Given a number $m\in\mathbb N$, let $\mathbb Z_m=\{0,1,\dots,m-1\}$ denote the ring of integers modulo $m$ (although we won't need multiplication, so any cyclic group of order $m$ will do). Given a second number $k\in\mathbb N$ I'm looking for subsets of $k$ elements from $\mathbb Z_m$ such that no pair-wise difference occurs more than once. More formally, I'm looking for some $A\subset\mathbb Z_m$ with $\lvert A\rvert=k$ and
$$\forall a,b,c\in A: \lvert\{a,b,c\}\rvert=3\rightarrow a+b-c\not\in A\pmod m$$
- Is there a name for such a kind of set? I guess it might be somehow related to Golomb rulers, but the cyclic nature is not common to rulers as far as I know.
- Is there a known method to efficiently enumerate such sets for given $m$ and $k$?
- Is there a known theorem concerning the existence of such subsets for specific $m$ and $k$?
I know I'm asking three distinct questions here, but I very much hope that someone may be able to address more than one, perhaps by providing a good reference. Knowing a name will probably help me locate suitable literature. But answers addressing only one of these questions are welcome as well.
Update: It seems that for the special case of $m=k^2-k+1$ where every possible difference has to actually occur exactly once, the kind of set I defined would be called an $(m,k,1)$ cyclic difference set. I'm still interested in a more general term for cases where some differences are missing.