A New Approach Towards the Golomb-Welch Conjecture

TL;DR

This paper introduces a new invariant approach to the Golomb-Welch conjecture, proving non-existence of certain linear Lee codes for small dimensions and constructing the first quasi-perfect Lee codes for dimension 3.

Contribution

It reformulates the Golomb-Welch conjecture using an invariant linked to abelian groups and demonstrates non-existence results and the first constructions of quasi-perfect Lee codes.

Findings

Proved non-existence of linear PL(n,2) codes for n ≤ 12.

Constructed the first quasi-perfect Lee codes for dimension n=3.

Showed only finitely many such codes exist over Z^n for fixed n.

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