12 1. FROM QUADRATIC FORMS TO SPHERE PACKINGS AND COVERINGS

d lattice Θd author(s)

2 A2 ∗ 1.209199 Kershner [151]

3 A3 ∗ 1.463505 Bambah [11]

4 A4 ∗ 1.765529 Delone and Ryshkov [78]

5 A5 ∗ 2.124286 Ryshkov and Baranovskii

[210], [21], [22], [215], [216]

Table 2. Known optimal lattice coverings.

The lattice optimality of the body centered cubic lattice A3 ∗ was first proven

by Bambah [11]. Later, Barnes [24] substantially simplified Bambah’s proof and

strengthened the result by showing that in dimensions 2 and 3 the lattice Ad

∗

is

the unique locally optimal lattice covering. A third proof was given by Few [110].

Nevertheless, it is still an open problem to determine

Θ3.∗

In [10] Bambah conjectured that the lattice A4

∗

gives the least dense four-

dimensional lattice covering. In [78] Delone and Ryshkov proved Bambah’s conjec-

ture. In [13], [14] Baranovskii gave an alternative proof of this fact. He determined

all locally optimal lattice coverings in dimension 4. Dickson [84] gave another al-

ternative proof of this fact.

In a series of papers [210], [21], [22], [215] Ryshkov and Baranovskii solved the

lattice covering problem in dimension 5. They prepared a 140-page long monograph

[216] based on their investigations.

In [208] Ryshkov raised the natural question of finding the lowest dimension d

for which there is a better lattice covering than the one given by Ad.

∗

In the same

paper he showed that Ad

∗

is not the most eﬃcient lattice covering for all even d ≥ 114

and for all odd d ≥ 201. Table 3 shows that by now it seems that Ad ∗ is likely to

give no best lattice coverings for any dimension d ≥ 6. Most of the results in the

table were obtained quite recently, a majority of them based on the theory to be

described in Chapter 4. We keep an updated list with additional information on

the involved lattices on the web page [227].

It may not surprise that the Leech lattice Λ24 yields the best known lattice

covering in dimension 24. Its covering density was computed by Conway, Parker

and Sloane (Chapter 23 of [69]). Knowing the comparatively low covering density

of the Leech lattice, Smith [239] was able to estimate the covering densities of

the dual laminated lattices Λ22

∗

and Λ23.

∗

For a definition and more information on

laminated lattices and their duals we refer to [69]. The lattices Ad,

r

with d ≥ 2

and r 1 divides d + 1, are the so-called Coxeter lattices (see Section 5.3.6 for a

definition). They give local lattice packing optima, but seem not to give locally

optimal lattice coverings. Nevertheless, some of them currently give the thinnest

known lattice coverings. Information on the lattices Ld,

c

some of them obtained by

modifying Coxeter lattices, can be found on the web page [227].

It is tempting to conjecture that the Leech lattice gives a thinnest lattice

covering in dimension 24. We show in Section 5.5 that the Leech lattice gives at

least a local optimum of the covering function among lattices. This may not be

surprising, but on the other hand the root lattice E8 does not have this property

(see Section 5.5). Even worse, E8 is “almost a local covering maximum”. Such

“pessima” and “real” covering maxima are treated in Section 5.6.