Let $k$ and $l$ be integers greater than $4$. I'm interested in set $S$ of $k$ elements in $\mathbb{Z}/2l\mathbb{Z}$ satisfying the following three conditions:

(1) if $a\in S$, then $a+l\not\in S$;

(2) for any $a\in S$, there exist $b\neq c\in S$ such that $2a=b+c$;

(3) for any $b\neq c\in S$ such that $b+c$ is even (this makes sense since $2l$ is even), there exists $a\in S$ such that $2a=b+c$.

Can we determine all such set $S$?

It is easy to verify that when $k$ is odd and $l=kn$, the set $\{t,t+2n,t+4n,\dots,t+2(k-1)n\}$ satisfies the three conditions for all $t\in\mathbb{Z}/2l\mathbb{Z}$. But I'm not aware of any other examples yet.



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