Reductions Between Code Equivalence Problems

TL;DR

This work systematically relates code equivalence variants by constructing polynomial-time Karp reductions between $\mathsf{PCE}$, $\mathsf{LCE}$, and $\mathsf{SPCE}$, and leveraging prior results to connect to the Lattice Isomorphism Problem ($\mathsf{LIP}$) for prime fields. The central technique augments input generator matrices into larger structured instances, preserving equivalence via block-structured witnesses and careful analysis of change-of-basis and permutation actions; crucially, the forward reductions demonstrate $\mathsf{PCE} \to \mathsf{LCE}$ and $\mathsf{PCE} \to \mathsf{SPCE}$ in time poly$(n,\log q)$, while the reverse direction shows $\mathsf{LCE} \Rightarrow \mathsf{PCE}$ through block-boundary arguments. Combined with BW24’s results, the paper establishes computational equivalence among $\mathsf{PCE}$, $\mathsf{SPCE}$, and $\mathsf{LCE}$ up to polynomial factors, and yields a reduction from $\mathsf{PCE}$ to $\mathsf{LIP}$ for prime fields. The findings illuminate the hardness landscape of CE variants and clarify pathways for reductions to lattice-based problems, with open questions about non-prime fields and potential log-time reductions remaining as future work.

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.

Figures (3)

• Figure 1: The matrix $A'$ obtained by \ref{['con:our-construction']}.
• Figure 2: The structure of matrix $\mathbf{M}'$.
• Figure 3: The structure of matrix $\mathbf{S}'$.

Theorems & Definitions (23)

• Theorem 1.1: $\mathsf{PCE}$ reduces to $\mathsf{LCE}$, informal
• Theorem 1.2: $\mathsf{PCE}$ reduces to $\mathsf{SPCE}$, informal
• Corollary 1.3: $\mathsf{PCE}$ reduces to $\mathsf{LIP}$
• Definition 2.2: $\mathsf{PCE}$
• Definition 2.3: $\mathsf{SPCE}$
• Definition 2.4: $\mathsf{LCE}$
• Theorem 3.1: $\mathsf{PCE}$ reduces to $\mathsf{LCE}$
• Theorem 3.2: $\mathsf{PCE}$ reduces to $\mathsf{SPCE}$
• Lemma 3.3
• proof