Bounds on the Length of Functional PIR and Batch codes
share
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. For k ≤ 4 we provide exact bounds on the number of servers. 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. 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 also a functional k-PIR, 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.
READ FULL TEXT