On Private Information Retrieval Array Codes

Given a database, the private information retrieval (PIR) protocol allows a user to make queries to several servers and retrieve a certain item of the database via the feedbacks without revealing the identity of the specific item to any single server. Classic <inline-formula> <tex-math notation="LaTeX">$k$ </tex-math></inline-formula>-server PIR protocols work on replicated databases, i.e., each of the <inline-formula> <tex-math notation="LaTeX">$k$ </tex-math></inline-formula> servers stores a whole copy of the database. Recently, new PIR models were proposed with coding techniques arising from the distributed storage system. In these new models, each server only stores a fraction <inline-formula> <tex-math notation="LaTeX">$1/s$ </tex-math></inline-formula> of the whole database, where <inline-formula> <tex-math notation="LaTeX">$s>1$ </tex-math></inline-formula> is the given rational number. The PIR array codes are recently proposed by Fazeli, Vardy, and Yaakobi to characterize the new models. The central problem in designing a PIR array code with <inline-formula> <tex-math notation="LaTeX">$m$ </tex-math></inline-formula> servers and the <inline-formula> <tex-math notation="LaTeX">$k$ </tex-math></inline-formula>-PIR property (which indicates that these <inline-formula> <tex-math notation="LaTeX">$m$ </tex-math></inline-formula> servers may emulate a classic <inline-formula> <tex-math notation="LaTeX">$k$ </tex-math></inline-formula>-server PIR protocol) is to maximize <inline-formula> <tex-math notation="LaTeX">$k/m$ </tex-math></inline-formula>, known as the virtual server rate. Our main contribution to this problem is twofold. First, for the case <inline-formula> <tex-math notation="LaTeX">$1 < s\le 2$ </tex-math></inline-formula>, although the PIR array codes with optimal rate have been constructed recently by Blackburn and Etzion, the number of servers in their construction is rather large. We determine the minimum number of servers admitting the existence of a PIR array code with an optimal rate for a certain range of parameters. Second, for the case <inline-formula> <tex-math notation="LaTeX">$s>2$ </tex-math></inline-formula>, a new upper bound on the rate of a PIR array code is presented. Besides, we also have some discussions on an asymptotically optimal construction by Blackburn and Etzion.