Editing Leech lattice (section)

==Constructions==

The Leech lattice can be constructed in a variety of ways. Like all lattices, it can be constructed by taking the [[integer|integral]] span of the columns of its [[generator matrix]], a 24&times;24 matrix with [[determinant]] 1.

{{hidden | Leech generator matrix |
A 24x24 generator (in row convention) for the Leech Lattice is given by the following matrix divided by <math>\sqrt{8}</math>:
 <nowiki>
 8 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
 4 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
 4 0 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
 4 0 0 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
 4 0 0 0 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
 4 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
 4 0 0 0 0 0 4 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 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
 4 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
 4 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0
 4 0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0 0 0 0
 2 2 2 2 0 0 0 0 2 2 2 2 0 0 0 0 0 0 0 0 0 0 0 0
 4 0 0 0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0 0
 2 2 0 0 2 2 0 0 2 2 0 0 2 2 0 0 0 0 0 0 0 0 0 0
 2 0 2 0 2 0 2 0 2 0 2 0 2 0 2 0 0 0 0 0 0 0 0 0
 2 0 0 2 2 0 0 2 2 0 0 2 2 0 0 2 0 0 0 0 0 0 0 0
 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0
 2 0 2 0 2 0 0 2 2 2 0 0 0 0 0 0 2 2 0 0 0 0 0 0
 2 0 0 2 2 2 0 0 2 0 2 0 0 0 0 0 2 0 2 0 0 0 0 0
 2 2 0 0 2 0 2 0 2 0 0 2 0 0 0 0 2 0 0 2 0 0 0 0
 0 2 2 2 2 0 0 0 2 0 0 0 2 0 0 0 2 0 0 0 2 0 0 0
 0 0 0 0 0 0 0 0 2 2 0 0 2 2 0 0 2 2 0 0 2 2 0 0
 0 0 0 0 0 0 0 0 2 0 2 0 2 0 2 0 2 0 2 0 2 0 2 0
−3 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
</nowiki>
}}<ref>{{citation | zbl=0915.52003 | last1=Conway | first1=J.H. | author1-link=John Horton Conway | last2=Sloane | first2=N.J.A. | author2-link=Neil Sloane | others=With contributions by Bannai, E.; Borcherds, R. E.; Leech, J.; Norton, S. P.; Odlyzko, A. M.; Parker, R. A.; Queen, L.; Venkov, B. B. | title=Sphere packings, lattices and groups | edition=Third | series=Grundlehren der Mathematischen Wissenschaften | volume=290 | location=New York, NY | publisher=[[Springer-Verlag]] | isbn=978-0-387-98585-5 | mr=662447 | year=1999 | url-access=registration | url=https://archive.org/details/spherepackingsla0000conw_b8u0 }}</ref>

===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>

===Using the Lorentzian lattice II<sub>25,1</sub>===

The Leech lattice can also be constructed as <math>w^\perp/w</math> where ''w'' is the Weyl vector:

:<math>(0,1,2,3,\dots,22,23,24; 70)</math>

in the 26-dimensional even Lorentzian [[unimodular lattice]] [[II25,1|II<sub>25,1</sub>]]. The existence of such an integral vector of Lorentzian norm zero relies on the fact that 1<sup>2</sup> + 2<sup>2</sup> + ... + 24<sup>2</sup> is a [[square number|perfect square]] (in fact 70<sup>2</sup>); the [[24 (number)|number 24]] is the only integer bigger than 1 with this property (see [[cannonball problem]]). This was conjectured by [[Édouard Lucas]], but the proof came much later, based on [[elliptic functions]].

The vector
<math>(0,1,2,3,\dots,22,23,24)</math>
in this construction is really the [[Weyl vector]] of the even sublattice ''D''<sub>24</sub> of the odd unimodular lattice ''I''<sup>25</sup>. More generally, if ''L'' is any positive definite unimodular lattice of dimension 25 with at least 4 vectors of norm 1, then the Weyl vector of its norm 2 roots has integral length, and there is a similar construction of the Leech lattice using ''L'' and this Weyl vector.

===Based on other lattices===

{{harvtxt|Conway|Sloane|1982}} described another 23 constructions for the Leech lattice, each based on a [[Niemeier lattice]]. It can also be constructed by using three copies of the [[E8 lattice]], in the same way that the binary Golay code can be constructed using three copies of the extended [[Hamming code]], H<sub>8</sub>. This construction is known as the '''Turyn''' construction of the Leech lattice.

===As a laminated lattice===

Starting with a single point, Λ<sub>0</sub>, one can stack copies of the lattice Λ<sub>n</sub> to form an (''n''&nbsp;+&nbsp;1)-dimensional lattice, Λ<sub>''n''+1</sub>, without reducing the minimal distance between points. Λ<sub>1</sub> corresponds to the [[integer lattice]], Λ<sub>2</sub> is to the [[hexagonal lattice]], and Λ<sub>3</sub> is the [[face-centered cubic]] packing. {{harvtxt|Conway|Sloane|1982b}} showed that the Leech lattice is the unique laminated lattice in 24 dimensions.

===As a complex lattice===

The Leech lattice is also a 12-dimensional lattice over the [[Eisenstein integers]]. This is known as the '''complex Leech lattice''', and is isomorphic to the 24-dimensional real Leech lattice. In the complex construction of the Leech lattice, the [[binary Golay code]] is replaced with the [[ternary Golay code]], and the [[Mathieu group M24|Mathieu group ''M''<sub>24</sub>]] is replaced with the [[Mathieu group M12|Mathieu group ''M''<sub>12</sub>]]. The ''E''<sub>6</sub> lattice, ''E''<sub>8</sub> lattice and [[Coxeter&ndash;Todd lattice]] also have constructions as complex lattices, over either the Eisenstein or [[Gaussian integers]].

===Using the icosian ring===

The Leech lattice can also be constructed using the ring of [[icosian]]s. The icosian ring is abstractly isomorphic to the [[E8 lattice]], three copies of which can be used to construct the Leech lattice using the Turyn construction.

===Witt's construction===

In 1972 Witt gave the following construction, which he said he found in 1940, on January 28. Suppose that ''H'' is an ''n'' by ''n'' [[Hadamard matrix]], where ''n''=4''ab''. Then the matrix <math>\begin{pmatrix} Ia&H/2\\H/2&Ib\end{pmatrix}</math> defines a bilinear form in 2''n'' dimensions, whose kernel has ''n'' dimensions. The quotient by this kernel is a nonsingular bilinear form taking values in (1/2)'''Z'''. It has 3 sublattices of index 2 that are integral bilinear forms. Witt obtained the Leech lattice as one of these three sublattices by taking ''a''=2, ''b''=3, and taking ''H'' to be the 24 by 24 matrix (indexed by '''Z'''/23'''Z''' ∪ ∞) with entries Χ(''m''+''n'') where Χ(∞)=1, Χ(0)=−1, Χ(''n'')=is the quadratic residue symbol mod 23 for nonzero ''n''. This matrix ''H'' is a [[Paley matrix]] with some insignificant sign changes.

===Using a Paley matrix===

{{harvtxt|Chapman|2001}} described a construction using a
[[Hadamard matrix#Skew Hadamard matrices|skew Hadamard matrix]] of [[Paley matrix|Paley]] type.
The [[Niemeier lattice]] with root system <math>D_{24}</math> can be made into a module
for the ring of integers of the field <math>\mathbb{Q}(\sqrt{-23})</math>. Multiplying this
Niemeier lattice by a non-principal ideal of the ring of integers gives the Leech lattice.

===Using higher power residue codes===
{{harvtxt|Raji|2005}} constructed the Leech lattice using higher power residue codes over the ring <math>Z_4</math>. A similar construction is used to construct some of the other lattices of rank 24.

===Using octonions===
If ''L'' is the set of [[octonion]]s with coordinates on the <math>E_8</math> [[E8_lattice|lattice]], then the Leech lattice is the set of triplets <math>(x,y,z)</math> such that

:<math>x,y,z \in L</math>

:<math>x+y+z \in Ls</math>

:<math>x+y,\ y+z,\ x+z \in L\bar{s}</math>

where <math>s= \frac 1 2 (-e_1 + e_2 + e_3 + e_4 + e_5 + e_6 + e_7)</math>.  This construction is due to {{harv|Wilson|2009}}.