On the nonexistence of linear perfect Lee codes

In 1968, Golomb and Welch conjectured that there does not exist perfect Lee code in $\mathbb{Z}^{n}$ with radius $r\ge2$ and dimension $n\ge3$. Besides its own interest in coding theory and discrete geometry, this conjecture is also strongly related to the degree-diameter problems of abelian Cayley graphs. Although there are many papers on this topic, the Golomb-Welch conjecture is far from being solved. In this paper, we prove the nonexistence of linear perfect Lee codes by introducing some new algebraic methods. Using these new methods, we show the nonexistence of linear perfect Lee codes of radii $r=2,3$ in $\mathbb{Z}^n$ for infinitely many values of the dimension $n$. In particular, there does not exist linear perfect Lee codes of radius $2$ in $\mathbb{Z}^n$ for all $3\le n\le 100$ except 8 cases.