Paper Info
Title
The set of dimensions for which there are no linear perfect 2-error-correcting Lee codes has positive density.
Abstract
The Golomb-Welch conjecture states that there are no perfect $e$-error-correcting Lee codes $mathbb{Z}^n$ ($PL(n,e)$-codes) whenever $ngeq 3$ and $egeq 2$. 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 $mathcal{N}$ of dimensions $ngeq 3$ for which there are no linear $PL(n,2)$-codes is infinite and $#{n in mathcal{N}: nleq x} geq frac{x}{3ln(x)/2} (1+o(1))$. In this paper we present a simple and elementary argument which allows to improve the above result to $#{n in mathcal{N}: nleq x} geq frac{4x}{25} (1+o(1))$. In particular, this implies that the set $mathcal{N}$ has positive (lower) density $mathbb{Z}^+$.
Year
Venue
Field
2018
arXiv: Information Theory
Discrete mathematics,Conjecture,Mathematics
DocType
Volume
Citations 
Journal
abs/1804.09290
1
PageRank 
References 
Authors
0.39
5
1
Name
Order
Citations
PageRank
Claudio Qureshi1104.48