Nearly Optimal Constructions of PIR and Batch Codes

In this paper, we study two families of codes with availability, namely, private information retrieval (PIR) codes and batch codes. While the former requires that every information symbol has k mutually disjoint recovering sets, the latter imposes this property for each multiset request of k information symbols. The main problem under this paradigm is to minimize the number of redundancy symbols. We denote this value by rP (n, k) and rB(n, k), for PIR codes and batch codes, respectively, where n is the number of information symbols. Previous results showed that for any constant k, rP (n, k) = θ ( √ n) and rB(n, k) = O(€ n log(n)). In this paper, we study the asymptotic behavior of these codes for non-constant k and specifically for k = € (n€ ). We also study the largest value of k such that the rate of the codes approaches 1 and show that for all € < 1, rP (n, n€ ) = o(n) and rB(n, n€ ) = o(n). Furthermore, several more results are proved for the case of fixed k.