
possibility of a bad agent successfully injecting a corrupted packet
into the multicast tree data path.
Another security concern is that some FEC information may be obtained
by receivers out-of-band in a session description, and if the session
description is forged or corrupted, then the receivers will not use
the correct protocol for decoding content from received packets. To
avoid these problems, it is RECOMMENDED that measures be taken to
prevent receivers from accepting incorrect session descriptions,
e.g., by using source authentication to ensure that receivers only
accept legitimate session descriptions from authorized senders.
7. IANA Considerations
Values of FEC Encoding IDs and FEC Instance IDs are subject to IANA
registration. For general guidelines on IANA considerations as they
apply to this document, see [RFC5052]. IANA has assigned the value 6
under the ietf:rmt:fec:encoding registry to "RaptorQ Code" as the
Fully-Specified FEC Encoding ID value associated with this
specification.
8. Acknowledgements
Thanks are due to Ranganathan (Ranga) Krishnan. Ranga Krishnan has
been very supportive in finding and resolving implementation details
and in finding the systematic indices. In addition, Habeeb Mohiuddin
Mohammed and Antonios Pitarokoilis, both from the Munich University
of Technology (TUM), and Alan Shinsato have done two independent
implementations of the RaptorQ encoder/decoder that have helped to
clarify and to resolve issues with this specification.
9. References
9.1. Normative References
[FIPS.180-3.2008]
National Institute of Standards and Technology, "Secure
Hash Standard", FIPS PUB 180-3, October 2008.
[RFC2119] Bradner, S., "Key words for use in RFCs to Indicate
Requirement Levels", BCP 14, RFC 2119, March 1997.
[RFC4082] Perrig, A., Song, D., Canetti, R., Tygar, J., and B.
Briscoe, "Timed Efficient Stream Loss-Tolerant
Authentication (TESLA): Multicast Source Authentication
Transform Introduction", RFC 4082, June 2005.
[RFC5052] Watson, M., Luby, M., and L. Vicisano, "Forward Error
Correction (FEC) Building Block", RFC 5052, August 2007.
9.2. Informative References
[LTCodes] Luby, M., "LT codes", Annual IEEE Symposium on Foundations
of Computer Science, pp. 271-280, November 2002.
[RFC3453] Luby, M., Vicisano, L., Gemmell, J., Rizzo, L., Handley,
M., and J. Crowcroft, "The Use of Forward Error Correction
(FEC) in Reliable Multicast", RFC 3453, December 2002.
[RFC5053] Luby, M., Shokrollahi, A., Watson, M., and T. Stockhammer,
"Raptor Forward Error Correction Scheme for Object
Delivery", RFC 5053, October 2007.
[RaptorCodes]
Shokrollahi, A. and M. Luby, "Raptor Codes", Foundations
and Trends in Communications and Information Theory: Vol.
6: No. 3-4, pp. 213-322, 2011.
Authors' Addresses
Michael Luby
Qualcomm Incorporated
3165 Kifer Road
Santa Clara, CA 95051
U.S.A.
EMail: luby@qualcomm.com
Amin Shokrollahi
EPFL
Laboratoire d'algorithmique
Station 14
Batiment BC
Lausanne 1015
Switzerland
EMail: amin.shokrollahi@epfl.ch
Mark Watson
Netflix Inc.
100 Winchester Circle
Los Gatos, CA 95032
U.S.A.
EMail: watsonm@netflix.com
Thomas Stockhammer
Nomor Research
Brecherspitzstrasse 8
Munich 81541
Germany
EMail: stockhammer@nomor.de
Lorenz Minder
Qualcomm Incorporated
3165 Kifer Road
Santa Clara, CA 95051
U.S.A.
EMail: lminder@qualcomm.com