Nonexistence of perfect $2$-error-correcting Lee codes in certain dimensions

The Golomb–Welch conjecture states that there are no perfect $e$-error-correcting codes in ${\mathbb{Z}}^n$ for $n\ge 3$ and $e\ge 2$. In this note, we prove the nonexistence of perfect $2$-error-correcting codes for a certain class of $n$, which is expected to be infinite. This result further substantiates the Golomb–Welch conjecture.