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

## TL;DR

This paper introduces weighted lifted Reed-Muller codes that enhance local decoding and privacy in PIR protocols, overcoming limitations of traditional codes by improving code dimension and collusion resistance.

## Contribution

It proposes weighted Reed-Muller codes and their lifting process, achieving better asymptotic parameters and practical PIR protocol robustness against collusions.

## Key findings

- Weighted codes enable PIR protocols with optimal complexity.
- Weighted lifting improves code dimension and asymptotic performance.
- Codes resist collusions of a small number of servers.

## Abstract

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 collusions of servers. In this paper, we propose a solution to this problem by considering \emph{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 weigthed 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.