PIR Array Codes with Optimal PIR Rates

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 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 $(1/\omegaR)$ of the bits of the database; here $\omegaR$ is a fixed rational number with $\omegaR > 1$. We study the maximum PIR rate of a PIR array code with the $k$-PIR property (which enables a $k$-server PIR protocol to be emulated on the $m$ servers), where the PIR rate is defined to be $k/m$. We present upper bounds on the achievable rate, some constructions, and ideas how to obtain PIR array codes with the highest possible PIR rate. In particular, we present constructions that asymptotically meet our upper bounds, and the exact largest PIR rate is obtained when $1 < \omegaR \leq 2$.