  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 investigated for $r\in\left(1-\frac{1}{n},1\right]$ in
dimensions $\geq2$ and for $r\in\left(\frac{1}{2},1\right]$ in dimension $3$.
For the $n$-dimensional cross-polytope, $2n$ 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
$r\in\left(\frac{1}{2},\frac{6}{11}\right]$.
