50 Years of 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.

arXiv comment28 pages, 2 figures

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Cited by in corpus (4)

• Corrigendum to "On subgroup perfect codes in Cayley graphs" [European J. Combin. 91 (2021) 103228]
• No lattice tiling of ${\mathbb{Z}}^n$ by Lee Sphere of radius 2
• On the nonexistence of linear perfect Lee codes
• On regular sets in Cayley graphs