Editing Leech lattice (section)

===Using the binary Golay code===

The Leech lattice can be explicitly constructed as the set of vectors of the form 2<sup>&minus;3/2</sup>(''a''<sub>1</sub>, ''a''<sub>2</sub>, ..., ''a''<sub>24</sub>) where the ''a''<sub>''i''</sub> are integers such that

:<math>a_1+a_2+\cdots+a_{24}\equiv 4a_1\equiv 4a_2\equiv\cdots\equiv4a_{24}\pmod 8</math>

and for each fixed residue class modulo 4, the 24 bit word, whose 1s correspond to the coordinates ''i'' such that ''a''<sub>''i''</sub> belongs to this residue class, is a word in the [[binary Golay code]]. The Golay code, together with the related Witt design, features in a construction for the 196560 minimal vectors in the Leech lattice.

Leech lattice (L mod 8) can be directly constructed by combination of the 3 following sets,

<math>L ~ = ~~ (4B + C)\otimes{1_{2^{12}}}  ~~ + ~~~  {1_{2^{24}}}\otimes 2G  ~~~ </math> ,      (<math>{1_{n}}</math> is a ones vector of size n),
* G - 24-bit Golay code
* B - Binary integer sequence
* C - [[Thue-Morse sequence|Thue-Morse Sequence]] or integer bit parity sum (that give chirality of the lattice)
<syntaxhighlight lang="c">
24-bit Golay  [2^12 codes]   24-bit integer[2^24 codes]   Parity   Leech Lattice [2^36 codes]
G =                          B =                          C =      L = (4B + C) ⊕ 2G
00000000 00000000 00000000   00000000 00000000 00000000   0        00000000 00000000 00000000
11111111 00000000 00000000   10000000 00000000 00000000   1        22222222 00000000 00000000
11110000 11110000 00000000   01000000 00000000 00000000   1        22220000 22220000 00000000
00001111 11110000 00000000   11000000 00000000 00000000   0        ...
11001100 11001100 00000000   00100000 00000000 00000000   1        51111111 11111111 11111111
00110011 11001100 00000000   10100000 00000000 00000000   0        73333333 11111111 11111111
00111100 00111100 00000000   01100000 00000000 00000000   0        ...
11000011 00111100 00000000   11100000 00000000 00000000   1        15111111 11111111 11111111
10101010 10101010 00000000   00010000 00000000 00000000   1        37333333 11111111 11111111
01010101 10101010 00000000   10010000 00000000 00000000   0        ...
01011010 01011010 00000000   01010000 00000000 00000000   0        44000000 00000000 00000000
10100101 01011010 00000000   11010000 00000000 00000000   1        66222222 00000000 00000000
...                          ...                          ...      ...
11111111 11111111 11111111   11111111 11111111 11111111   0        66666666 66666666 66666666
</syntaxhighlight>