  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 $n\leq 12$. 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 $Z^n$.
