I've been working on something with integers modulo $N$ and have sort of hit a roadblock where I'd like to have some references. The particular problem goes as follows. We have a system $\mathcal{S}$ of subsets of $\mathbb{Z}/N$ such that $\forall S \in \mathcal{S}, |S|=k$ and $s\ne 0, \forall s \in S$. Also, $2k < N, k \ge 3$. Further, for any two subsets $S_1, S_2 \in \mathcal{S}$, $1 \le |S_1 \cap S_2| \le k-2$. Also, most importantly, for every pair $S_1, S_2$ there exists a $d \in \mathbb{Z}/N$ such that $S_2 = \{s+d,s\in S_1\}$ where $+$ denotes addition modulo $N$.
Is there an upper bound on $|\mathcal{S}|$ given $k, N$? Are there papers discussing this problem or similar problems? I've looked at the Erdos-Ko-Rado type intersection theorems and a lot of works following it but they do not specifically consider the translation property that I've just stated which is inherent in my problem. Since I'm not a mathematician I'm not even sure if the aforementioned type of intersection theorems are what I'm supposed to be looking at. I'd be grateful if someone can provide resources/references which include this certain translation property.