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

## TL;DR

This paper proves the nonexistence of certain perfect 2-error-correcting Lee codes in specific dimensions, providing evidence supporting the Golomb--Welch conjecture that such codes do not exist for higher dimensions.

## Contribution

It establishes the nonexistence of perfect 2-error-correcting Lee codes for an infinite class of dimensions, advancing understanding of the Golomb--Welch conjecture.

## Key findings

- Proves nonexistence of perfect 2-error-correcting Lee codes in certain dimensions
- Supports the Golomb--Welch conjecture for higher dimensions
- Provides a new class of dimensions where perfect Lee codes cannot exist

## Abstract

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.