# Nearly Optimal Constructions of PIR and Batch Codes

## TL;DR

This paper investigates the minimal redundancy in PIR and batch codes with availability, especially for non-constant request sizes, providing nearly optimal constructions and asymptotic bounds.

## Contribution

It extends the understanding of PIR and batch codes for large request sizes, offering nearly optimal constructions and asymptotic bounds for the redundancy.

## Key findings

- For PIR codes, redundancy grows slower than linear for request sizes up to n^1.
- Batch codes maintain near-linear redundancy growth for request sizes up to n^{0.5}.
- The rate approaches 1 for certain ranges of request size, with specific bounds on redundancy.

## Abstract

In this work 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 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 $r_P(n,k), r_B(n,k)$, for PIR, batch codes, respectively, where $n$ is the number of information symbols. Previous results showed that for any constant $k$, $r_P(n,k) = \Theta(\sqrt{n})$ and $r_B(n,k)=O(\sqrt{n}\log(n)$. In this work we study the asymptotic behavior of these codes for non-constant $k$ and specifically for $k=\Theta(n^\epsilon)$. We also study the largest value of $k$ such that the rate of the codes approaches 1, and show that for all $\epsilon<1$, $r_P(n,n^\epsilon) = o(n)$, while for batch codes, this property holds for all $\epsilon< 0.5$.