线性码等价问题的搜索到判定归约
A search-to-decision reduction for the linear code equivalence problem
- University of South Florida（南佛罗里达大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文提出从线性码等价搜索问题到判定问题的多项式时间归约，通过预言机查询恢复线性等距，并给出置换与对角部分的确定性恢复算法。
AI中文摘要：
我们提出了从线性码等价问题的搜索变体（即寻找输入之间的线性等距）到其判定变体的多项式时间归约。更精确地说，给定两个线性等价的码 $\mathcal C_1,\mathcal C_2 \subseteq \mathbb{F}_q^n$，我们展示了如何通过向判定性线性码等价的预言机进行多项式次查询来恢复它们之间的线性等距。首先，我们证明搜索-置换码等价（search-PCE——寻找将 $\mathcal C_1$ 映射到 $\mathcal C_2$ 的置换 $\pi\in\mathcal S_n$ 的问题）在多项式时间内归约到 PCE（即判定是否存在从 $\mathcal C_1$ 到 $\mathcal C_2$ 的置换映射的问题），通过最多 $n^2$ 次对维度为 $k$、长度至多 $n^2(n+1)/2$ 的实例的预言机调用。然后，我们将此方法扩展到线性等价的码：我们通过最多 $n^2$ 次对相同大小实例的线性码等价（LCE）预言机调用来恢复线性等距的置换部分，并给出一个确定性多项式时间算法来在已知该置换后恢复对角部分。总之，这产生了一个从判定性 LCE 预言机恢复线性等距的多项式时间过程。从线性代数的角度来看，我们的结果提供了从对应轨道成员问题的预言机访问中显式重构两个矩阵表示之间的单项等价的构造。
英文摘要：
We present a polynomial-time reduction from the search variant of the linear code equivalence problem (i.e. the search for a linear isometry between the inputs) to its decisional variant. More precisely, given two linearly equivalent codes $\mathcal C_1,\mathcal C_2 \subseteq \mathbb{F}_q^n$, we show how to recover a linear isometry between them by making a polynomial number of queries to an oracle for decisional linear code equivalence. First, we prove that search-Permutation Code Equivalence (search-PCE -- the problem of finding a permutation $π\in\mathcal S_n$ mapping $\mathcal C_1$ to $\mathcal C_2$) reduces in polynomial time to PCE (i.e. the problem of deciding if there is a permutation map from $\mathcal C_1$ to $\mathcal C_2$) via at most $n^2$ oracle calls on instances of dimension $k$ and length at most $n^2(n+1)/2$. We then extend this approach to linearly equivalent codes: we recover the permutation part of a linear isometry via at most $n^2$ calls to a Linear Code Equivalence (LCE) oracle on instances of the same size, and we give a deterministic polynomial-time algorithm to recover the diagonal part once this permutation is known. Altogether, this yields a polynomial-time procedure to recover a linear isometry from an oracle for decisional LCE. From a linear-algebraic perspective, our results provide an explicit reconstruction of a monomial equivalence between two matrix representations from oracle access to the corresponding orbit membership problem.
