5 problems

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

Camarero–Martínez conjecture. The graph $\mathcal{G}_p$ is a Ramanujan graph for every prime $p$ such that $p\equiv 3 \pmod{4}$.

Let $S_{n,e}$ be the Lee sphere of radius $e$ in $\mathbb{Z}^n$, and let an optimal embedding of a finite number $k$ in $\mathbb{Z}^n$ mean an embedding realizing the corresponding…

Perfect distance-dominating set conjecture. For $k\geq 1$, there is no $r$-PDDS$[P_k]$ in $\Lambda_n$ for $n\geq 3$ and $r>1$, with the exception of a $2$-PDDS$[P_2]$ in…

Etzion's conjecture. There is no $DPL(n,d)$ code for $n\geq 3$ and $d>4$, with the exception of the pair $(n,d)=(3,6)$.