3 problems

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…

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