On unequal data demand private information retrieval codes

Abstract

A t-PIR code allows the recovery of an encoded data symbol from each of t disjoint collections of code word symbols. We consider a generalization where some data symbols are in higher demand than other data symbols. We refer to such codes as (t1, . . . , tk)-UDD PIR codes, where now the i-th data symbol can be recovered from each of ti disjoint collections of code word symbols, for i = 1, . . . , k. We generalize the Griesmer bound for the length of the shortest possible t-PIR code to the (t1, . . . , tk)-UDD PIR code case and provide two separate proofs for the bound. We show that the Griesmer bound is tight for k ≤ 3 but not in general for k = 4. We also generalize other known upper and lower bounds for the shortest possible t-PIR codes to the (t1, . . . , tk)-UDD PIR code case.