Non-Existence of Linear Perfect Lee Codes With Radius 2 for Infinitely Many Dimensions

The Golomb-Welch conjecture (1968) on the non-existence of perfect Lee codes in Z
ⁿ
with radius e ≥ 2 and dimensions n ≥ 3, widely believed to be true, has been up to now only proved for large radius in any dimension, for small dimensions, and for some small radii and specific n. The main result of this paper is that for radius e = 2, there are no perfect Lee linear codes in Z
ⁿ
for infinitely many values of n.