packing numbers and configuration spaces of the torus

Question

Let $S^1$ be the unit circle of radius $1$.

For any $k\geq 1$, let the $k$-dimensional torus $T^k= \underbrace{S^1\times S^1\times\cdots\times S^1}_k$ be the $k$-fold self-Cartesian product of $S^1$.

Equip $T^k$ with the product Riemannian metric.

For any real number $r>0$ and any integer $n\geq 1$, consider the (ordered) disk configuration space

$$ F_r(T^k,n)=\Big\{(x_1,x_2,\ldots,x_n)\in T^k~~\Big |~~ d(x_i,x_j)\geq 2r ~{\rm ~for~any~}~1\leq i<j\leq n \Big \}. $$

Fix $r$ and $k$.

If $n$ is large enough, then $F_r(T^k,n)=\emptyset$.

Denote the largest $n$ such that $F_r(T^k,n)\neq \emptyset$ by $N(k,r)$.

Question 1. For any integer $k\geq 1$ and any real number $r>0$, how to compute $N(k,r)$?

Question 2. For any real number $r>0$, how to compute the limit $\lim_{k\to\infty }N(k,r)^{\frac{1}{k}}$?

Are there any references?

Thank you very much.

Answer 1

If you rescale your circles to have circumference 2, then you are interested in packings of $N$ spheres in $n$-dimensional Euclidean space with periodicity $(2\mathbb{Z})^n$. The configuration space is usually written as $\mathbb{R}^n/(2\mathbb{Z})^n$, and the minimum distance between spheres, $2r$, would have the restriction $r\le 1$ so that the coset elements also respect the packing constraint. The parameter $r$ is the sphere radius.

For small $n$ and $r=1$ the maximum $N$ is achieved by Hamming distance 4 binary codes, where all the coordinates are 0 and 1. You can find the best known $N$ in the first column of the table on this website:

https://www.win.tue.nl/~aeb/codes/binary-1.html