Let F be a finite field. The Hamming weight of a vector is the number of nonzero entries. A multiset S of integers is called projection forcing if every linear map φ: Fn —> Fm, whose multiset of weight changes, {w(φ(v)−w(v)}, is S, is a coordinate projection up to permutation of entries. The MacWilliams Extension Theorem from coding theory says that S = {0, 0, ..., 0} is projection forcing.
In work with Josh Brown Kramer, we give a (super-polynomial) algorithm to determine whether or not a given set S is projection forcing. we also give a condition that can be checked in polynomial time that implies that S is projection forcing.