On local packings of the cross-polytope
The electronic journal of combinatorics, Tome 27 (2020) no. 3
The problem of finding the largest number of points in the unit cross-polytope such that the $l_{1}$-distance between any two distinct points is at least $2r$ is related to packings. For the $n$-dimensional cross-polytope, we show that $2n$ points can be placed when $r\in\left(1-\frac{1}{n},1\right]$. For the three-dimensional cross-polytope, $10$ and $12$ points can be placed if and only if $r\in\left(\frac{3}{5},\frac{2}{3}\right]$ and $r\in\left(\frac{4}{7},\frac{3}{5}\right]$ respectively, and no more than $14$ points can be placed when $r\in\left(\frac{1}{2},\frac{4}{7}\right]$. Also, constructive arrangements of points that attain the upper bounds of $2n$, $10$, and $12$ are provided, as well as $13$ points for dimension $3$ when $r\in\left(\frac{1}{2},\frac{6}{11}\right]$.
@article{10_37236_8990,
author = {Ji Hoon Chun},
title = {On local packings of the cross-polytope},
journal = {The electronic journal of combinatorics},
year = {2020},
volume = {27},
number = {3},
doi = {10.37236/8990},
zbl = {1446.52013},
url = {http://geodesic.mathdoc.fr/articles/10.37236/8990/}
}
Ji Hoon Chun. On local packings of the cross-polytope. The electronic journal of combinatorics, Tome 27 (2020) no. 3. doi: 10.37236/8990
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