Searching for partial match - 1 item to be searched in $n$ preprocessed items
If I understand the problem correctly, this looks like an instance of the Nearest Neighbor Search, under the Hamming distance.

In other words, the "distance" between ordered sets is just the number of mismatched positions. The goal is to preprocess a collection (dataset) of ordered sets, each of size n, so that given a "query" ordered set, the data structure reports an ordered set within Hamming distance n-m. In your case, n is the dimension, and n-m is the distance. (This variant is actually called near neighbor search (the nearest neighbor is one where one reports the closest dataset item).

If that's the problem, then the wiki page may be a good start for pointers on existing work.