Locally decodable codes (LDCs) are error-correcting codes $C : \Sigma^k \to \Sigma^n$ that admit a local decoding algorithm that recovers each individual bit of the message by querying only a few bits from a noisy codeword. An important question in this line of research is to understand the optimal trade-off between the query complexity of LDCs and their block length. Despite importance of these objects, the best known constructions of constant query LDCs have super-polynomial length, and there is a significant gap between the best constructions and the known lower bounds in terms of the block length. For many applications it suffices to consider the weaker notion of relaxed LDCs (RLDCs), which allows the local decoding algorithm to abort if by querying a few bits it detects that the input is not a codeword. This relaxation turned out to allow decoding algorithms with constant query complexity for codes with almost linear length. Specifically, [BGH+06] constructed an $O(q)$-query RLDC that encodes a message of length $k$ using a codeword of block length $n = O(k^{1+1/\sqrt{q}})$. In this work we improve the parameters of [BGH+06] by constructing an $O(q)$-query RLDC that encodes a message of length $k$ using a codeword of block length $O(k^{1+1/{q}})$. This construction matches (up to a multiplicative constant factor) the lower bounds of [KT00, Woo07] for constant query LDCs, thus making progress toward understanding the gap between LDCs and RLDCs in the constant query regime. In fact, our construction extends to the stronger notion of relaxed locally correctable codes (RLCCs), introduced in [GRR18], where given a noisy codeword the correcting algorithm either recovers each individual bit of the codeword by only reading a small part of the input, or aborts if the input is detected to be corrupt.

中文翻译：

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具有改进参数的宽松局部可校正代码

本地可解码代码 (LDC) 是纠错代码 $C : \Sigma^k \to \Sigma^n$ ，它采用本地解码算法，该算法通过仅从嘈杂的代码字中查询几位来恢复消息的每个单独位。这一系列研究中的一个重要问题是了解 LDC 的查询复杂性与其块长度之间的最佳权衡。尽管这些对象很重要，但常量查询 LDC 的最著名构造具有超多项式长度，并且在块长度方面，最佳构造与已知下界之间存在显着差距。对于许多应用，考虑宽松 LDC (RLDC) 的较弱概念就足够了，如果通过查询几位检测到输入不是代码字，则允许本地解码算法中止。事实证明，这种放宽允许解码算法对几乎线性长度的代码具有恒定的查询复杂度。具体来说，[BGH+06] 构造了一个 $O(q)$-query RLDC，它使用块长度 $n = O(k^{1+1/\sqrt{q} })$。在这项工作中，我们通过构造一个 $O(q)$-query RLDC 来改进 [BGH+06] 的参数，该 RLDC 使用块长度为 $O(k^{1+1/ {q}})$。这种构造匹配（高达乘法常数因子）恒定查询 LDC 的 [KT00, Woo07] 下界，从而在理解恒定查询机制中 LDC 和 RLDC 之间的差距方面取得了进展。事实上，我们的构造扩展到了 [GRR18] 中引入的宽松局部可纠正码 (RLCC) 的更强概念，