A locally decodable code (LDC) maps $K$ source symbols, each of size $L_{w}$ bits, to $M$ coded symbols, each of size $L_{x}$ bits, such that each source symbol can be decoded from $N \leq M$ coded symbols. A perfectly smooth LDC further requires that each coded symbol is uniformly accessed when we decode any one of the messages. The ratio $L_{w}/L_{x}$ is called the symbol rate of an LDC. The highest possible symbol rate for a class of LDCs is called the capacity of that class. It is shown that given $K, N$ , the maximum value of capacity of perfectly smooth LDCs, maximized over all code lengths $M$ , is $C^{*}=N\left ({1+1/N+1/N^{2}+\cdots +1/N^{K-1}}\right)^{-1}$ . Furthermore, given $K, N$ , the minimum code length $M$ for which the capacity of a perfectly smooth LDC is $C^{*}$ is shown to be $M = N^{K}$ . Both of these results generalize to a broader class of LDCs, called universal LDCs. The results are then translated into the context of PIR_max, i.e., Private Information Retrieval subject to maximum (rather than average) download cost metric. It is shown that the minimum upload cost of capacity achieving PIR_max schemes is $(K-1)\log N$ . The results also generalize to a variation of the PIR problem, known as Repudiative Information Retrieval (RIR).

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关于本地可译码的容量

本地可解码代码 (LDC) 映射 $K$ 源符号，每个大小 $L_{w}$ 位，到 百万美元 编码符号，每个大小 $L_{x}$ 位，这样每个源符号都可以从 $N \leq M$ 编码符号。完美平滑的 LDC 进一步要求在我们对任何一条消息进行解码时统一访问每个编码符号。比例 $L_{w}/L_{x}$ 称为 LDC 的符号率。一类最不发达国家的最高可能符号率称为该类的容量。表明给定 $K, N$ ，完全平滑的 LDC 容量的最大值，在所有代码长度上最大化 百万美元 ， 是 $C^{*}=N\left ({1+1/N+1/N^{2}+\cdots +1/N^{K-1}}\right)^{-1}$ . 此外，给定 $K, N$ , 最小码长 百万美元 一个完全光滑的 LDC 的容量是 $C^{*}$ 显示为 $M = N^{K}$ . 这两个结果都适用于更广泛的最不发达国家类别，称为普遍最不发达国家。然后将结果转换为 PIR _max的上下文，即受最大（而不是平均）下载成本度量约束的私人信息检索。结果表明，实现 PIR _max方案的容量的最小上传成本为 $(K-1)\log N$ . 结果还推广到 PIR 问题的一种变体，称为否认信息检索 (RIR)。