Abstract
In this work we study two families of codes with availability, namely private information retrieval (FIR) codes and batch codes. While the former requires that every information symbol has k mutually disjoint recovering sets, the latter asks this property for every 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), r B (n, k), for PIR, 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 r B (n, k) = O(√n log(n)). In this work we study the asymptotic behavior of these codes for non-constant k and specifically for k = Θ(n∊). We also study the largest value of k such that the rate of the codes approaches 1, and show that for all e P (n, n∊) = o(n), while for batch codes, this property holds for all ∊ < 0.5.
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