# 50 Years of the Golomb--Welch Conjecture

## TL;DR

This paper surveys the Golomb--Welch conjecture's history, progress, and related research, and presents new results on perfect Lee codes of large radii, discussing algebraic approaches for future work.

## Contribution

It provides a comprehensive survey of the Golomb--Welch conjecture and introduces new findings on perfect Lee codes with large radii, along with future algebraic methods.

## Key findings

- New results on perfect Lee codes of large radii
- Discussion of algebraic approaches to the conjecture
- Survey of research inspired by the Golomb--Welch conjecture

## Abstract

Since 1968, when the Golomb--Welch conjecture was raised, it has become the main motive power behind the progress in the area of the perfect Lee codes. Although there is a vast literature on the topic and it is widely believed to be true, this conjecture is far from being solved. In this paper, we provide a survey of papers on the Golomb--Welch conjecture. Further, new results on Golomb--Welch conjecture dealing with perfect Lee codes of large radii are presented. Algebraic ways of tackling the conjecture in the future are discussed as well. Finally, a brief survey of research inspired by the conjecture is given.