1.4. THE SPHERE PACKING-COVERING PROBLEM 15

d lattice packing covering constant γ ≈

6 L6

pc

1.411081

7 L7

pc

1.494983

8 E8 1.414213

24 Λ24 1.414213

Table 5. Lattice packing-covering records.

optimal in their dimensions. In Section 5.5 we show that the Leech lattice gives at

least a locally optimal lattice packing-covering constant. Despite some effort, it is

an open question whether or not the root lattice E8 gives a locally optimal lattice

packing-covering constant as well. The corresponding values of the lattice packing-

covering constant are shown in Table 5. The exact value for the smallest known

lattice packing-covering constant in dimension 6 is 2 2

√

798 − 56 (see Section 5.3).

In dimensions between 8 and 24 we still do not know enough to state any serious

conjectures. We believe that there is still a lot to discover.