Error-correcting codes that admit local decoding and correcting algorithms have been the focus of much recent research due to their numerous applications. An important goal is to obtain the best possible tradeoffs between the number of symbols of the codeword that the local decoding algorithm must examine (the locality), 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, but these constructions require strong trusted setup assumptions. We study variants of locally decodable and locally correctable codes in computationally bounded, adversarial channels, in a setting with no trusted setup. The only assumption we require is the selection of the public parameters (seed) for a collision-resistant hash function. 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, which compare favorably with their classical analogs, crucially employ collision-resistant hash functions and local expander graphs, extending ideas from recent cryptographic constructions of memory-hard functions.

中文翻译：

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计算有界通道中的宽松局部可校正码


由于其广泛的应用，允许本地解码和纠正算法的纠错码一直是最近研究的焦点。一个重要的目标是在本地解码算法必须检查的码字符号数量（局部性）和编码中的冗余量（信息率）之间获得最佳可能的权衡。在汉明的经典对抗性通道模型中，当前的权衡是戏剧性的，要么允许小局部性但超多项式块长度，要么允许小块长度但高局部性。然而，在 Lipton (STACS 1994) 提出的计算有界对抗信道模型中，本地可解码代码的构造突然表现出较小的局部性和较小的块长度，但这些构造需要强大的可信设置假设。我们在没有可信设置的环境中，在计算有限的对抗性通道中研究本地可解码和本地可校正代码的变体。我们需要的唯一假设是为抗冲突哈希函数选择公共参数（种子）。具体来说，我们在二进制字母表上提供宽松的局部可校正和宽松的局部可解码代码的构造，具有恒定的信息率和多对数局部性。我们的构造与经典类似物相比毫不逊色，关键是采用了抗碰撞哈希函数和局部扩展图，扩展了最近内存困难函数的密码构造的思想。