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