Projection-forcing multisets of weight changes

Let F be a finite field. A multiset S of integers is projection-forcing if for every linear function ϕ:Fn→Fm whose multiset of weight changes is S, ϕ is a coordinate projection up to permutation and scaling of entries. The MacWilliams Extension Theorem from coding theory says that S={0,0,…,0} is projection-forcing. We give a (super-polynomial) algorithm to determine whether or not a given S is projection-forcing. We also give a condition that can be checked in polynomial time that implies that S is projection-forcing. This result is a generalization of the MacWilliams Extension Theorem and work by the first author.