The Code Distortion Problem

2026-07-28

AI Summary

This study addresses the minimum-distortion linear mapping problem (CDP) between two linear error-correcting codes, a generalization of the linear code equivalence problem (LCE), by introducing a distortion metric to quantify structural differences between codes. The work formally defines this problem for the first time and establishes its computational hardness, proving it lies in the second level of the polynomial hierarchy (Σ₂^P) and is inapproximable within any constant factor. Methodologically, it innovatively adapts techniques from lattice-based cryptography—including reductions from the Shortest Vector Problem (SVP) to the Closest Vector Problem (CVP), successive minima bases, and ℓ⁰→ℓ⁰ norms of subspace matrices—to coding theory. The main contributions include a single-exponential-time k²-approximation algorithm and, for specific instances, a tighter (2k+1)²/9-approximation algorithm.

Abstract

Two linear error-correcting codes $\cal{C}_1, \cal{C}_2 \subseteq \mathbb{F}_q^n$ are called linearly equivalent if there is a linear isometry mapping $\cal{C}_1$ to $\cal{C}_2$. In this work, we generalize the notion of linear equivalence and study the minimum distortion $\cal{D}(\cal{C}_1, \cal{C}_2)$ of a linear mapping between codes $\cal{C}_1, \cal{C}_2 \subseteq \mathbb{F}_q^n$, which quantifies how similar $\cal{C}_1$ and $\cal{C}_2$ are. We introduce and study the Code Distortion Problem (CDP), which asks to find a minimum distortion mapping between two input codes $\cal{C}_1$ and $\cal{C}_2$. CDP generalizes the Linear Code Equivalence Problem (LCE), which is essentially the special case of CDP where $\cal{D}(\cal{C}_1, C_2) = 1$ and which is well-studied because of its role in cryptography. We prove that (decisional) CDP is $\mathsf{NP}$-hard to approximate to within any constant factor, and that it is in $Σ_2^P$. We also give a single-exponential-time $k^2$-approximation algorithm for CDP, where $k$ is the dimension of the input codes. Furthermore, we give a single-exponential-time $\big(\frac{2k + 1}{3})^2$-approximation algorithm for a natural special case of CDP, and we show that our analysis is tight in this case. We use techniques from analogous work on the Lattice Distortion Problem (LDP) by Bennett, Dadush, and Stephens-Davidowitz (ESA, 2016). We also introduce or study a number of additional concepts that might be of independent interest. These include an adaptation of the celebrated reduction of Goldreich, Micciancio, Safra, and Seifert (IPL, 1999) from the Shortest Vector Problem (SVP) to the Closest Vector Problem (CVP) on lattices to the analogous problems on codes; successive minima bases for codes; and the matrix $0 \to 0$ "norm" on subspaces.

Problem

• Code Distortion Problem
• Linear Code Equivalence
• Minimum Distortion
• NP-hardness
• Error-correcting Codes

Innovation

• Code Distortion Problem
• Linear Code Equivalence
• NP-hardness
• Approximation Algorithm
• Successive Minima Bases