High-Rate Locally Correctable Codes via Lifting
High-Rate Locally Correctable Codes via Lifting
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DOI：
10.1109/tit.2015.2503767
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发表时间：
2013-04
影响因子：
2.5
通讯作者：
Alan J. X. Guo
中科院分区：
文献类型：
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作者：
Alan J. X. Guo
我们提出了一个通用框架，用于校正高速公路校正代码，这些代码是基于对坐标的函数的提升代码，其方法通常使我们的方法提起，我们的方法通常将sublinear数量的查询数字化。仿射不变的代码（郭，科帕蒂和苏丹）及其概括自动化提升（在Ben-Sasson等人的工作中暗示了，但与学位提升不同），这取消了代数几何代码，相对于一组代码我们的自然态度是我们提升的自然替代品，它首先带有两个优势。 （例如，线性和不变性）基本代码，并且在代码的坐标上的功能集很少，我们通过提升代数几何代码来构建新的局部可校正代码Koppparty，Saraf，Yekhanin和Guo，Koppparty和Sudan的仿射式代码的多重性代码，我们的块长度n的代码可以达到n∈查询复杂性，而对于任何给定的∈，α> 0，而1 -α速率，而α> 0，而α> 0，而α> 0纠正误差的恒定分数，与Ben-Sasson等人的芦苇毛刺代码和度量较高的AG代码相比，该代码面对∈O的速率屏障（1/∈）。升起的AG代码，我们的代码在字母表上明显小于芦苇磨碎器代码，仿射式代码和多样性代码所获得的代码。
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 (alluded to in the work of Ben-Sasson et al., but distinct from their degree lifting), 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 requires no special properties (e.g. linearity and invariance) of the base code, and requires a 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, and Sudan, our codes of block length N can achieve N∈ query complexity and 1 - α rate for any given ∈, 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 ∈O(1/∈). 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.