9 problems

Gruslys–Leader–Tan conjecture. There is a number $d$ such that every tile $T$ in $\mathbb{Z}$ with $|T|=t$ tiles $\mathbb{Z}^d$.

Nonexistence conjecture. For $n>1$, there are no centrally symmetric perfect Delaunay polytopes whose translates tile $\mathbb{R}^n$.

Conjecture on APLL codes. The dimension must satisfy $n=1$ or $n=2$; equivalently, up to isometry, the only examples are the APLL code

Let $\Lambda_n=\mathbb{Z}^n$ and let a lattice $t$-PTMC be a lattice perfect total-multipacking code with parameter $t$. Nonexistence conjecture. There is no lattice $t$-PTMC in…

Semi-cross lattice tiling conjecture. Every tiling of $\mathbb{Z}^{p-1}$ by $V_{p-1}$ is lattice. The paper states that this special case is equivalent to the prime-tile lattice ti…

Let $V$ be a finite tile in $\a0\mathbb{Z}^n$ of prime size, written as … and suppose that $\mathbf{v}_1,\ldots,\mathbf{v}_{p-1}$ generate $\mathbb{Z}^n$ as an abelian group. Prime…

Conjecture. There are no linear perfect codes with parameters $(n,r,2)$ except when $r=1$ or

A tile $T\subset\mathbb{Z}$ is a finite subset of the integer lattice that tiles by translations, and $|T|$ denotes its cardinality. Uniform bounded-size tiling conjecture. For eve…

Let $T\subset\mathbb{Z}^n$ be a tile, meaning a finite subset of the integer lattice that tiles by translations. Chalcraft's conjecture. Every tile $T\subset\mathbb{Z}^n$ tiles…