I am curious if there exists a method for specifying a permutation $F_k: X \to X$ with a small(ish) $k$.
Something that comes very close to my goal is a block cipher, say AES. But block ciphers have fixed block size (e.g. 64, 128 bits), and I want something with parameterizable (and smaller) "block size" $\log_2 |X|$. I also don't need many characteristics of block ciphers (e.g. $k$ secrecy, etc.), only the permutation part.
Something else that is quite close is the Lehmer code. However, this takes into consideration all possible permutations $|X|!$, and would need the index (or "key") $k$ to be as large as $|X| \log_2 |X|$. I am happy with only $K << |X|!$ possible permutations (so long as $K$ is "big enough").