Hexagon

math.CO — Combinatorics

No lattice tiling of $\mathbb{Z}^{35}$ or $\mathbb{Z}^{805}$ by Lee balls of radius three

Contributed by Deep Bhattacharjee, Ushashi Bhattacharya

We prove that there is no lattice tiling of $\mathbb{Z}^{35}$ or of $\mathbb{Z}^{805}$ by Lee balls of radius three. These were the only cases left open by Xiao and Zhou among the Lee balls of dimension at least three and radius at least two with fewer than $2^{30}$ points, so the lattice Golomb--Welch conjecture holds for all of them.

A mix of human-written and AI-generated textHuman understanding: some partsGolomb-Welch conjectureLee metricgroup ringlattice tilingperfect code

Oct 17, 2026PDF ↗View work →

Matching works

No lattice tiling of $\mathbb{Z}^{35}$ or $\mathbb{Z}^{805}$ by Lee balls of radius three

Advanced search

math.AC — Commutative Algebra
math.AG — Algebraic Geometry
math.AP — Analysis of PDEs
math.AT — Algebraic Topology
math.CA — Classical Analysis and ODEs
math.CO — Combinatorics
math.CT — Category Theory
math.CV — Complex Variables
math.DG — Differential Geometry
math.DS — Dynamical Systems
math.FA — Functional Analysis
math.GM — General Mathematics
math.GN — General Topology
math.GR — Group Theory
math.GT — Geometric Topology
math.HO — History and Overview
math.IT — Information Theory
math.KT — K-Theory and Homology
math.LO — Logic
math.MG — Metric Geometry
math.NA — Numerical Analysis
math.NT — Number Theory
math.OA — Operator Algebras
math.OC — Optimization and Control
math.PR — Probability
math.QA — Quantum Algebra
math.RA — Rings and Algebras
math.RT — Representation Theory
math.SG — Symplectic Geometry
math.SP — Spectral Theory
math.ST — Statistics Theory
cs.AI — Artificial Intelligence
cs.CC — Computational Complexity
cs.CG — Computational Geometry
cs.DM — Discrete Mathematics
cs.DS — Data Structures and Algorithms
cs.FL — Formal Languages and Automata Theory
cs.IT — Information Theory
cs.LO — Logic in Computer Science
cs.MS — Mathematical Software
cs.NA — Numerical Analysis
cs.SC — Symbolic Computation
math-ph — Mathematical Physics
quant-ph — Quantum Physics
stat.TH — Statistics Theory

Matching works

No lattice tiling of Z35\mathbb{Z}^{35}Z35 or Z805\mathbb{Z}^{805}Z805 by Lee balls of radius three

No lattice tiling of $\mathbb{Z}^{35}$ or $\mathbb{Z}^{805}$ by Lee balls of radius three