On the Capacity of Locally Decodable Codes

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 PIRmax, 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 PIRmax schemes is $(K-1)\log N$ . The results also generalize to a variation of the PIR problem, known as Repudiative Information Retrieval (RIR).