Capacity-Achieving Private Information Retrieval Codes from MDS-Coded Databases with Minimum Message Size

We consider constructing capacity-achieving linear codes with minimum message size for private information retrieval (PIR) from $N$ non-colluding databases, where each message is coded using maximum distance separable (MDS) codes, such that it can be recovered from accessing the contents of any $T$ databases. It is shown that the minimum message size (sometimes also referred to as the sub-packetization factor) is significantly, in fact exponentially, lower than previously believed. More precisely, when $K>T/\textbf{gcd}(N,T)$ where $K$ is the total number of messages in the system and $\textbf{gcd}(\cdot,\cdot)$ means the greatest common divisor, we establish, by providing both novel code constructions and a matching converse, the minimum message size as $\textbf{lcm}(N-T,T)$, where $\textbf{lcm}(\cdot,\cdot)$ means the least common multiple. On the other hand, when $K$ is small, we show that it is in fact possible to design codes with a message size even smaller than $\textbf{lcm}(N-T,T)$.