SIAM Journal on Computing, Volume 50, Issue 2, Page 788-813, January 2021.
A locally decodable code (LDC) $C \colon \{0,1\}^k \to \{0,1\}^n$ is an error correcting code wherein individual bits of the message can be recovered by only querying a few bits of a noisy codeword. LDCs found a myriad of applications both in theory and in practice, ranging from probabilistically checkable proofs to distributed storage. However, despite nearly two decades of extensive study, the best known constructions of $O(1)$-query LDCs have superpolynomial blocklength. The notion of relaxed LDCs is a natural relaxation of LDCs, which aims to bypass the foregoing barrier by requiring local decoding of nearly all individual message bits, yet allowing decoding failure (but not error) on the rest. State of the art constructions of $O(1)$-query relaxed LDCs achieve blocklength $n = O\left(k^{1+ \gamma}\right)$ for an arbitrarily small constant $\gamma$. We prove a lower bound which shows that $O(1)$-query relaxed LDCs cannot achieve blocklength $n = k^{1+ o(1)}$. This resolves an open problem raised by Goldreich in 2004.

中文翻译：

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论宽松局部解码算法的威力

SIAM Journal on Computing，第 50 卷，第 2 期，第 788-813 页，2021 年 1 月。
本地可解码代码 (LDC) $C \colon \{0,1\}^k \to \{0,1\}^n$ 是一种纠错码，其中消息的各个位可以通过仅查询噪声码字的几位。最不发达国家在理论和实践中都发现了无数的应用，从概率可检查的证明到分布式存储。然而，尽管进行了近 20 年的广泛研究，最著名的 $O(1)$-query LDC 结构具有超多项式块长度。松弛 LDC 的概念是 LDC 的自然松弛，其目的是通过要求对几乎所有单个消息位进行本地解码来绕过上述障碍，但允许其余部分的解码失败（但不会出错）。$O(1)$-query 松弛 LDC 的最先进结构实现了块长度 $n = O\left(k^{1+ \gamma}\right)$ 对于任意小的常数 $\gamma$。我们证明了一个下界，它表明 $O(1)$-query 宽松的 LDCs 不能达到块长度 $n = k^{1+ o(1)}$。这解决了 Goldreich 在 2004 年提出的一个悬而未决的问题。