1.4. THE SPHERE PACKING-COVERING PROBLEM 13
d lattice covering density Θ in [69]
6 L6 c 2.4648 2.5511
7 L7 c 2.9000 3.0596
8 L8 c 3.1422 3.6658
9 L9 c 4.2685 4.3889
10 L10 c 5.1544 5.2517
11 L11 c 5.5182 6.2813
12 L12 c 7.4655 7.5101
13 L13 c 7.7621 8.9768
14 L14
c
8.8252 10.727
15 L15
c
11.0049 12.817
16 A16

15.3109 15.3109
17 A17
6
12.3574 18.2878
18 A18

21.8409 21.8409
19 A19
10
21.2292 26.0818
20 A20
7
20.3668 31.1434
21 A21
11
27.7731 37.1845
22 Λ22

27.8839 27.8839
23 Λ23

15.3218 15.3218
24 Λ24 7.903536 7.903536
Table 3. Least dense known (lattice) coverings up to dimension 24.
1.4. The sphere packing-covering problem
In the (simultaneous) packing-covering problem we ask for discrete sets Λ or
lattices minimizing the quotient Θ(Λ)/δ(Λ), respectively the packing-covering con-
stant
γ(Λ) = µ(Λ)/λ(Λ).
We denote by
γd = inf{γ(L) : L
Rd
lattice }
and
γd

= inf{γ(Λ) : Λ
Rd
a discrete set }
the smallest lattice packing-covering constant, respectively the smallest packing-
covering constant.
The (lattice) packing-covering problem has been studied in different contexts
and there are several different names and interpretations of the packing-covering
constants γd and γd
∗.
Lagarias and Pleasants [160] refer to them as the “De-
lone packing-covering constants”. Ryshkov [211] studied the equivalent problem of
minimizing the density of (r, R)-systems. An (r, R)-system is a discrete point set
X
Rd
where (1) the distance between any two points of X is at least r and (2)
the distance from any point in
Rd
to a point in X is at most R. Thus r = λ(X)/2
and R = µ(X).
Geometrically we may think of the lattice packing-covering problem as of max-
imizing the minimum distance between lattice points in a lattice covering with unit
spheres. Alternatively, we may think of it as minimizing the radius of a largest
sphere that could additionally be packed into a lattice packing of unit spheres (see
Figure 7). This minimal gap-radius is equal to γd 1. Therefore the problem
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