4 problems

Let $DPL(n,d)$ denote a diameter $d$ perfect Lee code in $\mathbb{Z}^{n}$. For odd $d$, this agrees with a perfect Lee code of radius $(d-1)/2$; for even $d$, it is defined using a…

Let $\mathcal A_q^n$ be the ambient Lee-metric space, and let a perfect $(n,M,d)_q$-Lee code be a Lee code with $M$ codewords whose packing and covering radii coincide. Golomb–Welc…

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 $(n,d)=(3,6)$.