If we denote \(SL(V)\) to be the group of all unimodular linear transformations on \(V\), we can now define \[\begin{align} c_{d} = \sup\left\{ \mu\left( g B_{r}(0)\right) \ | \ r> 0,~g \in SL(V) \text{ and } g B_{r}(0) \cap \Lambda_0 = \{ 0\}\right\}, \end{align}\]
The most optimal lattice packing density in \(d\)-dimensions is \(c_{d}/2^{d}\). Note that the most optimal sphere packing may not be a lattice packing for large \(d\). It is a common belief in the community that the optimal packings in large dimensions are probably aperiodic.
Here is a quote from [1]:
For \(d\ge 4\) the problem remains unsolved. Upper and lower bounds on the density are known, but they differ by an exponential factor as \(d \to \infty\). Each dimension seems to have its own peculiarities, and it does not seem likely that a single, simple construction will give the best packing in every dimension.
Also says that
Not every sphere packing is a lattice packing, and in fact it is plausible that in all sufficiently large dimensions, there are packings denser than every lattice packing. However, many important examples in low dimensions are lattice packings.
The exact value of \(c_d\) is known only for \(d \in \{1,2,3,4,5,6,7,8,24\}\).
Here are some known lower bounds. See Siegel Mean Value theorem for getting some of these bounds.
| Bound | Due to | Valid in range |
|---|---|---|
| \(c_{d} \ge 1\) | Minkowski (cf. [2]) | Any \(d \ge 1\) |
| \(c_{d} \ge 2(d-1)\) | Ball [3] | Any \(d \ge 1\) |
| \(c_{4n} \ge 8.8 n\) | Vance [4] | \(d = 4n, n \ge 1\) |
| \(c_{2 \varphi(n)} \ge n\) | Venkatesh [5] | \(d= 2\varphi(k)\), for some \(k \ge 1\) |
| \(c_{n} \ge 65963 n\) | Venkatesh [5] | \(d= n\) sufficiently large |
| \(c_{ \varphi(n)} \ge n\) | G. Viazovska [6] | \(d= \varphi(k)\), for some \(k \ge 1\) |
| \(c_{ n} \ge cn^2\) | Klartag [7] | Any \(n\) sufficiently large |
The upper bound given by Cohn-Elkies is a celebrated result. See [1]. In individual dimensions, this has lead to the resolution of the sphere packing problem in dimensions 8 and 24.
Other than that, there are Kabatiansky-Levenshtein bounds that have the statement \[\begin{equation} c_d \ll 2^{0.41.. d}. \end{equation}\] See Cohn-Zhao, for example.
This page was updated on October 1, 2025.
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