brotli/brotlispec.txt

J. Alakuijala
Z. Szabadka
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   DRAFT of
   Brotli Compression Algorithm Compressed Data Format Specification 1.0

Status of This Memo

   This memo provides information for the Internet community.  This memo
   does not specify an Internet standard of any kind.  Distribution of
   this memo is unlimited.

Notices

   Copyright (c) 2013  J. Alakuijala and Z. Szabadka

   Permission is granted to copy and distribute this document for any
   purpose and without charge, including translations into other
   languages and incorporation into compilations, provided that the
   copyright notice and this notice are preserved, and that any
   substantive changes or deletions from the original are clearly
   marked.

Abstract

   This specification defines a lossless compressed data format that
   compresses data using a combination of the LZ77 algorithm and Huffman
   coding, with efficiency comparable to the best currently available
   general-purpose compression methods.

1. Introduction

   1.1. Purpose

      The purpose of this specification is to define a lossless
      compressed data format that:
         * Is independent of CPU type, operating system, file system,
           and character set, and hence can be used for interchange;
         * Can be produced or consumed, even for an arbitrarily long
           sequentially presented input data stream, using only an a
           priori bounded amount of intermediate storage, and hence
           can be used in data communications or similar structures
           such as Unix filters;
         * Compresses data with efficiency comparable to the best
           currently available general-purpose compression methods,
           and in particular considerably better than the gzip program;
         * Decompresses much faster than the LZMA implementations.

      The data format defined by this specification does not attempt to:
         * Allow random access to compressed data;
         * Compress specialized data (e.g., raster graphics) as well
           as the best currently available specialized algorithms.

   1.2. Intended audience

      This specification is intended for use by implementors of software
      to compress data into "brotli" format and/or decompress data from
      "brotli" format.

      The text of the specification assumes a basic background in
      programming at the level of bits and other primitive data
      representations. Familiarity with the technique of Huffman coding
      is helpful but not required.

      This specification uses heavily the notations and terminology
      introduced in the DEFLATE format specification (RFC 1951, see
      reference [3] below). For the sake of completeness, we always
      include the whole text of the relevant parts of RFC 1951,
      therefore familiarity with the DEFLATE format is helpful but not
      required.

   1.3. Scope

      The specification specifies a method for representing a sequence
      of bytes as a (usually shorter) sequence of bits, and a method for
      packing the latter bit sequence into bytes.

   1.4. Compliance

      Unless otherwise indicated below, a compliant decompressor must be
      able to accept and decompress any data set that conforms to all
      the specifications presented here; a compliant compressor must
      produce data sets that conform to all the specifications presented
      here.

   1.5.  Definitions of terms and conventions used

      Byte: 8 bits stored or transmitted as a unit (same as an octet).
      For this specification, a byte is exactly 8 bits, even on machines
      which store a character on a number of bits different from eight.
      See below, for the numbering of bits within a byte.

      String: a sequence of arbitrary bytes.

      Bytes stored within a computer do not have a "bit order", since
      they are always treated as a unit.  However, a byte considered as
      an integer between 0 and 255 does have a most- and least-
      significant bit, and since we write numbers with the most-
      significant digit on the left, we also write bytes with the most-
      significant bit on the left.  In the diagrams below, we number the
      bits of a byte so that bit 0 is the least-significant bit, i.e.,
      the bits are numbered:

         +--------+
         |76543210|
         +--------+

      Within a computer, a number may occupy multiple bytes.  All
      multi-byte numbers in the format described here are stored with
      the least-significant byte first (at the lower memory address).
      For example, the decimal number 520 is stored as:

             0        1
         +--------+--------+
         |00001000|00000010|
         +--------+--------+
          ^        ^
          |        |
          |        + more significant byte = 2 x 256
          + less significant byte = 8


      1.5.1. Packing into bytes

         This document does not address the issue of the order in which
         bits of a byte are transmitted on a bit-sequential medium,
         since the final data format described here is byte- rather than
         bit-oriented.  However, we describe the compressed block format
         in below, as a sequence of data elements of various bit
         lengths, not a sequence of bytes.  We must therefore specify
         how to pack these data elements into bytes to form the final
         compressed byte sequence:

             * Data elements are packed into bytes in order of
               increasing bit number within the byte, i.e., starting
               with the least-significant bit of the byte.
             * Data elements other than Huffman codes are packed
               starting with the least-significant bit of the data
               element.
             * Huffman codes are packed starting with the most-
               significant bit of the code.

         In other words, if one were to print out the compressed data as
         a sequence of bytes, starting with the first byte at the
         *right* margin and proceeding to the *left*, with the most-
         significant bit of each byte on the left as usual, one would be
         able to parse the result from right to left, with fixed-width
         elements in the correct MSB-to-LSB order and Huffman codes in
         bit-reversed order (i.e., with the first bit of the code in the
         relative LSB position).

2. Compressed representation overview

   A compressed data set consists of a header and a series of meta-
   blocks, corresponding to successive meta-blocks of input data. The
   meta-block sizes are limited to bytes and the maximum meta-block size
   is 268,435,456 bytes.

   The header contains the size of a sliding window on the input data
   that is sufficient to keep on the intermediate storage at any given
   point during decoding the stream.

   Each meta-block is compressed using a combination of the LZ77
   algorithm (Lempel-Ziv 1977, see reference [2] below) and Huffman
   coding. The Huffman trees for each block are independent of those for
   previous or subsequent blocks; the LZ77 algorithm may use a
   reference to a duplicated string occurring in a previous meta-block,
   up to sliding window size input bytes before.

   Each meta-block consists of two parts: a meta-block header that
   describes the representation of the compressed data part, and a
   compressed data part. The compressed data consists of a series of
   commands. Each command consists of two parts: a sequence of literal
   bytes (of strings that have not been detected as duplicated within
   the sliding window), and a pointer to a duplicated string,
   represented as a pair <length, backward distance>.

   Each command in the compressed data is represented using three kinds
   of Huffman codes: one kind of code tree for the literal sequence
   lengths (also referred to as literal insertion lengths) and backward 
   copy lengths (that is, a single code word represents two lengths,
   one of the literal sequence and one of the backward copy), a separate
   kind of code tree for literals, and a third kind of code tree for
   distances. The code trees for each meta-block appear in a compact
   form just before the compressed data in the meta-block header.

   The sequence of each type of value in the representation of a command
   (insert-and-copy lengths, literals and distances) within a meta-
   block is further divided into blocks. In the "brotli" format, blocks
   are not contiguous chunks of compressed data, but rather the pieces
   of compressed data belonging to a block are interleaved with pieces
   of data belonging to other blocks. Each meta-block can be logically
   decomposed into a series of insert-and-copy length blocks, a series
   of literal blocks and a series of distance blocks. These are also
   called the three block categories: a meta-block has a series of
   blocks for each block category. Note that the physical structure of
   the meta-block is a series of commands, while the three series of
   blocks is the logical structure. Consider the following example:

      (IaC0, L0, L1, L2, D0)(IaC1, D1)(IaC2, L3, L4, D2)(IaC3, L5, D3)

   The meta-block here has 4 commands, and each three types of symbols
   within these commands can be rearranged into for example the
   following logical block structure:

      [IaC0, IaC1][IaC2, IaC3]  <-- block types 0 and 1

      [L0, L1][L2, L3, L4][L5]  <-- block types 0, 1, and 0

      [D0][D1, D2, D3]          <-- block types 0 and 1

   The subsequent blocks within each block category must have different
   block types, but blocks further away in the block sequence can have
   the same types. The block types are numbered from 0 to the maximum
   block type number of 255 and the first block of each block category
   must have type 0. The block structure of a meta-block is represented
   by the sequence of block-switch commands for each block category,
   where a block-switch command is a pair <block type, block length>.
   The block-switch commands are represented in the compressed data
   before the start of each new block using a Huffman code tree for
   block types and a separate Huffman code tree for block lengths for
   each block category. In the above example the physical layout of the
   meta-block is the following:

      IaC0 L0 L1 LBlockSwitch(1, 3) L2 D0 IaC1 DBlockSwitch(1, 1) D1
      IaCBlockSwitch(1, 2) IaC2 L3 L4 D2 IaC3 LBlockSwitch(0, 1) D3

   Note that the block switch commands for the first blocks are not part
   of the meta-block compressed data part, they are encoded in the meta-
   block header. The code trees for block types and lengths (total of
   six Huffman code trees) appear in a compact form in the meta-block
   header.

   Each type of value (insert-and-copy lengths, literals and distances) 
   can be encoded with any Huffman tree from a collection of Huffman
   trees of the same kind appearing in the meta-block header. The
   particual Huffman tree used can depend on two factors: the block type
   of the block the value appears in, and the context of the value. In
   the case of the literals, the context is the previous two bytes in
   the input data, and in the case of distances, the context is the copy
   length from the same command. For insert-and-copy lengths, no context
   is used and the Huffman tree depends only on the block type (in fact,
   the index of the Huffman tree is the block type number). In the case 
   of literals and distances, the context is mapped to a context id in
   the rage [0, 63] for literals and [0, 3] for distances and the matrix
   of the Huffman tree indices for each block type and context id,
   called the context map, is encoded in a compact form in the meta-
   block header.

   In addition to the parts listed above (Huffman code trees for insert-
   and-copy lengths, literals, distances, block types and block lengths
   and the context map), the meta-block header contains the number of
   input bytes in the meta-block and two additional parameters used in
   the representation of copy distances (number of "postfix bits" and
   number of direct distance codes).

3. Compressed representation of Huffman codes

   3.1. Introduction to prefix and Huffman coding

      Prefix coding represents symbols from an a priori known alphabet
      by bit sequences (codes), one code for each symbol, in a manner
      such that different symbols may be represented by bit sequences of
      different lengths, but a parser can always parse an encoded string
      unambiguously symbol-by-symbol.

      We define a prefix code in terms of a binary tree in which the two
      edges descending from each non-leaf node are labeled 0 and 1 and
      in which the leaf nodes correspond one-for-one with (are labeled
      with) the symbols of the alphabet; then the code for a symbol is
      the sequence of 0's and 1's on the edges leading from the root to 
      the leaf labeled with that symbol.  For example:

                       /\              Symbol    Code
                      0  1             ------    ----
                     /    \                A      00
                    /\     B               B       1
                   0  1                    C     011
                  /    \                   D     010
                 A     /\
                      0  1
                     /    \
                    D      C

      A parser can decode the next symbol from an encoded input stream
      by walking down the tree from the root, at each step choosing the 
      edge corresponding to the next input bit.

      Given an alphabet with known symbol frequencies, the Huffman
      algorithm allows the construction of an optimal prefix code (one
      which represents strings with those symbol frequencies using the
      fewest bits of any possible prefix codes for that alphabet). Such
      a code is called a Huffman code. (See reference [1] in Chapter 5,
      references for additional information on Huffman codes.)

      Note that in the "brotli" format, the Huffman codes for the
      various alphabets must not exceed certain maximum code lengths.
      This constraint complicates the algorithm for computing code
      lengths from symbol frequencies. Again, see Chapter 5, references
      for details.

   3.2. Use of Huffman coding in the "brotli" format

      The Huffman codes used for each alphabet in the "brotli" format
      are canonical Huffman codes, which have two additional rules:

         * All codes of a given bit length have lexicographically
           consecutive values, in the same order as the symbols they
           represent;

         * Shorter codes lexicographically precede longer codes.

      We could recode the example above to follow this rule as follows,
      assuming that the order of the alphabet is ABCD:

         Symbol  Code
         ------  ----
         A       10
         B       0
         C       110
         D       111

      I.e., 0 precedes 10 which precedes 11x, and 110 and 111 are
      lexicographically consecutive.

      Given this rule, we can define the canonical Huffman code for an
      alphabet just by giving the bit lengths of the codes for each
      symbol of the alphabet in order; this is sufficient to determine
      the actual codes. In our example, the code is completely defined
      by the sequence of bit lengths (2, 1, 3, 3). The following
      algorithm generates the codes as integers, intended to be read
      from most- to least-significant bit. The code lengths are
      initially in tree[I].Len; the codes are produced in tree[I].Code.

         1)  Count the number of codes for each code length.  Let
             bl_count[N] be the number of codes of length N, N >= 1.

         2)  Find the numerical value of the smallest code for each
             code length:

                code = 0;
                bl_count[0] = 0;
                for (bits = 1; bits <= MAX_BITS; bits++) {
                    code = (code + bl_count[bits-1]) << 1;
                    next_code[bits] = code;
                }

         3)  Assign numerical values to all codes, using consecutive
             values for all codes of the same length with the base
             values determined at step 2. Codes that are never used
             (which have a bit length of zero) must not be assigned a
             value.

                for (n = 0;  n <= max_code; n++) {
                    len = tree[n].Len;
                    if (len != 0) {
                        tree[n].Code = next_code[len];
                        next_code[len]++;
                    }
                }

      Example:

      Consider the alphabet ABCDEFGH, with bit lengths (3, 3, 3, 3, 3,
      2, 4, 4).  After step 1, we have:

         N      bl_count[N]
         -      -----------
         2      1
         3      5
         4      2

      Step 2 computes the following next_code values:

         N      next_code[N]
         -      ------------
         1      0
         2      0
         3      2
         4      14

      Step 3 produces the following code values:

         Symbol Length   Code
         ------ ------   ----
         A       3        010
         B       3        011
         C       3        100
         D       3        101
         E       3        110
         F       2         00
         G       4       1110
         H       4       1111

   3.3. Alphabet sizes

      Huffman codes are used for different purposes in the "brotli"
      format, and each purpose has a different alphabet size. For
      literal codes, the alphabet size is 256. For insert-and-copy
      length codes, the alphabet size is 704. For block length codes,
      the alphabet size is 26. For distance codes, block type codes and
      the Huffman codes used in compressing the context map, the
      alphabet size is dynamic and is based on other parameters.

   3.4. Simple Huffman codes

      The first bit of the compressed representation of each Huffman
      code distinguishes between simple and complex Huffman codes. If
      the first bit is 1, then a simple, otherwise a complex Huffman
      code follows.

      A simple Huffman code can have only up to four symbols with non-
      zero code length. The format of the simple Huffman code is as
      follows:

            1 bit:  1, indicating a simple Huffman code
            2 bits: NSYM - 1, where NSYM = # of symbols with non-zero
                    code length

            NSYM symbols, each encoded using ALPHABET_BITS bits

            1 bit:  tree-select, present only for NSYM = 4

      The value of ALPHABET_BITS depends on the alphabet of the Huffman
      code: it is the smallest number of bits that can represent all
      symbols in the alphabet. E.g. for the alphabet of literal bytes,
      ALPHABET_BITS is 8.

      The (non-zero) code lengths of the symbols can be reconstructed as
      follows:

         * if NSYM = 1, the code length for the one symbol is one at
           this stage, but only to distinguish it from the other zero
           code length symbols, when encoding this symbol in the
           compressed data stream using this Huffman code later, no
           actual bits are emitted. Similarly, when decoding a symbol
           using this Huffman code, no bits are read and the one symbol
           is returned.

        * if NSYM = 2, both symbols have code length 1.

        * if NSYM = 3, the code lengths for the symbols are 1, 2, 2 in
          the order they appear in the representation of the simple
          Huffman code.

        * if NSYM = 4, the code lengths (in order of symbols decoded)
          depend on the tree-select bit: 2, 2, 2, 2, (tree-select bit 0)
          or 1, 2, 3, 3 (tree-select bit 1).

   3.5. Complex Huffman codes

      A complex Huffman code is a canonical Huffman code, defined by the
      sequence of code lengths, as discussed in Paragraph 3.2, above.
      For even greater compactness, the code length sequences themselves
      are compressed using a Huffman code. The alphabet for code lengths
      is as follows:

            0 - 15: Represent code lengths of 0 - 15
                16: Copy the previous non-zero code length 3 - 6 times
                    The next 2 bits indicate repeat length
                          (0 = 3, ... , 3 = 6)
                    If this is the first code length, or all previous
                    code lengths are zero, a code length of 8 is
                    repeated 3 - 6 times
                    A repeated code length code of 16 modifies the
                    repeat count of the previous one as follows:
                       repeat count = (4 * (repeat count - 2)) +
                                      (3 - 6 on the next 2 bits)
                    Example:  Codes 7, 16 (+2 bits 11), 16 (+2 bits 10)
                              will expand to 22 code lengths of 7
                              (1 + 4 * (6 - 2) + 5)
                17: Repeat a code length of 0 for 3 - 10 times.
                    (3 bits of length)
                    A repeated code length code of 17 modifies the
                    repeat count of the previous one as follows:
                       repeat count = (8 * (repeat count - 2)) +
                                      (3 - 10 on the next 3 bits)

      A code length of 0 indicates that the corresponding symbol in the
      alphabet will not occur in the compressed data, and should not
      participate in the Huffman code construction algorithm given
      earlier.

      The bit lengths of the Huffman code over the code length alphabet
      are compressed with the following static Huffman code:

               Symbol   Code
               ------   ----
               0          00
               1        1010
               2         100
               3          11
               4          01
               5        1011

      We can now define the format of the complex Huffman code as
      follows:

            1 bit:  0, indicating a complex Huffman code
            4 bits: HCLEN, # of code length codes - 3
            1 bit : HSKIP, if 1, skip over first two code length codes

            (HCLEN + 3 - 2 * HSKIP) code lengths for symbols in the code
               length alphabet given just above, in the order: 1, 2, 3,
               4, 0, 17, 5, 6, 16, 7, 8, 9, 10, 11, 12, 13, 14, 15

               If HSKIP is 1, code lengths of code length symbols 1 and
               2 are implicit zeros. Code lengths of code length symbols
               beyond the (HCLEN + 4)th in the ordering above are also
               implicit zeros.

               The code lengths of code length symbols are between 0 and
               5 and they are represented with 2 - 5 bits according to
               the static Huffman code above. A code length of 0 means
               the corresponding code length symbol is not used.

            1 bit:  HLENINC, if 1, the number of code length symbols is
                    encoded next

          7-8 bits: HLEN, # of code length symbols, with the following
                    encoding: values 4 - 67 with bit pattern 0xxxxxx,
                    values 68 - 195 with bit pattern 1xxxxxxx, appears
                    only if HLENINC = 1

            Sequence of code lengths symbols, encoded using the code
               length Huffman code. The number of code length symbols
               is either HLEN (in case of HLENINC = 1), or as many as is
               needed to assign a code length to each symbol in the
               alphabet (i.e. the alphabet size minus the sum of all the
               repeat lengths defined by extra bits of code length
               symbols 16 and 17). In case of HLENINC = 1, all symbols
               not assigned a code length have implicit code length 0.

   3.6. Validity of the Huffman code

      There are two kinds of valid Huffman codes:
         * Huffman code that contains one symbol of length 1, and
         * Full canonical Huffman code.

      A decoder can check if the Huffman code is full by using integer
      arithmetic, by computing if the sum of (32768 right-shifted by
      code-length) over the non-zero code lengths leads to a total sum
      of 32768. However, if there is only one non-zero code-length, it
      shall have an implicit code length of one and the code is
      considered valid.

4. Encoding of distances

   As described in Section 2, one component of a compressed meta-block
   is a sequence of backward distances. In this section we provide the
   details to the encoding of distances.

   Each distance in the compressed data part of a meta-block is
   represented with a pair <distance code, extra bits>. The distance
   code and the extra bits are encoded back-to-back, the distance code
   is encoded using a Huffman code over the distance code alphabet,
   while the extra bits value is encoded as a fixed-width machine
   integer. The number of extra bits can be 0 - 24, and it is dependent
   on the distance code.

   To convert a distance code and associated extra bits to a backward
   distance, we need the sequence of past distances and two additonal
   parameters, the number of "postfix bits", denoted by NPOSTFIX, and
   the number of direct distance codes, denoted by NDIRECT. Both of
   these parameters are encoded in the meta-block header. We will also
   use the folowing derived parameter:

      POSTFIX_MASK = ((1 << NPOSTFIX) - 1)

   The first 16 distance codes are special short codes that reference
   past distances as follows:

         0: last distance
         1: second last distance
         2: third last distance
         3: fourth last distance
         4: last distance - 1
         5: last distance + 1
         6: last distance - 2
         7: last distance + 2
         8: last distance - 3
         9: last disatnce + 3
        10: second last distance - 1
        11: second last distance + 1
        12: second last distance - 2
        13: second last distance + 2
        14: second last distance - 3
        15: second last distance + 3

   The ring-buffer of four last distances is initialized by the values
   16, 15, 11 and 4 (i.e. the fourth last is set to 16, the third last
   to 15, the second last to 11 and the last distance to 4) at the
   beginning of the *stream* (as opposed to the beginning of the meta-
   block) and it is not reset at meta-block boundaries. When a distance
   code 0 appears, the distance it represents (i.e. the last distance
   in the sequence of distances) is not pushed to the ring-buffer of
   last distances, in other words, the expression "(second, third,
   fourth) last distance" means the (second, third, fourth) last
   distance that was not represented by a 0 distance code.

   The next NDIRECT distance codes, from 16 to 15 + NDIRECT, represent
   distances from 1 to NDIRECT. Neither the distance short codes, nor
   the NDIRECT direct distance codes have any extra bits.

   Distance codes 16 + NDIRECT and greater all have extra bits, the
   number of extra bits for a distance code "dcode" is given by the
   following formula:

      ndistbits = 1 + ((dcode - NDIRECT - 16) >> (NPOSTFIX + 1))

   The maximum number of extra bits is 24, therefore the size of the
   distance code alphabet is (16 + NDIRECT + (48 << NPOSTFIX)).

   Given a distance code "dcode" (>= 16 + NDIRECT), and extra bits
   "dextra", the backward distance is given by the following formula:

      hcode = (dcode - NDIRECT - 16) >> NPOSTFIX
      lcode = (dcode - NDIRECT - 16) & POSTFIX_MASK
      offset = ((2 + (hcode & 1)) << ndistbits) - 4;
      distance = ((offset + dextra) << NPOSTFIX) + lcode + NDIRECT + 1

5. Encoding of literal insertion lengths and copy lengths

   As described in Section 2, the literal insertion lengths and backward
   copy lengths are encoded using a single Huffman code. This section
   provides the details to this encoding.

   Each <insertion length, copy length> pair in the compressed data part
   of a meta-block is represented with the following triplet:

      <insert-and-copy length code, insert extra bits, copy extra bits>

   The insert-and-copy length code, the insert extra bits and the copy
   extra bits are encoded back-to-back, the insert-and-copy length code
   is encoded using a Huffman code over the insert-and-copy length code
   alphabet, while the extra bits values are encoded as fixed-width
   machine integers. The number of insert and copy extra bits can be
   0 - 24, and they are dependent on the insert-and-copy length code.

   Some of the insert-and-copy length codes also express the fact that
   the distance code of the distance in the same command is 0, i.e. the
   distance component of the command is the same as that of the previous
   command. In this case, the distance code and extra bits for the 
   distance are omitted from the compressed data stream.

   We describe the insert-and-copy length code alphabet in terms of the
   (not directly used) insert length code and copy length code
   alphabets. The symbols of the insert length code alphabet, along with
   the number of insert extra bits and the range of the insert lengths
   are as follows:

           Extra               Extra               Extra
      Code Bits Lengths   Code Bits Lengths   Code Bits Lengths
      ---- ---- ------    ---- ---- -------   ---- ---- -------
       0    0     0        8    2    10-13    16    6   130-193
       1    0     1        9    2    14-17    17    7   194-321
       2    0     2       10    3    18-25    18    8   322-527
       3    0     3       11    3    26-33    19    9   578-1089
       4    0     4       12    4    34-49    20   10   1090-2113
       5    0     5       13    4    50-65    21   12   2114-6209
       6    1    6,7      14    5    66-97    22   14   6210-22593
       7    1    8,9      15    5    98-129   23   24   22594-16799809

   The symbols of the copy length code alphabet, along with the number
   of copy extra bits and the range of copy lengths are as follows:

           Extra               Extra               Extra
      Code Bits Lengths   Code Bits Lengths   Code Bits Lengths
      ---- ---- ------    ---- ---- -------   ---- ---- -------
       0    0     2        8    1    10,11    16    5    70-101
       1    0     3        9    1    12,13    17    5   102-133
       2    0     4       10    2    14-17    18    6   134-197
       3    0     5       11    2    18-21    19    7   198-325
       4    0     6       12    3    22-29    20    8   326-581
       5    0     7       13    3    30-37    21    9   582-1093
       6    0     8       14    4    38-53    22   10   1094-2117
       7    0     9       15    4    54-69    23   24   2118-16779333

   To convert an insert-and-copy length code to an insert length code
   and a copy length code, the following table can be used:

          Insert
          length        Copy length code
          code       0-7       8-15     16-23
                 +---------+---------+
                 |         |         |
            0-7  |   0-63  |  64-127 | <--- distance code 0
                 |         |         |
                 +---------+---------+---------+
                 |         |         |         |
            0-7  | 128-191 | 192-255 | 383-447 |
                 |         |         |         |
                 +---------+---------+---------+
                 |         |         |         |
            8-15 | 256-319 | 320-383 | 512-575 |
                 |         |         |         |
                 +---------+---------+---------+
                 |         |         |         |
           16-23 | 448-551 | 576-639 | 640-703 |
                 |         |         |         |
                 +---------+---------+---------+

   First, look up the cell with the 64 value range containing the
   insert-and-copy length code, this gives the insert length code and
   and the copy length code ranges, both 8 values long. The copy length
   code within its range is determined by the lowest 3 bits of the
   insert-and-copy length code, and the insert length code within its
   range is determined by bits 3-5 (counted from the LSB) of the insert-
   and-copy length code. Given the insert length and copy length codes,
   the actual insert and copy lengths can be obtained by reading the
   number of extra bits given by the tables above.

   If the insert-and-copy length code is between 0 and 127, the distance
   code of the command is set to zero (the last distance reused).

6. Encoding of block switch commands

   As described in Section 2, a block-switch command is a pair
   <block type, block length>. These are encoded in the compressed data
   part of the meta-block, right before the start of each new block of a
   particular block category.

   Each block type in the compressed data is represented with a block
   type code, encoded using a Huffman code over the block type code
   alphabet. A block type code 0 means that the block type is the same
   as the type of the second last block from the same block category,
   while a block type code 1 means that the block type equals the last
   block type plus one. Block type codes 2 - 257 represent block types
   0 - 255. The second last and last block types are initialized with 0
   and 1, respectively, at the beginning of each meta-block.

   The first block type of each block category must be 0, and the block
   type of the first block switch command is therefore not encoded in
   the compressed data.

   The number of different block types in each block category, denoted
   by NBLTYPESL, NBLTYPESI, and NBLTYPESD for literals, insert-and-copy
   lengths and distances, respectively, is encoded in the meta-block
   header, and it must equal to the largest block type plus one in that
   block category. In other words, the set of literal, insert-and-copy
   length and distance block types must be [0..NBLTYPESL-1],
   [0..NBLTYPESI-1], and [0..NBLTYPESD-1], respectively. From this it
   follows that the alphabet size of literal, insert-and-copy length and
   distance block type codes is NBLTYPES + 2, NBLTYPESI + 2 and
   NBLTYPESD + 2, respectively.

   Each block length in the compressed data is represented with a pair
   <block length code, extra bits>. The block length code and the extra
   bits are encoded back-to-back, the block length code is encoded using
   a Huffman code over the block length code alphabet, while the extra
   bits value is encoded as a fixed-width machine integer. The number of
   extra bits can be 0 - 24, and it is dependent on the block length
   code. The symbols of the block length code alphabet, along with the
   number of extra bits and the range of block lengths are as follows:

           Extra               Extra               Extra
      Code Bits Lengths   Code Bits Lengths   Code Bits Lengths
      ---- ---- ------    ---- ---- -------   ---- ---- -------
       0    2    1-4       9    4    65-80    18    7   369-496
       1    2    5-8      10    4    81-96    19    8   497-752
       2    2    9-12     11    4    97-112   20    9   753-1264
       3    2   13-16     12    5   113-144   21   10   1265-2288
       4    3   17-24     13    5   145-176   22   11   2289-4336
       5    3   25-32     14    5   177-208   23   12   4337-8432
       6    3   33-40     15    5   209-240   24   13   8433-16624
       7    3   41-48     16    6   241-304   25   24   16625-16793840
       8    4   49-64     17    6   305-368

   The first block switch command of each block category is special in
   the sense that it is encoded in the meta-block header, and as
   described earlier the block type code is omitted, since it is an
   implicit zero.

7. Context modeling

   As described in Section 2, the Huffman tree used to encode a literal
   byte or a distance code depends on the context id and the block type.
   This section specifies how to compute the context id for a particular
   literal and distance code, and how to encode the context map that
   maps a <context id, block type> pair to the index of a Huffman
   tree in the array of literal and distance Huffman trees.

   7.1. Context modes and context id lookup for literals

      The context for encoding the next literal is defined by the last
      two bytes in the stream (p1, p2, where p1 is the most recent
      byte), regardless if these bytes are produced by backward
      references or by literal insertions.

      There are four methods, called context modes, to compute the
      Context ID:
         * MSB6, where the Context ID is the value of six most
           significant bits of p1,
         * LSB6, where the Context ID is the value of six least
           significant bits of p1,
         * UTF8, where the Context ID is a complex function of p1, p2,
           optimized for text compression, and
         * Signed, where Context ID is a complex function of p1, p2,
           optimized for compressing sequences of signed integers.

      The Context ID for the UTF8 and Signed context modes is computed
      using the following lookup tables Lut0, Lut1, and Lut2.

      Lut0 :=
         0,  0,  0,  0,  0,  0,  0,  0,  0,  4,  4,  0,  0,  4,  0,  0,
         0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,
         8, 12, 16, 12, 12, 20, 12, 16, 24, 28, 12, 12, 32, 12, 36, 12,
        44, 44, 44, 44, 44, 44, 44, 44, 44, 44, 32, 32, 24, 40, 28, 12,
        12, 48, 52, 52, 52, 48, 52, 52, 52, 48, 52, 52, 52, 52, 52, 48,
        52, 52, 52, 52, 52, 48, 52, 52, 52, 52, 52, 24, 12, 28, 12, 12,
        12, 56, 60, 60, 60, 56, 60, 60, 60, 56, 60, 60, 60, 60, 60, 56,
        60, 60, 60, 60, 60, 56, 60, 60, 60, 60, 60, 24, 12, 28, 12,  0,
         0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1,
         0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1,
         0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1,
         0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1,
         2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3,
         2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3,
         2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3,
         2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3

      Lut1 :=
         0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
         0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
         0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
         2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1,
         1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2,
         2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1,
         1, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3,
         3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1, 1, 0,
         0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
         0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
         0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
         0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
         0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
         0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
         2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2,
         2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2

      Lut2 :=
         0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
         2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2,
         2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2,
         2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2,
         3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3,
         3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3,
         3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3,
         3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3,
         4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4,
         4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4,
         4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4,
         4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4,
         5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5,
         5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5,
         5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5,
         6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 7

      Given p1 = last decoded byte, and p2 = second last decoded byte,
      the context ids can be computed as follows:

         For LSB6  :  Context ID = p1 & 0x3f
         For MSB6  :  Context ID = p1 >> 2
         For UTF8  :  Context ID = Lut0[p1] | Lut1[p2]
         For Signed:  Context ID = (Lut2[p1] << 3) | Lut2[p2]

      The context modes LSB6, MSB6, UTF8, and Signed are denoted by
      integers 0, 1, 2, 3.

      The context mode is defined for each literal block type, and they
      are stored in a consequtive array of bits in the meta-block
      header, always two bits per block type.

   7.2. Context id for distances

      The context for encoding a distance code is defined by the copy
      length corresponding to the distance. The context ids are 0, 1, 2,
      and 3 for copy lengths 2, 3, 4, and more than 4, respectively.

   7.3. Encoding of the context map

      There are two kinds of context maps, for literals and for
      distances. The size of the context map is 64 * NBLTYPESL for
      literals, and 4 * NBLTYPESD for distances. Each value in the
      context map is an integer between 0 and 255, indicating the index
      of the Huffman tree to be used when encoding the next literal or
      distance.

      The context map is encoded as a one-dimensional array,
      CMAPL[0..(64 * NBLTYPESL - 1)] and CMAPD[0..(4 * NBLTYPESD - 1)].

      The index of the Huffman tree for encoding a literal or distance
      code with context id "cid" and block type "bltype" is

         index of literal Huffman tree = CMAPL[bltype * 64 + cid]

         index of distance Huffman tree = CMAPD[bltype * 4 + cid]

      The values of the context map are encoded with the combination
      of run length encoding for zero values and Huffman coding. Let
      RLEMAX denote the number of run length codes and NTREES denote the
      maximum value in the context map plus one. NTREES must equal the
      number of different values in the context map, in other words,
      the different values in the context map must be the [0..NTREES-1]
      interval. The alphabet of the Huffman code has the following
      RLEMAX + NTREES symbols:

            0: value zero
            1: repeat a zero 2-3 times, read 1 bit for repeat length
            2: repeat a zero 4-7 times, read 2 bits for repeat length
            ...
            RLEMAX: repeat a zero (2^RLEMAX)-(2^(RLEMAX+1) - 1) times,
                    read RLEMAX bits for repeat length
            RLEMAX + 1: value 1
            ...
            RLEMAX + NTREES - 1: value NTREES - 1

      If RLEMAX = 0, the run length coding is not used, and the symbols
      of the alphabet are directly the values in the context map. We can
      now define the format of the context map (the same format is used 
      for literal and distance context maps):

          1-5 bits: RLEMAX, 0 is encoded with one 0 bit, and values
                    1 - 16 are encoded with bit pattern 1xxxx

            Huffman code with alphabet size NTREES + RLEMAX

            Context map size values encoded with the above Huffman code
               and run length coding for zero values

            1 bit:  IMTF bit, if set, we do an inverse move-to-front
                    transform on the values in the context map to get
                    the Huffman code indexes

      For the encoding of NTREES see Section 9.2.

8. Static dictionary

   At any given point during decoding the compressed data, a reference
   to a duplicated string in the output produced so far has a maximum
   backward distance value, which is the minumum of the window size and
   the number of output bytes produced. However, decoding a distance
   from the input stream, as described in section 4, can produce
   distances that are greater than this maximum allowed value. The
   difference between these distances and the first invalid distance
   value is treated as reference to a word in the static dictionary
   given in Appendix A. The maximum valid copy length for a static
   dictionary reference is 24. The static dictionary has three parts:

      * DICT[0..DICTSIZE], an array of bytes
      * DOFFSET[0..24], an array of byte offset values for each length
      * NDBITS[0..24], an array of bit-depth values for each length

   The number of static dictionary words for a given length is:

      NWORDS[length] = 0                       (if length < 3)
      NWORDS[length] = (1 << NDBITS[lengths])  (if length >= 3)

   DOFFSET and DICTSIZE are defined by the following recursion:

      DOFFSET[0] = 0
      DOFFSET[length + 1] = DOFFSET[length] + length * NWORDS[length]
      DICTSIZE = DOFFSET[24] + 24 * NWORDS[24]

   The offset of a word within the DICT array for a given length and
   index is:

      offset(length, index) = DOFFSET[length] + index * length

   Each static dictionary word has 64 different forms, given by applying
   a word transformation to a base word in the DICT array. The list of
   word transformations is given in Appendix B. The static dictionary
   word for a <length, distance> pair can be reconstructed as follows:

      word_id = distance - (max allowed distance + 1)
      index = word_id % NWORDS[length]
      base_word = DICT[offset(length, index)..offset(length, index+1))
      transform_id = word_id >> NBITS[length]

   The string copied to the output stream is computed by applying the 
   transformation to the base dictionary word. If transform_id is
   greater than 63 or length is greater than 24, the compressed data set
   is invalid and must be discarded.

9. Compressed data format

   In this section we describe the format of the compressed data set in
   terms of the format of the individual data items described in the
   previous secions.

   9.1. Format of the stream header

      The stream header has only the following one field:

          1-4 bits: WBITS, a value in the range 16 - 24, value 16 is
                    encoded with one 0 bit, and values 17 - 24 are
                    encoded with bit pattern 1xxx

      The size of the sliding window, which is the maximum value of any
      non-dictionary reference backward distance, is given by the
      following formula:

         window size = (1 << WBITS) - 16

   9.2. Format of the meta-block header

      A compliant compressed data set has at least one meta-block. Each
      meta-block contains a header, with information about the
      uncompressed length of the meta-block, and a bit signaling if the
      meta-block is the last one. The format of the meta-block header is
      the following:

            1 bit:  ISLAST, set to 1 if this is the last meta-block
            1 bit:  ISEMPTY, set to 1 if the meta-block is empty, this
                    field is only present if ISLAST bit is set, since
                    only the last meta-block can be empty
            2 bits: MNIBBLES, (# of nibbles to represent the length) - 4

            (MNIBBLES + 4) x 4 bits: MLEN - 1, where MLEN is the length
               of the meta-block in the input data in bytes

            1 bit:  ISUNCOMPRESSED, if set to 1, any bits of input up to
                    the next byte boundary are ignored, and the rest of
                    the meta-block contains MLEN bytes of literal data;
                    this field is only present if ISLAST bit is not set

         1-11 bits: NBLTYPESL, # of literal block types, encoded with
                    the following variable length code:

                          Value   Bit Pattern
                          -----   -----------
                            1      0
                            2      1000
                           3-4     1001x
                           5-8     1010xx
                           9-16    1011xxx
                          17-32    1100xxxx
                          33-64    1101xxxxx
                          65-128   1110xxxxxx
                         129-256   1111xxxxxxx

            Huffman code over the block type code alphabet for literal
               block types, appears only if NBLTYPESL >= 2

            Huffman code over the block length code alphabet for literal
               block lengths, appears only if NBLTYPESL >= 2

            Block length code + Extra bits for first literal block 
               length, appears only if NBLTYPESL >= 2

         1-11 bits: NBLTYPESI, # of insert-and-copy block types, encoded
                    with the same variable length code as above

            Huffman code over the block type code alphabet for insert-
               and-copy block types, only if NBLTYPESI >= 2

            Huffman code over the block length code alphabet for insert-
               and-copy block lengths, only if NBLTYPESI >= 2

            Block length code + Extra bits for first insert-and-copy
               block length, only if NBLTYPESI >= 2

         1-11 bits: NBLTYPESD, # of distance block types, encoded with
                    the same variable length code as above

            Huffman code over the block type code alphabet for distance
               block types, appears only if NBLTYPESD >= 2

            Huffman code over the block length code alphabet for
               distance block lengths, only if NBLTYPESD >= 2

            Block length code + Extra bits for first distance block
               length, only if NBLTYPESD >= 2

            2 bits: NPOSTFIX, parameter used in the distance coding

            4 bits: four most significant bits of NDIRECT, to get the
                    actual value of the parameter NDIRECT, left-shift
                    this four bit number by NPOSTFIX bits

            NBLTYPESL x 2 bits: context mode for each literal block type

         1-11 bits: NTREESL, # of literal Huffman trees, encoded with
                    the same variable length code as NBLTYPESL

            Literal context map, encoded as described in Paragraph 7.3,
               appears only if NTREESL >= 2, otherwise the context map
               has only zero values

         1-11 bits: NTREESD, # of distance Huffman trees, encoded with
                    the same variable length code as NBLTYPESD

            Distance context map, encoded as described in Paragraph 7.3,
               appears only if NTREESD >= 2, otherwise the context map
               has only zero values

            NTREESL Huffman codes for literals

            NBLTYPESI Huffman codes for insert-and-copy lengths

            NTREESD Huffman codes for distances

   9.3. Format of the meta-block data

      The compressed data part of a meta-block consists of a series of
      commands. Each command has the following format:

            Block type code for next insert-and-copy block type, appears
               only if NBLTYPESI >= 2 and the previous insert-and-copy
               block has ended

            Block length code + Extra bits for next insert-and-copy
               block length, appears only if NBLTYPESI >= 2 and the
               previous insert and-copy block has ended

            Insert-and-copy length, encoded as in section 5, using the
               insert-and-copy length Huffman code with the current
               insert-and-copy block type index

            Insert length number of literals, with the following format:

                  Block type code for next literal block type, appears
                     only if NBLTYPESL >= 2 and the previous literal
                     block has ended

                  Block length code + Extra bits for next literal block
                     length, appears only if NBLTYPESL >= 2 and the
                     previous literal block has ended

                  Next byte of the input data, encoded with the literal
                     Huffman code with the index determined by the
                     previuos two bytes of the input data, the current
                     literal block type and the context map, as
                     described in Paragraph 7.3.

            Block type code for next distance block type, appears only
               if NBLTYPESD >= 2 and the previous distance block has
               ended

            Block length code + Extra bits for next distance block
               length, appears only if NBLTYPESD >= 2 and the previous
               distance block has ended

            Distance code, encoded as in section 4, using the distance
               Huffman code with the current distance block type index,
               appears only if the distance code is not an implicit 0,
               as indicated by the insert-and-copy length code

      The number of commands in the meta-block is such that the sum of
      insert lengths and copy lengths over all the commands gives the
      uncompressed length, MLEN encoded in the meta-block header.

10. Decoding algorithm

   The decoding algorithm that produces the output data is as follows:

      read window size
      do
         read ISLAST bit
         if ISLAST
            read ISEMPTY bit
            if ISEMPTY
               break from loop
         read MLEN
         if not ISLAST
            read ISUNCOMPRESSED bit
            if ISUNCOMPRESSED
               skip any bits up to the next byte boundary
               copy MLEN bytes of input to the output stream
               continue to the next meta-block
         loop for each three block categories (i = L, I, D)
            read NBLTYPESi
            if NBLTYPESi >= 2
               read Huffman code for block types, HTREE_BTYPE_i
               read Huffman code for block lengths, HTREE_BLEN_i
               read block length, BLEN_i
               set block type, BTYPE_i to 0
               initialize second last and last block types to 0 and 1
            else
               set block type, BTYPE_i to 0
               set block length, BLEN_i to 268435456
         read NPOSTFIX and NDIRECT
         read array of literal context modes, CMODE[]
         read NTREESL
         if NTREESL >= 2
            read literal context map, CMAPL[]
         else
            fill CMAPL[] with zeros
         read NTREESD
         if NTREESD >= 2
            read distance context map, CMAPD[]
         else
            fill CMAPD[] with zeros
         read array of Huffman codes for literals, HTREEL[]
         read array of Huffman codes for insert-and-copy, HTREEI[]
         read array of Huffman codes for distances, HTREED[]
         do
            if BLEN_I is zero
               read block type using HTREE_BTYPE_I and set BTYPE_I
               read block length using HTREE_BLEN_I and set BLEN_I
            decrement BLEN_I
            read insert and copy length, ILEN, CLEN with HTREEI[BTYPE_I]
            loop for ILEN
               if BLEN_L is zero
                  read block type using HTREE_BTYPE_L and set BTYPE_L
                  read block length using HTREE_BLEN_L and set BLEN_L
               decrement BLEN_L
               look up context mode CMODE[BTYPE_L]
               compute context id, CIDL from last two bytes of output
               read literal using HTREEL[CMAPL[64 * BTYPE_L + CIDL]]
               copy literal to output stream
            if number of output bytes produced in the loop is MLEN
               break from loop
            if distance code is implicit zero from insert-and-copy code
               set backward distance to the last distance
            else
               if BLEN_D is zero
                  read block type using HTREE_BTYPE_D and set BTYPE_D
                  read block length using HTREE_BLEN_D and set BLEN_D
               decrement BLEN_D
               compute context id, CIDD from CLEN
               read distance code with HTREED[CMAPD[4 * BTYPE_D + CIDD]]
               compute distance by distance short code substitution
            move backwards distance bytes in the output stream, and
              copy CLEN bytes from this position to the output stream,
              or look up the static dictionary word and copy it to the
              output stram
         while number of output bytes produced in the loop < MLEN
      while not ISLAST

      Note that a duplicated string reference may refer to a string in a
      previous meta-block; i.e., the backward distance may cross one or
      more meta-block boundaries. However a backward copy distance
      cannot refer past the beginning of the output stream, and it can
      not be greater than the window size, any such distance must be
      interpreted as a reference to a static dictionary word. Note also
      that the referenced string may overlap the current position; for
      example, if the last 2 bytes decoded have values X and Y, a string
      reference with <length = 5, distance = 2> adds X,Y,X,Y,X to the
      output stream.

11. References

   [1] Huffman, D. A., "A Method for the Construction of Minimum
       Redundancy Codes", Proceedings of the Institute of Radio
       Engineers, September 1952, Volume 40, Number 9, pp. 1098-1101.

   [2] Ziv J., Lempel A., "A Universal Algorithm for Sequential Data
       Compression", IEEE Transactions on Information Theory, Vol. 23,
       No. 3, pp. 337-343.

   [3] Deutsch, P., "DEFLATE Compressed Data Format Specification
       version 1.3", RFC 1951, Aladdin Enterprises, May 1996.
       http://www.ietf.org/rfc/rfc1951.txt

12. Source code

   Source code for a C language implementation of a "brotli" compliant
   decompressor and a C++ language implementation of a compressor is
   available in the brotli/ directory within the font-compression-
   reference open-source project:
   https://code.google.com/p/font-compression-reference/source/browse/

Appendix A. List of dictionary words

   TO BE WRITTEN

Appendix B. List of word transformations

   TO BE WRITTEN