Bounds on the Length of Functional PIR and Batch Codes

A <italic>functional k-Private Information Retrieval (k</italic>-PIR) <italic>code</italic> of dimension <italic>s</italic> consists of <inline-formula> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula> servers storing linear combinations of <italic>s</italic> linearly independent information symbols. Any linear combination of the <italic>s</italic> information symbols can be recovered by <italic>k</italic> disjoint subsets of servers. The goal is to find the minimum number of servers for given <italic>k</italic> and <italic>s</italic>. We provide lower bounds on the minimum number of servers and constructions which yield upper bounds on this number. For <italic>k</italic> ≤ 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 <italic>k</italic> of linear combinations from the linearly independent information symbols can be recovered by <italic>k</italic> disjoint subset of servers, then the servers form a <italic>functional</italic> <italic>k</italic> <italic>-batch code</italic>. A functional <italic>k</italic>-batch code is a functional <italic>k</italic>-PIR code, where all the <italic>k</italic> linear combinations in the multiset are equal. We provide some bounds on the minimum number of servers for functional <italic>k</italic>-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.

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