Private Information Delivery with Coded Storage

In private information delivery (PID) problem, there are $K$ messages stored across $N$ servers, each capable of storing $M$ messages and a user. Servers want to convey one of the $K$ messages to the user without revealing the identity (index) of the message conveyed. The capacity of PID problem is defined as maximum number of bits of the desired message that can be conveyed privately, per bit of total communication, to the user. For the restricted case of replicated systems, where coded messages or splitting one message into several servers is not allowed, the capacity of PID has been characterized by Hua Sun in "Private Information Delivery, IEEE Transactions on Information Theory, December 2020" in terms of $K, N$ and $M.$ In this paper, we study the problem of PID with coded storage at the servers. For a class of problems called {\it bi-regular PID} we characterize the capacity for $N=K/M$ and for $N>K/M$ we provide an achievable scheme. In both the cases the rates achieved are more than the rates achievable with the replicated systems.