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 <inline-formula> <tex-math notation="LaTeX">$\mathbb {Z}^{n}$ </tex-math></inline-formula> with radius <inline-formula> <tex-math notation="LaTeX">$e\geq 2$ </tex-math></inline-formula> and dimensions <inline-formula> <tex-math notation="LaTeX">$n\geq 3$ </tex-math></inline-formula>, 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 <inline-formula> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula>. The main result of this paper is that for radius <inline-formula> <tex-math notation="LaTeX">$e = 2$ </tex-math></inline-formula>, there are no perfect Lee linear codes in <inline-formula> <tex-math notation="LaTeX">$\mathbb {Z}^{n}$ </tex-math></inline-formula> for infinitely many values of <inline-formula> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula>.

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