A new approach towards the Golomb–Welch conjecture

The Golomb–Welch conjecture deals with the existence of perfect ee-error correcting Lee codes of word length nn, PL(n,e)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)PL(n,2) codes for n≤12n≤12. Using this new approach we also construct the first quasi-perfect Lee codes for dimension n=3n=3, and show that, for fixed nn, there are only finitely many such codes over ZZ.