Relaxed Locally Correctable Codes in Computationally Bounded Channels
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Error-correcting codes that admit local decoding and correcting algorithms have been the focus of much recent research due to their numerous theoretical and practical applications. The goal is to obtain the best possible tradeoffs between the number of queries the algorithm makes to its oracle (the locality of the task), and the amount of redundancy in the encoding (the information rate). In Hamming's classical adversarial channel model, the current tradeoffs are dramatic, allowing either small locality, but superpolynomial blocklength, or small blocklength, but high locality. However, in the computationally bounded, adversarial channel model, proposed by Lipton (STACS 1994), constructions of locally decodable codes suddenly exhibit small locality and small blocklength. The first such constructions are due to Ostrovsky, Pandey and Sahai (ICALP 2007) who build private locally decodable codes under the assumption that one-way functions exist, and in the setting where the sender and receiver share a private key. We study variants of locally decodable and locally correctable codes in computationally bounded, adversarial channels, under the much weaker assumption that collision-resistant hash functions exist, and with no public-key or private-key cryptographic setup. Specifically, we provide constructions of relaxed locally correctable and relaxed locally decodable codes over the binary alphabet, with constant information rate, and poly-logarithmic locality. Our constructions compare favorably with existing schemes built under much stronger cryptographic assumptions, and with their classical analogues in the computationally unbounded, Hamming channel. Our constructions crucially employ collision-resistant hash functions and local expander graphs, extending ideas from recent cryptographic constructions of memory-hard functions.
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