Bounds on the Length of Functional PIR and Batch codes

TL;DR

This paper investigates the minimum number of servers needed for functional PIR and batch codes, providing bounds, exact results for small k, and constructions based on coding theory to optimize storage and retrieval efficiency.

Contribution

It establishes new bounds and exact results for the number of servers in functional PIR and batch codes, and introduces novel constructions using advanced coding techniques.

Findings

Exact bounds for k ≤ 4 in functional PIR codes.

Asymptotic bounds for large parameters.

New constructions using simplex, WOM, and RIO codes.

Abstract

A functional $k$-PIR code of dimension $s$ consists of $n$ servers storing linear combinations of $s$ linearly independent information symbols. Any linear combination of the $s$ information symbols can be recovered by $k$ disjoint subsets of servers. The goal is to find the smallest number of servers for given $k$ and $s$. We provide lower bounds on the number of servers and constructions which yield upper bounds on this number. For $k \leq 4$, exact bounds on the number of servers are proved. Furthermore, we provide some asymptotic bounds. The problem coincides with the well known private information retrieval problem based on a coded database to reduce the storage overhead, when each linear combination contains exactly one information symbol. If any multiset of size $k$ of linear combinations from the linearly independent information symbols can be recovered by $k$ disjoint subset of servers, then the servers form a functional $k$-batch code. A~functional $k$-batch code is a functional $k$-PIR code, where all the $k$ linear combinations in the multiset are equal. We provide some bounds on the number of servers for functional $k$-batch codes. In particular we present a random construction and a construction based on simplex codes, WOM codes, and RIO codes.