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.
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