For which natural number $n$ are there elements $\{x_i\}_{i=1}^n \subset \mathbb{Z}_n^2$ such that $x_i - x_j = x_k - x_l$ if and only if $i = k$ and $j=l$ (i.e. such that all differences between the elements are distinct)?
Such elements do not exist for $n=2$ and $n=3$. But they exist for example for $n=4,5,6$. E.g. for $n=4$ take the elements $\{(\bar{0}, \bar{0}), (\bar{1},\bar{0}), (\bar{2}, \bar{1}), (\bar{1}, \bar{3})\}$.
The problem of finding $n$ points with distinct differences in a given rectangular subset of $\mathbb{Z}^2$ has been investigated a lot. Unfortunately, this is not the case for the modular formulation presented above. The best reference I could find is this paper, section V. There, they construct similar subsets of $\mathbb{Z}_n^2$ using e.g. Golomb Rulers (also known as Sidon sets). The closest result to the question posed above is Theorem 20, but it is not strong enough.
Does anyone know where I could find answers to the question above?
Thank you!
Antropath