Projection-Forcing Multisets of Weight Changes

TL;DR

This paper characterizes multisets of weight changes that force linear functions over finite fields to be coordinate projections, providing algorithms to determine and verify this property, thus generalizing the MacWilliams Extension Theorem.

Contribution

It introduces a criterion and algorithms for identifying projection-forcing multisets of weight changes, extending the classical MacWilliams Extension Theorem.

Findings

Provides a super-polynomial algorithm to test projection-forcing multisets.

Introduces a polynomial-time check for sufficient conditions of being projection-forcing.

Generalizes the MacWilliams Extension Theorem to broader classes of multisets.

Abstract

Let $F$ be a finite field. A multiset $S$ of integers is projection-forcing if for every linear function $\phi : F^n \to F^m$ whose multiset of weight changes is $S$, $\phi$ 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.