Leech lattice

In

The Leech lattice Λ₂₄ is the unique lattice in 24-dimensional Euclidean space, E²⁴, with the following list of properties:

• It is unimodular; i.e., it can be generated by the columns of a certain 24×24 matrix with determinant 1.
• It is even; i.e., the square of the length of each vector in Λ₂₄ is an even integer.
• The length of every non-zero vector in Λ₂₄ is at least 2.

The last condition is equivalent to the condition that unit balls centered at the points of Λ₂₄ do not overlap. Each is tangent to 196,560 neighbors, and this is known to be the largest number of non-overlapping 24-dimensional unit balls that can simultaneously touch a single unit ball. This arrangement of 196,560 unit balls centred about another unit ball is so efficient that there is no room to move any of the balls; this configuration, together with its mirror-image, is the only 24-dimensional arrangement where 196,560 unit balls simultaneously touch another. This property is also true in 1, 2 and 8 dimensions, with 2, 6 and 240 unit balls, respectively, based on the integer lattice, hexagonal tiling and E₈ lattice, respectively.

It has no root system and in fact is the first unimodular lattice with no roots (vectors of norm less than 4), and therefore has a centre density of 1. By multiplying this value by the volume of a unit ball in 24 dimensions, ${\displaystyle {\tfrac {\pi ^{12}}{12!}}}$, one can derive its absolute density.

Conway (1983) showed that the Leech lattice is isometric to the set of simple roots (or the Dynkin diagram) of the reflection group of the 26-dimensional even Lorentzian unimodular lattice II_25,1. By comparison, the Dynkin diagrams of II_9,1 and II_17,1 are finite.

Applications

The

The Leech lattice can be constructed in a variety of ways. Like all lattices, it can be constructed by taking the integral span of the columns of its generator matrix, a 24×24 matrix with determinant 1.

 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

^[1]

Using the binary Golay code

The Leech lattice can be explicitly constructed as the set of vectors of the form 2^−3/2(a₁, a₂, ..., a₂₄) where the a_i are integers such that

${\displaystyle a_{1}+a_{2}+\cdots +a_{24}\equiv 4a_{1}\equiv 4a_{2}\equiv \cdots \equiv 4a_{24}{\pmod {8}}}$

and for each fixed residue class modulo 4, the 24 bit word, whose 1s correspond to the coordinates i such that a_i 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,

${\displaystyle L~=~~(4B+C)\otimes {1_{2^{12}}}~~+~~~{1_{2^{24}}}\otimes 2G~~~}$ , (${\displaystyle {1_{n}}}$ is a ones vector of size n),

• G - 24-bit Golay code
• B - Binary integer sequence
• C -
  Thue-Morse Sequence
  or integer bit parity sum (that give chirality of the lattice)

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

Using the Lorentzian lattice II_25,1

The Leech lattice can also be constructed as ${\displaystyle w^{\perp }/w}$ where w is the Weyl vector:

${\displaystyle (0,1,2,3,\dots ,22,23,24;70)}$

in the 26-dimensional even Lorentzian

The vector ${\displaystyle (0,1,2,3,\dots ,22,23,24)}$ in this construction is really the

Based on other lattices

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, H₈. This construction is known as the Turyn construction of the Leech lattice.

As a laminated lattice

Starting with a single point, Λ₀, one can stack copies of the lattice Λ_n to form an (n + 1)-dimensional lattice, Λ_n+1, without reducing the minimal distance between points. Λ₁ corresponds to the

The Leech lattice can also be constructed using the ring of icosians. 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=4ab. Then the matrix ${\displaystyle {\begin{pmatrix}Ia&H/2\\H/2&Ib\end{pmatrix}}}$ defines a bilinear form in 2n 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/23Z ∪ ∞) 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

Using a Paley matrix

Chapman (2001) described a construction using a

${\displaystyle D_{24}}$

${\displaystyle \mathbb {Q} ({\sqrt {-23}})}$

Using higher power residue codes

Raji (2005) constructed the Leech lattice using higher power residue codes over the ring ${\displaystyle Z_{4}}$. A similar construction is used to construct some of the other lattices of rank 24.

Using octonions

If L is the set of octonions with coordinates on the ${\displaystyle E_{8}}$ lattice, then the Leech lattice is the set of triplets ${\displaystyle (x,y,z)}$ such that

${\displaystyle x,y,z\in L}$

${\displaystyle x+y,y+z,x+z\in L{\bar {s}}}$

${\displaystyle x+y+z\in Ls}$

where ${\displaystyle s={\frac {1}{2}}(-e_{1}+e_{2}+e_{3}+e_{4}+e_{5}+e_{6}+e_{7})}$. This construction is due to (Wilson 2009).

Symmetries

The Leech lattice is highly symmetrical. Its

Despite having such a high rotational symmetry group, the Leech lattice does not possess any hyperplanes of reflection symmetry. In other words, the Leech lattice is

Conway, Parker & Sloane (1982) showed that the covering radius of the Leech lattice is ${\displaystyle {\sqrt {2}}}$; in other words, if we put a closed ball of this radius around each lattice point, then these just cover Euclidean space. The points at distance at least ${\displaystyle {\sqrt {2}}}$ from all lattice points are called the deep holes of the Leech lattice. There are 23 orbits of them under the automorphism group of the Leech lattice, and these orbits correspond to the 23

The Leech lattice has a density of ${\displaystyle {\tfrac {\pi ^{12}}{12!}}\approx 0.001930}$. Cohn & Kumar (2009) showed that it gives the densest lattice packing of balls in 24-dimensional space. Henry Cohn, Abhinav Kumar, and Stephen D. Miller et al. (2016) improved this by showing that it is the densest sphere packing, even among non-lattice packings.

The 196560 minimal vectors are of three different varieties, known as shapes:

• ${\displaystyle 1104={\binom {24}{2}}\cdot 2^{2}}$ vectors of shape (4²,0²²), for all permutations and sign choices;
• ${\displaystyle 97152=759\cdot 2^{8}\cdot {\frac {1}{2}}}$ vectors of shape (2⁸,0¹⁶), where the '2's correspond to an octad in the Golay code, and there are any even number of minus signs;
• ${\displaystyle 98304=2^{12}\cdot 24}$ vectors of shape (∓3,±1²³), where the lower sign is used for the '1's of any codeword of the Golay code, and the '∓3' can appear in any position.

The ternary Golay code, binary Golay code and Leech lattice give very efficient 24-dimensional spherical codes of 729, 4096 and 196560 points, respectively. Spherical codes are higher-dimensional analogues of Tammes problem, which arose as an attempt to explain the distribution of pores on pollen grains. These are distributed as to maximise the minimal angle between them. In two dimensions, the problem is trivial, but in three dimensions and higher it is not. An example of a spherical code in three dimensions is the set of the 12 vertices of the regular icosahedron.

Theta series

One can associate to any (positive-definite) lattice Λ a theta function given by

${\displaystyle \Theta _{\Lambda }(\tau )=\sum _{x\in \Lambda }e^{i\pi \tau \|x\|^{2}}\qquad \operatorname {Im} \tau >0.}$

The theta function of a lattice is then a holomorphic function on the upper half-plane. Furthermore, the theta function of an even unimodular lattice of rank n is actually a modular form of weight n/2 for the full modular group PSL(2,Z). The theta function of an integral lattice is often written as a power series in ${\displaystyle q=e^{2i\pi \tau }}$ so that the coefficient of qⁿ gives the number of lattice vectors of squared norm 2n. In the Leech lattice, there are 196560 vectors of squared norm 4, 16773120 vectors of squared norm 6, 398034000 vectors of squared norm 8 and so on. The theta series of the Leech lattice is

${\displaystyle {\begin{aligned}\Theta _{\Lambda _{24}}(\tau )&=E_{12}(\tau )-{\frac {65520}{691}}\Delta (\tau )\\[5pt]&=1+\sum _{m=1}^{\infty }{\frac {65520}{691}}\left(\sigma _{11}(m)-\tau (m)\right)q^{m}\\[5pt]&=1+196560q^{2}+16773120q^{3}+398034000q^{4}+\cdots ,\end{aligned}}}$

where ${\displaystyle E_{12}(\tau )}$ is the normalized Eisenstein series of weight 12, ${\displaystyle \Delta (\tau )}$ is the

${\displaystyle \sigma _{11}(n)}$

${\displaystyle \tau (n)}$

${\displaystyle {\frac {65520}{691}}\left(\sigma _{11}(m)-\tau (m)\right).}$

History

Many of the cross-sections of the Leech lattice, including the Coxeter–Todd lattice and Barnes–Wall lattice, in 12 and 16 dimensions, were found much earlier than the Leech lattice. O'Connor & Pall (1944) discovered a related odd unimodular lattice in 24 dimensions, now called the odd Leech lattice, one of whose two even neighbors is the Leech lattice. The Leech lattice was discovered in 1965 by John Leech (1967, 2.31, p. 262), by improving some earlier sphere packings he found (Leech 1964).

Witt (1941, p. 324), has a single rather cryptic sentence mentioning that he found more than 10 even unimodular lattices in 24 dimensions without giving further details. Witt (1998, p. 328–329) stated that he found 9 of these lattices earlier in 1938, and found two more, the Niemeier lattice with A²⁴
₁ root system and the Leech lattice (and also the odd Leech lattice), in 1940.

See also

• Sphere packing
• E₈ lattice

References