通过提升的高速率局部可纠正码
High rate locally correctable codes via lifting
摘要：
我们提出了一个一般的框架，用于构建高速率的纠错码，是本地可纠正的（因此，本地解码，如果线性）与次线性数量的查询，提升代码的基础上的坐标上的功能。我们的方法推广了郭，Kopparty和苏丹的仿射不变码的提升和其推广的自守提升，由Ben-Sasson等人提出，它提升了代数几何码相对于一组自同构的代码。我们的概念提升是一个自然的替代度提升的Ben-Sasson等人，它有两个优点。首先，它克服了学位提升所固有的速率障碍。其次，它非常灵活，不需要基本代码的特殊属性（例如线性，不变性），并且在代码坐标上的函数集上需要很少的结构。作为应用，我们利用提升代数几何码构造了新的显式局部可纠错码族。与Kopparty、Saraf、Yekhanin的多重性码和Guo、Kopparty、Sudan的仿射提升码一样，我们的块长度$N$码可以在纠正恒定错误分数的同时，实现对任何给定$ε, α> 0$的$N^ε$查询复杂度和$1-α$速率，与面临速率障碍$ε^{O(1/ε)}$的Reed-Muller码和Ben-Sasson等人的度提升AG码形成对比。然而，像度提升AG代码，我们的代码是在一个字母表显着小于里德-穆勒码，仿射提升码，和多重性代码。
英文摘要：
We present a general framework for constructing high rate error correcting codes that are locally correctable (and hence locally decodable if linear) with a sublinear number of queries, based on lifting codes with respect to functions on the coordinates. Our approach generalizes the lifting of affine-invariant codes of Guo, Kopparty, and Sudan and its generalization automorphic lifting, suggested by Ben-Sasson et al, which lifts algebraic geometry codes with respect to a group of automorphisms of the code. Our notion of lifting is a natural alternative to the degree-lifting of Ben-Sasson et al and it carries two advantages. First, it overcomes the rate barrier inherent in degree-lifting. Second, it is extremely flexible, requiring no special properties (e.g. linearity, invariance) of the base code, and requiring very little structure on the set of functions on the coordinates of the code. As an application, we construct new explicit families of locally correctable codes by lifting algebraic geometry codes. Like the multiplicity codes of Kopparty, Saraf, Yekhanin and the affine-lifted codes of Guo, Kopparty, Sudan, our codes of block-length $N$ can achieve $N^ε$ query complexity and $1-α$ rate for any given $ε, α> 0$ while correcting a constant fraction of errors, in contrast to the Reed-Muller codes and the degree-lifted AG codes of Ben-Sasson et al which face a rate barrier of $ε^{O(1/ε)}$. However, like the degree-lifted AG codes, our codes are over an alphabet significantly smaller than that obtained by Reed-Muller codes, affine-lifted codes, and multiplicity codes.
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