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Low degree Reed-Muller codes are known to satisfy local decoding propertieswhich find applications in private information retrieval (PIR) protocols, forinstance. However, their practical instantiation encounters a first barrier dueto their poor information rate in the low degree regime. This lead thecommunity 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 PIRprotocol resists collusions of servers. In this paper, we propose a solution tothis problem by considering \emph{weighted} Reed-Muller codes. We prove thatsuch codes allow us to build PIR protocols with optimal computation complexityand resisting to a small number of colluding servers.  In order to improve the dimension of the codes, we then introduce an analogueof the lifting process for weigthed degrees. With a careful analysis of theirdegree sets, we notably show that the weighted lifting of Reed-Solomon codesproduces families of codes with remarkable asymptotic parameters.

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