The Golomb--Welch weak conjecture for perfect Lee codes

For integers $n,e,q$, let a $PL(n,e,q)$-code mean a perfect $e$-error-correcting code in the Lee metric on $\mathbb{Z}_q^n$. Golomb--Welch weak conjecture. There is no $PL(n,e,q)$-code over large alphabets for $n\geq 3$ and $e\geq 2$. This is the finite-alphabet formulation of the Golomb--Welch problem; the paper surveys substantial partial results, including nonexistence in several dimensions and for sufficiently large error radius, but records the conjecture as unresolved in general.