The lattice packing problem in dimension 9 by Voronoi's algorithm

TL;DR

This work completes Voronoi's algorithm for the lattice packing problem in dimension $9$ by exhaustively enumerating all perfect forms, totaling $2237251040$ non-similar forms and confirming that the laminated lattice $\Lambda_9$ is the densest lattice packing in dimension $9$ with Hermite constant $\gamma_9=2$. The authors advance the state of the art with substantial algorithmic and implementation innovations, including a canonical-form framework, a dual-description strategy under symmetry, and a Recursive Adjacency Decomposition Method, enabling scalable parallel processing and data sharing. They also deliver a precise classification of kissing numbers, characterize high-incidence forms, and analyze extreme and dual-extreme structures, culminating in insights such as the Bergé-Martinet invariant landscape in dimension $9$. The results have immediate implications for lattice packing theory, computational discrete geometry, and related number-theoretic lattice invariants, and the authors provide public data and software to enable further exploration.

Abstract

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 $2 \cdot \{ 1, \ldots, 91, 99, 120, \ldots, 129, 136 \}$.

Figures (5)

• Figure 1: The density of a lattice packing is determined by its first minimum $\lambda_1(\mathcal{L})$ and its (co)volume $\det(\mathcal{L})$.
• Figure 2: On the left, part of Ryshkov polyhedron inside the cone of $2$-dimensional positive definite quadratic forms. Its vertices are perfect forms, such as the red and green vertices on the right. The group $\mathop{\mathrm{GL}}\nolimits_d(\mathbb{Z})$ acts as a symmetry group on the Ryshkov polyhedron. The red vertices are neighbouring perfect forms of the green vertex, indicating an exploration step of Voronoi's algorithm.
• Figure 3: This figure shows an abstract view of (part of) the face lattice of a pointed polyhedral cone $P$ of dimension $n$. Face inclusions are indicated with solid lines. The facet $F$ is adjacent to $F_1$ and $F_2$ as they share ridges $R_1$ and $R_2$ respectively. The facet $F$ defines a subcone indicated in red with extreme rays $v_2, v_3 \in F$. The ridges $R_1$ and $R_2$ of $P$ are facets of $F$.
• Figure 4: Part of Voronoi graph showing all perfect forms that are only connected via high-incidence perfect forms with kissing number at least $2 \cdot 61$. $Q_{a,b}$ indicates a perfect form with kissing number $2a$ and $b$ the size of its automorphism group or a label indicating a specific known lattice. Their properties are listed in \ref{['tab:special_forms']}.
• Figure 5: Part of face (poset) lattice of $\mathop{\mathrm{Vor}}\nolimits(Q_{99}), \mathop{\mathrm{Vor}}\nolimits(Q_{129})$ and $\mathop{\mathrm{Vor}}\nolimits(Q_{\mathsf{\Lambda}_9})$, with combinatorially equivalent faces merged. The nodes are labelled with the incidence of the face. On the left the dimension of the face is denoted.

Theorems & Definitions (20)

• Theorem 1
• Corollary 1
• Theorem 2
• Definition 1: Lattice
• Definition 2: Packing density
• Example 1
• Definition 3: Hermite invariant and constant
• Definition 4: Ryshkov polyhedron ryshkov1970polyhedron
• Lemma 1: Convex optimization
• proof