Proceedings of the Fourteenth Annual ACM-SIAM Symposium on Discrete Algorithms
Proceedings of the Fourteenth Annual ACM-SIAM Symposium on Discrete Algorithms
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第十四届 ACM-SIAM 离散算法年度研讨会论文集
DOI：
10.1137/1.9781611975994.84
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发表时间：
2020
期刊：
影响因子：
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通讯作者：
Chiesa A
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文献类型：
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作者：
Chiesa A
本地可纠正码（LCC）是一种纠错码，它允许本地算法通过少量的查询来纠正损坏码字的任何单个符号。对于系统码，这个概念比局部可解码码（LDCs）更强，局部可解码码的目标是只恢复消息的单个符号。算法编码理论的核心问题之一是构造查询最小块长度的LCC和LDC。唉，尽管在过去二十年中受到了广泛关注，但最先进的此类代码需要超多项式块长度来接纳查询算法以进行本地纠正和解码。对松弛LCC和LDC的研究，允许校正算法在一小部分位置上中止（但不会出错），提供了一种绕过这一障碍的方法。这种放松原来允许常数查询纠正和解码算法的代码与多项式块长度。专注于本地校正，Gur，Ramnarayan和Rothblum [第九届理论计算机科学创新会议论文集，ITCS'18，2018，第10页。1-27]表明，存在查询松弛LCC，实现近四次块长度，对于任意小的常数。我们构造了一个查询松弛LCC与近线性块长度，为一个任意小的常数。这显著地缩小了下限之间的差距，该下限表明存在具有块长度的无查询松弛LCC。特别地，我们的构造与Ben-Sasson等人[SIAM J. Comput.，36（2006），pp. 889-974]，他用同样的参数构建了宽松的最不发达国家。这解决了Gur，Ramnarayan和Rothblum提出的一个悬而未决的问题[第九届理论计算机科学创新会议论文集，ITCS'18，2018，第10页]。1-27]。
Locally correctable codes (LCCs) are error correcting codeswhich admit local algorithms that correct any individual symbol of a corruptedcodewordvia a minuscule number of queries. For systematic codes, this notion is stronger than that of locally decodable codes (LDCs), where the goal is to only recover individual symbols of themessage. One of the central problems in algorithmic coding theory is to construct-query LCCs and LDCs with minimal block length. Alas, state-of-the-art of such codes requires super-polynomial block length to admit-query algorithms for local correction and decoding, despite much attention during the last two decades. The study ofrelaxedLCCs and LDCs, which allow the correction algorithm toabort(but not err) on a small fraction of the locations, provides a way to circumvent this barrier. This relaxation turned out to allow constant-query correcting and decoding algorithms for codes with polynomial block length. Focusing on local correction, Gur, Ramnarayan, and Rothblum [Proceedings of the 9th Innovations in Theoretical Computer Science Conference, ITCS’18, 2018, pp. 1–27] showed that there exist-query relaxed LCCs that achieve nearly-quartic block length, for an arbitrarily small constant. We construct an-query relaxed LCC withnearly-linearblock length, for an arbitrarily small constant. This significantly narrows the gap between the lower bound which states that there are no-query relaxed LCCs with block length. In particular, our construction matches the parameters achieved by Ben-Sasson et al. [SIAM J. Comput., 36 (2006), pp. 889–974], who constructed relaxed LDCs with the same parameters. This resolves an open problem raised by Gur, Ramnarayan, and Rothblum [Proceedings of the 9th Innovations in Theoretical Computer Science Conference, ITCS’18, 2018, pp. 1–27].