We consider the task of locally correcting, and locally list-correcting, multivariate linear functions over the domain $\{0,1\}^n$ over arbitrary fields and more generally Abelian groups. Such functions form error-correcting codes of relative distance $1/2$ and we give local-correction algorithms correcting up to nearly $1/4$-fraction errors making $\widetilde{\mathcal{O}}(\log n)$ queries. This query complexity is optimal up to $\mathrm{poly}(\log\log n)$ factors. We also give local list-correcting algorithms correcting $(1/2 - \varepsilon)$-fraction errors with $\widetilde{\mathcal{O}}_{\varepsilon}(\log n)$ queries. These results may be viewed as natural generalizations of the classical work of Goldreich and Levin whose work addresses the special case where the underlying group is $\mathbb{Z}_2$. By extending to the case where the underlying group is, say, the reals, we give the first non-trivial locally correctable codes (LCCs) over the reals (with query complexity being sublinear in the dimension (also known as message length)). The central challenge in constructing the local corrector is constructing ``nearly balanced vectors'' over $\{-1,1\}^n$ that span $1^n$ -- we show how to construct $\mathcal{O}(\log n)$ vectors that do so, with entries in each vector summing to $\pm1$. The challenge to the local-list-correction algorithms, given the local corrector, is principally combinatorial, i.e., in proving that the number of linear functions within any Hamming ball of radius $(1/2-\varepsilon)$ is $\mathcal{O}_{\varepsilon}(1)$. Getting this general result covering every Abelian group requires integrating a variety of known methods with some new combinatorial ingredients analyzing the structural properties of codewords that lie within small Hamming balls.

翻译：我们考虑在任意域以及更一般的阿贝尔群上，针对定义域$\{0,1\}^n$的多变量线性函数进行局部校正和局部列表校正的问题。此类函数构成相对距离为$1/2$的纠错码，我们提出的局部校正算法能纠正接近$1/4$比例的误差，且查询复杂度为$\widetilde{\mathcal{O}}(\log n)$。该查询复杂度在$\mathrm{poly}(\log\log n)$倍数因子内达到最优。同时，我们给出可纠正$(1/2 - \varepsilon)$比例误差的局部列表校正算法，其查询复杂度为$\widetilde{\mathcal{O}}_{\varepsilon}(\log n)$。这些结果可视为Goldreich与Levin经典工作的自然推广——该经典工作仅处理底层群为$\mathbb{Z}_2$的特例。通过将底层群扩展至实数域等情形，我们首次在实数域上构造出非平凡的局部可纠错码（LCC），其查询复杂度在维度（也称消息长度）上呈次线性。构造局部校正器的核心挑战在于：在$\{-1,1\}^n$上构造能张成向量$1^n$的“近平衡向量”——我们证明可利用$\mathcal{O}(\log n)$个向量实现该目标，且每个向量各分量之和为$\pm1$。在给定局部校正器后，局部列表校正算法的挑战主要在于组合层面：即需证明任意汉明半径$(1/2-\varepsilon)$的球内线性函数数量为$\mathcal{O}_{\varepsilon}(1)$。为获得覆盖所有阿贝尔群的通用结论，需将多种已知方法与分析小汉明球内码字结构特性的新组合工具相结合。