Slight extension of cross posting from https://cstheory.stackexchange.com/questions/7408/lee-metric-gilbert-varshamov-and-hamming-bounds-for-larger-relative-distance-rang (closed there)

The following link provides a Gilbert-Varshamov lower bound and a Hamming upper bound for the Lee metric when the distance between codewords is smaller than the length of the code (captured by $r = \delta n$ where $0 \le \delta \le 0.37$ in the paper).

ftp://ftp.cs.brown.edu/pub/techreports/91/cs91-29.pdf

Consider the alphabet is of form $2k + 1$ where $k$ is any non-negative integer.

Is there a reference which provides the corresponding lower/upper bound for ranges above this? Atleast is there a lower bound (preferably as powerful as the Gilbert-Varshamov bound) and a upper bound(preferably tighter than the Hamming bound) for the case $\frac{2 \delta n + 1}{n} = \frac{n + 1}{n}$, that is $\delta = \frac{1}{2}$?

Are there any constructive techniques known for the case $\delta = 0.5$ for all/any even/any odd $n$ to build good codes?

Are there any spectral/fourier analytic/algebraic geometric techniques that has studied this problem? In particular are there connections to domino tiling that are studied anywhere?