There has been much recent interest in Private information Retrieval (PIR) in
models where a database is stored across several servers using coding
techniques from distributed storage, rather than being simply replicated. In
particular, a recent breakthrough result of Fazelli, Vardy and Yaakobi
introduces the notion of a PIR code and a PIR array code, and uses this notion
to produce efficient PIR protocols.
In this paper we are interested in designing PIR array codes. We consider the
case when we have m servers, with each server storing a fraction (1/s) of
the bits of the database; here s is a fixed rational number with s>1. A
PIR array code with the k-PIR property enables a k-server PIR protocol
(with k≤m) to be emulated on m servers, with the overall storage
requirements of the protocol being reduced. The communication complexity of a
PIR protocol reduces as k grows, so the virtual server rate, defined to be
k/m, is an important parameter. We study the maximum virtual server rate of a
PIR array code with the k-PIR property. We present upper bounds on the
achievable virtual server rate, some constructions, and ideas how to obtain PIR
array codes with the highest possible virtual server rate. In particular, we
present constructions that asymptotically meet our upper bounds, and the exact
largest virtual server rate is obtained when 1<s≤2.
A k-PIR code (and similarly a k-PIR array code) is also a locally
repairable code with symbol availability k−1. Such a code ensures k
parallel reads for each information symbol. So the virtual server rate is very
closely related to the symbol availability of the code when used as a locally
repairable code. The results of this paper are discussed also in this context,
where subspace codes also have an important role