摊销松弛局部可解码码
Amortized Relaxed Locally Decodable Codes
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中文总结 AI 辅助
本文提出摊销松弛局部可解码码（aRLDC），在完全信息论模型下实现常数码率、常数容错性和常数摊销局部性，且恢复长块时每个符号探测少于两次。
中文摘要 AI 辅助
局部可解码码（LDCs）使得通过仅探测可能损坏的码字中的少量位置即可恢复任何消息符号。LDC 的核心参数是其码率、局部性和容错性。理想情况下，人们希望这三个参数均为常数。然而，经典下界表明此类码不可能存在。近期一系列工作引入了摊销局部可解码码（aLDCs），其中解码器被要求恢复一整块连续的消息符号，而非单个符号。虽然先前的工作获得了具有常数码率、常数容错性和常数摊销局部性的理想 aLDCs，但这些构造依赖于对信道隐藏的共享随机性或限制信道的计算假设。另一个被广泛研究的松弛是松弛局部可解码码（RLDC）的概念，其中解码器可以输出一个特殊的失败符号 $\bot$，而不是冒险解码错误。在本工作中，我们引入了摊销松弛局部可解码码（aRLDC）的概念，将摊销解码与松弛解码范式相结合。与先前的理想 aLDC 构造不同，我们的模型是完全信息论意义上的，并且不对共享随机性或对抗性信道的计算限制做任何假设。我们构造了第一个具有常数码率、常数容错性和常数摊销局部性的 aRLDC。此外，对于任何长度为 $\Omega(\mathrm{polylog}(k))$ 的块，我们的解码器实现了摊销局部性 $1+\delta^{1 - o(1)}$，其中 $\delta$ 是容错参数。因此，渐近地，恢复一个长块每个恢复的消息符号所需的码字探测本质上少于两次。相比之下，没有摊销，任何 RLDC 都无法同时实现常数码率、常数容错性和常数局部性。
英文摘要
Locally decodable codes (LDCs) enable recovery of any message symbol by probing only a small number of positions in a possibly corrupted codeword. The central parameters of an LDC are its rate, locality, and error tolerance. Ideally, one would like all three parameters to be constant. However, classical lower bounds show that such codes cannot exist. A recent line of work introduced amortized locally decodable codes (aLDCs), in which the decoder is tasked with recovering an entire block of consecutive message symbols rather than a single symbol. While prior work obtained ideal aLDCs with constant rate, constant error tolerance, and constant amortized locality, those constructions relied on either shared randomness hidden from the channel or computational assumptions restricting the channel. Another well-studied relaxation is the notion of a relaxed locally decodable code (RLDC), in which the decoder may output a special failure symbol $\bot$ rather than risk decoding incorrectly. In this work, we introduce the notion of an amortized relaxed locally decodable code (aRLDC), combining amortized decoding with the relaxed decoding paradigm. Unlike prior ideal aLDC constructions, our model is fully information-theoretic and makes no assumptions about shared randomness or computational limitations of the adversarial channel. We construct the first aRLDC with constant rate, constant error tolerance, and constant amortized locality. Moreover, for any block of length $Ω(\mathrm{polylog}(k))$, our decoder achieves amortized locality $1+δ^{1 - o(1)}$, where $δ$ is the error tolerance parameter. Thus, asymptotically, recovering a long block requires essentially less than two codeword probe per message symbol recovered. By contrast, without amortization no RLDC can simultaneously achieve constant rate, constant error tolerance, and constant locality.
