Reductions Between Code Equivalence Problems

In this paper, we present two reductions between variants of the Code Equivalence problem. We give polynomialtime 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) shown by Bennett and Win (2024), our second result implies a reduction from PCE to LIP.