  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.
