RFC 5053

Raptor Forward Error Correction Scheme for Object Delivery

📅 published October 2007, from ietf

RFC 5053: Raptor Forward Error Correction Scheme for Object Delivery 🎀 (Part 1 of 11)

okay bestie, let's talk about RFC 5053 💖 It's a Standards Track doc from October 2007, from the Network Working Group. The authors are M. Luby (Digital Fountain), A. Shokrollahi (EPFL), M. Watson (Digital Fountain), and T. Stockhammer (Nomor Research). It's RFC number 5053 and the title is "Raptor Forward Error Correction Scheme for Object Delivery." The title block doesn't list anything it obsoletes or updates ✨

Status of This Memo

This doc specifies an Internet standards track protocol for the Internet community, and it's asking for discussion and suggestions for improvements 🫶 For the standardization state and status of this protocol, check the current edition of the "Internet Official Protocol Standards" (STD 1). Distribution of this memo is unlimited 🦋

Abstract

This doc describes a Fully-Specified Forward Error Correction (FEC) scheme, which is FEC Encoding ID 1 💅 It's for the Raptor forward error correction code and how it's used for reliable delivery of data objects 🔥

Raptor is a fountain code, which is so iconic 🌊 That means the encoder can make as many encoding symbols as needed on the fly from the source symbols of a source block of data. The decoder can recover the source block from any set of encoding symbols that's only slightly more in number than the number of source symbols 🤯

The Raptor code in here is a systematic code, which means all the source symbols are among the encoding symbols that can be generated 💜

   1.  Introduction . . . . . . . . . . . . . . . . . . . . . . . . .  3
   2.  Requirements Notation  . . . . . . . . . . . . . . . . . . . .  3
   3.  Formats and Codes  . . . . . . . . . . . . . . . . . . . . . .  3
     3.1.  FEC Payload IDs  . . . . . . . . . . . . . . . . . . . . .  3
     3.2.  FEC Object Transmission Information (OTI)  . . . . . . . .  4
       3.2.1.  Mandatory  . . . . . . . . . . . . . . . . . . . . . .  4
       3.2.2.  Common . . . . . . . . . . . . . . . . . . . . . . . .  4
       3.2.3.  Scheme-Specific  . . . . . . . . . . . . . . . . . . .  5
   4.  Procedures . . . . . . . . . . . . . . . . . . . . . . . . . .  5
     4.1.  Content Delivery Protocol Requirements . . . . . . . . . .  5
     4.2.  Example Parameter Derivation Algorithm . . . . . . . . . .  6
   5.  Raptor FEC Code Specification  . . . . . . . . . . . . . . . .  8
     5.1.  Definitions, Symbols, and Abbreviations  . . . . . . . . .  8
       5.1.1.  Definitions  . . . . . . . . . . . . . . . . . . . . .  8
       5.1.2.  Symbols  . . . . . . . . . . . . . . . . . . . . . . .  9
       5.1.3.  Abbreviations  . . . . . . . . . . . . . . . . . . . . 11
     5.2.  Overview . . . . . . . . . . . . . . . . . . . . . . . . . 11
     5.3.  Object Delivery  . . . . . . . . . . . . . . . . . . . . . 12
       5.3.1.  Source Block Construction  . . . . . . . . . . . . . . 12
       5.3.2.  Encoding Packet Construction . . . . . . . . . . . . . 14
     5.4.  Systematic Raptor Encoder  . . . . . . . . . . . . . . . . 15
       5.4.1.  Encoding Overview  . . . . . . . . . . . . . . . . . . 15
       5.4.2.  First Encoding Step: Intermediate Symbol Generation  . 16
       5.4.3.  Second Encoding Step: LT Encoding  . . . . . . . . . . 20
       5.4.4.  Generators . . . . . . . . . . . . . . . . . . . . . . 21
     5.5.  Example FEC Decoder  . . . . . . . . . . . . . . . . . . . 23
       5.5.1.  General  . . . . . . . . . . . . . . . . . . . . . . . 23
       5.5.2.  Decoding a Source Block  . . . . . . . . . . . . . . . 23
     5.6.  Random Numbers . . . . . . . . . . . . . . . . . . . . . . 28
       5.6.1.  The Table V0 . . . . . . . . . . . . . . . . . . . . . 28
       5.6.2.  The Table V1 . . . . . . . . . . . . . . . . . . . . . 29
     5.7.  Systematic Indices J(K)  . . . . . . . . . . . . . . . . . 30
   6.  Security Considerations  . . . . . . . . . . . . . . . . . . . 43
   7.  IANA Considerations  . . . . . . . . . . . . . . . . . . . . . 43
   8.  Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . 44
   9.  References . . . . . . . . . . . . . . . . . . . . . . . . . . 44
     9.1.  Normative References . . . . . . . . . . . . . . . . . . . 44
     9.2.  Informative References . . . . . . . . . . . . . . . . . . 44

1. Introduction ✨

okay so this doc specifies an FEC Scheme for the Raptor forward error correction code, used for object delivery apps 🎀 The idea of an FEC Scheme is defined in [RFC5052], and this doc follows the format that one prescribes and uses its terminology 💅 Raptor Codes were introduced in [Raptor]. If you want an overview, check out [CCNC] 🧚

The Raptor FEC Scheme is a Fully-Specified FEC Scheme, and it's FEC Encoding ID 1 👑

Raptor is a fountain code again, so the encoder can generate as many encoding symbols as needed on the fly from the source symbols of a block. The decoder can recover the source block from any set of encoding symbols that's only slightly more in number than the number of source symbols 🌸

The code here is a systematic code 🍓 That means the original source symbols can be sent unmodified from sender to receiver, along with some repair symbols. For more background on using Forward Error Correction codes in reliable multicast, see [RFC3453] 🦋

The code described here is identical to the one described in [MBMS] 💖

2. Requirements Notation 🫶

Okay, this part is important, so listen up 👀 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. FEC Payload IDs 🎀

The FEC Payload ID MUST be a 4 octet field, and it's laid out like this 💅

        0                   1                   2                   3
        0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1
       +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
       |     Source Block Number       |      Encoding Symbol ID       |
       +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+

Figure 1: FEC Payload ID format

Source Block Number (SBN), (16 bits): an integer identifier for the source block that the encoding symbols inside the packet belong to 💜

Encoding Symbol ID (ESI), (16 bits): an integer identifier for the encoding symbols inside the packet 🌸

How the Source Block Number and Encoding Symbol Identifier are interpreted is defined in Section 5 ✨

3.2. FEC Object Transmission Information (OTI) 🍓

3.2.1. Mandatory 👑

The value of the FEC Encoding ID MUST be 1 (one), as assigned by IANA. See Section 7 for that 🦋

3.2.2. Common 🧚

The Common FEC Object Transmission Information elements this FEC Scheme uses are:

Transfer Length (F)
Encoding Symbol Length (T)

The Transfer Length is a non-negative integer less than 2^^45. The Encoding Symbol Length is a non-negative integer less than 2^^16 💖

The encoded Common FEC Object Transmission Information 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                          |
      +                               +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
      |                               |           Reserved            |
      +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
      |    Encoding Symbol Length     |
      +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+

Figure 2: Encoded Common FEC OTI for Raptor FEC Scheme

NOTE 1: The limit of 2^^45 on the transfer length is a consequence of the limitation on the symbol size to 2^^16-1, the limitation on the number of symbols in a source block to 2^^13, and the limitation on the number of source blocks to 2^^16. However, the Transfer Length is encoded as a 48-bit field for simplicity 🙈

3.2.3. Scheme-Specific 🌷

The following parameters are carried in the Scheme-Specific FEC Object Transmission Information element for this FEC Scheme:

The number of source blocks (Z)
The number of sub-blocks (N)
A symbol alignment parameter (Al)

All of these parameters are non-negative integers. The encoded Scheme-Specific Object Transmission Information is a 4-octet field made of Z (2 octets), N (1 octet), and Al (1 octet), as shown in Figure 3 💅

        0                   1                   2                   3
        0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1
       +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
       |             Z                 |      N        |       Al      |
       +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+

Figure 3: Encoded Scheme-Specific FEC Object Transmission Information

The encoded FEC Object Transmission Information is a 14-octet field, made by putting together the encoded Common FEC Object Transmission Information and the encoded Scheme-Specific FEC Object Transmission Information 🎀

These three parameters define the source block partitioning, as described in Section 5.3.1.2 🦋

4. Procedures 💜

4.1. Content Delivery Protocol Requirements 🫶

This section describes the info exchange between the Raptor FEC Scheme and any Content Delivery Protocol (CDP) that uses the Raptor FEC Scheme for object delivery 🌈

For object delivery, the Raptor encoder and decoder need this info from the CDP:

The transfer length of the object, F, in bytes
A symbol alignment parameter, Al
The symbol size, T, in bytes, which MUST be a multiple of Al
The number of source blocks, Z
The number of sub-blocks in each source block, N

The Raptor encoder for object delivery also needs:

The object to be encoded, F bytes

For each packet to be sent, the Raptor encoder gives the CDP the following info 🧚

Source Block Number (SBN)
Encoding Symbol ID (ESI)
Encoding symbol(s)

The CDP MUST communicate this information to the receiver 🔥

4.2. Example Parameter Derivation Algorithm ✨

okay so this section is the recipe for picking the three transport parameters, T, Z, and N 💖 it's based on these inputs:

F the transfer length of the object, in bytes 📦
W a target on the sub-block size, in bytes 🎯
P the maximum packet payload size, in bytes, which is assumed to be a multiple of Al 🌸
Al the symbol alignment parameter, in bytes 📏
Kmax the maximum number of source symbols per source block.

Note: Section 5.1.2 defines Kmax to be 8192.

Kmin a minimum target on the number of symbols per source block 🍓
Gmax a maximum target number of symbols per packet 🧚

Once you have those, the transport parameters get calculated like this:

Let

   G = min{ceil(P*Kmin/F), P/Al, Gmax}

   T = floor(P/(Al*G))*Al

   Kt = ceil(F/T)

   Z = ceil(Kt/Kmax)

   N = min{ceil(ceil(Kt/Z)*T/W), T/Al}

Here's the vibe: G is the max number of symbols you can squeeze into a single packet 📦✨ and Kt is the total number of symbols needed to represent the source data of the object 🫶 The G and N values you get from this are lowbkey treated as lower bounds 👀 It may be advantageous to bump them up, for example, to the nearest power of two 💅

Also, the algorithm above does not guarantee that the symbol size, T, divides the maximum packet size, P, so it may not be possible to use packets of size exactly P 😭 BUT if you pick G so that it divides P/Al, then T will be a divisor of P and you can use packets of size P 🎀 iykyk

The algorithm above and that defined in Section 5.3.1.2 ensure that the sub-symbol sizes are a multiple of the symbol alignment parameter, Al. This is useful because the XOR operations used for encoding and decoding are generally performed several bytes at a time, for example, at least 4 bytes at a time on a 32-bit processor. Thus, the encoding and decoding can be performed faster if the sub-symbol sizes are a multiple of this number of bytes. 🔥

Recommended settings for the input parameters Al, Kmin, and Gmax are: Al = 4, Kmin = 1024, Gmax = 10 🌈

The parameter W 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 W depends on the implementation, but it is possible to implement decoding using working memory only slightly larger than W 💜

5. Raptor FEC Code Specification 🦋

5.1. Definitions, Symbols, and Abbreviations 🎀

5.1.1. Definitions

Okay bestie, here are the terms that apply for this specification 📖

Source block: a block of K source symbols that are considered together for Raptor encoding purposes.
Source symbol: the smallest unit of data used during the encoding process. All source symbols within a source block have the same size.
Encoding symbol: a symbol that is included in a data packet. The encoding symbols consist of the source symbols and the repair symbols. Repair symbols generated from a source block have the same size as the source symbols of that source block.
Systematic code: a code in which all the source symbols may be included as part of the encoding symbols sent for a source block.
Repair symbol: the encoding symbols sent for a source block that are not the source symbols. The repair symbols are generated based on the source symbols.
Intermediate symbols: symbols generated from the source symbols using an inverse encoding process . 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 included in data packets.
Symbol: a unit of data. The size, in bytes, of a symbol is known as the symbol size.
Encoding symbol group: a group of encoding symbols that are sent together, i.e., within the same packet whose relationship to the source symbols can be derived from a single Encoding Symbol ID.
Encoding Symbol ID: information that defines the relationship between the symbols of an encoding symbol group and the source symbols.
Encoding packet: data packets that contain encoding symbols
Sub-block: a source block is sometimes broken into sub-blocks, each of which is sufficiently small to be decoded in working memory. For a source block consisting of K source symbols, each sub-block consists of K sub-symbols, each symbol of the source block being composed of one sub-symbol from each sub-block.
Sub-symbol: part of a symbol. Each source symbol is composed of as many sub-symbols as there are sub-blocks in the source block.
Source packet: data packets that contain source symbols.
Repair packet: data packets that contain repair symbols.

5.1.2. Symbols

Now the symbol soup 🍲 (copied exactly so nothing gets lost in translation):

i, j, x, h, a, b, d, v, m  represent positive integers.

ceil(x)  denotes the smallest positive integer that is greater than
         or equal to x.

choose(i,j)  denotes the number of ways j objects can be chosen from
             among i objects without repetition.

floor(x)  denotes the largest positive integer that is less than or
          equal to x.

i % j  denotes i modulo j.

X ^ Y  denotes, for equal-length bit strings X and Y, the bitwise
       exclusive-or of X and Y.

Al   denotes a symbol alignment parameter.  Symbol and sub-symbol
     sizes are restricted to be multiples of Al.

A    denotes a matrix over GF(2).

Transpose[A]  denotes the transposed matrix of matrix A.

A^^-1  denotes the inverse matrix of matrix A.

K    denotes the number of symbols in a single source block.

Kmax denotes the maximum number of source symbols that can be in a
     single source block.  Set to 8192.

L    denotes the number of pre-coding symbols for a single source
     block.

S    denotes the number of LDPC symbols for a single source block.

H    denotes the number of Half symbols for a single source block.

C    denotes an array of intermediate symbols, C[0], C[1], C[2],...,
     C[L-1].

C'   denotes an array of source symbols, C'[0], C'[1], C'[2],...,
     C'[K-1].

X    a non-negative integer value

V0, V1  two arrays of 4-byte integers, V0[0], V0[1],..., V0[255] and
        V1[0], V1[1],..., V1[255]

Rand[X, i, m]  a pseudo-random number generator

Deg[v]  a degree generator

LTEnc[K, C ,(d, a, b)]  a LT encoding symbol generator

Trip[K, X]  a triple generator function

G    the number of symbols within an encoding symbol group

GF(n)  the Galois field with n elements.

N    the number of sub-blocks within a source block

T    the symbol size in bytes.  If the source block is partitioned
     into sub-blocks, then T = T'*N.

T'   the sub-symbol size, in bytes.  If the source block is not
     partitioned into sub-blocks, then T' is not relevant.

F    the transfer length of an object, in bytes

I    the sub-block size in bytes

P    for object delivery, the payload size of each packet, in bytes,
     that is used in the recommended derivation of the object
     delivery transport parameters.

Q    Q = 65521, i.e., Q is the largest prime smaller than 2^^16

Z    the number of source blocks, for object delivery

J(K) the systematic index associated with K
I_S  denotes the SxS identity matrix.

0_SxH  denotes the SxH zero matrix.

a ^^ b  a raised to the power b

5.1.3. Abbreviations

For the purposes of the present document, the following abbreviations apply, and yes they're short but they're doing a LOT 💅

ESI       Encoding Symbol ID

LDPC      Low Density Parity Check

LT        Luby Transform

SBN       Source Block Number

SBL       Source Block Length (in units of symbols)

5.2. Overview 🌸

Alright, the main event: the principal component of the systematic Raptor code is the basic encoder described in Section 5.4 💖 First, it's described how to derive values for a set of intermediate symbols from the original source symbols, so that knowing the intermediate symbols is enough to rebuild the source symbols ✨ Second, the encoder makes repair symbols, and each one is the exclusive OR of a number of the intermediate symbols 🫶 The encoding symbols are the combo of the source and repair symbols. The repair symbols are made in such a way that the intermediate symbols, and therefore also the source symbols, can be recovered from any sufficiently large set of encoding symbols. It's giving resilience 🛡️👑

This document specifies the systematic Raptor code encoder. A number of possible decoding algorithms are possible. An efficient decoding algorithm is provided in Section 5.5 🧚

The construction of the intermediate and repair symbols is based in part on a pseudo-random number generator described in Section 5.4.4.1. This generator is based on a fixed set of 512 random numbers that MUST be available to both sender and receiver. These are provided in Section 5.6 🎲 (the MUST is serious bestie, both sides need the same numbers 🙈)

Finally, the construction of the intermediate symbols from the source symbols is governed by a 'systematic index', values of which are provided in Section 5.7 for source block sizes from 4 source symbols to Kmax = 8192 source symbols 🍓

5.3. Object Delivery 📦

5.3.1. Source Block Construction

5.3.1.1. General

Okay so here's how the object gets chopped up 🔪 The object may be broken into Z >= 1 blocks, known as source blocks. The Raptor encoder is applied independently to each source block. Each source block is identified by a unique integer 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 bytes each. Each source symbol is identified by a unique integer Encoding Symbol Identifier (ESI), where the first source symbol of a source block has ESI zero, the second has ESI one, etc. 🫧

Each source block with K source symbols is divided into N >= 1 sub-blocks, which are small enough to be decoded in the working memory 🧠 Each sub-block is divided into K sub-symbols of size T'.

Note that the value of K is not necessarily the same for each source block of an object, and the value of T' may not necessarily be the same for each sub-block of a source block. However, the symbol size T is the same for all source blocks of an object, and the number of symbols, K, is the same for every sub-block of a source block. Exact partitioning of the object into source blocks and sub-blocks is described in Section 5.3.1.2 below. 💜✨

5.3.1.2. Source Block and Sub-Block Partitioning

Okay bestie, this is where we chop the object into bite-sized pieces 🍰🔪 Building source blocks and sub-blocks depends on five input parameters, F, Al, T, Z, and N, plus a helper function called Partition[]. Here's what each one means:

F the transfer length of the object, in bytes
Al a symbol alignment parameter, in bytes
T the symbol size, in bytes, which MUST be a multiple of Al
Z the number of source blocks
N the number of sub-blocks in each source block

These parameters MUST be set so that ceil(ceil(F/T)/Z) <= Kmax. ✨ Tips for picking them are in Section 4.2.

Partition[] takes a pair of integers (I, J) and spits out four integers (IL, IS, JL, JS). The output for 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. It's basically splitting a block of size I into J roughly equal pieces: JL blocks of length IL, and JS blocks of length IS. 🧁

The source object MUST be partitioned into source blocks and sub-blocks as follows:

Kt = ceil(F/T)

(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 have length KL*T bytes, and the remaining ZS source blocks each have KS*T bytes. 💅

If Kt*T > F, then for encoding purposes, the last symbol MUST be padded at the end with Kt*T - F zero bytes.

Next, each source block MUST be divided into N = NL + NS contiguous sub-blocks. The first NL sub-blocks each consist of K contiguous sub-symbols of size TL*Al, and the remaining NS sub-blocks each consist of K contiguous sub-symbols of size TS*Al. The symbol alignment parameter Al makes sure sub-symbols are always a multiple of Al bytes. 🎀

Finally, the m-th symbol of a source block is the concatenation of the m-th sub-symbol from each of the N sub-blocks.

Plot twist: when N > 1, a symbol is NOT a contiguous portion of the object. Wild, right? 🙈

5.3.2. Encoding Packet Construction

Every encoding packet carries three things: 📦

Source Block Number (SBN)
Encoding Symbol ID (ESI)
encoding symbol(s)

Each source block gets encoded independently of the others, and source blocks are numbered consecutively starting from zero. 🌸

Encoding Symbol ID values from 0 to K-1 identify the source symbols of a source block in sequential order, where K is the number of symbols in the source block. Encoding Symbol IDs from K onwards identify repair symbols. 🦋

Each encoding packet is either entirely source symbols (source packet) or entirely repair symbols (repair packet). A packet can hold any number of symbols from the same source block. If the last source symbol in a source packet has padding bytes added for FEC encoding purposes, those bytes don't need to be included in the packet. Otherwise, only whole symbols MUST be included. 💖

The Encoding Symbol ID, X, carried in each source packet is the ID of the first source symbol in that packet. The following source symbols in the packet have IDs X+1 to X+G-1, in sequential order, where G is the number of symbols in the packet. 🧚

It works the same way for repair packets. The Encoding Symbol ID, X, placed into a repair packet is the ID of the first repair symbol in that packet, and the subsequent repair symbols have IDs X+1 to X+G-1 in sequential order, where G is the number of symbols in the packet. ✨

Note that it is not necessary for the receiver to know the total number of repair packets. 👑

Each symbol is associated with a triple of integers, (d, a, b).

The G repair symbol triples (d[0], a[0], b[0]), ..., (d[G-1], a[G-1], b[G-1]) for the repair symbols placed into a repair packet with ESI X are computed using the Triple generator defined in Section 5.4.4.4, as follows:

For each i = 0, ..., G-1, (d[i], a[i], b[i]) = Trip[K,X+i]

The G repair symbols to be placed in the repair packet with ESI X are calculated from the repair symbol triples, as described in Section 5.4, using the intermediate symbols C and the LT encoder LTEnc[K, C, (d[i], a[i], b[i])]. 🔥

5.4. Systematic Raptor Encoder

5.4.1. Encoding Overview

The systematic Raptor encoder generates repair symbols from a source block made up of K source symbols. 🌈

Symbols are the fundamental data units of encoding and decoding. For each source block (sub-block), all symbols (sub-symbols) are the same size. The only atomic operation on symbols (sub-symbols), for both encoding and decoding, is exclusive-or. 💜

Let C'[0],..., C'[K-1] denote the K source symbols.

Let C[0],..., C[L-1] denote L intermediate symbols.

The first step of encoding is to generate a number, L > K, of intermediate symbols from the K source symbols. In this step, K source symbol triples (d[0], a[0], b[0]), ..., (d[K-1], a[K-1], b[K-1]) are generated using the Trip[] generator as described in Section 5.4.2.2. The K source symbol triples are associated with the K source symbols, and they're then used to work out the L intermediate symbols C[0],..., C[L-1] from the source symbols, using an inverse encoding process. This process can be realized by a Raptor decoding process. 🧠

Certain "pre-coding relationships" MUST hold within the L intermediate symbols. Section 5.4.2.3 describes these relationships and how the intermediate symbols are generated from the source symbols. 🫶

Once the intermediate symbols exist, repair symbols get produced, and one or more repair symbols are placed as a group into a single data packet. Each repair symbol group is associated with an Encoding Symbol ID (ESI) and a number, G, of repair symbols. The ESI is used to generate a triple of three integers, (d, a, b), for each repair symbol, again using the Trip[] generator as described in Section 5.4.4.4. Then each (d,a,b)-triple is used to generate the corresponding repair symbol from the intermediate symbols, using the LTEnc[K, C[0],..., C[L-1], (d,a,b)] generator described in Section 5.4.4.3. 🦄

5.4.2. First Encoding Step: Intermediate Symbol Generation

5.4.2.1. General

The first encoding step is a pre-coding step that generates the L intermediate symbols C[0], ..., C[L-1] from the source symbols C'[0], ..., C'[K-1]. 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 triples. The generation of the source symbol triples is defined in Section 5.4.2.2, using the Trip[] generator described in Section 5.4.4.4.
A set of pre-coding relationships holds within the intermediate symbols themselves. These are defined in Section 5.4.2.3.

The generation of the L intermediate symbols is then defined in Section 5.4.2.4. 🌷

5.4.2.2. Source Symbol Triples

Each of the K source symbols is associated with a triple (d[i], a[i], b[i]) for 0 <= i < K. The source symbol triples are determined using the Triple generator defined in Section 5.4.4.4, as:

For each i, 0 <= i < K

   (d[i], a[i], b[i]) = Trip[K, i]

5.4.2.3. Pre-Coding Relationships

The pre-coding relationships amongst the L intermediate symbols are defined by expressing the last L-K intermediate symbols in terms of the first K intermediate symbols. 🧩

The last L-K intermediate symbols C[K],...,C[L-1] consist of S LDPC symbols and H Half symbols. The values of S and H are determined from K as described below. Then L = K+S+H.

Let:

X be the smallest positive integer such that X*(X-1) >= 2*K.
S be the smallest prime integer such that S >= ceil(0.01*K) + X.
H be the smallest integer such that choose(H,ceil(H/2)) >= K + S.
H' = ceil(H/2)
L = K+S+H
C[0],...,C[K-1] denote the first K intermediate symbols.
C[K],...,C[K+S-1] denote the S LDPC symbols, initialised to zero.
C[K+S],...,C[L-1] denote the H Half symbols, initialised to zero.

The S LDPC symbols are defined to be the values of C[K],...,C[K+S-1] at the end of the following process:

For i = 0,...,K-1 do

   a = 1 + (floor(i/S) % (S-1))

   b = i % S

   C[K + b] = C[K + b] ^ C[i]

   b = (b + a) % S

   C[K + b] = C[K + b] ^ C[i]

   b = (b + a) % S

   C[K + b] = C[K + b] ^ C[i]

The H Half symbols are defined as follows. Let:

g[i] = i ^ (floor(i/2)) for all positive integers i

Note: g[i] is the Gray sequence, in which each element differs from the previous one in a single bit position. 🌙

m[k] denotes the subsequence of g[.] whose elements have exactly k non-zero bits in their binary representation.
m[j,k] denotes the jth element of the sequence m[k], where j=0, 1, 2, ...

Then the Half symbols are defined as the values of C[K+S],...,C[L-1] after the following process:

For h = 0,...,H-1 do

   For j = 0,...,K+S-1 do

      If bit h of m[j,H'] is equal to 1 then C[h+K+S] = C[h+K+S] ^ C[j].

5.4.2.4. Intermediate Symbols

5.4.2.4.1. Definition

Given the K source symbols C'[0], C'[1],..., C'[K-1], the L intermediate symbols C[0], C[1],..., C[L-1] are the uniquely defined symbol values that satisfy the following conditions: 🎯

The K source symbols C'[0], C'[1],..., C'[K-1] satisfy the K constraints

C'[i] = LTEnc[K, (C[0],..., C[L-1]), (d[i], a[i], b[i])], for all i, 0 <= i < K.

The L intermediate symbols C[0], C[1],..., C[L-1] satisfy the pre-coding relationships defined in Section 5.4.2.3.

5.4.2.4.2. Example Method for Calculation of Intermediate Symbols ✨

okay so this subsection gives a possible way to work out the L intermediate symbols C[0], C[1],..., C[L-1] that satisfy the constraints from Section 5.4.2.4.1 💅🧮

First, the generator matrix. For a code that makes N output symbols from K input symbols, it's an NxK matrix over GF(2). Each row is one output symbol and each column is one input symbol. The ith output symbol is the sum of the input symbols whose column has a non-zero entry in row i. It's giving linear algebra with a little bit of drama 🦋

Then the L intermediate symbols can be calculated like this:

Let

C be the column vector of the L intermediate symbols, C[0], C[1],..., C[L-1].
D be the column vector made of S+H zero symbols, followed by the K source symbols C'[0], C'[1], ..., C'[K-1].

A*C = D

The above constraints define an LxL matrix over GF(2), called A. So the whole thing is basically "A times C equals D" 🌸

Here's how you build A. Let:

G_LDPC be the S x K generator matrix of the LDPC symbols. So,

G_LDPC * Transpose[(C[0],...., C[K-1])] = Transpose[(C[K], ...,
C[K+S-1])]

G_Half be the H x (K+S) generator matrix of the Half symbols. So,

G_Half * Transpose[(C[0], ..., C[S+K-1])] = Transpose[(C[K+S],
..., C[K+S+H-1])]

I_S be the S x S identity matrix.
I_H be the H x H identity matrix.
0_SxH be the S x H zero matrix.
G_LT be the KxL generator matrix of the encoding symbols generated by the LT Encoder. So,

G_LT * Transpose[(C[0], ..., C[L-1])] =
Transpose[(C'[0],C'[1],...,C'[K-1])]

i.e., G_LT(i,j) = 1 if and only if C[j] is included in the symbols that are XORed to produce LTEnc[K, (C[0], ..., C[L-1]), (d[i], a[i], b[i])].

Then, the rows of A go like this 🎀

The first S rows of A are equal to G_LDPC | I_S | 0_SxH.
The next H rows of A are equal to G_Half | I_H.
The remaining K rows of A are equal to G_LT.

Here's the matrix A, depicted in Figure 4 below 🍓

              K               S       H
  +-----------------------+-------+-------+
  |                       |       |       |
S |        G_LDPC         |  I_S  | 0_SxH |
  |                       |       |       |
  +-----------------------+-------+-------+
  |                               |       |
H |        G_Half                 |  I_H  |
  |                               |       |
  +-------------------------------+-------+
  |                                       |
  |                                       |
K |                 G_LT                  |
  |                                       |
  |                                       |
  +---------------------------------------+

                       Figure 4: The matrix A

Once you have A, the intermediate symbols are just:

C = (A^^-1)*D

The source symbol triples are generated so that, for any K matrix, A has full rank and is therefore invertible. Iconic behaviour 👑 This calculation can be done by running a Raptor decoding process on the K source symbols C'[0], C'[1],..., C'[K-1] to produce the L intermediate symbols C[0], C[1],..., C[L-1].

To generate the intermediate symbols from the source symbols efficiently, it's recommended to use an efficient decoder implementation like the one described in Section 5.5. The source symbol triples are designed to make decoding with that algorithm efficient. Designed with intention, bestie 🫶

5.4.3. Second Encoding Step: LT Encoding 🦋

In the second encoding step, the repair symbol with ESI X is generated by applying the generator LTEnc[K, (C[0], C[1],...,C[L-1]), (d, a, b)], defined in Section 5.4.4.3, to the L intermediate symbols C[0], C[1],..., C[L-1]. The triple used is (d, a, b)=Trip[K,X], generated according to Section 5.3.2. Main character energy ✨

5.4.4. Generators 🌈

5.4.4.1. Random Generator

The random number generator Rand[X, i, m] is defined like this, where X is a non-negative integer, i is a non-negative integer, and m is a positive integer. The value it produces is an integer between 0 and m-1. Let V0 and V1 be arrays of 256 entries each, where each entry is a 4-byte unsigned integer. These arrays are provided in Section 5.6. 🧚

Then:

Rand[X, i, m] = (V0[(X + i) % 256] ^ V1[(floor(X/256)+ i) % 256])
% m

5.4.4.2. Degree Generator

The degree generator Deg[v] is defined as follows, where v is an integer that is at least 0 and less than 2^^20 = 1048576. 🍓

In Table 1, find the index j such that f[j-1] <= v < f[j]

Then, Deg[v] = d[j]

Here's Table 1, the degree distribution for encoding symbols:

+---------+---------+------+
| Index j | f[j]    | d[j] |
+---------+---------+------+
| 0       | 0       | --   |
| 1       | 10241   | 1    |
| 2       | 491582  | 2    |
| 3       | 712794  | 3    |
| 4       | 831695  | 4    |
| 5       | 948446  | 10   |
| 6       | 1032189 | 11   |
| 7       | 1048576 | 40   |
+---------+---------+------+

Table 1: Defines the degree distribution for encoding symbols

5.4.4.3. LT Encoding Symbol Generator

The encoding symbol generator LTEnc[K, (C[0], C[1],..., C[L-1]), (d, a, b)] takes these inputs 🎀

K is the number of source symbols (or sub-symbols) for the source block (sub-block). Let L be derived from K as described in Section 5.4.2.3, and let L' be the smallest prime integer greater than or equal to L.
(C[0], C[1],..., C[L-1]) is the array of L intermediate symbols (sub-symbols) generated as described in Section 5.4.2.4.
(d, a, b) is a source triple determined using the Triple generator defined in Section 5.4.4.4, whereby:
- d is an integer denoting an encoding symbol degree.
- a is an integer between 1 and L'-1 inclusive.
- b is an integer between 0 and L'-1 inclusive.

The encoding symbol generator produces a single encoding symbol as output, using this algorithm 💖

While (b >= L) do b = (b + a) % L'

Let result = C[b].

For j = 1,...,min(d-1,L-1) do

   b = (b + a) % L'

   While (b >= L) do b = (b + a) % L'

   result = result ^ C[b]

Return result

5.4.4.4. Triple Generator

The triple generator Trip[K,X] takes these inputs 🦋

K - The number of source symbols.
X - An encoding symbol ID.

Let:

L be determined from K as described in Section 5.4.2.3.
L' be the smallest prime that is greater than or equal to L.
Q = 65521, the largest prime smaller than 2^^16.
J(K) be the systematic index associated with K, as defined in Section 5.7.

The output of the triple generator is a triple, (d, a, b), determined as follows ✨

A = (53591 + J(K)*997) % Q

B = 10267*(J(K)+1) % Q

Y = (B + X*A) % Q

v = Rand[Y, 0, 2^^20]

d = Deg[v]

a = 1 + Rand[Y, 1, L'-1]

b = Rand[Y, 2, L']

5.5. Example FEC Decoder 🔥

5.5.1. General 💅

This section describes an efficient decoding algorithm for the Raptor codes described in this specification. Each received encoding symbol can be thought of as the value of an equation among the intermediate symbols. From these simultaneous equations, plus the known pre-coding relationships among the intermediate symbols, any algorithm for solving simultaneous equations can successfully decode the intermediate symbols, and therefore the source symbols. ngl the algorithm you pick has a big effect on how efficient the decoding is 🧠

5.5.2. Decoding a Source Block 🌸

5.5.2.1. General ✨

okay so the decoder already knows the structure of the source block it's decoding, including the symbol size T and the number K of symbols in the block 🧚

from the algorithms in Section 5.4, the Raptor decoder can work out the total L = K+S+H of pre-coding symbols and figure out how they were generated from the source block. in this description we assume the received encoding symbols for the block get passed to the decoder. ngl there's a catch 👀: as Section 5.3.2 explains, the last source symbol of a source packet may have padding bytes added for FEC encoding purposes. those padding bytes might not actually be in the packet that gets sent, so they have to be put back in at the receiver before the symbol goes to the decoder.

for each encoding symbol, we also assume the decoder is given the number and set of intermediate symbols whose exclusive-or equals that encoding symbol. for source symbols, the source symbol triples from Section 5.4.2.2 tell you the number and set of intermediate symbols that add up to each source symbol 💅

let N >= K be the number of received encoding symbols for a source block, and let M = S+H+N. from the info given to the decoder, we can build this M by L bit matrix A. let C be the column vector of the L intermediate symbols, and let D be the column vector of M symbols whose values the receiver already knows. the first S+H of those M symbols are zero-valued symbols standing in for the LDPC and Half symbols. they're check symbols for the LDPC and Half symbols, not the LDPC and Half symbols themselves. the remaining N of the M symbols are the received encoding symbols for the source block. then A is the bit matrix that satisfies A*C = D, where * means matrix multiplication over GF[2]. specifically, A[i,j] = 1 if the intermediate symbol for index j gets exclusive-ORed into the LDPC, Half, or encoding symbol for index i in the encoding, or if index i is an LDPC or Half symbol and index j is that same LDPC or Half symbol. for every other i and j, A[i,j] = 0 🦋

so decoding a source block is basically the same as decoding C from the known A and D. it's clear that C can be decoded if and only if the rank of A over GF[2] is L 🔥 once C is decoded, any missing source symbols can be found by using the source symbol triples to work out the number and set of intermediate symbols that MUST be exclusive-ORed to get each missing source symbol 🌸

the first step in decoding C is forming a decoding schedule. here A gets turned into the L by L identity matrix using Gaussian elimination (row operations plus row and column reorderings), and after throwing away M - L rows. the decoding schedule is the sequence of row operations and row and column reorderings during that elimination, and it only depends on A, not on D. decoding C from D can happen at the same time as forming the schedule, or it can happen afterwards using the schedule.

here's how the schedule and the decoding of C line up. start with c[0] = 0, c[1] = 1, ..., c[L-1] = L-1 and d[0] = 0, d[1] = 1, ..., d[M-1] = M-1:

every time row i of A is exclusive-ORed into row i' in the decoding schedule, the decoding process exclusive-ORs symbol D[d[i]] into symbol D[d[i']].
every time row i is swapped with row i' in the schedule, the decoding process swaps the values of d[i] and d[i'].
every time column j is swapped with column j' in the schedule, the decoding process swaps the values of c[j] and c[j'].

so the total number of symbol exclusive-ORs while decoding the source block is the number of row operations (not swaps) in the Gaussian elimination. since A is the L by L identity matrix after elimination and after discarding the last M - L rows, once decoding succeeds it's clear that the L symbols D[d[0]], D[d[1]], ..., D[d[L-1]] are the values of the L symbols C[c[0]], C[c[1]], ..., C[c[L-1]] ✨

the order you do Gaussian elimination in doesn't change whether decoding succeeds. but it hugely affects how fast decoding is 🏃‍♀️ (keeping a sparse representation of A is also crucial, though that isn't described here). the rest of this section walks through an order that's relatively efficient.

5.5.2.2. First Phase

the first phase of Gaussian elimination splits the matrix A into conceptual submatrices. their sizes are set by non-negative integers i and u, both starting at 0. the submatrices of A are:

the submatrix I, where the first i rows and first i columns intersect. this is the identity matrix at the end of each step in the phase.
the submatrix where the first i rows intersect all the columns except the first i and the last u columns. every entry here is zero.
the submatrix where the first i columns intersect all the rows except the first i. every entry here is zero.
the submatrix U, where all the rows intersect the last u columns.
the submatrix V, where all the columns except the first i and the last u columns intersect all the rows except the first i.

Figure 5 shows the submatrices of A. at the start of the first phase, V = A. in each step, a row of A gets chosen 🎀

+-----------+-----------------+---------+
|           |                 |         |
|     I     |    All Zeros    |         |
|           |                 |         |
+-----------+-----------------+    U    |
|           |                 |         |
|           |                 |         |
| All Zeros |       V         |         |
|           |                 |         |
|           |                 |         |
+-----------+-----------------+---------+

               Figure 5: Submatrices of A in the first phase

the graph defined by the structure of V decides which row of A gets chosen. the columns that intersect V are the nodes of the graph, and the rows with exactly 2 ones in V are the edges that connect the two columns (nodes) where those ones sit. a component is a maximal set of nodes (columns) and edges (rows) where there's a path between every pair of nodes and edges. the size of a component is the number of nodes (columns) in it 🧠

there are at most L steps in the first phase. the phase ends successfully when i + u = L, meaning V and the all-zeroes submatrix above V have disappeared, and A consists of I, the all-zeroes submatrix below I, and U. the phase ends unsuccessfully, which is a decoding failure, if at some step before V disappears there's no non-zero row in V to choose. whenever there are non-zero rows in V, the next step starts by choosing a row of A like this:

let r be the minimum integer such that at least one row of A has exactly r ones in V.
- if r != 2, choose a row with exactly r ones in V that has the minimum original degree among all such rows.
- if r = 2, choose any row with exactly 2 ones in V that is part of a maximum size component in the graph defined by V.

once the row is chosen, the first row of A that intersects V gets swapped with the chosen row, so the chosen row is now the first row that intersects V. the columns of A that intersect V get reordered so one of the r ones in the chosen row sits in the first column of V, and the remaining r-1 ones sit in the last columns of V. then the chosen row is exclusive-ORed into all the other rows below it that have a one in the first column of V. finally, i goes up by 1 and u goes up by r-1, and that completes the step 💖

5.5.2.3. Second Phase

the submatrix U gets split further into the first i rows, U_upper, and the remaining M - i rows, U_lower. Gaussian elimination runs on U_lower in the second phase, either to figure out that its rank is less than u (decoding failure) or to turn it into a matrix whose first u rows form the identity matrix (success for the second phase). call that u by u identity matrix I_u. the M - L rows of A that intersect U_lower - I_u get thrown away. after this phase, A has L rows and L columns 🌈

5.5.2.4. Third Phase

after the second phase, the only part of A that still needs zeroing out to turn it into the L by L identity matrix is U_upper. the number of rows i in the submatrix U_upper is usually way bigger than the number of columns u in it. to zero out U_upper efficiently, a precomputation matrix U' gets built from I_u in the third phase, and then U' gets used in the fourth phase to zero out U_upper 🧚

the u rows of I_u get split into ceil(u/8) groups of 8 rows each. then, for each group of 8 rows, every non-zero combination of those 8 rows is computed, giving 2^^8 - 1 = 255 rows. this can be done with 2^^8-8-1 = 247 exclusive-ORs of rows per group, since the combinations with Hamming weight one that show up in I_u don't need to be recomputed. so the precomputation matrix U' ends up with ceil(u/8)*255 rows and u columns. note that U' isn't formally part of matrix A, but it'll be used in the fourth phase to zero out U_upper 🍓

5.5.2.5. Fourth Phase

for each of the first i rows of A, and for each group of 8 columns in that row's U_upper submatrix, check whether the set of 8 column entries in U_upper are all zero. if they aren't all zero, the row of the precomputation matrix U' that matches that 8-column pattern gets exclusive-ORed into the row. that zeroes out those 8 columns in the row, at the cost of exclusive-ORing one row of U' into it 💅

after this phase, A is the L by L identity matrix and a complete decoding schedule has been formed successfully. then, as explained in Section 5.5.2.1, the matching decoding (exclusive-ORing known encoding symbols) can be run to recover the intermediate symbols based on the schedule. the triples for all source symbols get computed according to Section 5.4.2.2. the triples for received source symbols are used in the decoding. the triples for missing source symbols are used to figure out which intermediate symbols need to be exclusive-ORed to recover the missing source symbols 🫶

5.6. Random Numbers

the two tables V0 and V1 described in Section 5.4.4.1 are shown below. each entry is a 32-bit integer in decimal representation 🔢

5.6.1. The Table V0 🎀

Okay bestie, this is where the lookup tables start. Here's 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

5.6.2. The Table V1 🦋

Next up is Table V1, and it's just as iconic ✨

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

5.7. Systematic Indices J(K) 👑

Okay so here's the vibe: for every value of K, the systematic index J(K) is designed so that the source symbol triples (d[0], a[0], b[0]), ..., (d[L-1], a[L-1], b[L-1]) lock in the L intermediate symbols, meaning they're uniquely defined. In other words, the matrix A in Section 5.4.2.4.2 has full rank and is therefore invertible. That's the whole goal, and it's so satisfying 🥹💅

Here's the list of systematic indices for values of K between 4 and 8192 inclusive 🌈

okay so this part is just the big random number table continuing, and it is giving vibes 🧚✨

   18, 14, 61, 46, 14, 22, 20, 40, 48, 1, 29, 40, 43, 46, 18, 8, 20, 2,
   61, 26, 13, 29, 36, 19, 58, 5, 58, 0, 54, 56, 24, 14, 5, 67, 39, 31,
   25, 29, 24, 19, 14, 56, 49, 49, 63, 30, 4, 39, 2, 1, 20, 19, 61, 4,
   54, 70, 25, 52, 9, 26, 55, 69, 27, 68, 75, 19, 64, 57, 45, 3, 37, 31,
   100, 41, 25, 41, 53, 23, 9, 31, 26, 30, 30, 46, 90, 50, 13, 90, 77,
   61, 31, 54, 54, 3, 21, 66, 21, 11, 23, 11, 29, 21, 7, 1, 27, 4, 34,
   17, 85, 69, 17, 75, 93, 57, 0, 53, 71, 88, 119, 88, 90, 22, 0, 58,
   41, 22, 96, 26, 79, 118, 19, 3, 81, 72, 50, 0, 32, 79, 28, 25, 12,
   25, 29, 3, 37, 30, 30, 41, 84, 32, 31, 61, 32, 61, 7, 56, 54, 39, 33,
   66, 29, 3, 14, 75, 75, 78, 84, 75, 84, 25, 54, 25, 25, 107, 78, 27,
   73, 0, 49, 96, 53, 50, 21, 10, 73, 58, 65, 27, 3, 27, 18, 54, 45, 69,
   29, 3, 65, 31, 71, 76, 56, 54, 76, 54, 13, 5, 18, 142, 17, 3, 37,
   114, 41, 25, 56, 0, 23, 3, 41, 22, 22, 31, 18, 48, 31, 58, 37, 75,
   88, 3, 56, 1, 95, 19, 73, 52, 52, 4, 75, 26, 1, 25, 10, 1, 70, 31,
   31, 12, 10, 54, 46, 11, 74, 84, 74, 8, 58, 23, 74, 8, 36, 11, 16, 94,
   76, 14, 57, 65, 8, 22, 10, 36, 36, 96, 62, 103, 6, 75, 103, 58, 10,
   15, 41, 75, 125, 58, 15, 10, 34, 29, 34, 4, 16, 29, 18, 18, 28, 71,
   28, 43, 77, 18, 41, 41, 41, 62, 29, 96, 15, 106, 43, 15, 3, 43, 61,
   3, 18, 103, 77, 29, 103, 19, 58, 84, 58, 1, 146, 32, 3, 70, 52, 54,
   29, 70, 69, 124, 62, 1, 26, 38, 26, 3, 16, 26, 5, 51, 120, 41, 16, 1,
   43, 34, 34, 29, 37, 56, 29, 96, 86, 54, 25, 84, 50, 34, 34, 93, 84,
   96, 29, 29, 50, 50, 6, 1, 105, 78, 15, 37, 19, 50, 71, 36, 6, 54, 8,
   28, 54, 75, 75, 16, 75, 131, 5, 25, 16, 69, 17, 69, 6, 96, 53, 96,
   41, 119, 6, 6, 88, 50, 88, 52, 37, 0, 124, 73, 73, 7, 14, 36, 69, 79,
   6, 114, 40, 79, 17, 77, 24, 44, 37, 69, 27, 37, 29, 33, 37, 50, 31,
   69, 29, 101, 7, 61, 45, 17, 73, 37, 34, 18, 94, 22, 22, 63, 3, 25,
   25, 17, 3, 90, 34, 34, 41, 34, 41, 54, 41, 54, 41, 41, 41, 163, 143,
   96, 18, 32, 39, 86, 104, 11, 17, 17, 11, 86, 104, 78, 70, 52, 78, 17,
   73, 91, 62, 7, 128, 50, 124, 18, 101, 46, 10, 75, 104, 73, 58, 132,
   34, 13, 4, 95, 88, 33, 76, 74, 54, 62, 113, 114, 103, 32, 103, 69,
   54, 53, 3, 11, 72, 31, 53, 102, 37, 53, 11, 81, 41, 10, 164, 10, 41,
   31, 36, 113, 82, 3, 125, 62, 16, 4, 41, 41, 4, 128, 49, 138, 128, 74,
   103, 0, 6, 101, 41, 142, 171, 39, 105, 121, 81, 62, 41, 81, 37, 3,
   81, 69, 62, 3, 69, 70, 21, 29, 4, 91, 87, 37, 79, 36, 21, 71, 37, 41,
   75, 128, 128, 15, 25, 3, 108, 73, 91, 62, 114, 62, 62, 36, 36, 15,
   58, 114, 61, 114, 58, 105, 114, 41, 61, 176, 145, 46, 37, 30, 220,
   77, 138, 15, 1, 128, 53, 50, 50, 58, 8, 91, 114, 105, 63, 91, 37, 37,
   13, 169, 51, 102, 6, 102, 23, 105, 23, 58, 6, 29, 29, 19, 82, 29, 13,
   36, 27, 29, 61, 12, 18, 127, 127, 12, 44, 102, 18, 4, 15, 206, 53,
   127, 53, 17, 69, 69, 69, 29, 29, 109, 25, 102, 25, 53, 62, 99, 62,
   62, 29, 62, 62, 45, 91, 125, 29, 29, 29, 4, 117, 72, 4, 30, 71, 71,
   95, 79, 179, 71, 30, 53, 32, 32, 49, 25, 91, 25, 26, 26, 103, 123,
   26, 41, 162, 78, 52, 103, 25, 6, 142, 94, 45, 45, 94, 127, 94, 94,
   94, 47, 209, 138, 39, 39, 19, 154, 73, 67, 91, 27, 91, 84, 4, 84, 91,
   12, 14, 165, 142, 54, 69, 192, 157, 185, 8, 95, 25, 62, 103, 103, 95,
   71, 97, 62, 128, 0, 29, 51, 16, 94, 16, 16, 51, 0, 29, 85, 10, 105,
   16, 29, 29, 13, 29, 4, 4, 132, 23, 95, 25, 54, 41, 29, 50, 70, 58,
   142, 72, 70, 15, 72, 54, 29, 22, 145, 29, 127, 29, 85, 58, 101, 34,
   165, 91, 46, 46, 25, 185, 25, 77, 128, 46, 128, 46, 188, 114, 46, 25,
   45, 45, 114, 145, 114, 15, 102, 142, 8, 73, 31, 139, 157, 13, 79, 13,
   114, 150, 8, 90, 91, 123, 69, 82, 132, 8, 18, 10, 102, 103, 114, 103,
   8, 103, 13, 115, 55, 62, 3, 8, 154, 114, 99, 19, 8, 31, 73, 19, 99,
   10, 6, 121, 32, 13, 32, 119, 32, 29, 145, 30, 13, 13, 114, 145, 32,
   1, 123, 39, 29, 31, 69, 31, 140, 72, 72, 25, 25, 123, 25, 123, 8, 4,
   85, 8, 25, 39, 25, 39, 85, 138, 25, 138, 25, 33, 102, 70, 25, 25, 31,
   25, 25, 192, 69, 69, 114, 145, 120, 120, 8, 33, 98, 15, 212, 155, 8,
   101, 8, 8, 98, 68, 155, 102, 132, 120, 30, 25, 123, 123, 101, 25,
   123, 32, 24, 94, 145, 32, 24, 94, 118, 145, 101, 53, 53, 25, 128,
   173, 142, 81, 81, 69, 33, 33, 125, 4, 1, 17, 27, 4, 17, 102, 27, 13,
   25, 128, 71, 13, 39, 53, 13, 53, 47, 39, 23, 128, 53, 39, 47, 39,
   135, 158, 136, 36, 36, 27, 157, 47, 76, 213, 47, 156, 25, 25, 53, 25,
   53, 25, 86, 27, 159, 25, 62, 79, 39, 79, 25, 145, 49, 25, 143, 13,
   114, 150, 130, 94, 102, 39, 4, 39, 61, 77, 228, 22, 25, 47, 119, 205,
   122, 119, 205, 119, 22, 119, 258, 143, 22, 81, 179, 22, 22, 143, 25,
   65, 53, 168, 36, 79, 175, 37, 79, 70, 79, 103, 70, 25, 175, 4, 96,
   96, 49, 128, 138, 96, 22, 62, 47, 95, 105, 95, 62, 95, 62, 142, 103,
   69, 103, 30, 103, 34, 173, 127, 70, 127, 132, 18, 85, 22, 71, 18,
   206, 206, 18, 128, 145, 70, 193, 188, 8, 125, 114, 70, 128, 114, 145,
   102, 25, 12, 108, 102, 94, 10, 102, 1, 102, 124, 22, 22, 118, 132,
   22, 116, 75, 41, 63, 41, 189, 208, 55, 85, 69, 8, 71, 53, 71, 69,
   102, 165, 41, 99, 69, 33, 33, 29, 156, 102, 13, 251, 102, 25, 13,
   109, 102, 164, 102, 164, 102, 25, 29, 228, 29, 259, 179, 222, 95, 94,
   30, 30, 30, 142, 55, 142, 72, 55, 102, 128, 17, 69, 164, 165, 3, 164,
   36, 165, 27, 27, 45, 21, 21, 237, 113, 83, 231, 106, 13, 154, 13,
   154, 128, 154, 148, 258, 25, 154, 128, 3, 27, 10, 145, 145, 21, 146,
   25, 1, 185, 121, 0, 1, 95, 55, 95, 95, 30, 0, 27, 95, 0, 95, 8, 222,
   27, 121, 30, 95, 121, 0, 98, 94, 131, 55, 95, 95, 30, 98, 30, 0, 91,
   145, 66, 179, 66, 58, 175, 29, 0, 31, 173, 146, 160, 39, 53, 28, 123,
   199, 123, 175, 146, 156, 54, 54, 149, 25, 70, 178, 128, 25, 70, 70,
   94, 224, 54, 4, 54, 54, 25, 228, 160, 206, 165, 143, 206, 108, 220,
   234, 160, 13, 169, 103, 103, 103, 91, 213, 222, 91, 103, 91, 103, 31,
   30, 123, 13, 62, 103, 50, 106, 42, 13, 145, 114, 220, 65, 8, 8, 175,
   11, 104, 94, 118, 132, 27, 118, 193, 27, 128, 127, 127, 183, 33, 30,
   29, 103, 128, 61, 234, 165, 41, 29, 193, 33, 207, 41, 165, 165, 55,
   81, 157, 157, 8, 81, 11, 27, 8, 8, 98, 96, 142, 145, 41, 179, 112,
   62, 180, 206, 206, 165, 39, 241, 45, 151, 26, 197, 102, 192, 125,
   128, 67, 128, 69, 128, 197, 33, 125, 102, 13, 103, 25, 30, 12, 30,
   12, 30, 25, 77, 12, 25, 180, 27, 10, 69, 235, 228, 343, 118, 69, 41,
   8, 69, 175, 25, 69, 25, 125, 41, 25, 41, 8, 155, 146, 155, 146, 155,
   206, 168, 128, 157, 27, 273, 211, 211, 168, 11, 173, 154, 77, 173,
   77, 102, 102, 102, 8, 85, 95, 102, 157, 28, 122, 234, 122, 157, 235,
   222, 241, 10, 91, 179, 25, 13, 25, 41, 25, 206, 41, 6, 41, 158, 206,
   206, 33, 296, 296, 33, 228, 69, 8, 114, 148, 33, 29, 66, 27, 27, 30,
   233, 54, 173, 108, 106, 108, 108, 53, 103, 33, 33, 33, 176, 27, 27,
   205, 164, 105, 237, 41, 27, 72, 165, 29, 29, 259, 132, 132, 132, 364,
   71, 71, 27, 94, 160, 127, 51, 234, 55, 27, 95, 94, 165, 55, 55, 41,
   0, 41, 128, 4, 123, 173, 6, 164, 157, 121, 121, 154, 86, 164, 164,
   25, 93, 164, 25, 164, 210, 284, 62, 93, 30, 25, 25, 30, 30, 260, 130,
   25, 125, 57, 53, 166, 166, 166, 185, 166, 158, 94, 113, 215, 159, 62,
   99, 21, 172, 99, 184, 62, 259, 4, 21, 21, 77, 62, 173, 41, 146, 6,
   41, 128, 121, 41, 11, 121, 103, 159, 164, 175, 206, 91, 103, 164, 72,
   25, 129, 72, 206, 129, 33, 103, 102, 102, 29, 13, 11, 251, 234, 135,
   31, 8, 123, 65, 91, 121, 129, 65, 243, 10, 91, 8, 65, 70, 228, 220,
   243, 91, 10, 10, 30, 178, 91, 178, 33, 21, 25, 235, 165, 11, 161,
   158, 27, 27, 30, 128, 75, 36, 30, 36, 36, 173, 25, 33, 178, 112, 162,
   112, 112, 112, 162, 33, 33, 178, 123, 123, 39, 106, 91, 106, 106,
   158, 106, 106, 284, 39, 230, 21, 228, 11, 21, 228, 159, 241, 62, 10,
   62, 10, 68, 234, 39, 39, 138, 62, 22, 27, 183, 22, 215, 10, 175, 175,
   353, 228, 42, 193, 175, 175, 27, 98, 27, 193, 150, 27, 173, 17, 233,
   233, 25, 102, 123, 152, 242, 108, 4, 94, 176, 13, 41, 219, 17, 151,
   22, 103, 103, 53, 128, 233, 284, 25, 265, 128, 39, 39, 138, 42, 39,
   21, 86, 95, 127, 29, 91, 46, 103, 103, 215, 25, 123, 123, 230, 25,
   193, 180, 30, 60, 30, 242, 136, 180, 193, 30, 206, 180, 60, 165, 206,
   193, 165, 123, 164, 103, 68, 25, 70, 91, 25, 82, 53, 82, 186, 53, 82,
   53, 25, 30, 282, 91, 13, 234, 160, 160, 126, 149, 36, 36, 160, 149,
   178, 160, 39, 294, 149, 149, 160, 39, 95, 221, 186, 106, 178, 316,
   267, 53, 53, 164, 159, 164, 165, 94, 228, 53, 52, 178, 183, 53, 294,
   128, 55, 140, 294, 25, 95, 366, 15, 304, 13, 183, 77, 230, 6, 136,
   235, 121, 311, 273, 36, 158, 235, 230, 98, 201, 165, 165, 165, 91,
   175, 248, 39, 185, 128, 39, 39, 128, 313, 91, 36, 219, 130, 25, 130,
   234, 234, 130, 234, 121, 205, 304, 94, 77, 64, 259, 60, 60, 60, 77,
   242, 60, 145, 95, 270, 18, 91, 199, 159, 91, 235, 58, 249, 26, 123,
   114, 29, 15, 191, 15, 30, 55, 55, 347, 4, 29, 15, 4, 341, 93, 7, 30,
   23, 7, 121, 266, 178, 261, 70, 169, 25, 25, 158, 169, 25, 169, 270,
   270, 13, 128, 327, 103, 55, 128, 103, 136, 159, 103, 327, 41, 32,
   111, 111, 114, 173, 215, 173, 25, 173, 180, 114, 173, 173, 98, 93,
   25, 160, 157, 159, 160, 159, 159, 160, 320, 35, 193, 221, 33, 36,
   136, 248, 91, 215, 125, 215, 156, 68, 125, 125, 1, 287, 123, 94, 30,
   184, 13, 30, 94, 123, 206, 12, 206, 289, 128, 122, 184, 128, 289,
   178, 29, 26, 206, 178, 65, 206, 128, 192, 102, 197, 36, 94, 94, 155,
   10, 36, 121, 280, 121, 368, 192, 121, 121, 179, 121, 36, 54, 192,
   121, 192, 197, 118, 123, 224, 118, 10, 192, 10, 91, 269, 91, 49, 206,
   184, 185, 62, 8, 49, 289, 30, 5, 55, 30, 42, 39, 220, 298, 42, 347,
   42, 234, 42, 70, 42, 55, 321, 129, 172, 173, 172, 13, 98, 129, 325,
   235, 284, 362, 129, 233, 345, 175, 261, 175, 60, 261, 58, 289, 99,
   99, 99, 206, 99, 36, 175, 29, 25, 432, 125, 264, 168, 173, 69, 158,
   273, 179, 164, 69, 158, 69, 8, 95, 192, 30, 164, 101, 44, 53, 273,
   335, 273, 53, 45, 128, 45, 234, 123, 105, 103, 103, 224, 36, 90, 211,
   282, 264, 91, 228, 91, 166, 264, 228, 398, 50, 101, 91, 264, 73, 36,
   25, 73, 50, 50, 242, 36, 36, 58, 165, 204, 353, 165, 125, 320, 128,
   298, 298, 180, 128, 60, 102, 30, 30, 53, 179, 234, 325, 234, 175, 21,
   250, 215, 103, 21, 21, 250, 91, 211, 91, 313, 301, 323, 215, 228,
   160, 29, 29, 81, 53, 180, 146, 248, 66, 159, 39, 98, 323, 98, 36, 95,
   218, 234, 39, 82, 82, 230, 62, 13, 62, 230, 13, 30, 98, 0, 8, 98, 8,
   98, 91, 267, 121, 197, 30, 78, 27, 78, 102, 27, 298, 160, 103, 264,
   264, 264, 175, 17, 273, 273, 165, 31, 160, 17, 99, 17, 99, 234, 31,
   17, 99, 36, 26, 128, 29, 214, 353, 264, 102, 36, 102, 264, 264, 273,
   273, 4, 16, 138, 138, 264, 128, 313, 25, 420, 60, 10, 280, 264, 60,
   60, 103, 178, 125, 178, 29, 327, 29, 36, 30, 36, 4, 52, 183, 183,
   173, 52, 31, 173, 31, 158, 31, 158, 31, 9, 31, 31, 353, 31, 353, 173,
   415, 9, 17, 222, 31, 103, 31, 165, 27, 31, 31, 165, 27, 27, 206, 31,
   31, 4, 4, 30, 4, 4, 264, 185, 159, 310, 273, 310, 173, 40, 4, 173, 4,
   173, 4, 250, 250, 62, 188, 119, 250, 233, 62, 121, 105, 105, 54, 103,
   111, 291, 236, 236, 103, 297, 36, 26, 316, 69, 183, 158, 206, 129,
   160, 129, 184, 55, 179, 279, 11, 179, 347, 160, 184, 129, 179, 351,
   179, 353, 179, 129, 129, 351, 11, 111, 93, 93, 235, 103, 173, 53, 93,
   50, 111, 86, 123, 94, 36, 183, 60, 55, 55, 178, 219, 253, 321, 178,
   235, 235, 183, 183, 204, 321, 219, 160, 193, 335, 121, 70, 69, 295,
   159, 297, 231, 121, 231, 136, 353, 136, 121, 279, 215, 366, 215, 353,
   159, 353, 353, 103, 31, 31, 298, 298, 30, 30, 165, 273, 25, 219, 35,
   165, 259, 54, 36, 54, 54, 165, 71, 250, 327, 13, 289, 165, 196, 165,
   165, 94, 233, 165, 94, 60, 165, 96, 220, 166, 271, 158, 397, 122, 53,
   53, 137, 280, 272, 62, 30, 30, 30, 105, 102, 67, 140, 8, 67, 21, 270,
   298, 69, 173, 298, 91, 179, 327, 86, 179, 88, 179, 179, 55, 123, 220,
   233, 94, 94, 175, 13, 53, 13, 154, 191, 74, 83, 83, 325, 207, 83, 74,
   83, 325, 74, 316, 388, 55, 55, 364, 55, 183, 434, 273, 273, 273, 164,
   213, 11, 213, 327, 321, 21, 352, 185, 103, 13, 13, 55, 30, 323, 123,
   178, 435, 178, 30, 175, 175, 30, 481, 527, 175, 125, 232, 306, 232,

okay bestie, this is the rest of the V1 table, still just pure numbers, so no cap it's the one part you can't paraphrase 🎀 ✨

   206, 306, 364, 206, 270, 206, 232, 10, 30, 130, 160, 130, 347, 240,
   30, 136, 130, 347, 136, 279, 298, 206, 30, 103, 273, 241, 70, 206,
   306, 434, 206, 94, 94, 156, 161, 321, 321, 64, 161, 13, 183, 183, 83,
   161, 13, 169, 13, 159, 36, 173, 159, 36, 36, 230, 235, 235, 159, 159,
   335, 312, 42, 342, 264, 39, 39, 39, 34, 298, 36, 36, 252, 164, 29,
   493, 29, 387, 387, 435, 493, 132, 273, 105, 132, 74, 73, 206, 234,
   273, 206, 95, 15, 280, 280, 280, 280, 397, 273, 273, 242, 397, 280,
   397, 397, 397, 273, 397, 280, 230, 137, 353, 67, 81, 137, 137, 353,
   259, 312, 114, 164, 164, 25, 77, 21, 77, 165, 30, 30, 231, 234, 121,
   234, 312, 121, 364, 136, 123, 123, 136, 123, 136, 150, 264, 285, 30,
   166, 93, 30, 39, 224, 136, 39, 355, 355, 397, 67, 67, 25, 67, 25,
   298, 11, 67, 264, 374, 99, 150, 321, 67, 70, 67, 295, 150, 29, 321,
   150, 70, 29, 142, 355, 311, 173, 13, 253, 103, 114, 114, 70, 192, 22,
   128, 128, 183, 184, 70, 77, 215, 102, 292, 30, 123, 279, 292, 142,
   33, 215, 102, 468, 123, 468, 473, 30, 292, 215, 30, 213, 443, 473,
   215, 234, 279, 279, 279, 279, 265, 443, 206, 66, 313, 34, 30, 206,
   30, 51, 15, 206, 41, 434, 41, 398, 67, 30, 301, 67, 36, 3, 285, 437,
   136, 136, 22, 136, 145, 365, 323, 323, 145, 136, 22, 453, 99, 323,
   353, 9, 258, 323, 231, 128, 231, 382, 150, 420, 39, 94, 29, 29, 353,
   22, 22, 347, 353, 39, 29, 22, 183, 8, 284, 355, 388, 284, 60, 64, 99,
   60, 64, 150, 95, 150, 364, 150, 95, 150, 6, 236, 383, 544, 81, 206,
   388, 206, 58, 159, 99, 231, 228, 363, 363, 121, 99, 121, 121, 99,
   422, 544, 273, 173, 121, 427, 102, 121, 235, 284, 179, 25, 197, 25,
   179, 511, 70, 368, 70, 25, 388, 123, 368, 159, 213, 410, 159, 236,
   127, 159, 21, 373, 184, 424, 327, 250, 176, 176, 175, 284, 316, 176,
   284, 327, 111, 250, 284, 175, 175, 264, 111, 176, 219, 111, 427, 427,
   176, 284, 427, 353, 428, 55, 184, 493, 158, 136, 99, 287, 264, 334,
   264, 213, 213, 292, 481, 93, 264, 292, 295, 295, 6, 367, 279, 173,
   308, 285, 158, 308, 335, 299, 137, 137, 572, 41, 137, 137, 41, 94,
   335, 220, 36, 224, 420, 36, 265, 265, 91, 91, 71, 123, 264, 91, 91,
   123, 107, 30, 22, 292, 35, 241, 356, 298, 14, 298, 441, 35, 121, 71,
   63, 130, 63, 488, 363, 71, 63, 307, 194, 71, 71, 220, 121, 125, 71,
   220, 71, 71, 71, 71, 235, 265, 353, 128, 155, 128, 420, 400, 130,
   173, 183, 183, 184, 130, 173, 183, 13, 183, 130, 130, 183, 183, 353,
   353, 183, 242, 183, 183, 306, 324, 324, 321, 306, 321, 6, 6, 128,
   306, 242, 242, 306, 183, 183, 6, 183, 321, 486, 183, 164, 30, 78,
   138, 158, 138, 34, 206, 362, 55, 70, 67, 21, 375, 136, 298, 81, 298,
   298, 298, 230, 121, 30, 230, 311, 240, 311, 311, 158, 204, 136, 136,
   184, 136, 264, 311, 311, 312, 312, 72, 311, 175, 264, 91, 175, 264,
   121, 461, 312, 312, 238, 475, 350, 512, 350, 312, 313, 350, 312, 366,
   294, 30, 253, 253, 253, 388, 158, 388, 22, 388, 22, 388, 103, 321,
   321, 253, 7, 437, 103, 114, 242, 114, 114, 242, 114, 114, 242, 242,
   242, 306, 242, 114, 7, 353, 335, 27, 241, 299, 312, 364, 506, 409,
   94, 462, 230, 462, 243, 230, 175, 175, 462, 461, 230, 428, 426, 175,
   175, 165, 175, 175, 372, 183, 572, 102, 85, 102, 538, 206, 376, 85,
   85, 284, 85, 85, 284, 398, 83, 160, 265, 308, 398, 310, 583, 289,
   279, 273, 285, 490, 490, 211, 292, 292, 158, 398, 30, 220, 169, 368,
   368, 368, 169, 159, 368, 93, 368, 368, 93, 169, 368, 368, 443, 368,
   298, 443, 368, 298, 538, 345, 345, 311, 178, 54, 311, 215, 178, 175,
   222, 264, 475, 264, 264, 475, 478, 289, 63, 236, 63, 299, 231, 296,
   397, 299, 158, 36, 164, 164, 21, 492, 21, 164, 21, 164, 403, 26, 26,
   588, 179, 234, 169, 465, 295, 67, 41, 353, 295, 538, 161, 185, 306,
   323, 68, 420, 323, 82, 241, 241, 36, 53, 493, 301, 292, 241, 250, 63,
   63, 103, 442, 353, 185, 353, 321, 353, 185, 353, 353, 185, 409, 353,
   589, 34, 271, 271, 34, 86, 34, 34, 353, 353, 39, 414, 4, 95, 95, 4,
   225, 95, 4, 121, 30, 552, 136, 159, 159, 514, 159, 159, 54, 514, 206,
   136, 206, 159, 74, 235, 235, 312, 54, 312, 42, 156, 422, 629, 54,
   465, 265, 165, 250, 35, 165, 175, 659, 175, 175, 8, 8, 8, 8, 206,
   206, 206, 50, 435, 206, 432, 230, 230, 234, 230, 94, 299, 299, 285,
   184, 41, 93, 299, 299, 285, 41, 285, 158, 285, 206, 299, 41, 36, 396,
   364, 364, 120, 396, 514, 91, 382, 538, 807, 717, 22, 93, 412, 54,
   215, 54, 298, 308, 148, 298, 148, 298, 308, 102, 656, 6, 148, 745,
   128, 298, 64, 407, 273, 41, 172, 64, 234, 250, 398, 181, 445, 95,
   236, 441, 477, 504, 102, 196, 137, 364, 60, 453, 137, 364, 367, 334,
   364, 299, 196, 397, 630, 589, 589, 196, 646, 337, 235, 128, 128, 343,
   289, 235, 324, 427, 324, 58, 215, 215, 461, 425, 461, 387, 440, 285,
   440, 440, 285, 387, 632, 325, 325, 440, 461, 425, 425, 387, 627, 191,
   285, 440, 308, 55, 219, 280, 308, 265, 538, 183, 121, 30, 236, 206,
   30, 455, 236, 30, 30, 705, 83, 228, 280, 468, 132, 8, 132, 132, 128,
   409, 173, 353, 132, 409, 35, 128, 450, 137, 398, 67, 432, 423, 235,
   235, 388, 306, 93, 93, 452, 300, 190, 13, 452, 388, 30, 452, 13, 30,
   13, 30, 306, 362, 234, 721, 635, 809, 784, 67, 498, 498, 67, 353,
   635, 67, 183, 159, 445, 285, 183, 53, 183, 445, 265, 432, 57, 420,
   432, 420, 477, 327, 55, 60, 105, 183, 218, 104, 104, 475, 239, 582,
   151, 239, 104, 732, 41, 26, 784, 86, 300, 215, 36, 64, 86, 86, 675,
   294, 64, 86, 528, 550, 493, 565, 298, 230, 312, 295, 538, 298, 295,
   230, 54, 374, 516, 441, 54, 54, 323, 401, 401, 382, 159, 837, 159,
   54, 401, 592, 159, 401, 417, 610, 264, 150, 323, 452, 185, 323, 323,
   185, 403, 185, 423, 165, 425, 219, 407, 270, 231, 99, 93, 231, 631,
   756, 71, 364, 434, 213, 86, 102, 434, 102, 86, 23, 71, 335, 164, 323,
   409, 381, 4, 124, 41, 424, 206, 41, 124, 41, 41, 703, 635, 124, 493,
   41, 41, 487, 492, 124, 175, 124, 261, 600, 488, 261, 488, 261, 206,
   677, 261, 308, 723, 908, 704, 691, 723, 488, 488, 441, 136, 476, 312,
   136, 550, 572, 728, 550, 22, 312, 312, 22, 55, 413, 183, 280, 593,
   191, 36, 36, 427, 36, 695, 592, 19, 544, 13, 468, 13, 544, 72, 437,
   321, 266, 461, 266, 441, 230, 409, 93, 521, 521, 345, 235, 22, 142,
   150, 102, 569, 235, 264, 91, 521, 264, 7, 102, 7, 498, 521, 235, 537,
   235, 6, 241, 420, 420, 631, 41, 527, 103, 67, 337, 62, 264, 527, 131,
   67, 174, 263, 264, 36, 36, 263, 581, 253, 465, 160, 286, 91, 160, 55,
   4, 4, 631, 631, 608, 365, 465, 294, 427, 427, 335, 669, 669, 129, 93,
   93, 93, 93, 74, 66, 758, 504, 347, 130, 505, 504, 143, 505, 550, 222,
   13, 352, 529, 291, 538, 50, 68, 269, 130, 295, 130, 511, 295, 295,
   130, 486, 132, 61, 206, 185, 368, 669, 22, 175, 492, 207, 373, 452,
   432, 327, 89, 550, 496, 611, 527, 89, 527, 496, 550, 516, 516, 91,
   136, 538, 264, 264, 124, 264, 264, 264, 264, 264, 535, 264, 150, 285,
   398, 285, 582, 398, 475, 81, 694, 694, 64, 81, 694, 234, 607, 723,
   513, 234, 64, 581, 64, 124, 64, 607, 234, 723, 717, 367, 64, 513,
   607, 488, 183, 488, 450, 183, 550, 286, 183, 363, 286, 414, 67, 449,
   449, 366, 215, 235, 95, 295, 295, 41, 335, 21, 445, 225, 21, 295,
   372, 749, 461, 53, 481, 397, 427, 427, 427, 714, 481, 714, 427, 717,
   165, 245, 486, 415, 245, 415, 486, 274, 415, 441, 456, 300, 548, 300,
   422, 422, 757, 11, 74, 430, 430, 136, 409, 430, 749, 191, 819, 592,
   136, 364, 465, 231, 231, 918, 160, 589, 160, 160, 465, 465, 231, 157,
   538, 538, 259, 538, 326, 22, 22, 22, 179, 22, 22, 550, 179, 287, 287,
   417, 327, 498, 498, 287, 488, 327, 538, 488, 583, 488, 287, 335, 287,
   335, 287, 41, 287, 335, 287, 327, 441, 335, 287, 488, 538, 327, 498,
   8, 8, 374, 8, 64, 427, 8, 374, 417, 760, 409, 373, 160, 423, 206,
   160, 106, 499, 160, 271, 235, 160, 590, 353, 695, 478, 619, 590, 353,
   13, 63, 189, 420, 605, 427, 643, 121, 280, 415, 121, 415, 595, 417,
   121, 398, 55, 330, 463, 463, 123, 353, 330, 582, 309, 582, 582, 405,
   330, 550, 405, 582, 353, 309, 308, 60, 353, 7, 60, 71, 353, 189, 183,
   183, 183, 582, 755, 189, 437, 287, 189, 183, 668, 481, 384, 384, 481,
   481, 481, 477, 582, 582, 499, 650, 481, 121, 461, 231, 36, 235, 36,
   413, 235, 209, 36, 689, 114, 353, 353, 235, 592, 36, 353, 413, 209,
   70, 308, 70, 699, 308, 70, 213, 292, 86, 689, 465, 55, 508, 128, 452,
   29, 41, 681, 573, 352, 21, 21, 648, 648, 69, 509, 409, 21, 264, 21,
   509, 514, 514, 409, 21, 264, 443, 443, 427, 160, 433, 663, 433, 231,
   646, 185, 482, 646, 433, 13, 398, 172, 234, 42, 491, 172, 234, 234,
   832, 775, 172, 196, 335, 822, 461, 298, 461, 364, 1120, 537, 169,
   169, 364, 694, 219, 612, 231, 740, 42, 235, 321, 279, 960, 279, 353,
   492, 159, 572, 321, 159, 287, 353, 287, 287, 206, 206, 321, 287, 159,
   321, 492, 159, 55, 572, 600, 270, 492, 784, 173, 91, 91, 443, 443,
   582, 261, 497, 572, 91, 555, 352, 206, 261, 555, 285, 91, 555, 497,
   83, 91, 619, 353, 488, 112, 4, 592, 295, 295, 488, 235, 231, 769,
   568, 581, 671, 451, 451, 483, 299, 1011, 432, 422, 207, 106, 701,
   508, 555, 508, 555, 125, 870, 555, 589, 508, 125, 749, 482, 125, 125,
   130, 544, 643, 643, 544, 488, 22, 643, 130, 335, 544, 22, 130, 544,
   544, 488, 426, 426, 4, 180, 4, 695, 35, 54, 433, 500, 592, 433, 262,
   94, 401, 401, 106, 216, 216, 106, 521, 102, 462, 518, 271, 475, 365,
   193, 648, 206, 424, 206, 193, 206, 206, 424, 299, 590, 590, 364, 621,
   67, 538, 488, 567, 51, 51, 513, 194, 81, 488, 486, 289, 567, 563,
   749, 563, 338, 338, 502, 563, 822, 338, 563, 338, 502, 201, 230, 201,
   533, 445, 175, 201, 175, 13, 85, 960, 103, 85, 175, 30, 445, 445,
   175, 573, 196, 877, 287, 356, 678, 235, 489, 312, 572, 264, 717, 138,
   295, 6, 295, 523, 55, 165, 165, 295, 138, 663, 6, 295, 6, 353, 138,
   6, 138, 169, 129, 784, 12, 129, 194, 605, 784, 445, 234, 627, 563,
   689, 627, 647, 570, 627, 570, 647, 206, 234, 215, 234, 816, 627, 816,
   234, 627, 215, 234, 627, 264, 427, 427, 30, 424, 161, 161, 916, 740,
   180, 616, 481, 514, 383, 265, 481, 164, 650, 121, 582, 689, 420, 669,
   589, 420, 788, 549, 165, 734, 280, 224, 146, 681, 788, 184, 398, 784,
   4, 398, 417, 417, 398, 636, 784, 417, 81, 398, 417, 81, 185, 827,
   420, 241, 420, 41, 185, 185, 718, 241, 101, 185, 185, 241, 241, 241,
   241, 241, 185, 324, 420, 420, 1011, 420, 827, 241, 184, 563, 241,
   183, 285, 529, 285, 808, 822, 891, 822, 488, 285, 486, 619, 55, 869,
   39, 567, 39, 289, 203, 158, 289, 710, 818, 158, 818, 355, 29, 409,
   203, 308, 648, 792, 308, 308, 91, 308, 6, 592, 792, 106, 106, 308,
   41, 178, 91, 751, 91, 259, 734, 166, 36, 327, 166, 230, 205, 205,
   172, 128, 230, 432, 623, 838, 623, 432, 278, 432, 42, 916, 432, 694,
   623, 352, 452, 93, 314, 93, 93, 641, 88, 970, 914, 230, 61, 159, 270,
   159, 493, 159, 755, 159, 409, 30, 30, 836, 128, 241, 99, 102, 984,
   538, 102, 102, 273, 639, 838, 102, 102, 136, 637, 508, 627, 285, 465,
   327, 327, 21, 749, 327, 749, 21, 845, 21, 21, 409, 749, 1367, 806,
   616, 714, 253, 616, 714, 714, 112, 375, 21, 112, 375, 375, 51, 51,
   51, 51, 393, 206, 870, 713, 193, 802, 21, 1061, 42, 382, 42, 543,
   876, 42, 876, 382, 696, 543, 635, 490, 353, 353, 417, 64, 1257, 271,
   64, 377, 127, 127, 537, 417, 905, 353, 538, 465, 605, 876, 427, 324,
   514, 852, 427, 53, 427, 557, 173, 173, 7, 1274, 563, 31, 31, 31, 745,
   392, 289, 230, 230, 230, 91, 218, 327, 420, 420, 128, 901, 552, 420,
   230, 608, 552, 476, 347, 476, 231, 159, 137, 716, 648, 716, 627, 740,
   718, 679, 679, 6, 718, 740, 6, 189, 679, 125, 159, 757, 1191, 409,
   175, 250, 409, 67, 324, 681, 605, 550, 398, 550, 931, 478, 174, 21,
   316, 91, 316, 654, 409, 425, 425, 699, 61, 699, 321, 698, 321, 698,
   61, 425, 699, 321, 409, 699, 299, 335, 321, 335, 61, 698, 699, 654,
   698, 299, 425, 231, 14, 121, 515, 121, 14, 165, 81, 409, 189, 81,
   373, 465, 463, 1055, 507, 81, 81, 189, 1246, 321, 409, 886, 104, 842,
   689, 300, 740, 380, 656, 656, 832, 656, 380, 300, 300, 206, 187, 175,
   142, 465, 206, 271, 468, 215, 560, 83, 215, 83, 215, 215, 83, 175,
   215, 83, 83, 111, 206, 756, 559, 756, 1367, 206, 559, 1015, 559, 559,

okay so the table is still going, bestie 🎀 these are more of the V1 values, and they stay in exactly the same order, no skipping 💅✨

   946, 1015, 548, 559, 756, 1043, 756, 698, 159, 414, 308, 458, 997,
   663, 663, 347, 39, 755, 838, 323, 755, 323, 159, 159, 717, 159, 21,
   41, 128, 516, 159, 717, 71, 870, 755, 159, 740, 717, 374, 516, 740,
   51, 148, 335, 148, 335, 791, 120, 364, 335, 335, 51, 120, 251, 538,
   251, 971, 1395, 538, 78, 178, 538, 538, 918, 129, 918, 129, 538, 538,
   656, 129, 538, 538, 129, 538, 1051, 538, 128, 838, 931, 998, 823,
   1095, 334, 870, 334, 367, 550, 1061, 498, 745, 832, 498, 745, 716,
   498, 498, 128, 997, 832, 716, 832, 130, 642, 616, 497, 432, 432, 432,
   432, 642, 159, 432, 46, 230, 788, 160, 230, 478, 46, 693, 103, 920,
   230, 589, 643, 160, 616, 432, 165, 165, 583, 592, 838, 784, 583, 710,
   6, 583, 583, 6, 35, 230, 838, 592, 710, 6, 589, 230, 838, 30, 592,
   583, 6, 583, 6, 6, 583, 30, 30, 6, 375, 375, 99, 36, 1158, 425, 662,
   417, 681, 364, 375, 1025, 538, 822, 669, 893, 538, 538, 450, 409,
   632, 527, 632, 563, 632, 527, 550, 71, 698, 550, 39, 550, 514, 537,
   514, 537, 111, 41, 173, 592, 173, 648, 173, 173, 173, 1011, 514, 173,
   173, 514, 166, 648, 355, 161, 166, 648, 497, 327, 327, 550, 650, 21,
   425, 605, 555, 103, 425, 605, 842, 836, 1011, 636, 138, 756, 836,
   756, 756, 353, 1011, 636, 636, 1158, 741, 741, 842, 756, 741, 1011,
   677, 1011, 770, 366, 306, 488, 920, 920, 665, 775, 502, 500, 775,
   775, 648, 364, 833, 207, 13, 93, 500, 364, 500, 665, 500, 93, 295,
   183, 1293, 313, 272, 313, 279, 303, 93, 516, 93, 1013, 381, 6, 93,
   93, 303, 259, 643, 168, 673, 230, 1261, 230, 230, 673, 1060, 1079,
   1079, 550, 741, 741, 590, 527, 741, 741, 442, 741, 442, 848, 741,
   590, 925, 219, 527, 925, 335, 442, 590, 239, 590, 590, 590, 239, 527,
   239, 1033, 230, 734, 241, 741, 230, 549, 548, 1015, 1015, 32, 36,
   433, 465, 724, 465, 73, 73, 73, 465, 808, 73, 592, 1430, 250, 154,
   154, 250, 538, 353, 353, 353, 353, 353, 175, 194, 206, 538, 632,
   1163, 960, 175, 175, 538, 452, 632, 1163, 175, 538, 960, 194, 175,
   194, 632, 960, 632, 94, 632, 461, 960, 1163, 1163, 461, 632, 960,
   755, 707, 105, 382, 625, 382, 382, 784, 707, 871, 559, 387, 387, 871,
   784, 559, 784, 88, 36, 570, 314, 1028, 975, 335, 335, 398, 573, 573,
   573, 21, 215, 562, 738, 612, 424, 21, 103, 788, 870, 912, 23, 186,
   757, 73, 818, 23, 73, 563, 952, 262, 563, 137, 262, 1022, 952, 137,
   1273, 442, 952, 604, 137, 308, 384, 913, 235, 325, 695, 398, 95, 668,
   776, 713, 309, 691, 22, 10, 364, 682, 682, 578, 481, 1252, 1072,
   1252, 825, 578, 825, 1072, 1149, 592, 273, 387, 273, 427, 155, 1204,
   50, 452, 50, 1142, 50, 367, 452, 1142, 611, 367, 50, 50, 367, 50,
   1675, 99, 367, 50, 1501, 1099, 830, 681, 689, 917, 1089, 453, 425,
   235, 918, 538, 550, 335, 161, 387, 859, 324, 21, 838, 859, 1123, 21,
   723, 21, 335, 335, 206, 21, 364, 1426, 21, 838, 838, 335, 364, 21,
   21, 859, 920, 838, 838, 397, 81, 639, 397, 397, 588, 933, 933, 784,
   222, 830, 36, 36, 222, 1251, 266, 36, 146, 266, 366, 581, 605, 366,
   22, 966, 681, 681, 433, 730, 1013, 550, 21, 21, 938, 488, 516, 21,
   21, 656, 420, 323, 323, 323, 327, 323, 918, 581, 581, 830, 361, 830,
   364, 259, 364, 496, 496, 364, 691, 705, 691, 475, 427, 1145, 600,
   179, 427, 527, 749, 869, 689, 335, 347, 220, 298, 689, 1426, 183,
   554, 55, 832, 550, 550, 165, 770, 957, 67, 1386, 219, 683, 683, 355,
   683, 355, 355, 738, 355, 842, 931, 266, 325, 349, 256, 1113, 256,
   423, 960, 554, 554, 325, 554, 508, 22, 142, 22, 508, 916, 767, 55,
   1529, 767, 55, 1286, 93, 972, 550, 931, 1286, 1286, 972, 93, 1286,
   1392, 890, 93, 1286, 93, 1286, 972, 374, 931, 890, 808, 779, 975,
   975, 175, 173, 4, 681, 383, 1367, 173, 383, 1367, 383, 173, 175, 69,
   238, 146, 238, 36, 148, 888, 238, 173, 238, 148, 238, 888, 185, 925,
   925, 797, 925, 815, 925, 469, 784, 289, 784, 925, 797, 925, 925,
   1093, 925, 925, 925, 1163, 797, 797, 815, 925, 1093, 784, 636, 663,
   925, 187, 922, 316, 1380, 709, 916, 916, 187, 355, 948, 916, 187,
   916, 916, 948, 948, 916, 355, 316, 316, 334, 300, 1461, 36, 583,
   1179, 699, 235, 858, 583, 699, 858, 699, 1189, 1256, 1189, 699, 797,
   699, 699, 699, 699, 427, 488, 427, 488, 175, 815, 656, 656, 150, 322,
   465, 322, 870, 465, 1099, 582, 665, 767, 749, 635, 749, 600, 1448,
   36, 502, 235, 502, 355, 502, 355, 355, 355, 172, 355, 355, 95, 866,
   425, 393, 1165, 42, 42, 42, 393, 939, 909, 909, 836, 552, 424, 1333,
   852, 897, 1426, 1333, 1446, 1426, 997, 1011, 852, 1198, 55, 32, 239,
   588, 681, 681, 239, 1401, 32, 588, 239, 462, 286, 1260, 984, 1160,
   960, 960, 486, 828, 462, 960, 1199, 581, 850, 663, 581, 751, 581,
   581, 1571, 252, 252, 1283, 264, 430, 264, 430, 430, 842, 252, 745,
   21, 307, 681, 1592, 488, 857, 857, 1161, 857, 857, 857, 138, 374,
   374, 1196, 374, 1903, 1782, 1626, 414, 112, 1477, 1040, 356, 775,
   414, 414, 112, 356, 775, 435, 338, 1066, 689, 689, 1501, 689, 1249,
   205, 689, 765, 220, 308, 917, 308, 308, 220, 327, 387, 838, 917, 917,
   917, 220, 662, 308, 220, 387, 387, 220, 220, 308, 308, 308, 387,
   1009, 1745, 822, 279, 554, 1129, 543, 383, 870, 1425, 241, 870, 241,
   383, 716, 592, 21, 21, 592, 425, 550, 550, 550, 427, 230, 57, 483,
   784, 860, 57, 308, 57, 486, 870, 447, 486, 433, 433, 870, 433, 997,
   486, 443, 433, 433, 997, 486, 1292, 47, 708, 81, 895, 394, 81, 935,
   81, 81, 81, 374, 986, 916, 1103, 1095, 465, 495, 916, 667, 1745, 518,
   220, 1338, 220, 734, 1294, 741, 166, 828, 741, 741, 1165, 1371, 1371,
   471, 1371, 647, 1142, 1878, 1878, 1371, 1371, 822, 66, 327, 158, 427,
   427, 465, 465, 676, 676, 30, 30, 676, 676, 893, 1592, 93, 455, 308,
   582, 695, 582, 629, 582, 85, 1179, 85, 85, 1592, 1179, 280, 1027,
   681, 398, 1027, 398, 295, 784, 740, 509, 425, 968, 509, 46, 833, 842,
   401, 184, 401, 464, 6, 1501, 1501, 550, 538, 883, 538, 883, 883, 883,
   1129, 550, 550, 333, 689, 948, 21, 21, 241, 2557, 2094, 273, 308, 58,
   863, 893, 1086, 409, 136, 1086, 592, 592, 830, 830, 883, 830, 277,
   68, 689, 902, 277, 453, 507, 129, 689, 630, 664, 550, 128, 1626,
   1626, 128, 902, 312, 589, 755, 755, 589, 755, 407, 1782, 589, 784,
   1516, 1118, 407, 407, 1447, 589, 235, 755, 1191, 235, 235, 407, 128,
   589, 1118, 21, 383, 1331, 691, 481, 383, 1129, 1129, 1261, 1104,
   1378, 1129, 784, 1129, 1261, 1129, 947, 1129, 784, 784, 1129, 1129,
   35, 1104, 35, 866, 1129, 1129, 64, 481, 730, 1260, 481, 970, 481,
   481, 481, 481, 863, 481, 681, 699, 863, 486, 681, 481, 481, 55, 55,
   235, 1364, 944, 632, 822, 401, 822, 952, 822, 822, 99, 550, 2240,
   550, 70, 891, 860, 860, 550, 550, 916, 1176, 1530, 425, 1530, 916,
   628, 1583, 916, 628, 916, 916, 628, 628, 425, 916, 1062, 1265, 916,
   916, 916, 280, 461, 916, 916, 1583, 628, 1062, 916, 916, 677, 1297,
   924, 1260, 83, 1260, 482, 433, 234, 462, 323, 1656, 997, 323, 323,
   931, 838, 931, 1933, 1391, 367, 323, 931, 1391, 1391, 103, 1116,
   1116, 1116, 769, 1195, 1218, 312, 791, 312, 741, 791, 997, 312, 334,
   334, 312, 287, 287, 633, 1397, 1426, 605, 1431, 327, 592, 705, 1194,
   592, 1097, 1118, 1503, 1267, 1267, 1267, 618, 1229, 734, 1089, 785,
   1089, 1129, 1148, 1148, 1089, 915, 1148, 1129, 1148, 1011, 1011,
   1229, 871, 1560, 1560, 1560, 563, 1537, 1009, 1560, 632, 985, 592,
   1308, 592, 882, 145, 145, 397, 837, 383, 592, 592, 832, 36, 2714,
   2107, 1588, 1347, 36, 36, 1443, 1453, 334, 2230, 1588, 1169, 650,
   1169, 2107, 425, 425, 891, 891, 425, 2532, 679, 274, 274, 274, 325,
   274, 1297, 194, 1297, 627, 314, 917, 314, 314, 1501, 414, 1490, 1036,
   592, 1036, 1025, 901, 1218, 1025, 901, 280, 592, 592, 901, 1461, 159,
   159, 159, 2076, 1066, 1176, 1176, 516, 327, 516, 1179, 1176, 899,
   1176, 1176, 323, 1187, 1229, 663, 1229, 504, 1229, 916, 1229, 916,
   1661, 41, 36, 278, 1027, 648, 648, 648, 1626, 648, 646, 1179, 1580,
   1061, 1514, 1008, 1741, 2076, 1514, 1008, 952, 1089, 427, 952, 427,
   1083, 425, 427, 1089, 1083, 425, 427, 425, 230, 920, 1678, 920, 1678,
   189, 189, 953, 189, 133, 189, 1075, 189, 189, 133, 1264, 725, 189,
   1629, 189, 808, 230, 230, 2179, 770, 230, 770, 230, 21, 21, 784,
   1118, 230, 230, 230, 770, 1118, 986, 808, 916, 30, 327, 918, 679,
   414, 916, 1165, 1355, 916, 755, 733, 433, 1490, 433, 433, 433, 605,
   433, 433, 433, 1446, 679, 206, 433, 21, 2452, 206, 206, 433, 1894,
   206, 822, 206, 2073, 206, 206, 21, 822, 21, 206, 206, 21, 383, 1513,
   375, 1347, 432, 1589, 172, 954, 242, 1256, 1256, 1248, 1256, 1256,
   1248, 1248, 1256, 842, 13, 592, 13, 842, 1291, 592, 21, 175, 13, 592,
   13, 13, 1426, 13, 1541, 445, 808, 808, 863, 647, 219, 1592, 1029,
   1225, 917, 1963, 1129, 555, 1313, 550, 660, 550, 220, 660, 552, 663,
   220, 533, 220, 383, 550, 1278, 1495, 636, 842, 1036, 425, 842, 425,
   1537, 1278, 842, 554, 1508, 636, 554, 301, 842, 792, 1392, 1021, 284,
   1172, 997, 1021, 103, 1316, 308, 1210, 848, 848, 1089, 1089, 848,
   848, 67, 1029, 827, 1029, 2078, 827, 1312, 1029, 827, 590, 872, 1312,
   427, 67, 67, 67, 67, 872, 827, 872, 2126, 1436, 26, 2126, 67, 1072,
   2126, 1610, 872, 1620, 883, 883, 1397, 1189, 555, 555, 563, 1189,
   555, 640, 555, 640, 1089, 1089, 610, 610, 1585, 610, 1355, 610, 1015,
   616, 925, 1015, 482, 230, 707, 231, 888, 1355, 589, 1379, 151, 931,
   1486, 1486, 393, 235, 960, 590, 235, 960, 422, 142, 285, 285, 327,
   327, 442, 2009, 822, 445, 822, 567, 888, 2611, 1537, 323, 55, 1537,
   323, 888, 2611, 323, 1537, 323, 58, 445, 593, 2045, 593, 58, 47, 770,
   842, 47, 47, 842, 842, 648, 2557, 173, 689, 2291, 1446, 2085, 2557,
   2557, 2291, 1780, 1535, 2291, 2391, 808, 691, 1295, 1165, 983, 948,
   2000, 948, 983, 983, 2225, 2000, 983, 983, 705, 948, 2000, 1795,
   1592, 478, 592, 1795, 1795, 663, 478, 1790, 478, 592, 1592, 173, 901,
   312, 4, 1606, 173, 838, 754, 754, 128, 550, 1166, 551, 1480, 550,
   550, 1875, 1957, 1166, 902, 1875, 550, 550, 551, 2632, 551, 1875,
   1875, 551, 2891, 2159, 2632, 3231, 551, 815, 150, 1654, 1059, 1059,
   734, 770, 555, 1592, 555, 2059, 770, 770, 1803, 627, 627, 627, 2059,
   931, 1272, 427, 1606, 1272, 1606, 1187, 1204, 397, 822, 21, 1645,
   263, 263, 822, 263, 1645, 280, 263, 605, 1645, 2014, 21, 21, 1029,
   263, 1916, 2291, 397, 397, 496, 270, 270, 1319, 264, 1638, 264, 986,
   1278, 1397, 1278, 1191, 409, 1191, 740, 1191, 754, 754, 387, 63, 948,
   666, 666, 1198, 548, 63, 1248, 285, 1248, 169, 1248, 1248, 285, 918,
   224, 285, 1426, 1671, 514, 514, 717, 514, 51, 1521, 1745, 51, 605,
   1191, 51, 128, 1191, 51, 51, 1521, 267, 513, 952, 966, 1671, 897, 51,
   71, 592, 986, 986, 1121, 592, 280, 2000, 2000, 1165, 1165, 1165,
   1818, 222, 1818, 1165, 1252, 506, 327, 443, 432, 1291, 1291, 2755,
   1413, 520, 1318, 227, 1047, 828, 520, 347, 1364, 136, 136, 452, 457,
   457, 132, 457, 488, 1087, 1013, 2225, 32, 1571, 2009, 483, 67, 483,
   740, 740, 1013, 2854, 866, 32, 2861, 866, 887, 32, 2444, 740, 32, 32,
   866, 2225, 866, 32, 1571, 2627, 32, 850, 1675, 569, 1158, 32, 1158,
   1797, 2641, 1565, 1158, 569, 1797, 1158, 1797, 55, 1703, 42, 55,
   2562, 675, 1703, 42, 55, 749, 488, 488, 347, 1206, 1286, 1286, 488,
   488, 1206, 1286, 1206, 1286, 550, 550, 1790, 860, 550, 2452, 550,
   550, 2765, 1089, 1633, 797, 2244, 1313, 194, 2129, 194, 194, 194,
   818, 32, 194, 450, 1313, 2387, 194, 1227, 2387, 308, 2232, 526, 476,
   278, 830, 830, 194, 830, 194, 278, 194, 714, 476, 830, 714, 830, 278,
   830, 2532, 1218, 1759, 1446, 960, 1747, 187, 1446, 1759, 960, 105,
   1446, 1446, 1271, 1446, 960, 960, 1218, 1446, 1446, 105, 1446, 960,
   488, 1446, 427, 534, 842, 1969, 2460, 1969, 842, 842, 1969, 427, 941,
   2160, 427, 230, 938, 2075, 1675, 1675, 895, 1675, 34, 129, 1811, 239,
   749, 1957, 2271, 749, 1908, 129, 239, 239, 129, 129, 2271, 2426,
   1355, 1756, 194, 1583, 194, 194, 1583, 194, 1355, 194, 1628, 2221,
   1269, 2425, 1756, 1355, 1355, 1583, 1033, 427, 582, 30, 582, 582,
   935, 1444, 1962, 915, 733, 915, 938, 1962, 767, 353, 1630, 1962,
   1962, 563, 733, 563, 733, 353, 822, 1630, 740, 2076, 2076, 2076, 589,

6. Security Considerations 🔥

okay bestie, this is the scary section 😭 Data delivery can get hit by denial-of-service attacks, where attackers send corrupted packets and receivers accept them as legit 💀. This is extra worrying for multicast, because a bad packet slipped in near the root of the multicast tree ends up at tons of receivers at once 🙈. And with this code, even ONE corrupted packet carrying encoding data can make the decoder spit out a totally corrupted, unusable object. Yikes 😭.

So it is RECOMMENDED that source authentication and integrity checking are applied to decoded objects before they get handed to an application 🫶. For example, a SHA-1 hash [SHA1] of an object may be appended before transmission. The hash gets computed and checked after decoding but before the app sees the object ✨. Source authentication SHOULD be provided, for example by including a digital signature that the receiver can verify, 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 as they arrive 🦋. This method can also be used to provide source authentication. And finally, it is RECOMMENDED that Reverse Path Forwarding checks be enabled in all network routers and switches along the path from the sender to the receivers, to limit the chance of a bad agent successfully sneaking a corrupted packet into the multicast tree data path 💅.

Another concern is that some FEC information may be picked up by receivers out-of-band, from a session description. If that session description is forged or corrupted, the receivers won't use the correct protocol for decoding content from the packets they receive 😭. To avoid these problems, it is RECOMMENDED that measures be taken to stop receivers from accepting incorrect session descriptions, e.g., by using source authentication so 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, check out [RFC5052]. This document assigns the Fully-Specified FEC Encoding ID 1 under the ietf:rmt:fec:encoding name-space to "Raptor Code" 👑✨.

8. Acknowledgements 🫶

Numerous editorial improvements and clarifications were made to this spec during the review process within 3GPP 🙌. Huge thanks to the members of 3GPP Technical Specification Group SA, Working Group 4, for those 💖.

9. References 📚

9.1. Normative References

[RFC2119]  Bradner, S., "Key words for use in RFCs to Indicate
           Requirement Levels", BCP 14, RFC 2119, March 1997.

[RFC4082]  Perrig, A., Song, D., Canetti, R., Tygar, J., and B.
           Briscoe, "Timed Efficient Stream Loss-Tolerant
           Authentication (TESLA): Multicast Source Authentication
           Transform Introduction", RFC 4082, June 2005.

[RFC5052]  Watson, M., Luby, M., and L. Vicisano, "Forward Error
           Correction (FEC) Building Block", RFC 5052, August 2007.

9.2. Informative References

[CCNC]     Luby, M., Watson, M., Gasiba, T., Stockhammer, T., and W.
           Xu, "Raptor Codes for Reliable Download Delivery in
           Wireless Broadcast Systems", CCNC 2006, Las Vegas, NV ,
           Jan 2006.

[MBMS]     3GPP, "Multimedia Broadcast/Multicast Service (MBMS);
           Protocols and codecs", 3GPP TS 26.346 6.1.0, June 2005.

[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.

[Raptor]   Shokrollahi, A., "Raptor Codes", IEEE Transactions on
           Information Theory no. 6, June 2006.

[SHA1]     "Secure Hash Standard", Federal Information Processing
           Standards Publication (FIPS PUB) 180-1, April 2005.

Authors' Addresses 🦋

Michael Luby
Digital Fountain
39141 Civic Center Drive
Suite 300
Fremont, CA  94538
U.S.A.

EMail: luby@digitalfountain.com


Amin Shokrollahi
EPFL
Laboratory of Algorithmic Mathematics
IC-IIF-ALGO
PSE-A
Lausanne  1015
Switzerland

EMail: amin.shokrollahi@epfl.ch


Mark Watson
Digital Fountain
39141 Civic Center Drive
Suite 300
Fremont, CA  94538
U.S.A.

EMail: mark@digitalfountain.com


Thomas Stockhammer
Nomor Research
Brecherspitzstrasse 8
Munich  81541
Germany

EMail: stockhammer@nomor.de

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