Number Theory: An Introduction to Mathematics

350 VIII The Geometry of Numbers

Table 1.Indecomposable root lattices

An={x=(ξ 0 ,ξ 1 ,...,ξn)∈Zn+^1 :ξ 0 +ξ 1 +···+ξn= 0 }(n≥ 1 );

Dn={x=(ξ 1 ,...,ξn)∈Zn:ξ 1 +···+ξneven}(n≥ 3 );

E 8 =D 8 ∪D† 8 ,whereD† 8 =( 1 / 2 , 1 / 2 ,..., 1 / 2 )+D 8 ;

E 7 ={x=(ξ 1 ,...,ξ 8 )∈E 8 :ξ 7 =−ξ 8 };

E 6 ={x=(ξ 1 ,...,ξ 8 )∈E 8 :ξ 6 =ξ 7 =−ξ 8 }.

they are all listed in Table 1. We give also their minimal vectors in terms of the canon-
ical basise 1 ,...,enofRn.
The lattice Anhasn(n+ 1 )minimal vectors, namely the vectors±(ej−ek)
( 0 ≤ j<k ≤n), and the vectorse 0 −e 1 ,e 1 −e 2 ,...,en− 1 −enform a simple
basis. By calculating the determinant ofBtB,whereBis the(n+ 1 )×nmatrix whose
columns are the vectors of this simple basis, it may be seen that the determinant of the
latticeAnis(n+ 1 )^1 /^2.
The lattice Dnhas 2n(n− 1 )minimal vectors, namely the vectors±ej±ek
( 1 ≤j<k ≤n). The vectorse 1 −e 2 ,e 2 −e 3 ,...,en− 1 −en,en− 1 +enform a
simple basis and hence the latticeDnhas determinant 2.
The latticeE 8 has 240 minimal vectors, namely the 112 vectors±ej±ek( 1 ≤
j<k≤ 8 )and the 128 vectors(±e 1 ±···±e 8 )/2 with an even number of minus
signs. The vectors

v 1 =(e 1 −e 2 −···−e 7 +e 8 )/ 2 ,v 2 =e 1 +e 2 ,

v 3 =e 2 −e 1 ,v 4 =e 3 −e 2 ,..., v 8 =e 7 −e 6 ,

form a simple basis and hence the lattice has determinant 1.
The latticeE 7 has 126 minimal vectors, namely the 60 vectors±ej±ek( 1 ≤j<

k≤ 6 ), the vectors±(e 7 −e 8 )and the 64 vectors±

(∑

6

i= 1 (±ei)−e^7 +e^8

)

/2 with

an odd number of minus signs in the sum. The vectorsv 1 ,...,v 7 form a simple basis
and the lattice has determinant

√

2.

The latticeE 6 has 72 minimal vectors, namely the 40 vectors±ej±ek( 1 ≤j<

k≤ 5 )and the 32 vectors±

(∑

5

i= 1 (±ei)−e^6 −e^7 +e^8

)

/2 with an even number of

minus signs in the sum. The vectorsv 1 ,...,v 6 form a simple basis and the lattice has
determinant

√

3.

We now return to lattice packings of balls. The densest lattices forn≤8aregiven
in Table 2. These lattices were shown to be densest by Lagrange (1773) forn=2,
by Gauss (1831) forn=3, by Korkine and Zolotareff (1872,1877) forn= 4 ,5and
by Blichfeldt (1925,1926,1934) forn= 6 , 7 ,8.
Although the densest lattice inRnis unknown for everyn>8, there are plausible
candidates in some dimensions. In particular, a lattice discovered by Leech (1967) is
believed to be densest in 24 dimensions. This lattice may be constructed in the follow-
ing way. Letpbe a prime such thatp≡3 mod 4 and letHnbe the Hadamard matrix
of ordern=p+1 constructed by Paley’s method (see Chapter V,§2). The columns
of the matrix