Request for Comments: 6330
Category: Standards Track
ISSN: 20701721
Qualcomm Incorporated
A. Shokrollahi
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 FullySpecified 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.rfceditor.org/info/rfc6330.
Copyright Notice

Copyright © 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/licenseinfo) 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. SchemeSpecific . . . . . . . . . . . . . . . . . . . 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 FullySpecified 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 4octet 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): 8bit unsigned integer. A nonnegative integer identifier for the source block that the encoding symbols within the packet relate to. o Encoding Symbol ID (ESI): 24bit unsigned integer. A nonnegative 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

The Common FEC Object Transmission Information elements used by this FEC scheme are:
o Transfer Length (F): 40bit unsigned integer. A nonnegative integer that is at most 946270874880. This is the transfer length of the object in units of octets.
 Symbol Size (T): 16bit 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^^161, 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. SchemeSpecific

The following parameters are carried in the SchemeSpecific FEC Object Transmission Information element for this FEC scheme:
 The number of source blocks (Z): 8bit unsigned integer.
 The number of subblocks (N): 16bit unsigned integer.
 A symbol alignment parameter (Al): 8bit unsigned integer.
These parameters are all positive integers. The encoded Scheme specific Object Transmission Information is a 4octet 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 SchemeSpecific FEC Object Transmission Information
The encoded FEC Object Transmission Information is a 12octet field consisting of the concatenation of the encoded Common FEC Object Transmission Information and the encoded Schemespecific 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:
 F: the transfer length of the object, in octets
 Al: the symbol alignment parameter
 T: the symbol size in octets, which MUST be a multiple of Al
 Z: the number of source blocks
 N: the number of subblocks 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:
 Source Block Number (SBN)
 Encoding Symbol ID (ESI)
 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:
 F: the transfer length of the object, in octets
 WS: the maximum size block that is decodable in working memory, in octets
 P': the maximum payload size in octets, which is assumed to be a multiple of Al
 Al: the symbol alignment parameter, in octets
 SS: a parameter where the desired lower bound on the subsymbol size is SS*Al
 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
 T = P'
 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))))
 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 subsymbol 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 32bit 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 subblock is divided into K subsymbols 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 subblock 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 subblock of a source block. Exact partitioning of the object into source blocks and subblocks is described in Section 4.4.1.2 below.
4.4.1.2. Source Block and SubBlock Partitioning

The construction of source blocks and subblocks 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:
 F: the transfer length of the object, in octets
 Al: a symbol alignment parameter, in octets
 T: the symbol size, in octets, which MUST be a multiple of Al
 Z: the number of source blocks
 N: the number of subblocks 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 equalsized 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
 Kt = ceil(F/T),
o (KL, KS, ZL, ZS) = Partition[Kt, Z],
 (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*TF zero octets.
Next, each source block with K source symbols MUST be divided into N = NL + NS contiguous subblocks, the first NL subblocks each consisting of K contiguous subsymbols of size of TL*Al octets and the remaining NS subblocks each consisting of K contiguous sub symbols of size of TS*Al octets. The symbol alignment parameter Al ensures that subsymbols are always a multiple of Al octets.
Finally, the mth symbol of a source block consists of the concatenation of the mth subsymbol from each of the N subblocks. 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:
 Source Block Number (SBN)
 Encoding Symbol ID (ESI)
 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 K1 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+G1 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+G1 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 subblock 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 subblocks 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 subblocks associated with the source block, where subsymbols within each subblock 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 subsymbols of the source symbols so that all consecutive subsymbols of the first subblock are first, followed by all consecutive subsymbols of the second subblock, etc., followed by all consecutive subsymbols of the Nth subblock. This strategy is especially applicable if the receiver has enough RAM to decode an entire source block.
Another strategy is to separately recover the subblocks of a source block. For example, a receiver may demultiplex and store subsymbols associated with each subblock separately as packets containing encoding symbols arrive, and then use the stored subsymbols received for a subblock to recover that subblock 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

 Source block: a block of K source symbols that are considered together for RaptorQ encoding and decoding purposes.
 Extended Source Block: a block of K' source symbols, where K' >= K, constructed from a source block and zero or more padding symbols.
 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.
 Source symbol: the smallest unit of data used during the encoding process. All source symbols within a source block have the same size.
 Padding symbol: a symbol with all zero bits that is added to the source block to form the extended source block.
 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.
 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.
 Intermediate symbols: symbols generated from the source symbols using an inverse encoding process based on precoding 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.
 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.
 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".
 HDPC symbols: there is a small subset of the intermediate symbols that are HDPC symbols. Each HDPC symbol has a precoding relationship with a large fraction of the other intermediate symbols. HDPC means "High Density Parity Check".
 LDPC symbols: there is a moderatesized subset of the intermediate symbols that are LDPC symbols. Each LDPC symbol has a precoding relationship with a small fraction of the other intermediate symbols. LDPC means "Low Density Parity Check".
 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.
 Encoding Symbol ID (ESI): information that uniquely identifies each encoding symbol associated with a source block for sending and receiving purposes.
 Internal Symbol ID (ISI): information that uniquely identifies each symbol associated with an extended source block for encoding and decoding purposes.
 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 nonnegative 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 nonnegative 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 equallength bit strings u and v, the bitwise exclusiveor 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 nonHDPC intermediate symbols that are PI symbols. C denotes an array of intermediate symbols, C[0], C[1], C[2], ..., C[L1]. C' denotes an array of the symbols of the extended source block, where C'[0], C'[1], C'[2], ..., C'[K1] 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 32bit 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 pseudorandom 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 exclusiveor 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 RaptorQcompliant 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 pseudorandom 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, ..., K1, 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, ..., K1, 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'[K1] 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[L1] from the source symbols using an inverse encoding process. This process can be realized by a RaptorQ decoding process.
Certain "precoding 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 precoding step to generate the L intermediate symbols C[0], ..., C[L1] 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:
 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.
 A number of precoding 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. PreCoding Relationships

The precoding 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 = PH of the other K' intermediate symbols. The W LT symbols consist of the S LDPC symbols together with WS 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
 S = S(K')
 H = H(K')
 W = W(K')
o L = K' + S + H o P = L  W
 P1 denote the smallest prime number greater than or equal to P.
o U = P  H o B = W  S o C[0], ..., C[B1] denote the intermediate symbols that are LT symbols but not LDPC symbols. o C[B], ..., C[B+S1] denote the S LDPC symbols that are also LT symbols. o C[W], ..., C[W+U1] denote the intermediate symbols that are PI symbols but not HDPC symbols. o C[LH], ..., C[L1] denote the H HDPC symbols that are also PI symbols.
The first set of precoding relations, called LDPC relations, is described below and requires that at the end of this process the set of symbols D[0] , ..., D[S1] are all zero:
 Initialize the symbols D[0] = C[B], ..., D[S1] = C[B+S1].
o For i = 0, ..., B1 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, ..., S1 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[B1])] + G_LDPC,2 * Transpose(C[W], ..., C[W+P1]) + Transpose[(C[B], ..., C[B+S1])] = 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[L1] are the HDPC relations and they are defined as follows:
Let
 alpha denote the octet represented by integer 2 as defined in Section 5.7.
 MT denote an H x (K' + S) matrix of octets, where for j=0, ..., K'+S2, 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,H1] + 1) % H, and MT[i,j] is the zero element for all other values of i, and for j=K'+S1, MT[i,j] = alpha^^i for i=0, ..., H1.
o GAMMA denote a (K'+S) x (K'+S) matrix of octets, where GAMMA[i,j] = alpha ^^ (ij) for i >= j,


 otherwise.

Then, the relationship between the first K'+S intermediate symbols C[0], ..., C[K'+S1] and the H HDPC symbols C[K'+S], ..., C[K'+S+H1] is given by: Transpose[C[K'+S], ..., C[K'+S+H1]] + MT * GAMMA * Transpose[C[0], ..., C[K'+S1]] = 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[L1] 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[L1]), (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[L1] satisfy the precoding 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[L1] satisfying the constraints in Section 5.3.3.4.1.
The L intermediate symbols can be calculated as follows:
Let
 C denote the column vector of the L intermediate symbols, C[0], C[1], ..., C[L1].
 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
 G_LDPC,1 and G_LDPC,2 be S x B and S x P matrices as defined in Section 5.3.3.3.
 G_HDPC be the H x (K'+S) matrix such that
G_HDPC * Transpose(C[0], ..., C[K'+S1]) = Transpose(C[K'+S], ..., C[L1]), 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[L1])] = 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[L1]), (d[i], a[i], b[i], d1[i], a1[i], b1[i])] and G_ENC[i,j] = 0 otherwise. Then
 The first S rows of A are equal to G_LDPC,1  I_S  G_LDPC,2.
 The next H rows of A are equal to G_HDPC  I_H.
 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[L1].
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[L1]), (d, a, b, d1, a1, b1)] defined in Section 5.3.5.3 to the L intermediate symbols C[0], C[1], ..., C[L1] 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 nonnegative integer, i is a nonnegative integer less than 256, and m is a positive integer, and the value produced is an integer between 0 and m1. 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[d1] <= v < f[d], and set Deg[v] = min(d, W2). 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[L1]), (d, a, b, d1, a1, b1)] takes the following inputs:
 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[L1]) is the array of L intermediate symbols (subsymbols) generated as described in Section 5.3.3.4.
 (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 W1 inclusive
 b is a nonnegative integer between 0 and W1 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 P11 inclusive
 b1 is a nonnegative integer between 0 and P11 inclusive
The encoding symbol generator produces a single encoding symbol as output (referred to as result), according to the following algorithm:
 result = C[b]
o For j = 1, ..., d1 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, ..., d11 do * b1 = (b1 + a1) % P1 * While (b1 >= P) do b1 = (b1+a1) % P1 * result = result + C[W+b1]
 Return result
5.3.5.4. Tuple Generator

The tuple generator Tuple[K',X] takes the following inputs:
 K': the number of source symbols in the extended source block
 X: an ISI
Let
 L be determined from K' as described in Section 5.3.3.3
 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 }
 B = 10267*(J+1)
o y = (B + X*A) % 2^^32 o v = Rand[y, 0, 2^^20]
 d = Deg[v]
o a = 1 + Rand[y, 1, W1] 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, P11] 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 precoding 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[L1] = L1 and d[0] = 0, d[1] = 1, ..., d[M1] = M1 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']].
 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.
 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'].
 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[L1]] are the values of the L symbols C[c[0]], C[c[1]], ..., C[c[L1]].
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 nonnegative integers i and u, which are initialized to 0 and P, the number of PI symbols, respectively. The submatrices of A are:
 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.
 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.
 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.
 The submatrix U defined by the intersection of all the rows and the last u columns.
 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 LP 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:
 If all entries of V are zero, then no row is chosen and decoding fails.
 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 nonHDPC 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 r1 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 r1, 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:
 If A[j,j] is not one, then divide row j of A by A[j,j].
2. For l from 1 to j1, 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 32bit 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 nonnegative 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 exclusiveor 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 nonnegative integers, and OCT_EXP maps nonnegative 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 righthand 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 nonnegative 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 nonnegative 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[T1] and v[0], ..., v[T1], respectively, the sum of symbols U + V is defined to be the componentwise sum of octets, i.e., equal to the symbol D formed by the octets d[0], ..., d[T1], 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 RaptorQcompliant 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 RaptorQcompliant 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.
 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.
 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.
 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.,
 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
 it fulfills the mandatory recovery properties from above.
6. Security Considerations

Data delivery can be subject to denialofservice 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 SHA256 hash [FIPS.1803.2008] of an object may be appended before transmission, and the SHA256 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 outofband 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 FullySpecified 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.1803.2008]



National Institute of Standards and Technology, "Secure Hash Standard", FIPS PUB 1803, 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 LossTolerant 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. 271280, 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. 34, pp. 213322, 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