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，对定义在域$\{0,1\}^n$上的多元线性函数进行局部修正和局部列表修正的任务。这类函数构成相对距离为$1/2$的纠错码，我们给出使用约$\widetilde{\mathcal{O}}(\log n)$次查询即可纠正近$1/4$分数错误的局部修正算法。该查询复杂度在$\mathrm{poly}(\log\log n)$因子内达到最优。我们还给出使用$\widetilde{\mathcal{O}}_{\varepsilon}(\log n)$次查询即可纠正$(1/2 - \varepsilon)$分数错误的局部列表修正算法。这些结果可视为Goldreich和Levin经典工作的自然推广，其工作仅处理底层群为$\mathbb{Z}_2$的特殊情形。通过将底层群推广至实数域等情形，我们给出了首个非平凡的实数域局部可纠错码（LCC）（其查询复杂度在维度（即消息长度）上呈亚线性）。构造局部修正器的核心挑战在于构建$\{-1,1\}^n$上"近乎平衡向量"以张成$1^n$——我们展示了如何构造$\mathcal{O}(\log n)$个满足各向量元素之和为$\pm1$的这类向量。在给定局部修正器后，局部列表修正算法的挑战主要在于组合方面，即需证明任意半径为$(1/2-\varepsilon)$的汉明球内线性函数的数量为$\mathcal{O}_{\varepsilon}(1)$。要获得覆盖所有阿贝尔群的这一通用结论，需要将多种已知方法整合到新的组合工具中，分析位于小汉明球内码字的结构性质。