Reductions Between Code Equivalence Problems

TL;DR

This paper establishes polynomial-time reductions between various code equivalence problems, linking permutation, linear, signed permutation, and lattice isomorphism problems, thereby unifying their computational complexities.

Contribution

It introduces new polynomial-time reductions from Permutation Code Equivalence to Linear Code Equivalence and Signed Permutation Code Equivalence, and from Signed Permutation Code Equivalence to Lattice Isomorphism.

Findings

PCE reduces to LCE in polynomial time

PCE reduces to SPCE in polynomial time

SPCE reduces to LIP in polynomial time

Abstract

In this paper we present two reductions between variants of the Code Equivalence problem. We give polynomial-time Karp reductions from Permutation Code Equivalence (PCE) to both Linear Code Equivalence (LCE) and Signed Permutation Code Equivalence (SPCE). Along with a Karp reduction from SPCE to the Lattice Isomorphism Problem (LIP) proved in a paper by Bennett and Win (2024), our second result implies a reduction from PCE to LIP.