On almost perfect linear Lee codes of packing radius 2

More than 50 years ago, Golomb and Welch conjectured that there is no perfect Lee codes C of packing radius xmlns:xlink="http://www.w3.org/1999/xlink">r in Zn for ≥ 2 xmlns:xlink="http://www.w3.org/1999/xlink">n 3. Recently, Leung the second author proved if linear, then Golomb-Welch conjecture valid = In this paper, we consider classification linear with second-best possibility, density lattice by spheres xmlns:xlink="http://www.w3.org/1999/xlink">S ( xmlns:xlink="http://www.w3.org/1999/xlink">n, r ) equals | xmlns:xlink="http://www.w3.org/1999/xlink">n,r )|/| )|+1 . We show that, ≡ 0, 3, 4 (mod 6), can never be achieved.