This is in continuation with this post: Geometric/Analytic techniques for constructive and asymptotic bounds in the Lee metric

Codes over alphabet $\mathbb{Z}_{q}$ of length $n$ for the Lee metric seems to be connected to spaced points on the $n$-torus. Are there any references which talks about their connection rigorously?

From wiki:

In coding theory, the "Lee distance" is a distance between two strings $x_{1} x_{2} \dots x_{n}$ and $y_{1} y_{2} \dots y_{n}$ of equal length $n$ over the $q$-ary alphabet $\{0,1,\cdots,q-1\}$ of size $q\ge2$. It is a metric, defined as

$\sum_{i=1}^n \min(|x_i-y_i|,q-|x_i-y_i|)$

If $q=2$ or $3$, the Lee distance coincides with the Hamming distance.

The metric space induced by the Lee distance is a discrete analog of the Elliptic geometry|elliptic space.