AI 中文总结
本研究提出代码失真问题（CDP），证明其为 NP 难近似问题且属于 Σ₂^P，给出两类近似算法，推广了线性码等价问题并引入相关码论新概念。
AI 中文摘要
两个线性纠错码 $\boldsymbol{\textit{C}}_1, \boldsymbol{\textit{C}}_2 \text{是} \boldsymbol{\textit{F}}_q^n \text{的子集}$，若存在线性等距映射将 $\boldsymbol{\textit{C}}_1$ 映射到 $\boldsymbol{\textit{C}}_2$，则称二者线性等价。本研究推广了线性等价的概念，探讨线性纠错码 $\boldsymbol{\textit{C}}_1, \boldsymbol{\textit{C}}_2 \text{是} \boldsymbol{\textit{F}}_q^n \text{的子集}$ 间线性映射的最小失真 $\boldsymbol{\textit{D}}(\boldsymbol{\textit{C}}_1, \boldsymbol{\textit{C}}_2)$，该指标用于量化两码的相似程度。我们提出并研究代码失真问题（CDP），旨在找到输入两码间的最小失真映射。CDP 推广了线性码等价问题（LCE），当 $\boldsymbol{\textit{D}}(\boldsymbol{\textit{C}}_1, \boldsymbol{\textit{C}}_2)=1$ 时即为 LCE，因在密码学中的作用而被广泛研究。我们证明（判定型）CDP 无法在任何常数因子内近似，且属于复杂度类 $\boldsymbol{\textit{\text{Σ}}}_2^P$；还给出 CDP 的单指数时间 $k^2$ 近似算法，其中 $k$ 为输入码的维度。此外，针对 CDP 的一类自然特例，我们给出单指数时间 $\boldsymbol{\textit{\text{（}}}\frac{2k+1}{3}\boldsymbol{\textit{\text{）}}}^2$ 近似算法，并证明该分析在此类特例中是紧的。我们采用了 Bennett、Dadush 和 Stephens-Davidowitz 在《格失真问题（LDP）》（ESA, 2016）中类似工作的技术，还引入或研究了若干可能具有独立意义的概念，包括将 Goldreich、Micciancio、Safra 和 Seifert（IPL, 1999）提出的从格上最短向量问题（SVP）到最近向量问题（CVP）的著名归约适配为码上的类似问题、码的逐次极小基，以及子空间上的矩阵 $0 \to 0$“范数”。
英文摘要
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.
CommentsAPPROX 2026
