  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).
