Let $S$ be a set containing some permutations of the alphabet $\{0, 1, \ldots, L-1\}$. Now someone randomly draws one permutation $s_i$ from $S$, randomly chooses $k \in \{ 0, 1, \ldots, L - 1 \}$, $e \in \{ 1, 2, \ldots, D \}$ (where $D < L$) and provides me with the elements $r_1 = s_i(k)$ and $r_2 = s_i(k + e \bmod L)$. I only get to know these two elements and get to know which is which, I get to see neither $k$ nor $e$. Based on $r_1$ and $r_2$ I want to be able to uniquely identify which permutation $s_i$ they come from.
How do I find a set $S$ such that $N = |S|$ is large for given $D$ and given $L$?
Trivial example: Ascending and Descending Sequence Let $L = 11$ and $D = 5$. Then $S = \{ (0, 1, \ldots, 9, 10), (10, 9, \ldots, 1, 0) \}$ (permutations written in one-line notation, think of them as sequences) is one solution with the largest possible $|S|$, namely $|S| = 2$. For instance, if I get $r_1 = 5$ and $r_2 = 3$, I know that $r_1$ and $r_2$ were drawn from the descending sequence. If it were $r_1 = 10$ and $r_2 = 0$, then it would have had to be the ascending sequence that was chosen.
For $L = 11$ and smaller $D$, it should be possible to find larger sets though.
Any hints are appreciated, especially links to possibly existing research into this question!