1.4. THE SPHERE PACKING-COVERING PROBLEM 15
d lattice packing covering constant γ ≈
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.