If I have a DFA $D$ that accepts some regular language $L$, is it possible to construct some finite-state transducer $T$ that takes in an index $i$, and puts out the $i$'th word in $L$, ordered lexicographically?
I know from this paper that you can construct such a transducer that maps from a given word in $L$ to the next word in $L$, but the paper only briefly mentions the variant of the problem where an index is given instead.
