Weighted Lifted Codes: Local Correctabilities and Application to Robust Private Information Retrieval

Low degree Reed-Muller codes are known to satisfy local decoding properties which find applications in private information retrieval (PIR) protocols, for instance. However, their practical instantiation encounters a first barrier due to their poor information rate in the low degree regime. This lead the community to design codes with similar local properties but larger dimension, namely the lifted Reed-Solomon codes. However, a second practical barrier appears when one requires that the PIR protocol resists collusion of servers. In this paper, we propose a solution to this problem by considering weighted Reed-Muller codes. We prove that such codes allow us to build PIR protocols with optimal computation complexity and resisting to a small number of colluding servers. In order to improve the dimension of the codes, we then introduce an analogue of the lifting process for weighted degrees. With a careful analysis of their degree sets, we notably show that the weighted lifting of Reed-Solomon codes produces families of codes with remarkable asymptotic parameters.