Bounds on the Length of Functional PIR and Batch Codes

A functional k-Private Information Retrieval (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 minimum number of servers for given k and s. We provide lower bounds on the minimum number of servers and constructions which yield upper bounds on this number. For k ≤ 4, exact bounds on this number are proved. Furthermore, we provide some asymptotic bounds. The problem coincides with the well known PIR 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 minimum number of servers for functional k-batch codes. In particular we present a random construction and a construction based on simplex codes, Write-Once Memory (WOM) codes, and Random I/O (RIO) codes.