On the non-existence of 3-dimensional tiling in the Lee metric
European Journal of Combinatorics
On Perfect Codes and Related Concepts
Designs, Codes and Cryptography
Fast decoding of quasi-perfect Lee distance codes
Designs, Codes and Cryptography
Nonexistence of face-to-face four-dimensional tilings in the Lee metric
European Journal of Combinatorics
Optimal Lee-Type Local Structures in Cartesian Products of Cycles and Paths
SIAM Journal on Discrete Mathematics
European Journal of Combinatorics
Non-periodic Tilings of ℝn by Crosses
Discrete & Computational Geometry
Quasi-perfect Lee distance codes
IEEE Transactions on Information Theory
Graphs, tessellations, and perfect codes on flat tori
IEEE Transactions on Information Theory
Product Constructions for Perfect Lee Codes
IEEE Transactions on Information Theory
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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@?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.