The set of dimensions for which there are no linear perfect 2-error-correcting Lee codes has positive density

04/24/2018

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by   Claudio Qureshi, et al.

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The Golomb-Welch conjecture states that there are no perfect e-error-correcting Lee codes in Z^n (PL(n,e)-codes) whenever n≥ 3 and e≥ 2. A special case of this conjecture is when e=2. In a recent paper of A. Campello, S. Costa and the author of this paper, it is proved that the set N of dimensions n≥ 3 for which there are no linear PL(n,2)-codes is infinite and #{n ∈N: n≤ x}≥x/3(x)/2 (1+o(1)). In this paper we present a simple and elementary argument which allows to improve the above result to #{n ∈N: n≤ x}≥4x/25 (1+o(1)). In particular, this implies that the set N has positive (lower) density in Z^+.