A New Approach Towards the Golomb-Welch Conjecture

Abstract

The Golomb-Welch conjecture deals with the existence of perfect $e$% -error correcting Lee codes of word length $n,$ $PL(n,e)$ codes. Although there are many papers on the topic, the conjecture is still far from being solved. In this paper we initiate the study of an invariant connected to abelian groups that enables us to reformulate the conjecture, and then to prove the non-existence of linear PL(n,2) codes for $n\le 12$. Using this new approach we also construct the first quasi-perfect Lee codes for dimension $n=3,$ and show that, for fixed $n$, there are only finitely many such codes over $Z^n$.

References in corpus (2)

• Product Constructions for Perfect Lee Codes
• Diameter Perfect Lee Codes

Cited by in corpus (8)

• 50 Years of the Golomb--Welch Conjecture
• Nonexistence of perfect $2$-error-correcting Lee codes in certain dimensions
• Quasi-perfect Lee Codes of Radius 2 and Arbitrarily Large Dimension
• On the nonexistence of linear perfect Lee codes
• No lattice tiling of ${\mathbb{Z}}^n$ by Lee Sphere of radius 2
• Quasi-perfect codes in the ${\ell }_p$ metric
• Perfect codes in the lp metric
• The set of dimensions for which there are no linear perfect 2-error-correcting Lee codes has positive density