The lattice packing problem in dimension 9 by Voronoi's algorithm
Fuente:
arXiv
Enregistré dans:
| Auteurs principaux: | , |
|---|---|
| Format: | Preprint |
| Publié: |
2025
|
| Sujets: | |
| Accès en ligne: | |
| Tags: |
Ajouter un tag
Pas de tags, Soyez le premier à ajouter un tag!
|
| _version_ | 1866911429666275328 |
|---|---|
| author | Sikirić, Mathieu Dutour van Woerden, Wessel |
| author_facet | Sikirić, Mathieu Dutour van Woerden, Wessel |
| contents | In 1908, Voronoi introduced an algorithm that solves the lattice packing problem in any dimension in finite time. Voronoi showed that any lattice with optimal packing density must be a so-called perfect lattice, and his algorithm enumerates the finitely many perfect lattices up to similarity in a fixed dimension. However, due to the high complexity of the algorithm this enumeration had, until now, only been completed up to dimension 8.
In this work we compute all 2237251040 perfect lattices in dimension 9 via Voronoi's algorithm. As a corollary, this shows that the laminated lattice $Λ_9$ gives the densest lattice packing in dimension 9. Equivalently, we show that the Hermite constant $γ_9$ in dimension 9 equals $2$. Furthermore, we extend a result by Watson (1971) and show that the set of possible kissing numbers in dimension 9 is precisely $2 \cdot \{ 1, \ldots, 91, 99, 120, \ldots, 129, 136 \}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_20719 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The lattice packing problem in dimension 9 by Voronoi's algorithm Sikirić, Mathieu Dutour van Woerden, Wessel Number Theory 11H31, 11H55, 05E18 In 1908, Voronoi introduced an algorithm that solves the lattice packing problem in any dimension in finite time. Voronoi showed that any lattice with optimal packing density must be a so-called perfect lattice, and his algorithm enumerates the finitely many perfect lattices up to similarity in a fixed dimension. However, due to the high complexity of the algorithm this enumeration had, until now, only been completed up to dimension 8. In this work we compute all 2237251040 perfect lattices in dimension 9 via Voronoi's algorithm. As a corollary, this shows that the laminated lattice $Λ_9$ gives the densest lattice packing in dimension 9. Equivalently, we show that the Hermite constant $γ_9$ in dimension 9 equals $2$. Furthermore, we extend a result by Watson (1971) and show that the set of possible kissing numbers in dimension 9 is precisely $2 \cdot \{ 1, \ldots, 91, 99, 120, \ldots, 129, 136 \}$. |
| title | The lattice packing problem in dimension 9 by Voronoi's algorithm |
| topic | Number Theory 11H31, 11H55, 05E18 |
| url | https://arxiv.org/abs/2508.20719 |