An MDS-PIR Capacity-Achieving Protocol for Distributed Storage Using Non-MDS Linear Codes

TL;DR

This paper introduces a PIR protocol for distributed storage with non-MDS linear codes, providing conditions for achieving maximum capacity and demonstrating that certain codes like cyclic and Reed-Muller codes meet these conditions.

Contribution

It derives a PIR rate expression for arbitrary linear codes and establishes necessary and sufficient conditions for codes to achieve MDS PIR capacity.

Findings

Cyclic and Reed-Muller codes achieve MDS-PIR capacity.

Necessary condition based on generalized Hamming weights.

Sufficient condition based on code automorphisms.

Abstract

We propose a private information retrieval (PIR) protocol for distributed storage systems with noncolluding nodes where data is stored using an arbitrary linear code. An expression for the PIR rate, i.e., the ratio of the amount of retrieved data per unit of downloaded data, is derived, and a necessary and a sufficient condition for codes to achieve the maximum distance separable (MDS) PIR capacity are given. The necessary condition is based on the generalized Hamming weights of the storage code, while the sufficient condition is based on code automorphisms. We show that cyclic codes and Reed-Muller codes satisfy the sufficient condition and are thus MDS-PIR capacity-achieving.