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标题：Explicit Uniquely Decodable Codes for Space Bounded Channels That Achieve List-Decoding Capacity
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作者：Ronen Shaltiel ; Jad Silbak
期刊名称：Electronic Colloquium on Computational Complexity
印刷版ISSN：1433-8092
出版年度：2020
卷号：2020
页码：1-68
出版社：Universität Trier, Lehrstuhl für Theoretische Computer-Forschung
摘要：We consider codes for space bounded channels. This is a model for communication under noise that was introduced by Guruswami and Smith (J. ACM 2016) and lies between the Shannon (random) and Hamming (adversarial) models. In this model, a channel is a space bounded procedure that reads the codeword in one pass, and modifies at most a p fraction of the bits of the codeword. • Explicit uniquely decodable codes for space bounded channels: Our main result is that for every 0 ≤ p 0 and a uniquely decodable code that is explicit (meaning that encoding and decoding are in poly-time) and has rate 1 − H(p) for channels with space n δ . This improves upon previous explicit codes by Guruswami and Smith, and Kopparty, Shaltiel and Silbak (FOCS 2019). Specifically, we obtain the same space and rate as earlier works, even though prior work gave only list-decodable codes (rather than uniquely decodable codes). • Complete characterization of the capacity of space bounded channels: Together with a result by Guruswami and Smith showing the impossibility of unique decoding for p ≥ 1 4 , our techniques also give a complete characterization of the capacity R(p) of space n 1−o(1) channels, specifically: R(p) = ( 1 − H(p) 0 ≤ p p0 ≈ 0.0804, there is a separation between the capacities of space bounded channels and casual channels, and the capacity of the former is strictly larger than that of the latter. Our approach builds on previous explicit list decodable codes for space bounded channels. We introduce and study a notion of “evasivenss” of codes, which is concerned with whether a decoding algorithm rejects a word that is obtained when a channel induces few errors to a uniformly chosen string. We use evasiveness (as well as several additional new ideas related to coding theory and pseudorandomness) to identify the “correct” message in the list. Loosely speaking, this is achieved by arguing that on “incorrect messages” the decoding algorithm cannot distinguish the codeword from a uniform string.