Projection-forcing multisets of weight changes

Let F be a finite field. A multiset S of integers is projectionforcing if for every linear function φ : F → F whose multiset of weight changes is S, φ 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. 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 an author of this paper.